Astroparticle Physics

Astroparticle Physics

Claus GrupenAstroparticle PhysicsWith contributions from Glen Cowan, Simon Eidelman,and Tilo StrohABC

Prof. Dr. Claus GrupenUniversity of SiegenDepartment of PhysicsWalter-Flex-Strasse 357068 SiegenGermanye-mail: grupen@hep.physik.uni-siegen.deWith contributions from:Dr. Glen CowanRoyal Holloway, University of LondonPhysics DepartmentEnglande-mail: G.Cowan@rhul.ac.ukDipl. Phys. Tilo StrohUniversity of SiegenDepartment of PhysicsGermanye-mail: stroh@sirs02.physik.uni-siegen.deProf. Dr. Simon EidelmanBudker Institute of Nuclear PhysicsNovosibirskRussiae-mail: Simon.Eidelman@cern.chLibrary of Congress Control Number: 2005924544ISBN -10ISBN -133-540-25312-2 Springer Berlin Heidelberg New York978-3-540-25312-9 Springer Berlin Heidelberg New YorkThis work is subject to copyright. All rights are reserved, whether the whole or part of the material isconcerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting,reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publicationor parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer. Violationsare liable for prosecution under the German Copyright Law.Springer is a part of Springer Science+Business Mediaspringeronline.comc○ Springer-Verlag Berlin Heidelberg 2005Printed in GermanyThe use of general descriptive names, registered names, trademarks, etc. in this publication does not imply,even in the absence of a specific statement, that such names are exempt from the relevant protective lawsand regulations and therefore free for general use.Typesetting: by the authors and Tilo Stroh using a Springer TEX macro packageCover design: design & production GmbH, Heidelberg based on an idea of the authorImages: ArmbrustDesign, Siegen, GermanyPrinted on acid-free paper SPIN: 10817996 56/3141/jvg 543210

VPreface“The preface is the most important part of a book. Even reviewersread a preface.”Philip GuedallaPreface to the English TranslationThis book on astroparticle physics is the translation of the book on ‘Astroteilchenphysik’published in German by Vieweg, Wiesbaden, in the year 2000. It is not only a translation,however, but also an update. The young field of astroparticle physics is developing sorapidly, in particular with respect to ‘new astronomies’ such as neutrino astronomy and thedetailed measurements of cosmic background radiation, that these new experimental resultsand also new theoretical insights need to be included.The details of the creation of the universe are not fully understood yet and it is stillnot completely clear how the world will end, but recent results from supernovae observationsand precise measurement of the primordial blackbody radiation seem to indicate withincreasing reliability that we are living in a flat Euclidean universe which expands in anaccelerated fashion.In the last couple of years cosmology has matured from a speculative science to a fieldof textbook knowledge with precision measurements at the percent level.The updating process has been advanced mainly by my colleague Dr. Glen Cowan whois lecturing on astroparticle physics at Royal Holloway College, London, and by myself.The chapter on ‘Cosmology’ has been rewritten, and chapters on ‘The Early Universe’,‘Big Bang Nucleosynthesis’, ‘The Cosmic Microwave Background’, and ‘Inflation’ as wellas a section on gravitational astronomy have been added. The old chapter on ‘UnsolvedProblems’ was moved into a new chapter on ‘Dark Matter’, and part of it went into chapterson primary and secondary cosmic rays.The book has been extended by a large number of problems related to astroparticlephysics. Full solutions to all problems are given. To ease the understanding of theoreticalaspects and the interpretation of physics data, a mathematical appendix is offered wheremost of the formulae used are presented and/or derived. In addition, details on the thermodynamicsof the early universe have been treated in a separate appendix.Prof. Dr. Simon Eidelman from the Budker Institute of Nuclear Physics in Novosibirskand Dipl.Phys. Tilo Stroh have carefully checked the problems and proposed new ones.Dr. Ralph Kretschmer contributed some interesting and very intricate problems. I have alsoreceived many comments from my colleagues and students in Siegen.The technical aspects of producing the English version lay in the hands of Ms. UteSmolik, Lisa Hoppe, and Ms. Angelika Wied (text), Dipl.Phys. Stefan Armbrust (updatedthe figures), Dr. Glen Cowan and Ross Richardson (polished my own English translation),

VIPrefaceand M.Sc. Mehmet T. Kurt (helped with the editing). The final appearance of the bookincluding many comments on the text, the figures, and the layout was accomplished byDipl.Phys. Tilo Stroh and M.Sc. Nadir Omar Hashim.Without the help of these people, it would have been impossible for me to complete thetranslation in any reasonable time, if at all. In particular, I would like to thank my colleagueProf. Dr. Torsten Fließbach, an expert on Einstein’s theory of general relativity, for his criticalassessment of the chapter on cosmology and for proposing significant improvements.Also the contributions by Dr. Glen Cowan on the new insights into the evolution of the earlyuniverse and related subjects are highly appreciated. Dr. Cowan has really added essentialingredients with the last chapters of the book. Finally, Prof. Dr. Simon Eidelman, Dr. ArminBöhrer, and Dipl.Phys. Tilo Stroh read the manuscript with great care and made invaluablecomments. I thank all my friends for their help in creating this English version of my book.Siegen, February 2005Claus Grupen

PrefaceVIIPreface to the German EditionThe field of astroparticle physics is not really a new one. Up until 1960, the physics ofcosmic rays essentially represented this domain. Elementary particle physics in acceleratorshas evolved from the study of elementary-particle processes in cosmic radiation. Amongothers, the first antiparticles (positrons) and the members of the second lepton generation(muons) were discovered in cosmic-ray experiments.The close relationship between cosmology and particle physics was, however, recognizedonly relatively recently. Hubble’s discovery of the expanding universe indicates thatthe cosmos originally must have had a very small size. At such primeval times, the universewas a microworld that can only be described by quantum-theoretical methods of elementaryparticle physics. Today, particle physicists try to recreate the conditions that existed in theearly universe by using electron–positron and proton–antiproton collisions at high energiesto simulate ‘mini Big Bangs’.The popular theories of elementary particle physics attempt to unify the various types ofinteractions in the Standard Model. The experimental confirmation of the existence of heavyvector bosons that mediate weak interactions (W + ,W − ,Z 0 ), and progress in the theoreticalunderstanding of strong interactions seem to indicate that one may be able to understand thedevelopment of the universe just after the Big Bang. The high temperatures or energiesthat existed at the time of the Big Bang will, however, never be reached in earthboundlaboratories. This is why a symbiosis of particle physics, astronomy, and cosmology is onlytoo natural. Whether this new field is named astroparticle physics or particle astrophysics ismore or less a matter of taste or the background of the author. This book will deal both withastrophysics and elementary particle physics aspects. We will equally discuss the conceptsof astrophysics focusing on particles and particle physics using astrophysical methods. Theguiding line is physics with astroparticles. This is why I preferred the term astroparticlephysics over particle astrophysics.After a relatively detailed historical introduction (Chap. 1) in which the milestones ofastroparticle physics are mentioned, the basics of elementary particle physics (Chap. 2),particle interactions (Chap. 3), and measurement techniques (Chap. 4) are presented. Astronomicalaspects prevail in the discussion of acceleration mechanisms (Chap. 5) and primarycosmic rays (Chap. 6). In these fields, new disciplines such as neutrino and gamma-ray astronomyrepresent a close link to particle physics. This aspect is even more pronounced inthe presentation of secondary cosmic rays (Chap. 7). On the one hand, secondary cosmicrays have been a gold mine for discoveries in elementary particle physics. On the other hand,however, they sometimes represent an annoying background in astroparticle observations.The highlight of astroparticle physics is surely cosmology (Chap. 8) in which the theoryof general relativity, which describes the macrocosm, is united with the successes of elementaryparticle physics. Naturally, not all questions have been answered; therefore a finalchapter is devoted to open and unsolved problems in astroparticle physics (Chap. 9).The book tries to bridge the gap between popular presentations of astroparticle physicsand textbooks written for advanced students. The necessary basics from elementary particlephysics, quantum physics, and special relativity are carefully introduced and applied,without rigorous derivation from appropriate mathematical treatments. It should be possibleto understand the calculations presented with the knowledge of basic A-level mathematics.

VIIIPrefaceOn top of that, the basic ideas discussed in this book can be followed without referring tospecial mathematical derivations.I owe thanks to many people for their help during the writing of this book. Dr. ArminBöhrer read the manuscript with great care. Ms. Ute Bender and Ms. Angelika Wied wrotethe text, and Ms. Claudia Hauke prepared the figures that were finalized by Dipl.Phys. StefanArmbrust. I owe special thanks to Dr. Klaus Affholderbach and Dipl.Phys. Olaf Kraselwho created the computer layout of the whole book in the LATEX style. I am especially indebtedto Dipl.Phys. Tilo Stroh for his constant help, not only as far as physics questionsare concerned, but in particular for applying the final touch to the manuscript with his inimitable,masterful eye for finding the remaining flaws in the text and the figures. Finally, I owemany thanks to the Vieweg editors, Ms. Christine Haite and Dipl.Math. Wolfgang Schwarz.Geneva, July 2000

IXTable of Contents“The most technologically efficient machine that man has inventedis the book.”Northrop FryePreface ..............................................................V1 Historical Introduction ............................................. 11.1 Discoveriesinthe20thCentury.................................... 31.2 DiscoveriesofNewElementaryParticles............................ 81.3 Start of the Satellite Era .......................................... 111.4 OpenQuestions................................................. 171.5 Problems ...................................................... 192 The Standard Model of Elementary Particles .......................... 212.1 ExamplesofInteractionProcesses ................................. 262.2 Problems ...................................................... 333 Kinematics and Cross Sections ...................................... 353.1 ThresholdEnergies.............................................. 373.2 Four-Vectors ................................................... 393.3 LorentzTransformation .......................................... 443.4 CrossSections.................................................. 463.5 Problems ...................................................... 474 Physics of Particle and Radiation Detection ........................... 494.1 InteractionsofAstroparticles...................................... 504.2 InteractionProcessesUsedforParticleDetection..................... 514.3 PrinciplesoftheAtmosphericAirCherenkovTechnique............... 554.4 SpecialAspectsofPhotonDetection ............................... 564.5 Cryogenic Detection Techniques . . . ................................ 584.6 Propagation and Interactions of Astroparticles in GalacticandExtragalacticSpace .......................................... 584.7 CharacteristicFeaturesofDetectors ................................ 604.8 Problems ...................................................... 61

XTable of Contents5 Acceleration Mechanisms .......................................... 635.1 Cyclotron Mechanism . . .......................................... 645.2 Acceleration by Sunspot Pairs ..................................... 655.3 Shock Acceleration . . . . .......................................... 665.4 Fermi Mechanism ............................................... 685.5 Pulsars ........................................................ 695.6 Binaries ....................................................... 715.7 EnergySpectraofPrimaryParticles ................................ 735.8 Problems ...................................................... 766 Primary Cosmic Rays .............................................. 776.1 Charged Component of Primary Cosmic Rays . ...................... 786.2 Neutrino Astronomy . . . .......................................... 866.2.1 AtmosphericNeutrinos .................................... 886.2.2 SolarNeutrinos........................................... 946.2.3 SupernovaNeutrinos ...................................... 1006.2.4 High-EnergyGalacticandExtragalacticNeutrinos ............. 1046.3 Gamma Astronomy . . . . .......................................... 1086.3.1 Introduction. . . . .......................................... 1086.3.2 Production Mechanisms for γ Rays.......................... 1106.3.3 Measurement of γ Rays ................................... 1136.3.4 Observation of γ -RayPointSources ......................... 1176.3.5 γ Burster................................................ 1206.4 X-Ray Astronomy . .............................................. 1236.4.1 Introduction. . . . .......................................... 1236.4.2 Production Mechanisms for X Rays .......................... 1246.4.3 DetectionofXRays....................................... 1266.4.4 ObservationofX-RaySources.............................. 1286.5 Gravitational-Wave Astronomy . . . . ................................ 1336.6 Problems ...................................................... 1367 Secondary Cosmic Rays ............................................ 1417.1 PropagationintheAtmosphere.................................... 1427.2 CosmicRaysatSeaLevel ........................................ 1477.3 Cosmic Rays Underground . . ..................................... 1517.4 ExtensiveAirShowers ........................................... 1567.5 NatureandOriginoftheHighest-EnergyCosmicRays................ 1637.6 Problems ...................................................... 169

Table of ContentsXI8 Cosmology ....................................................... 1718.1 The Hubble Expansion . .......................................... 1738.2 TheIsotropicandHomogeneousUniverse........................... 1758.3 TheFriedmannEquationfromNewtonianGravity.................... 1778.4 TheFriedmannEquationfromGeneralRelativity..................... 1798.5 TheFluidEquation.............................................. 1828.6 The Acceleration Equation . . . ..................................... 1838.7 NatureofSolutionstotheFriedmannEquation....................... 1838.8 ExperimentalEvidencefortheVacuumEnergy ...................... 1868.9 Problems ...................................................... 1899 The Early Universe ................................................ 1919.1 ThePlanckScale................................................ 1919.2 Thermodynamics of the Early Universe . . ........................... 1939.2.1 Energy and Number Densities . . . ........................... 1949.2.2 TheTotalEnergyDensity .................................. 1959.2.3 EquationsofState ........................................ 1989.2.4 RelationbetweenTemperatureandScaleFactor ............... 1999.3 SolvingtheFriedmannEquation................................... 1999.3.1 Digression on Thermal Equilibrium . . . . ...................... 2029.4 Thermal History of the First Ten Microseconds ...................... 2039.5 TheBaryonAsymmetryoftheUniverse ............................ 2059.5.1 ExperimentalEvidenceofBaryonAsymmetry................. 2069.5.2 SizeoftheBaryonAsymmetry.............................. 2089.5.3 TheSakharovConditions .................................. 2099.6 Problems ...................................................... 21110 Big Bang Nucleosynthesis .......................................... 21310.1 SomeIngredientsforBBN........................................ 21410.2 StartoftheBBNEra............................................. 21510.3 TheNeutron-to-ProtonRatio...................................... 21610.4 Neutrino Decoupling, Positron Annihilation, and Neutron Decay . ....... 21810.5 SynthesisofLightNuclei......................................... 22010.6 DetailedBBN .................................................. 22210.7 Constraints on the Number of Neutrino Families ..................... 22610.8 Problems ...................................................... 228

XIITable of Contents11 The Cosmic Microwave Background ................................. 22911.1 Prelude: Transition to a Matter-Dominated Universe . ................. 22911.2 DiscoveryandBasicPropertiesoftheCMB ......................... 23111.3 FormationoftheCMB........................................... 23311.4 CMBAnisotropies .............................................. 23511.5 The Monopole and Dipole Terms . . ................................ 23611.5.1 Small-AngleAnisotropy ................................... 23711.6 DeterminationofCosmologicalParameters.......................... 23811.7 Problems ...................................................... 24412 Inflation ......................................................... 24512.1 TheHorizonProblem............................................ 24612.2 TheFlatnessProblem............................................ 24612.3 The Monopole Problem .......................................... 24812.4 HowInflationWorks............................................. 25112.5 Mechanisms for Inflation ......................................... 25312.6 SolutiontotheFlatnessProblem................................... 25612.7 SolutiontotheHorizonProblem................................... 25812.8 Solution to the Monopole Problem . ................................ 25912.9 InflationandtheGrowthofStructure............................... 26012.10 OutlookonInflation ............................................. 26212.11 Problems ...................................................... 26413 Dark Matter ..................................................... 26513.1 Large-ScaleStructureoftheUniverse .............................. 26513.2 MotivationforDarkMatter ....................................... 26613.2.1 DarkStars ............................................... 26913.2.2 NeutrinosasDarkMatter .................................. 27213.2.3 WeaklyInteractingMassiveParticles(WIMPs) ................ 27513.2.4 Axions.................................................. 27813.2.5 TheRôleoftheVacuumEnergyDensity...................... 27913.2.6 GalaxyFormation......................................... 28113.2.7 ResuméonDarkMatter.................................... 28313.3 Problems ...................................................... 28514 Astrobiology ..................................................... 28714.1 Problems ...................................................... 29015 Outlook ......................................................... 29315.1 Problems ...................................................... 29716 Glossary ......................................................... 29917 Solutions......................................................... 337

Table of ContentsXIIIA Mathematical Appendix ............................................ 381A.1 SelectedFormulae............................................... 381A.2 Mathematics for Angular Variations of the CMB ..................... 385B Results from Statistical Physics: Thermodynamics of the Early Universe ... 389B.1 Statistical Mechanics Review ..................................... 390B.2 Number and Energy Densities ..................................... 396B.3 EquationsofState............................................... 398C Definition of Equatorial and Galactic Coordinates ...................... 401Important Constants for Astroparticle Physics ............................. 403References ........................................................... 405Further Reading ...................................................... 407Photo Credits ......................................................... 412Index ................................................................ 415

11 Historical Introduction“Look into the past as guidance for thefuture.”Robert Jacob GoodkinThe field of astroparticle physics, or particle astrophysics isrelatively new. It is therefore not easy to describe the historyof this branch of research. The selection of milestones inthis book is necessarily subject to a certain arbitrariness andpersonal taste.Historically, astroparticle physics is based on opticalastronomy. As detector techniques improved, this observationalscience matured into astrophysics. This researchtopic involves many subfields of physics, like mechanicsand electrodynamics, thermodynamics, plasma physics, nuclearphysics, and elementary particle physics, as well asspecial and general relativity. Precise knowledge of particlephysics is necessary to understand many astrophysical contexts,particularly since comparable experimental conditionscannot be prepared in the laboratory. The astrophysical environmenttherefore constitutes an important laboratory forhigh energy physicists.The use of the term astroparticle physics is certainly justified,since astronomical objects have been observed in the‘light’ of elementary particles. Of course, one could arguethat X-ray or gamma-ray astronomy is more closely relatedto astronomy rather than to astroparticle physics. To be onthe safe side, the new term astroparticle physics, should berestricted to ‘real’ elementary particles. The observations ofour Sun in the light of neutrinos in the Homestake Mine(Davis experiment) in 1967, constitutes the birth of astroparticlephysics, even though the first measurements ofsolar neutrinos by this radiochemical experiment were performedwithout directional correlation. It is only since theKamiokande 1 experiment of 1987, that one has been able to‘see’ the Sun in real time, whilst additionally being able tomeasure the direction of the emitted neutrinos. Nature wasalso kind enough to explode a supernova in the Large Mag-1 Kamiokande – Kamioka Nucleon Decay Experimentastrophysics as laboratoryfor high energy physicsjustificationof the nomenclatureDavis experimentKamiokande experimentSN 1987A

2 1 Historical IntroductionFig. 1.1Crab Nebula {1}Vela supernovaVela X1Crab Nebulanorthern lights(aurora borealis)ellanic Cloud in 1987 (SN 1987A), whose neutrino burstcould be recorded in the large water Cherenkov detectors ofKamiokande and IMB 2 and in the scintillator experiment atBaksan.Presently, the fields of gamma and neutrino astronomyare expanding rapidly. Astronomy with charged particles,however, is a different matter. Irregular interstellar and intergalacticmagnetic fields randomize the directions of chargedcosmic rays. Only particles at very high energies travel alongapproximately straight lines through magnetic fields. Thismakes astronomy with charged particles possible, if the intensityof energetic primaries is sufficiently high.Actually, there are hints that the highest-energy cosmicrays (> 10 19 eV) have a non-uniform distribution and possiblyoriginate from the supergalactic plane. This plane is anaccumulation of galaxies in a disk-like fashion, in a similarway that stars form the Milky Way. Other possible sources,however, are individual galactic nuclei (M87?) at cosmologicaldistances.The milestones which have contributed to the new disciplineof astroparticle physics shall be presented in chronologicalorder. For that purpose, the relevant discoveries in astronomy,cosmic rays, and elementary particle physics willbe considered in a well-balanced way. It is, of course, truethat this selection is subject to personal bias.It is interesting to point out the observations of the Velasupernova by the Sumerians 6000 years ago. This supernovaexploded in the constellation Vela at a distance of 1500 lightyears.Today the remnant of this explosion is visible, e.g., inthe X-ray and gamma range. Vela X1 is a binary, one componentof which is the Vela pulsar. With a rotational periodof 89 ms the Vela pulsar is one of the ‘slowest’ pulsars so farobserved in binaries. The naming scheme of X-ray sourcesis such that Vela X1 denotes the strongest (̂= ‘the first’) X-ray source in the constellation Vela.The second spectacular supernova explosion was observedin China in 1054. The relic of this outburst is the CrabNebula, whose remnant also emits X rays and gamma rayslike Vela X1. Because of its time-independent brightness theCrab is often used as a ‘standard candle’ in gamma-ray astronomy(Fig. 1.1).The observation of the northern lights (Gassendi 1621and Halley 1716) as the aurora borealis (‘northern dawn’)2 IMB – Irvine Michigan Brookhaven collaboration

1.1 Discoveries in the 20th Century 3lead Mairan, in 1733, to the idea that this phenomenon mightbe of solar origin. Northern and southern lights are causedby solar electrons and protons incident in the polar regionstraveling on helical trajectories along the Earth’s magneticfield lines. At high latitudes, the charged particles essentiallyfollow the magnetic field lines. This allows them topenetrate much deeper into the atmosphere, compared toequatorial latitudes where they have to cross the field linesperpendicularly (Fig. 1.2).It is also worth mentioning that the first correct interpretationof nebulae, as accumulations of stars which formgalaxies, was given by a philosopher (Kant 1775) rather thanby an astronomer.Earth´smagnetic fieldparticle trajectoryFig. 1.2Helical trajectory of an electron inthe Earth’s magnetic field1.1 Discoveries in the 20th Century“Astronomy is perhaps the sciencewhose discoveries owe least to chance,in which human understanding appearsin its whole magnitude, and throughwhich man can best learn how small heis.”Georg Christoph LichtenbergThe discovery of X rays (Röntgen 1895, Nobel Prize 1901),radioactivity (Becquerel 1896, Nobel Prize 1903), and theelectron (Thomson 1897, Nobel Prize 1906) already indicateda particle physics aspect of astronomy. At the turn ofthe century Wilson (1900) and Elster & Geitel (1900) wereconcerned with measuring the remnant conductivity of air.Rutherford realized in 1903 that shielding an electroscopereduced the remnant conductivity (Nobel Prize 1908 for investigationson radioactive elements). It was only natural toassume that the radioactivity of certain ores present in theEarth’s crust, as discovered by Becquerel, was responsiblefor this effect.In 1910, Wulf measured a reduced intensity in an electrometerat the top of the Eiffel tower, apparently confirmingthe terrestrial origin of the ionizing radiation. Measurementsby Hess (1911/1912, Nobel Prize 1936) with balloons at altitudesof up to 5 km showed that, in addition to the terrestrialcomponent, there must also be a source of ionizing radiationwhich becomes stronger with increasing altitude (Figs. 1.3and 1.4).astronomyand particle physicscosmic rays

4 1 Historical IntroductionFig. 1.3Victor Hess at a balloon ascent formeasuring cosmic radiation {2}Fig. 1.4Robert Millikan at a take-off ofballoon experiments in Bismarck,North Dakota (1938) {3}This extraterrestrial component was confirmed by Kohlhörstertwo years later (1914). By developing the cloudchamber in 1912, Wilson made it possible to detect and followthe tracks left by ionizing particles (Nobel Prize 1927,Fig. 1.5).The extraterrestrial cosmic radiation that increases withaltitude (‘Höhenstrahlung’) has numerous experimental possibilities(Fig. 1.6) and is of special importance to the developmentof astroparticle physics.Fig. 1.5Tracks of cosmic particles in acloud chamber {4}Fig. 1.6Possibilities for experiments in thefield of cosmic rays

1.1 Discoveries in the 20th Century 5In parallel to these experimental observations, Einsteindeveloped his theories of special and general relativity (1905and 1916). The theory of special relativity is of paramountimportance for particle physics, while the prevailing domainof general relativity is cosmology. Einstein received the NobelPrize in 1921 not, however, for his fundamental theorieson relativity and gravitation, but for the correct quantummechanicalinterpretation of the photoelectric effect and theexplanation of Brownian motion. Obviously the Nobel committeein Stockholm was not aware of the outstanding importanceof the theories of relativity or possibly not evensure about the correctness of their predictions. This occurredeven though Schwarzschild had already drawn correct conclusionsfor the existence of black holes as early as 1916,and Eddington had verified the predicted gravitational bendingof light passing near the Sun during the solar eclipsein 1919. The experimental observation of the deflection oflight in gravitational fields also constituted the discovery ofgravitational lensing. This is when the image of a star appearsto be displaced due to the gravitational lensing of lightthat passes near a massive object. This effect can also lead todouble, multiple, or ring-shaped images of a distant star orgalaxy if there is a massive object in the line of sight betweenthe observer on Earth and the star (Fig. 1.7). It was only in1979 that multiple images of a quasar (double quasar) couldbe observed. This was followed in 1988 by an Einstein ringin a radio galaxy, as predicted by Einstein in 1936.theories of relativityblack holesgravitational lensingapparent positiondouble imagesEinstein ringobserversunstarFig. 1.7Gravitational lensing by a massiveobject:a) deflection of light,b) double images,c) Einstein ringIn the field of astronomy, stars are classified accordingto their brightness and colour of the spectrum (Hertzsprung–Russell diagram 1911). This scheme allowed a better understandingof the stellar evolution of main-sequence stars tostellar evolutionHertzsprung–Russelldiagram

6 1 Historical Introductionintensity [10 cm s sr ]-3 -2 -1 -1expansion of the universepenetration of cosmic raysinto the atmosphere87latitude effectred giants and white dwarves. In 1924 Hubble was able toconfirm Kant’s speculation that ‘nebulae’ are accumulationsof stars in galaxies, by resolving individual stars in the AndromedaNebula. Only a few years later (1929), he observedthe redshift of the spectral lines of distant galaxies, therebydemonstrating experimentally that the universe is expanding.In the meantime, a clearer picture about the nature ofcosmic rays had emerged. Using new detector techniquesin 1926, Hoffmann observed particle multiplication underabsorbing layers (‘Hoffmann’s collisions’). In 1927, Claydemonstrated the dependence of the cosmic-ray intensity onthe geomagnetic latitude. This was a clear indication of thecharged-particle nature of cosmic rays, since photons wouldnot have been influenced by the Earth’s magnetic field.Primary cosmic rays can penetrate deep into the atmo-sphere at the Earth’s poles, by traveling parallel to the magneticfield lines. At the equator they would feel the full componentof the Lorentz force (F = e(v × B); F –Lorentzforce, v – velocity of the cosmic-ray particle, B –Earth’smagnetic field, e – elementary charge: at the poles v ‖ Bholds with the consequence of F = 0, while at the equatorone has v ⊥ B which leads to |F |=evB). This latitudeeffect was controversial at the time, because expeditionsstarting from medium latitudes (≈ 50 ◦ north) to the equatordefinitely showed this effect, whereas expeditions to thenorth pole observed no further increase in cosmic-ray inten-sity. This result could be explained by the fact that chargedcosmic-ray particles not only have to overcome the magneticcutoff, but also suffer a certain ionization energy loss in theatmosphere. This atmospheric cutoff of about 2 GeV preventsa further increase in the cosmic-ray intensity towardsthe poles (Fig. 1.8). In 1929 Bothe and Kohlhörster could finallyconfirm the charged-particle character of cosmic raysat sea level by using coincidence techniques.In as early as 1930, Störmer calculated trajectories ofcharged particles through the Earth’s magnetic field to betterunderstand the geomagnetic effects. In these calculations,he initially used positions far away from the Earth as startingpoints for the cosmic-ray particles. He soon realized,however, that most particles failed to reach sea level dueto the action of the magnetic field. The low efficiency ofthis approach led him to the idea of releasing antiparticles90° 60° 30° 0°north poleequatorgeomagnetic latitudeFig. 1.8Latitude effect: geomagnetic andatmospheric cutofftrajectoriesof charged particlesin the Earth’s magnetic field

1.1 Discoveries in the 20th Century 7from sea level to discover where the Earth’s magnetic fieldwould guide them. In these studies, he observed that particleswith certain momenta could be trapped by the magneticfield, which caused them to propagate back and forth fromone magnetic pole to the other in a process called ‘magneticmirroring’. The accumulated particles form radiation belts,which were discovered in 1958 by Van Allen with experimentson board the Explorer I satellite (Fig. 1.9, see alsoFig. 7.2).Van Allen beltsFig. 1.9Van Allen beltsThe final proof that primary cosmic rays consist predominantlyof positively charged particles was establishedby the observation of the east–west effect (Johnson and Alvarez& Compton, Nobel Prize Alvarez 1968, Nobel PrizeCompton 1927). Considering the direction of incidence ofcosmic-ray particles at the north pole, one finds a higher intensityfrom the west compared to the east. The origin of thisasymmetry relates to the fact that some possible trajectoriesof positively charged particles from easterly directions donot reach out into space (dashed tracks in Fig. 1.10). Therefore,the intensity from these directions is reduced.In 1933, Rossi showed in a coincidence experiment thatsecondary cosmic rays at sea level initiate cascades in a leadabsorber of variable thickness (‘Rossi curve’). The absorptionmeasurements in his apparatus also indicated that cosmicrays at sea level consist of a soft and a penetrating component.observation pointFig. 1.10East–west effect

8 1 Historical Introduction1.2 Discoveries of New Elementary Particles“If I would remember the names of allthese particles, I’d be a botanist.”Enrico Fermidiscovery of the positronneutrino postulateneutrino astronomyUp to the thirties, only electrons, protons (as part of thenucleus), and photons were known as elementary particles.The positron was discovered in a cloud chamber by Andersonin 1932 (Nobel Prize 1936). This was the antiparticleof the electron, which was predicted by Dirac in 1928 (NobelPrize 1933). This, and the discovery of the neutron byChadwick in 1932 (Nobel Prize 1935), started a new chapterin elementary particle and astroparticle physics. Additionallyin 1930, Pauli postulated the existence of a neutral,massless spin- 1 2particle to restore the validity of the energy,momentum, and angular-momentum conservation laws thatappeared to be violated in nuclear beta decay (Nobel Prize1945). This hypothetical enigmatic particle, the neutrino,could only be shown to exist in a reactor experiment in 1956(Cowan & Reines, Nobel Prize 1995). It eventually lead to acompletely new branch of astronomy; neutrino astronomy isa classic example of a perfect interplay between elementaryparticle physics and astronomy.It was reported that Landau (Nobel Prize 1962), withinseveral hours of hearing about the discovery of the neutron,predicted the existence of cold, dense stars which consistedmainly of neutrons. In 1967, the existence of rotating neu-tron stars (pulsars) was confirmed by observing radio signals(Hewish and Bell, Nobel Prize for Hewish 1975).Neutrons in a neutron star do not decay. This is becausethe Pauli exclusion principle (1925) forbids neutrons to decayinto occupied electron states. The Fermi energy of remnantelectrons in a neutron star is at around several 100 MeV,while the maximum energy transferred to electrons in neutrondecay is 0.77 MeV. There are therefore no vacant electronlevels available.After discovering the neutron, the second building blockof the nucleus, the question of how atomic nuclei could sticktogether arose. Although neutrons are electrically neutral,the protons would electrostatically repel each other. Basedon the range of the nuclear force and Heisenberg’s uncertaintyprinciple (1927, Nobel Prize 1932), Yukawa conjecturedin 1935 that unstable mesons of 200-fold electron masscould possibly mediate nuclear forces (Nobel Prize 1949).neutron starspulsarsstability of a neutron starYukawa particle

1.2 Discoveries of New Elementary Particles 9Initially it appeared that the muon discovered by Andersonand Neddermeyer in a cloud chamber in 1937, had the requiredproperties of the hypothetical Yukawa particle. Themuon, however, has no strong interactions with matter, andit soon became clear that the muon was a heavy counterpartof the electron. The fact that another electron-like particleexisted caused Rabi (Nobel Prize 1944) to remark: “Whoordered this?” Rabi’s question remains unanswered to thisday. The situation became even more critical when Perl (NobelPrize 1995) discovered another, even heavier lepton, thetau, in 1975.The discovery of the strongly interacting charged pions(π ± ) in 1947 by Lattes, Occhialini, Powell, and Muirhead,using nuclear emulsions exposed to cosmic rays at mountainaltitudes, solved the puzzle about the Yukawa particles(Nobel Prize 1950 to Cecil Powell for his development ofthe photographic method of studying nuclear processes andhis discoveries regarding mesons made with this method).The pion family was supplemented in 1950 by the discoveryof the neutral pion (π 0 ). Since 1949, pions can also beproduced in particle accelerators.Up to this time, elementary particles were predominantlydiscovered in cosmic rays. In addition to the muon(µ ± ) and the pions (π + ,π − ,π 0 ), tracks of charged andneutral kaons were observed in cloud-chamber events. Neutralkaons revealed themselves through their decay into twocharged particles. This made the K 0 appear as an upsidedown ‘V’, because only the ionization tracks of the chargeddecay products of the K 0 were visible in the cloud chamber(Rochester & Butler 1947, Fig. 1.11).In 1951, part of the Vs were recognized as Lambdabaryons, which also decayed relatively quickly into twocharged secondaries (Λ 0 → p + π − ). In addition, the Ξand Σ hyperons were discovered in cosmic rays (Ξ: Armenteroset al., 1952; Σ: Tomasini et al., 1953).Apart from studying local interactions of cosmic-rayparticles, their global properties were also investigated. Theshowers observed under lead plates by Rossi were alsofound in the atmosphere (Pfotzer, 1936). The interactions ofprimary cosmic rays in the atmosphere initiate extensive airshowers, see Sect. 7.4, (Auger, 1938). These showers leadto a maximum intensity of cosmic rays at altitudes of 15 kmabove sea level (‘Pfotzer maximum’, Fig. 1.12).One year earlier (1937), Bethe and Heitler, and at thesame time Carlson and Oppenheimer, developed the theorydiscovery of muonsdiscovery of the taudiscovery of pionsFig. 1.11Decays of neutral kaons in a cloudchamber {4}V particlesbaryons and hyperonsFig. 1.12Intensity profile of cosmic particlesin the atmosphere

10 1 Historical Introductionsolar windenergy source of starscompressedmagnetic fieldnuclear fusionsolar windtail of Earth´smagnetic fieldFig. 1.13Influence of the solar wind on theEarth’s magnetic fieldFermiacceleration mechanismchemical compositionof primary cosmic raysdiscovery of antiparticlesof electromagnetic cascades, which was successfully used todescribe the extensive air showers.In 1938, Bethe together with Weizsäcker, solved thelong-standing mystery of the energy generation in stars. Thefusion of protons leads to the production of helium nuclei,in which the binding energy of 6.6 MeV per nucleon is released,making the stars shine (Nobel Prize 1967).In 1937, Forbush realised that a significant decrease ofthe cosmic-ray intensity correlated with an increased solaractivity. The active Sun appears to create some sort of solarwind which consists of charged particles whose flux generatesa magnetic field in addition to the geomagnetic field.The solar activity thereby modulates the galactic componentof cosmic rays (Fig. 1.13).The observation that the tails of comets always pointaway from the Sun led Biermann to conclude in 1951, thatsome kind of solar wind must exist. This more or less continuousparticle flux was first directly observed by the Mariner2 space probe in 1962. The solar wind consists predominantlyof electrons and protons, with a small admixture of αparticles. The particle intensities at a distance of one astronomicalunit (the distance from Sun to Earth) are 2 × 10 8ions/(cm 2 s). This propagating solar plasma carries part ofthe solar magnetic field with it, thereby preventing some primarycosmic-ray particles to reach the Earth.In 1949 it became clear that primary cosmic rays consistedmainly of protons. Schein, Jesse, and Wollan used balloonexperiments to identify protons as the carriers of cosmicradiation.Fermi (Nobel Prize in 1938 for experiments on radioactivityand the theory of nuclear beta decay) investigated theinteractions of cosmic-ray particles with atmospheric atomicnuclei and with the solar and terrestrial magnetic fields. Byas early as 1949, he also had considered possible mechanismsthat accelerated cosmic-ray particles to very high energies.Meanwhile, it had been discovered that in addition toelectrons, protons, and α particles, the whole spectrum ofheavy nuclei existed in cosmic radiation (Freier, Bradt, Peters,1948). In 1950, ter Haar discussed supernova explosionsas the possible origin of cosmic rays, an idea that waslater confirmed by simulations and measurements.After discovering the positron in 1932, the antiproton,the second known antiparticle, was found in an accelerator

1.3 Start of the Satellite Era 11experiment by Chamberlain and Segrè in 1955 (Nobel Prizein 1959). Positrons (Meyer & Vogt, Earl, 1961) and antiprotons(Golden, 1979) were later observed in primary cosmicrays. It is, however, assumed that these cosmic-ray antiparticlesdo not originate from sources consisting of antimatter,but are produced in secondary interactions between primarycosmic rays and the interstellar gas or in the upper layers ofthe atmosphere.1.3 Start of the Satellite Era“Having probes in space was like havinga cataract removed.”Hannes AlfvénThe launch of the first artificial satellite (Sputnik, October4th, 1957) paved the way for developments that providedcompletely new opportunities in astroparticle physics. Theatmosphere represents an absorber with a thickness of ≈ 25radiation lengths. The observation of primary X rays andgamma radiation was previously impossible due to their absorptionin the upper layers of the atmosphere. This electromagneticradiation can only be investigated – undisturbedby atmospheric absorption – at very high altitudes near the‘top’ of the atmosphere. It still took some time until the firstX-ray satellites (e.g., 1970 Uhuru, 1978 Einstein Observatory,1983 Exosat; Nobel Prize for R. Giacconi 2002) andgamma satellites (e.g., 1967 Vela, 1969 OSO-3, 1972 SAS-2, 1975 COS-B) 3 were launched. They provided a wealthof new data in a hitherto unaccessible spectral range. Thegalactic center was found to be bright in X rays and gammarays, and the first point sources of high-energy astroparticlescould also be detected (Crab Nebula, Vela X1, Cygnus X3,...).With the discovery of quasistellar radio sources (quasars,1960), mankind advanced as far as to the edge of the universe.Quasars appear to outshine whole galaxies if they arereally located at cosmological distances. Their distance isdetermined from the redshift of their spectral lines. The mostdistant quasar currently known, was discovered in 2001 andhas a redshift of z = λ−λ 0λ 0= 6.28. An object even fartheraway is the galaxy Abell 1835 IR 1916 with a redshift3 OSO – Orbiting Solar ObservatorySAS – Small Astronomy SatelliteX-ray satellitesγ satellitequasars

12 1 Historical IntroductionBigBangtheorysteady-state universecosmic background radiationof z = 10. Its discovery was made possible through lightamplification by a factor of about 50 resulting from stronggravitational lensing by a very massive galactic cluster in theline of sight to the distant galaxy [1]. As will be discussedin Chap. 8, this implies that the quasar is seen in a statewhen the universe was less than 5% of its present age. Consequently,this quasar resides at a distance of 13 billion lightyears.4 Initially, there was some controversy about whetherthe observed quasar redshifts were of gravitational or cosmologicalorigin. Today, there is no doubt that the observedlarge redshifts are a consequence of the Hubble expansionof the universe.The expansion of the universe implies that it began ina giant explosion, some time in the past. Based on this BigBang hypothesis, one arrives at the conclusion that this musthave occurred about 15 billion years ago. The Big Bangmodel was in competition with the idea of a steady-stateuniverse for quite some time. The steady-state model wasbased on the assumption that the universe as a whole wastime independent with new stars being continuously createdwhile old stars died out. On the other hand, Gamowhad been speculating since the forties that there should bea residual radiation from the Big Bang. According to hisestimate, the temperature of this radiation should be in therange of a few Kelvin. Penzias and Wilson (Nobel Prize1978) detected this echo of the Big Bang by chance in 1965,while they were trying to develop low-noise radio antennae(Fig. 1.14). 5 With this discovery, the Big Bang model finallygained general acceptance. The exact temperature ofthis blackbody radiation was measured by the COBE 6 satellitein 1992 as 2.726 ± 0.005 Kelvin. 74 It has become common practice in the scientific literature thatthe number 10 9 is called a billion, while in other countries thebillion is 10 12 . Throughout this book the notation that a billionis equal to a thousand millions is used.5 The excrements of pigeons presented a severe problem duringan attempt to reduce the noise of their horn antenna. When,after a thorough cleaning of the whole system, a residual noisestill remained, Arno Penzias was reported to have said: “Eitherwe have seen the birth of the universe, or we have seen a pileof pigeon shit.”6 COBE – COsmic ray Background Explorer7 The presently (2004) most accurate value of the blackbodytemperature is 2.725 ± 0.001 K.

1.3 Start of the Satellite Era 13COBE also found spatial asymmetries of the 2.7 Kelvinblackbody radiation at a level of T /T ≈ 10 −5 .Thisimpliesthat the early universe had a lumpy structure, whichcan be considered as a seed for galaxy formation.In parallel with the advance of cosmology, the famoustwo-neutrino experiment of Lederman, Schwartz, and Steinbergerin 1962 (Nobel Prize 1988) represented an importantstep for the advancement of astroparticle physics. This experimentdemonstrated that the neutrino emitted in nuclearbeta decay is not identical with the neutrino occurring inpion decay (ν µ ̸= ν e ). At present, three generations of neutrinosare known (ν e , ν µ ,andν τ ). The direct observationof the tau neutrino was established only relatively recently(July 2000) by the DONUT 8 experiment.The observation of solar neutrinos by the Davis experimentin 1967 marked the beginning of the disciplineof neutrino astronomy (Nobel Prize for R. Davis 2002).In fact, Davis measured a deficit in the flux of solar neutrinos,which was confirmed by subsequent experiments,GALLEX 9 ,SAGE 10 , and Kamiokande (Nobel Prize for M.Koshiba 2002). It is considered unlikely that a lack of understandingof solar physics is responsible for the solar neutrinoproblem. In 1958 Pontecorvo highlighted the possibilityof neutrino oscillations. Such oscillations (ν e → ν µ )are presently generally accepted as explanation of the solarneutrino deficit. This would imply that neutrinos havea very small non-vanishing mass. In the framework of theelectroweak theory (Glashow, Salam, Weinberg 1967; NobelPrize 1979) that unifies electromagnetic and weak interactions,a non-zero neutrino mass was not foreseen. Theintroduction of quarks as fundamental constituents of matter(Gell-Mann and Zweig 1964, Nobel Prize for Gell-Mann1969), and their description by the theory of quantum chromodynamicsextended the electroweak theory to the StandardModel of elementary particles (Veltman, t’Hooft; NobelPrize 1999).In this model, the masses of elementary particles cannotbe calculated a priori. Therefore, small non-zero neutrinomasses should not represent a real problem for the standardmodel, especially since it contains 18 free parameters that8 DONUT – Direct Observation of NU Tau (ν τ )9 GALLEX – German–Italian GALLium EXperiment10 SAGE – Soviet American Gallium ExperimentFig. 1.14Penzias and Wilson in front of theirhorn antenna used for measuring ofthe blackbody radiation {5}neutrino oscillationelectroweak theoryquarksStandard Model

14 1 Historical Introductioncharmed mesonsfourth quarkCygnus X1 andgravitational singularitiesblack hole in Cygnus X1?Hawking radiationevaporation of black holesFig. 1.15Decay of an excited charm particle(ψ ′ → ψ + π + + π − , with thesubsequent decay ψ → µ + + µ − )have to be determined by experimental information. However,three neutrino generations with non-zero mass wouldadd another 7 parameters (three for the masses and four mixingparameters). It is generally believed that the standardmodel will not be the final word of the theoreticians.The discovery of charmed mesons in cosmic rays (Niu etal. 1971) and the confirmation, by accelerator experiments,for the existence of a fourth quark (Richter & Ting 1974,Nobel Prize 1976, Fig. 1.15) extended the standard modelof Gell-Mann and Zweig (up, down, strange, and charm).The theory of general relativity and Schwarzschild’sideas on the formation of gravitational singularities weresupported in 1970 by precise investigations of the strong X-ray source Cygnus X1. Optical observations of Cygnus X1indicated that this compact X-ray source is ten times moremassive than our Sun. The rapid variation in the intensityof X rays from this object leads to the conclusion that thissource only has a diameter of about 10 km. A typical neutronstar has a similar diameter to this, but is only three times asheavy as the Sun. An object that was as massive as CygnusX1 would experience such a large gravitational contraction,which would overcome the Fermi pressure of degenerateneutrons. This leads to the conclusion that a black hole mustreside at the center of Cygnus X1.By 1974, Hawking had already managed to unify someaspects of the theory of general relativity and quantumphysics. He was able to show that black holes could evap-orate by producing fermion pairs from the gravitational energyoutside the event horizon. If one of the fermions escapedfrom the black hole, its total energy and thereby itsmass would be decreased (Hawking radiation). The timeconstants for the evaporation process of massive black holes,however, exceed the age of the universe by many orders ofmagnitude.There were some hopes that gravitational waves, whichwould be measured on Earth, could resolve questions on theformation of black holes and other cosmic catastrophes.These hopes were boosted by gravitational-wave experimentsby Weber in 1969. The positive signals of these earlyexperiments have, so far, not been confirmed. It is generallybelieved that the findings of Weber were due to mundaneexperimental backgrounds.In contrast, Taylor and Hulse succeeded in providing indirectevidence for the emission of gravitational waves in

1.3 Start of the Satellite Era 151974, by observing a binary star system that consisted ofa pulsar and a neutron star (Nobel Prize 1993). They wereable to precisely test the predictions of general relativity usingthis binary star system. The rotation of the orbital ellipse(periastron rotation) of this system is ten thousand timeslarger than the perihelion rotation of the planet Mercury. Thedecreasing orbital period of the binary is directly related tothe energy loss by the emission of gravitational radiation.The observed speeding-up rate of the orbital velocities ofthe partners of the binary system and the slowing-down rateof the orbital period agree with the prediction based on thetheory of general relativity to better than 1‰.It is to be expected that there are processes occurring inthe universe, which lack an immediate explanation. This wasunderlined by the discovery of gamma-ray bursters (GRB)in 1967. It came as a surprise when gamma-ray detectorson board military reconnaissance satellites, which were inorbit to check possible violations of the test-ban treaty onthermonuclear explosions, observed γ bursts. This discoverywas withheld for a while due to military secrecy. However,when it became clear that the γ bursts did not originatefrom Earth but rather from outer space, the results were published.Gamma-ray bursters light up only once and are veryshort-lived, with burst durations lasting from 10 ms to a fewseconds. It is conceivable that γ bursts are caused by supernovaexplosions or by collisions between neutron stars.It might appear that the elementary-particle aspect of astroparticlephysics has been completed by the discovery ofthe b quark (Lederman 1977) and t quark (CDF collaboration1995). There are now six known leptons (ν e , e − ; ν µ ,µ − ; ν τ , τ − ) along with their antiparticles (¯ν e , e + ; ¯ν µ , µ + ;¯ν τ , τ + ). These are accompanied by six quarks (up, down;charm, strange; top, bottom) and their corresponding six antiquarks.These matter particles can be arranged in threefamilies or ‘generations’. Measurements of the primordialdeuterium, helium, and lithium abundance in astrophysicshad already given some indication that there may be onlythree families with light neutrinos. This astrophysical resultwas later confirmed beyond any doubt by experiments at theelectron–positron collider LEP 11 in 1989 (see also Fig. 2.1).The standard model of elementary particles, with its threefermion generations, was also verified by the discovery ofgluons, the carriers of the strong force (DESY 12 , 1979), and11 LEP – Large Electron–Positron collider at CERN in Genevaγ burstertop and bottom quarksgluons

16 1 Historical IntroductionW + ,W − ,ZFig. 1.16Supernova explosion SN 1987A inthe Tarantula Nebula {6}ROSATHubble telescopeCGROthe bosons of the weak interaction (W + ,W − ,Z;CERN 131983; Nobel Prize for Rubbia and van der Meer 1984). Thediscovery of asymptotic freedom of quarks in the theory ofthe strong interaction by Gross, Politzer, and Wilczek washonored by the Nobel Prize in 2004.The observation of the supernova explosion 1987A,along with the burst of extragalactic neutrinos, representedthe birth of real astroparticle physics. The measurement ofonly 20 neutrinos out of a possible 10 58 emitted, allowedelementary particle physics investigations that were hithertoinaccessible in laboratory experiments. The dispersion ofarrival times enabled physicists to derive an upper limit ofthe neutrino mass (m νe < 10 eV). The mere fact that theneutrino source was 170 000 light-years away in the LargeMagellanic Cloud, allowed a lower limit on the neutrinolifetime to be estimated. The gamma line emission from SN1987A gave confirmation that heavy elements up to iron,cobalt, and nickel were synthesized in the explosion, inagreement with predictions of supernova models. As the firstoptically visible supernova since the discovery of the telescope,SN 1987A marked an ideal symbiosis of astronomy,astrophysics, and elementary particle physics (Fig. 1.16).The successful launch of the high-resolution X-ray satelliteROSAT 14 in 1990, paved the way for the discovery ofnumerous X-ray sources. The Hubble telescope, which wasstarted in the same year, provided optical images of stars andgalaxies in hitherto unprecedented quality, once the slightlydefocusing mirror had been adjusted by a spectacular repairin space. The successful mission of ROSAT was followedby the X-ray satellites Chandra (named after SubrahmanyanChandrasekhar, Nobel Prize 1983) and XMM 15both launched in 1999.The Compton Gamma-Ray Observatory (CGRO,launched in 1991) opened the door for GeV gammaastronomy. Ground-based atmospheric air Cherenkov telescopesand extensive air-shower experiments were able toidentify TeV point sources in our Milky Way (Crab Nebula,12 DESY – Deutsches Elektronen Synchrotron in Hamburg13 CERN – Conseil Européen pour la Recherche Nucléaire14 ROSAT – ROentgen SATellite of the Max-Planck Institute forExtraterrestrial Physics, Munich15 XMM – X-ray Multi-Mirror mission, renamed Newton Observatoryin 2002

1.4 Open Questions 171989) and at extragalactical distances (1992, Markarian421, Markarian 501). The active galactic nuclei of theMarkarian galaxies are also considered excellent candidatesources of high-energy hadronic charged cosmic rays.1.4 Open Questions“We will first understand how simplethe universe is, when we realize, howstrange it is.”AnonymousA still unsolved question of astroparticle physics is the problemof dark matter and dark energy. From the observationof orbital velocities of stars in our Milky Way and the velocitiesof galaxies in galactic clusters, it is clear that the energydensity of the visible matter in the universe is insufficient tocorrectly describe the dynamics (Fig. 1.17).Since the early nineties, the MACHO 16 and EROS 17 experimentshave searched for compact, non-luminous, Jupiterlikeobjects in the halo of our Milky Way, using the techniqueof microlensing. Some candidates have been found,but their number is nowhere near sufficient to explain themissing dark matter in the universe. One can conjecture thatexotic, currently unknown particles (supersymmetric particles,WIMPs 18 , ...), or massive neutrinos may contributeto solve the problem of the missing dark matter. A nonvanishingvacuum energy density of the universe is alsoknown to play a decisive rôle in the dynamics and evolutionof the universe.In 1998 the Super-Kamiokande experiment found evidencefor a non-zero neutrino mass by studying the relativeabundances of atmospheric electron and muon neutrinos.The observed deficit of atmospheric muon neutrinosis most readily and elegantly explained by the assumptionthat neutrinos oscillate from one lepton flavour to another(ν µ → ν τ ). This is only possible if neutrinos have mass.The presently favoured mass of 0.05 eV for ν τ ,however,isinsufficient to explain the dynamics of the universe alone.16 MACHO – search for MAssive Compact Halo Objects17 EROS – Expérience pour la Recherche d’Objets Sombres18 WIMP – Weakly Interacting Massive ParticlesFig. 1.17Orbital velocities of stars in theMilky Way in comparison withKeplerian trajectoriesMACHO, EROSsupersymmetric particlesWIMPsnon-zero neutrino masses

18 1 Historical IntroductionHiggs particleThe oscillation scenario for solar neutrinos was confirmedin 2001 by the SNO 19 experiment by showing thatthe total flavour-independent neutrino flux from the Sun arrivingat Earth (ν e ,ν µ ,ν τ ) was consistent with solar-modelexpectations, demonstrating that some of the solar electronneutrinos had oscillated into a different neutrino flavour.The generation of the masses for elementary particles isstill an open question. In the standard model of electroweakand strong interactions, the mass generation is believed tocome about by a spontaneous symmetry breaking, thesocalledHiggs mechanism. This process favours the existenceof at least one additional massive neutral boson. Whether theLEP experiments at CERN have seen this enigmatic particleat the kinematic limit of LEP, with a mass of about 115 GeV,needs to be confirmed by future hadron colliders.A very recent, equally exciting discovery, is the measurementof the acceleration parameter of the universe.Based on the ideas of the classical Big Bang, one would assumethat the initial thrust of the explosion would be sloweddown by gravitation. Observations on distant supernova ex-plosions (1998) however, appeared to indicate that in earlycosmological epochs, the rate of expansion was smaller thantoday. The finding of an accelerating universe –whichisnow generally accepted – has important implications forcosmology. It suggests that the largest part of the missingdark matter is stored as dark energy in a dynamical vacuum(‘quintessence’).Finally, it should be highlighted that the discovery of ex-trasolar planets (Mayor and Queloz 1995) has led to the resumptionof discussions on the existence of extraterrestrialintelligence. So far, the nearest extrasolar planet (‘Millennium’)has been observed in the Tau Boötis solar system,by the Herschel telescope on the Canary Islands. The planetis twice as large as Jupiter and eight times as heavy, and issituated at a distance of 55 light-years. Until now about 100extrasolar planets have been discovered. Possibly we are notthe only intelligent beings in the universe pursuing astroparticlephysics.accelerating universe?extrasolar planets19 SNO – Sudbury Neutrino Observatory

1.5 Problems 191.5 Problems1. Work out thea) velocity of an Earth satellite in a low-altitude orbit,b) the escape velocity from Earth,c) the altitude of a geostationary satellite above groundlevel. Where can such a geostationary satellite bepositioned?2. What is the bending radius of a solar particle (proton,1 MeV kinetic energy) in the Earth’s magnetic field(0.5 Gauss) for vertical incidence with respect to thefield? Use the relation between the centrifugal force andthe Lorentz force (6.1), and argue whether a classicalcalculation or a relativistic calculation is appropriate.3. Estimate the average energy loss of a muon in the atmosphere(production altitude 20 km, muon energy ≈10 GeV; check with Fig. 4.2).4. What is the ratio of intensities of two stars which differby one unit in magnitude only (for the definition of themagnitude see the Glossary)?5. Small astronomical objects like meteorites and asteroidsare bound by solid-state effects while planets are boundby gravitation. Estimate the minimum mass from whereon gravitational binding starts to dominate as bindingforce. Gravitational binding dominates if the potentialgravitational energy exceeds the total binding energy ofthe solid material, where the latter is taken to be proportionalto the number of atoms in the object. The averageatomic number is A, from which together with the Bohrradius r B the average density can be estimated.6. There is a statement in this chapter that a quasar at aredshift of z = 6.68 gives us information on the universewhen it was only about 3% of its present age. Can youconvert the redshift into the age of the universe?

212 The Standard Modelof Elementary Particles“Most basic ideas of science are essentiallysimple and can usually be expressed in alanguage that everyone understands.”Albert EinsteinOver the last years a coherent picture of elementary particleshas emerged. Since the development of the atomic model,improvements in experimental resolution have allowed scientiststo investigate smaller and smaller structures. Even theatomic nucleus, which contains practically the total mass ofthe atom, is a composite object. Protons and neutrons, thebuilding blocks of the nucleus, have a granular structure thatbecame obvious in electron–nucleon scattering experiments.In the naïve quark parton model, a nucleon consists of threequarks. The onion-type phenomenon of ever smaller constituentsof particles that were initially considered to be fundamentaland elementary, may have come to an end withthe discovery of quarks and their dynamics. While atoms,atomic nuclei, protons, and neutrons can be observed as freeparticles in experiments, quarks can never escape from theirhadronic prison. In spite of an intensive search by numerousexperiments, nobody has ever been able to find free quarks.Quantum chromodynamics, which describes the interactionof quarks, only allows the asymptotic freedom of quarks athigh momenta. Bound quarks that are inside nucleons typicallyhave low momenta and are subject to ‘infrared slavery’.This confinement does not allow the quarks to separatefrom each other.Quarks are constituents of strongly interacting hadronicmatter. The size of quarks is below 10 −17 m. In additionto quarks, there are leptons that interact weakly and electromagnetically.With the resolution of the strongest microscopes(accelerators and storage rings), quarks and leptonsappear to be pointlike particles, having no internal structure.Three different types of leptons are known: electrons,muons, and taus. Each charged lepton has a separate neutrino:ν e , ν µ , ν τ . Due to the precise investigations of the Zparticle, which is the neutral carrier of weak interactions, itis known that there are exactly three particle families withcomposition of atomic nucleicomposition of nucleonsquark confinementquarks and leptons

22 2 The Standard Model of Elementary Particleslight neutrinos (Fig. 2.1). This result was obtained fromthe measurement of the total Z decay width. According toHeisenberg’s uncertainty principle, the resolution of complementaryquantities is intrinsically limited by Planck’sconstant (h = 6.626 0693×10 −34 J s). The relation betweenthe complementary quantities of energy and time isE t ≥ ¯h/2 (¯h = h/2π) . (2.1)Fig. 2.1Determination of the number ofneutrino generations from Z decayThese decays can occur even if the charged leptons l x associatedwith the respective generation are too heavy to beproduced in Z decay. A large number of different light neu-trinos will consequently reduce the Z lifetime, thereby increasingits decay width. The exact measurement of the Zdecay width took place at the LEP storage ring (Large Electron–Positroncollider) in 1989, enabling the total numberof neutrino generations to be determined: there are exactlythree lepton generations with light neutrinos.The measurement of the primordial helium abundancehad already allowed physicists to derive a limit for the numberof neutrino generations. The nucleosynthesis in the earlyuniverse was essentially determined by the number of relativisticparticles, which were able to cool down the universeafter the Big Bang. At temperatures of ≈ 10 10 K, whichcorrespond to energies where nucleons start to bind in nuclei(≈ 1 MeV), these relativistic particles would have consistedof protons, neutrons, electrons, and neutrinos. If manydifferent neutrino flavours exist, a large amount of energywould have escaped from the original fireball, owing to thelow interaction probability of neutrinos. This has the consequencethat the temperature would have decreased quickly.A rapidly falling temperature means that the time taken forneutrons to reach nuclear binding energies would have beenvery short, and consequently they would have had very littletime to decay (lifetime τ n = 885.7 s). If there were manyneutrons that did not decay, they would have been able toform helium together with stable protons. The primordialhelium abundance is therefore an indicator of the numberneutrino generationsprimordialhelium abundancenucleosynthesisIf t = τ is the lifetime of the particle, relation (2.1) impliesthat the decay width E = Γ is larger when τ is shorter. Ifthere are many generations of light neutrinos, the Z particlecan decay into all these neutrinos,Z → ν x +¯ν x . (2.2)

2 The Standard Model of Elementary Particles 25are compared in Table 2.3. It is apparent that gravitation canbe completely neglected in the microscopic domain, becauseits strength in relation to strong interactions is only 10 −40 .weakness of gravitationinteraction→ gravitation electroweak strongpropertyelectromagneticweak↓acts on mass–energy flavouraffectedparticlesexchangeparticlerelativestrengthallquarks,leptonselectricchargeallchargedparticlescolourchargequarks,gluonsgraviton G W + , W − , Z γ gluons g10 −40 10 −5 10 −2 1range ∞ ≈ 10 −3 fm ∞ ≈ 1fmexamplesystemEarth–Moonβ decayatomicbindingnuclearbindingTable 2.3Properties of interactionsIn the primitive quark model, all strongly interactingparticles (hadrons) are composed of valence quarks. Abaryon is a three-quark system, whereas a meson consistsof a quark and an antiquark. Examples of baryons includethe proton, which is a uud system, and the neutron is a uddcomposite. Correspondingly, an example of a meson is thepositively charged pion, which is a u ¯d system. The existenceof baryons consisting of three identical quarks with parallelspin (Ω − = (sss),spin 3 2 ¯h) indicates that quarks must havea hidden quantum number, otherwise the Pauli exclusionprinciple would be violated. This hidden quantum number iscalled colour. Electron–positron interactions show that thereare exactly three different colours. Each quark thereforecomes in three colours, however all observed hadrons haveneutral colour. If the three degrees of freedom in colourare denoted by red (r), green (g), and blue (b), the protonis a composite object made up from u red u green d blue . Inaddition to valence quarks, there is also a sea of virtualquark–antiquark pairs in hadrons.valence quarkshidden quantum numberscolour of quarkssea quarks

}28 2 The Standard Model of Elementary ParticlesMuon decay can be described in a similar fashion (Fig. 2.8).The muon transfers its charge to a W − , thereby transform-ing itself into the neutral lepton of the second family, the ν µ .The W − in turn decays again into in e − ¯ν e .muon decayFig. 2.7Neutron decayFig. 2.8Muon decaynuu dd}pd uW –e – ee – – W – e }Finally, pion decay will be discussed (Fig. 2.9). In prin-ciple, the W + can also decay in this case, into an e + ν estate. Helicity reasons, however, strongly suppress this decay:as a spin-0 particle, the pion decays into two leptonsthat must have antiparallel spins due to angular-momentumconservation. The helicity is the projection of the spin ontothe momentum vector, and it is fixed for the neutrino (formassless particles the spin is either parallel or antiparallel tothe momentum). Particles normally carry negative helicity(spin ‖−p, left-handed) so that the positron, as an antiparticle(spin ‖ p, right-handed), must take on an unnaturalhelicity (Fig. 2.10). The probability of carrying an abnormalhelicity is proportional to 1 − v c(where v is velocityof the charged lepton). Owing to the relatively high mass ofthe muon (m µ ≫ m e ), it takes on a much smaller velocitycompared to the electron in pion decay, i.e., v(µ) ≪ v(e).The consequence of this is that the probability for the decaymuon to take on an unnatural helicity is much larger comparedto the positron. For this reason, the π + → e + ν e decayis strongly suppressed compared to the π + → µ + ν µ decay(the suppression factor is 1.23 × 10 −4 ).The various elementary particles are characterized byquantum numbers. In addition to the electric charge, themembership of a quark generation (quark flavour) or leptongeneration (lepton number) is introduced as a quantum num-ber. Leptons are assigned the lepton number +1 intheirrespectivegeneration, whereas antileptons are given the leptonnumber −1. Lepton numbers for the different lepton families(L e , L µ , L τ ) are separately conserved, as is shown inthe example of the muon decay:pion decay +uW + + dFig. 2.9Pion decay espin e + e +spin e +Fig. 2.10Helicity conservation in π + decayquantum numberslepton numberµ − → ν µ + e − + ¯ν eL µ 1 1 0 0 . (2.4)L e 0 0 1 −1

2.1 Examples of Interaction Processes 29The parity transformation P is the space inversion of aphysical state. Parity is conserved in strong and electromagneticinteractions, however, in weak interactions it is maximallyviolated. This means that the mirror state of a weakprocess does not correspond to a physical reality. Nature distinguishesbetween the right and left in weak interactions.The operation of charge conjugation C applied to a physicalstate changes all the charges, meaning that particlesand antiparticles are interchanged, whilst leaving quantitieslike momentum or spin untouched. Charge conjugation isalso violated in weak interactions. In β decay, for example,left-handed electrons (negative helicity) and right-handedpositrons (positive helicity) are favoured. Even though thesymmetry operations P and C are not conserved individually,their combination CP, which is the application of spaceinversion (parity operation P ) with subsequent interchangeof particles and antiparticles (charge conjugation C) isawell-respected symmetry. This symmetry, however, is stillbroken in certain decays (K 0 and B 0 decays), but it is a commonbelief that the CPT symmetry (CP symmetry with additionaltime inversion) is conserved under all circumstances.Some particles, like kaons, exhibit very strange behaviour.They are produced copiously, but decay relativelyslowly. These particles are produced in strong interactions,but they decay via weak interactions. This property is accountedfor by introducing the quantum number strangeness,which is conserved in strong interactions, but violated inweak decays. Owing to the conservation of strangeness instrong interactions, only the associate production of strangeparticles, i.e., the combined production of hadrons one ofwhich contains a strange and the other an anti-strange quark,is possible, such asπ − + p → K + + Σ − . (2.5)In this process, the ¯s quark in the K + (= u¯s) receives thestrangeness +1, whilst the s quark in the Σ − (= dds)is assigned the strangeness −1. In the weak decay of theK + → π + π 0 , the strangeness is violated, since pions donot contain strange quarks (s).Certain particles that behave in an identical way understrong interactions, but differ in their charge state, are integratedinto isospin multiplets. Protons and neutrons are nucleonsthat form an isospin doublet of I = 1/2. When thenucleon isospin is projected onto the z axis, the state withI z =+1/2 corresponds to a proton whereas the I z =−1/2parityparity violationcharge conjugationCP conservationin weak interactions?CP violationCPT symmetrystrange particlesstrangenessisospin multipletisospin doublet of nucleons

30 2 The Standard Model of Elementary Particlesisospin triplet of pionsbaryon numberconservation lawsof particle physicsstate relates to the neutron. The three pions (π + ,π − ,π 0 )combine to form an isospin triplet with I = 1. In this case,I z =−1 corresponds to the π − , I z =+1istheπ + , whilstI z = 0 relates to the π 0 . The particle multiplicity m in anisospin multiplet is related to the isospin via the equationm = 2I + 1 . (2.6)Finally, the baryon number should be mentioned. Quarksare assigned the baryon number 1/3, and antiquarks aregiven −1/3. All baryons consisting of three quarks aretherefore assigned the baryon number 1, whereas all otherparticles get the baryon number 0.The properties of the conservation laws for the differentinteraction types in elementary particle physics are compiledin Table 2.4.Table 2.4Conservation laws of particlephysics (conserved: +; violated: –)physicalinteractionquantity strong electromagnetic weakmomentum + + +energy (incl. mass) + + +ang. momentum + + +electric charge + + +quark flavour + + –lepton number ∗ ./. + +parity + + –charge conjugation + + –strangeness + + –isospin + – –baryon number + + +∗ the lepton number is not relevant for strong interactions}uudd}ps uW –Fig. 2.11Lambda decay: Λ → p + π −ud} –Unfortunately, there is a small but important complicationin the quark sector. As can be seen from Table 2.1, thereis a complete symmetry between leptons and quarks. Leptons,however, participate in interactions as free particles,whereas quarks do not. Due to quark confinement, spectatorquarks always participate in the interactions in some way.For charged leptons, there is a strict law of lepton-numberconservation: The members of different generations do notmix with each other. For the quarks, it was seen that weakprocesses can change the strangeness. In Λ decay, the squark belonging to the second generation can transform intoa u quark of the first generation. This would otherwise onlybe allowed to happen to a d quark (Fig. 2.11).

2.1 Examples of Interaction Processes 31It appears as if the s quark can sometimes behave likethe d quark. It is, in fact, the d ′ and s ′ quarks that couple toweak interactions, rather than the d and s quarks. The d ′ ands ′ quarks can be described as a rotation with respect to the dand s quarks. This rotation is expressed byd ′ = d cos θ C + s sin θ C ,s ′ =−d sin θ C + s cos θ C ,(2.7)where θ C is the mixing angle (Cabibbo angle).The reason that angles are used for weighting is based onthe fact that the sum of the squares of the weighting factors,cos 2 θ + sin 2 θ = 1, automatically guarantees the correctnormalization. θ C has been experimentally obtained to beapproximately 13 degrees (sin θ C ≈ 0.2235). Since cos θ C ≈0.9747, the d ′ quark predominantly behaves like the d quark,albeit with a small admixture of the s quark.The quark mixing originally introduced by Cabibbo wasextended by Kobayashi and Maskawa to all three quark families,such that d ′ , s ′ ,andb ′ are obtained from d, s, b by a rotationmatrix. This matrix is called the Cabibbo–Kobayashi–Maskawa matrix (CKM matrix),⎛ ⎞ ⎛⎝ d′s ′ ⎠ = U ⎝ d ⎞s ⎠ . (2.8)b ′ bThe elements on the main diagonal of the (3 × 3) matrixU are very close to unity. The off-diagonal elements indicatethe strength of the quark-flavour violation. A similarcomplication in the neutrino sector will be discussed later,where the eigenstates of the mass are not identical with theeigenstates of the weak interaction (see Sect. 6.2.1).The Standard Model of electroweak and strong interactionscannot be the final theory. The model contains toomany free parameters, which have to be adjusted by hand. Inaddition, the masses of all fundamental fermions are initiallyzero. They only get their masses by a mechanism of spontaneoussymmetry breaking (Higgs mechanism). Another veryimportant point to note is that gravitation is not consideredin this model at all, whereas it is the dominant force in theuniverse as a whole. There have been many attempts to formulatea Theory of Everything (TOE) that unites all interactions.A very promising candidate for such a global descriptionis the string theory. String theory is based on theassumption that elementary particles are not point-like, butCabibbo angleCKM matrixlimits of the Standard ModelHiggs mechanismTheory of Everythingstring theory

32 2 The Standard Model of Elementary Particlessupersymmetryweakness of gravityare one-dimensional strings. Different string excitations oroscillations correspond to different particles. In addition,certain string theories are supersymmetric. They establisha symmetry between fermions and bosons. String theories,and in particular superstring theories, are constructed in ahigher-dimensional space. Out of the original 11 dimensionsin the so-called M superstring theory, 7 must be compactedto a very small size, because they are not observed in nature.String theories are presently considered as best candidatesto unite quantum field theories and general relativity.They might even solve the problem of the three generationsof elementary particles. In the framework of string theoriesin eleven dimensions the weakness of gravity might be re-lated to the fact that part of the gravitational force is leakinginto extra dimensions, while, e.g., electromagnetism, in contrast,is confined to the familiar four dimensions.If gravity were really leaking into extra dimensions, theenergy sitting there could give rise to dark energy influencingthe structure of the universe (see Chap.13 on Dark Matter).Gravitational matter in extra dimensions would only bevisible by its gravitational interactions.It is also conceivable that we live in a holographic universein the sense that all informations from a higher-dimen-sional space could be coded into a lower-dimensional space,just like a three-dimensional body can be represented by atwo-dimensional hologram.In Fig. 2.12, an overview of the historical successes ofthe unification of different theories is displayed with a projectioninto the future. One assumes that with increasingtemperature (̂= energy), nature gets more and more sym-holographic universegravitationelectricityelectromagnetismSUGRAFig. 2.12Unification of all differentinteractions into a Theory ofEverything(GUT – Grand Unified Theory,SUGRA – Super Gravitation)magnetismweakinteractionstronginteractionelectroweaktheoryGUT

2.2 Problems 33metric. At very high temperatures, as existed at the time ofthe Big Bang, the symmetry was so perfect that all interactionscould be described by one universal force. The reasonthat increasingly large accelerators with higher energies arebeing constructed is to track down this universal descriptionof all forces.According to present beliefs, the all-embracing theory ofsupergravity (SUGRA) is embedded in the M theory, an 11-dimensional superstring theory. The smallest constituents ofthis superstring theory are p-dimensional objects (‘branes’)of the size of the Planck length L P = √¯hG/c3 (where Gis the gravitational constant, ¯h is Planck’s constant, and cis the velocity of light). Seven of the ten spatial dimensionsare compacted into a Calabi–Yau space. According to taste,the ‘M’ in the M theory stands for ‘membrane’, ‘matrix’,‘mystery’, or ‘mother (of all theories)’.universal forceMtheory2.2 Problems1. Which of the following reactions or decays are allowed?a) µ − → e − + γ ,b) µ + → e + + ν e +¯ν µ + e + + e − ,c) π 0 → γ + e + + e − ,d) π + → µ + + e − ,e) Λ → p + K − ,f) Σ + → n + π + ,g) K + → π + + π − + π + ,h) K + → π 0 + π 0 + e + + ν e .2. What is the minimum kinetic energy of a cosmic-raymuon to survive to sea level from a production altitudeof 20 km (τ µ = 2.197 03 µs, m µ = 105.658 37 MeV)?For this problem one should assume that all muons havethe given lifetime in their rest frame.3. Work out the Coulomb force and the gravitational forcebetween two singly charged particles of the Planck massat a distance of r = 1 fm!4. In a fixed-target experiment positrons are fired at a targetof electrons at rest. What positron energy is required toproduce a Z (m Z = 91.188 GeV)?

353 Kinematics and Cross Sections“The best way to escape a problem is tosolve it.”Alan SaportaIn astroparticle physics the energies of participating particlesare generally that high, that relativistic kinematics mustbe used. In this field of science it becomes obvious that massand energy are only different facets of the same thing. Massis a particularly compact form of energy, which is related tothe total energy of a particle by the famous Einstein relationrelativistic kinematicsE = mc 2 . (3.1)In this equation m is the mass of a particle, which moveswith the velocity v, andc is the velocity of light in vacuum.The experimental result that the velocity of light in vacuumis the maximum velocity in all inertial systems leads tothe fact that particles with velocity near the velocity of lightdo not get much faster when accelerated, but mainly onlybecome heavier,relativistic mass increasem =m 0√1 − β 2 = γm 0 . (3.2)In this equation m 0 is the rest mass, β = v/c is the particlevelocity, normalized to the velocity of light, andγ =1√1 − β 2(3.3)is the Lorentz factor. Using this result, (3.1) can also be writtenasLorentz factorE = γm 0 c 2 , (3.4)where m 0 c 2 is the rest energy of a particle. The momentumof a particle can be expressed asp = mv = γm 0 βc . (3.5)Using (3.3), the difference

36 3 Kinematics and Cross SectionsE 2 − p 2 c 2 = m2 0 c41 − β 2 (1 − β2 ) = m 2 0 c4 . (3.6)This result shows that E 2 −p 2 c 2 is a Lorentz-invariant quan-tity. This quantity is the same in all systems and it equals thesquare of the rest energy. Consequently, the total energy ofa relativistic particle can be expressed by√E = c p 2 + m 2 0 c2 . (3.7)This equation holds for all particles. For massless particlesor, more precisely, particles with rest mass zero, one obtainsE = cp . (3.8)Particles of total energy E without rest mass are also subjectto gravitation, because they acquire a mass according tom = E/c 2 . (3.9)The transition from relativistic kinematics to classical(Newtonian) mechanics (p ≪ m 0 c) can also be derivedfrom (3.7) by series expansion. The kinetic energy of a particleis obtained to√E kin = E − m 0 c 2 = c p 2 + m 2 0 c2 − m 0 c 2invariant massmass equivalentclassical approximationE 2 − p 2 c 2 = γ 2 m 2 0 c4 − γ 2 m 2 0 β2 c 4can be written as( ) p 2= m 0 c√1 2 + − m 0 c 2m 0 c(≈ m 0 c 2 1 + 1 ( ) )p2− m 0 c 22 m 0 c= p22m 0= 1 2 m 0v 2 , (3.10)in accordance with classical mechanics. Using (3.4) and(3.5), the velocity can be expressed byorv =p = c2 pγm 0 Eβ = cp E . (3.11)In relativistic kinematics it is usual to set c = 1. This simplifiesall formulae. If, however, numerical quantities haveto be calculated, the actual value of the velocity of light hasto be considered.

3.1 Threshold Energies 373.1 Threshold Energies“Energy has mass and mass representsits energy.”Albert EinsteinIn astroparticle physics frequently the problem occurs to determinethe threshold energy for a certain process of particleproduction. This requires that in the center-of-mass systemof the collision at least the masses of all particles in thefinal state of the reaction have to be provided. In storagerings the center-of-mass system is frequently identical withthe laboratory system so that, for example, the creation of aparticle of mass M in an electron–positron head-on collision(e + and e − have the same total energy E) requiresthreshold energy2E ≥ M. (3.12)If, on the other hand, a particle of energy E interacts witha target at rest as it is characteristic for processes in cosmicrays, the center-of-mass energy for such a process must firstbe calculated.For the general case of a collision of two particles withtotal energy E 1 and E 2 and momenta p 1 and p 2 the Lorentzinvariantcenter-of-mass energy E CMS can be determined using(3.7) and (3.11) in the following way:E CMS = √ s={(E 1 + E 2 ) 2 − (p 1 + p 2 ) 2} 1/2{} 1/2= E1 2 − p2 1 + E2 2 − p2 2 + 2E 1 E 2 − 2p 1 · p 2{} 1/2= m 2 1 + m2 2 + 2E 1 E 2 (1 − β 1 β 2 cos θ) . (3.13)determinationof the center-of-mass energyIn this equation θ is the angle between p 1 and p 2 . For highenergies (β 1 ,β 2 → 1andm 1 ,m 2 ≪ E 1 ,E 2 ) and not toosmall angles θ (3.13) simplifies toE CMS = √ s ≈ {2E 1 E 2 (1 − cos θ)} 1/2 . (3.14)If one particle (for example, the particle of the mass m 2 )isat rest (laboratory system E 2 = m 2 , p 2 = 0), (3.13) leads to√ s ={m21 + m 2 2 + 2E 1 m 2 }1/2 . (3.15)Using the relativistic approximation (m 2 1 ,m2 2 ≪ 2E 1 m 2 )one gets

38 3 Kinematics and Cross Sections√ s ≈√2E1 m 2 . (3.16)In such a reaction only particles with total masses M ≤ √ scan be produced.p ¯p productionExample 1: Let us assume that a high-energy cosmic-rayproton (energy E p ,momentump, rest mass m p )producesa proton–antiproton pair on a target proton at rest:p + p → p + p + p +¯p. (3.17)According to (3.13) the center-of-mass energy can becalculated as follows:√ s ={(E p + m p ) 2 − (p − 0) 2} 1/2={Ep 2 + 2m p E p + m2 p − p2} 1/2{1/2= 2m p E p + 2mp} 2 . (3.18)For the final state, consisting of three protons and oneantiproton (the mass of the antiproton is equal to themass of the proton), one has√ s ≥ 4mp . (3.19)e + e − productionFrom this the threshold energy of the incident proton canbe derived to be2m p E p + 2m 2 p ≥ 16 m2 p ,E p ≥ 7m p (= 6.568 GeV) , (3.20)Ep kin = E p − m p ≥ 6m p .For the equivalent process of e + e − pair production byan energetic electron on an electron target at rest,e − + e − → e − + e − + e + + e − , (3.21)one would get the corresponding result Ee kin ≥ 6m e .Example 2: Let us consider the photoproductionof an elec-tron–positron pair on a target electron at rest,photo pair productionγ + e − → e − + e + + e − ; (3.22)√ s ={m2e + 2E γ m e } 1/2 ≥ 3m e ,E γ ≥ 4m e ,E γ ≥ 2.04 MeV . (3.23)

3.2 Four-Vectors 39Example 3: Consider the photoproductionof a neutral pion(mass m π 0 ≈ 135 MeV) on a target proton at rest (massm p ):π 0 productionγ + p → p + π 0 ; (3.24)√ s ={m2p + 2E γ m p } 1/2 ≥ m p + m π 0 ,m 2 p + 2E γ m p ≥ m2 p + m2 π 0 + 2m p m π 0 ,E γ ≥ 2m p m π 0 + m 2 π 02m p= m π 0 + m2 π 02m p(3.25)≥ m π 0 + 9.7MeV≈ 145 MeV .3.2 Four-Vectors“The physicist in preparing for his workneeds three things, mathematics, mathematics,and mathematics.”Wilhelm Conrad RöntgenFor calculations of this kind it is practical to introduceLorentz-invariant four-vectors. In the same way as time tand the position vector s = (x,y,z) can be combined toform a four-vector, also a four-momentum vector( Eq = with p = (p x ,p y ,p z ) (3.26)p)can be introduced. Because ofLorentz-invariantfour-vectorse +e – *( ) E 2q 2 = = E 2 − p 2 = m 2 0 (3.27)pthe square of the four-momentum is equal to the square ofthe rest mass. For photons one hasq 2 = E 2 − p 2 = 0 . (3.28)Those particles, which fulfill (3.27) are said to lie on themass shell. On-shell particles are also called real. Apart fromthat, particles can also borrow energy for a short time fromthe vacuum within the framework of Heisenberg’s uncertaintyprinciple. Such particles are called virtual. They arenot on the mass shell. In interaction processes virtual particlescan only occur as exchange particles.nucleusnucleus´Fig. 3.1The processγ + nucleus → e + + e − + nucleus ′mass shell

40 3 Kinematics and Cross Sectionse –e + e − pair productionin the Coulomb fieldof a nucleuse − p scattering e –Example 4: Photoproduction of an electron–positron pairin the Coulomb field of a nucleusIn this example the incoming photon γ is real, while thephoton γ ∗ exchanged between the electron and the nucleusis virtual (Fig. 3.1).Example 5: Electron–proton scattering (Fig. 3.2)The virtuality of the exchanged photon γ ∗ can easilybe determined from the kinematics based on the fourmomentumvectors of the electron and proton. The fourmomentumvectors are defined in the following way: incomingelectron q e = ( E e)p e , final-state electron q′e =( E ′ )ep , incoming proton ′ qp = ( E p)ep p , final-state protonq p ′ = ( E p′ )p . Since energy and momentum are conserved,′ palso four-momentum conservation holds:q e + q p = q ′ e + q′ p . (3.29)ppThe four-momentum squared of the exchanged virtualphoton qγ 2 ∗ is determined to beFig. 3.2The process e − + p → e − + p qγ 2 ∗ = (q e − q′ e )2(Ee − E=e′ ) 2p e − p ′ = (E e − E e ′ )2 − (p e − p ′ e )2e= Ee 2 − p2 e + E′2 e − p′2 e − 2E e E′ e + 2p e · p′ ee + += 2m 2 e − 2E e E′ e (1 − β e β′ e cos θ) , (3.30)e – –(3.30) is simplified to *where β e and β e ′ are the velocities of the incoming andoutgoing electron and θ is the angle between p e and p ′ e .For high energies and not too small scattering anglesFig. 3.3The process e + e − → µ + µ − q2 γ ∗ =−2E e E′ e (1 − cos θ)=−4E e E ′ e sin2 θ 2 . (3.31)If sin θ 2 can be approximated by θ 2, one gets for not toosmall anglesq 2 γ ∗ =−E e E′ e θ 2 . (3.32)space-like photonsThe mass squared of the exchanged photon in this caseis negative! This means that the mass of γ ∗ is purelyimaginary. Such photons are called space-like.

3.2 Four-Vectors 41Example 6: Muon pair production in e + e − interactions(Fig. 3.3)Assuming that electrons and positrons have the sametotal energy E and opposite momentum (p e + =−p e −),one has()qγ 2 ∗ = (q e + + q E + E 2e −)2 =p e + + (−p e +)= 4E 2 . (3.33)In this case the mass of the exchanged photon is 2E,which is positive. Such a photon is called time-like. Themuon pair in the final state can be created if 2E ≥ 2m µ .The elegant formalism of four-momentum vectors forthe calculation of kinematical relations can be also extendedto decays of elementary particles. In a two-body decay ofan elementary particle at rest the two decay particles getwell-defined discrete energies because of momentum conservation.Example 7: The decay π + → µ + + ν µFour-momentum conservation yieldsµ pair productiontime-like photonstwo-body decayq 2 π = (q µ + q ν ) 2 = m 2 π . (3.34)In the rest frame of the pion the muon and neutrino areemitted in opposite directions, p µ =−p νµ ,(Eµ + E νp µ + p νµ) 2= (E µ + E ν ) 2 = m 2 π . (3.35)Neglecting a possible non-zero neutrino mass for thisconsideration, one hasE ν = p νµwith the resultE µ + p µ = m π .Rearranging this equation and squaring it givesEµ 2 + m2 π − 2E µ m π = p2 µ ,2E µ m π = m 2 π + m2 µ ,E µ = m2 π + m2 µ2m π. (3.36)

42 3 Kinematics and Cross Sectionsdecay kinematicsπ 0 decayFor m µ = 105.658 369MeV and m π ± = 139.570 18MeV one gets Eµ kin = E µ − m µ = 4.09 MeV. Forthe two-body decay of the kaon, K + → µ + + ν µ ,(3.36) gives Eµ kin = E µ − m µ = 152.49 MeV (m K ± =493.677 MeV).Due to helicity conservation the decay π + → e + + ν eis strongly suppressed (see Fig. 2.10). Using (3.36) thepositron would get in this decay a kinetic energy of) 2= E e + −m e = m π2+ m2 e2m π−m e = m π1 − m em π ≈69.3 MeV, which is approximately half the pion mass.This is not a surprise, since the ‘heavy’ pion decays intotwo nearly massless particles.Example 8: The decay π 0 → γ + γE kine +2(The kinematics of the π 0 decay at rest is extremelysimple. Each decay photon gets as energy one half ofthe pion rest mass. In this example also the decay of aπ 0 in flight will be considered. If the photon is emittedin the direction of flight of the π 0 , it will get ahigher energy compared with the emission opposite tothe flight direction. The decay of a π 0 in flight (Lorentzfactor γ = E π 0/m π 0) yields a flat spectrum of photonsbetween a maximum and minimum energy. Fourmomentumconservationq π 0 = q γ1 + q γ2leads toq 2 π 0 = m 2 π 0 = q 2 γ 1+ q 2 γ 2+ 2q γ1q γ2. (3.37)Since the masses of real photons are zero, the kinematiclimits are obtained from the relation2q γ1q γ2= m 2 π 0 . (3.38)In the limit of maximum or minimum energy transfer tothe photons they are emitted parallel or antiparallel tothe direction of flight of the π 0 .Thisleadstop γ1 ‖−p γ2 . (3.39)Using this, (3.38) can be expressed as2(E γ1 E γ2 − p γ1 · p γ2 ) = 4E γ1 E γ2 = m 2 π 0 . (3.40)

3.2 Four-Vectors 43Because of E γ2 = E π 0−E γ1 (3.40) leads to the quadraticequationEγ 2 1− E γ1 E π 0 + m2 π 0= 0 (3.41)4with the symmetric solutionsEγ max1= 1 2 (E π 0 + p π 0),Eγ min1= 1 (3.42)2 (E π 0 − p π 0).photon spectrumfrom π 0 decayBecause of E π 0 = γm π 0 and p π 0 = γm π 0β (3.42) canalso be expressed asE maxγ 1E minγ 1= 1 2 γm π 0(1 + β) = 1 2 m π 0 √1 + β1 − β ,= 1 2 γm π 0(1 − β) = 1 2 m π 0 √1 − β1 + β . (3.43)In the relativistic limit (γ ≫ 1, β ≈ 1) a photon emittedin the direction of flight of the π 0 gets the energyEγmax = E π 0 = γm π 0 and the energy of the backwardemittedphoton is zero.From (3.43) it is clear that for any energy of a neutralpion, a range of possible photon energies containsm π 0/2. If one has a spectrum of neutral pions, the energyspectra of the decay photons are superimposed insuch a way that the resulting spectrum has a maximumat half the π 0 mass.Much more difficult is the treatment of a three-body decay.Such a process is going to be explained for the exampleof the muon decay:three-body decayµ − → e − +¯ν e + ν µ . (3.44)Let us assume that the muon is originally at rest (E µ = m µ ).Four-momentum conservationq µ = q e + q¯νe + q νµ (3.45)can be rephrased as(q µ − q e ) 2 = (q¯νe + q νµ ) 2 ,

44 3 Kinematics and Cross Sections( )( )qµ 2 + q2 e − 2q µ q e = m2 µ + m2 e − 2 mµ Ee0 p e= (q¯νe + q νµ ) 2 ,E e = m2 µ + m2 e − (q¯ν e+ q νµ ) 2. (3.46)2m µThe electron energy is largest, if (q¯νe + q νµ ) 2 takes on aminimum value. For vanishing neutrino masses this meansthat the electron gets a maximum energy, ifq¯νe q νµ = E¯νe E νµ − p ¯νe · p νµ = 0 . (3.47)Equation (3.47) is satisfied for p ¯νe ‖ p νµ . This yieldsE maxe= m2 µ + m2 e≈ m µ= 52.83 MeV . (3.48)2m µ 2In this configuration the electron momentum p e is antiparallelto both neutrino momenta which in turn are parallel toeach other.If the spins of all participating particles and the struc-ture of weak interactions are taken into consideration, oneobtains for the electron spectrum, using the shorthand x =2E e /m µ ≈ E e /Ee max ,electron spectrumin muon decayN(x) = const x 2 (1.5 − x) . (3.49)Just as in nuclear beta decay (n → p + e − +¯ν e )theavailable decay energy in a three-body decay is distributedcontinuously among the final-state particles (Fig. 3.4).Fig. 3.4Energy spectrum of electrons frommuon decayIf interaction or decay processes are treated, it is fully suffi-cient to consider the process in the center-of-mass system. Ina different system (for example, the laboratory system) thetransformationbetween laboratoryand center-of-mass system3.3 Lorentz Transformation“We have learned something about thelaws of nature, their invariance withrespect to the Lorentz transformation,and their validity for all inertial systemsmoving uniformly, relative to each other.We have the laws but do not know theframe to which to refer them.”Albert Einstein

3.3 Lorentz Transformation 45energies and momenta are obtained by a Lorentz transformation.IfEand p are energy and momentum in the centerof-masssystem and if the laboratory system moves with thevelocity β relative to p ‖ , the transformed quantities E ∗ andp‖ ∗ in this system are calculated to be (compare Fig. 3.5)( E∗)( )( )γ −γβ E=,p−γβ γ⊥ ∗ p = p ⊥ . (3.50)‖p ∗ ‖The transverse momentum component is not affected by thistransformation. Instead of using the matrix notation, (3.50)can be written asE ∗ = γE− γβp ‖ ,p‖ ∗ =−γβE + γp (3.51)‖ .For β = 0 and correspondingly γ = 1 one trivially obtainsE ∗ = E and p‖ ∗ = p ‖ .Fig. 3.5Illustration of a LorentztransformationA particle of energy E = γ 2 m 0 , seen from a systemwhich moves with β 1 relative to the particle parallel to themomentum p, gets in this system the energyE ∗ = γ 1 E − γ 1 β 1 p ‖√γ1 2= γ 1 γ 2 m 0 − γ − 11√(γ 2 m 0 )γ 2 − m 2 01√= γ 1 γ 2 m 0 − m 0 γ1 √γ 2 − 1 2 2 − 1 . (3.52)If γ 1 = γ 2 = γ (for a system that moves along with aparticle) one naturally obtainsE ∗ = γ 2 m 0 − m 0 (γ 2 − 1) = m 0 .

46 3 Kinematics and Cross Sections3.4 Cross Sections“Physicists are, as a general rule, highbrows.They think and talk in long, Latinwords, and when they write anythingdown they usually include at least onepartial differential and three Greek letters.”Stephen Whitecross sectionApart from the kinematics of interaction processes the crosssection for a reaction is of particular importance. In the mostsimple case the cross section can be considered as an effectivearea which the target particle represents for the collisionwith a projectile. If the target has an area of πrT 2 and the projectilesize corresponds to πrP 2 , the geometrical cross sectionfor a collision is obtained to beσ = π(r T + r P ) 2 . (3.53)interaction lengthIn most cases the cross section also depends on other parameters,for example, on the energy of the particle. The atomiccross section σ A , measured in cm 2 , is related to the interactionlength λ according toAλ {cm} =N A {g −1 } ϱ {g/cm 3 } σ A {cm 2 }(3.54)absorption coefficientinteraction rates(N A – Avogadro number; A – atomic mass of the target, ϱ– density). Frequently, the interaction length is expressed by(λ ϱ) {g/cm 2 }. Correspondingly, the absorption coefficientis defined to beµ {cm −1 }= N A ϱσ A= 1 A λ ; (3.55)equivalently, the absorption coefficient can also be expressedby (µ/ϱ) {(g/cm 2 ) −1 }.The absorption coefficient also provides a useful relationfor the determination of interaction probabilities or rates,φ {(g/cm 2 ) −1 }= µ ϱ = N AA σ A . (3.56)If σ N is a cross section per nucleon, one hasφ {(g/cm 2 ) −1 }=σ N N A . (3.57)

3.5 Problems 47If j is the particle flux per cm 2 and s, the number of particlesdN scattered through an angle θ into the solid angle dΩ perunit time isdN(θ) = jσ(θ)dΩ , (3.58)whereσ(θ) = dσdΩis the differential scattering cross section, describing theprobability of scattering into the solid-angle element dΩ,wheredifferential scatteringcross sectiondΩ = sin θ dθ dϕ (3.59)(ϕ – azimuthal angle, θ – polar angle).For azimuthal symmetry one hasdΩ = 2π sin θ dθ =−2π d(cos θ) . (3.60)Apart from the angular dependence the cross section canalso depend on other quantities, so that a large number ofdifferential cross sections are known, for example,dσdE ,dσdp ,or even double differential cross sections such asd 2 σdE dθ . (3.61)Apart from the mentioned characteristic quantities there isquite a large number of other kinematical variables whichare used for the treatment of special processes and decays.double differentialcross section3.5 Problems1. What is the threshold energy for a photon, E γ1 ,toproducea µ + µ − pair in a collision with a blackbody photonof energy 1 meV?2. The mean free path λ (in g/cm 2 ) is related to the nuclearcross section σ N (in cm 2 )byλ = 1N A σ N,

48 3 Kinematics and Cross Sectionswhere N A is the Avogadro number, i.e., the number ofnucleons per g, and σ N is the cross section per nucleon.The number of particles penetrating a target x unaffectedby interactions isN = N 0 e −x/λ .How many collisions happen in a thin target of thicknessx (N A = 6.022 × 10 23 g −1 , σ N = 1b,N 0 = 10 8 , x =0.1g/cm 2 )?3. The neutrino was discovered in the reaction¯ν e + p → n + e + ,where the target proton was at rest. What is the minimumneutrino energy to induce this reaction?4. The scattering of a particle of charge z on a target ofnuclear charge Z is mediated by the electromagnetic interaction.Work out the momentum transfer p b , perpendicularto the momentum of the incoming particle foran impact parameter b (distance of closest approach)!For the calculation assume the particle track to be undisturbed,i.e., the scattering angle to be small.5. The scattering of an electron of momentum p on a targetnucleus of charge Z was treated in Problems 4 underthe assumption that the scattering angle is small. Workout the general expression for the transverse momentumusing the Rutherford scattering formulatan ϑ 2 = Zr ebβ 2 , (3.62)where ϑ is the scattering angle.

494 Physics of Particleand Radiation Detection“Every physical effect can be used as a basisfor a detector.”AnonymousThe measurement techniques relevant to astroparticle physicsare rather diverse. The detection of astroparticles is usuallya multistep process. In this field of research, particle detectionis mostly indirect. It is important to identify the natureof the astroparticle in a suitable interaction process. Thetarget for interactions is, in many cases, not identical withthe detector that measures the interaction products. Cosmicraymuon neutrinos, for example, interact via neutrino–nucleon interactions in the antarctic ice or in the ocean,subsequently producing charged muons. These muons sufferenergy losses from electromagnetic interactions with theice (water), which produces, among others, Cherenkov radiation.The Cherenkov light is recorded, via the photoelectriceffect, by photomultipliers. This is then used to reconstructthe energy and the direction of incidence of the muon, whichis approximately identical to the direction of incidence of theprimary neutrino.In this chapter, the primary interaction processes willfirst be described. The processes which are responsible forthe detection of the interaction products in the detector willthen be presented.The cross sections for the various processes depend onthe particle nature, the particle energy, and the target material.A useful relation to determine the interaction probabilityφ and the event rate is obtained from the atomic- (σ A )ornuclear-interaction cross section (σ N ) according toφ {(g/cm 2 ) −1 }= N AA σ A = N A {g −1 } σ N {cm 2 } , (4.1)where N A is Avogadro’s number, A is the atomic mass ofthe target, and σ A is the atomic cross section in cm 2 /atom(σ N in cm 2 /nucleon), see also (3.56) and (3.57). If the targetrepresents an area density d {g/cm 2 } and if the flux ofprimary particles is F {s −1 }, the event rate R is obtained asR = φ {(g/cm 2 ) −1 } d {(g/cm 2 )} F {(s −1 )} . (4.2)indirect particle detectioncross sectionevent rate

50 4 Physics of Particle and Radiation Detection4.1 Interactions of Astroparticles“Observations are meaningless withouta theory to interpret them.”Raymond A. Lyttletonmeasurementof primary nucleiFig. 4.1Cross section for proton–airinteractionsmean free pathλ =A ≈ 93 g/cm 2 . (4.3)N A σ AThis means that the first interaction of protons occurs in theupper part of the atmosphere. If the primary particles are notprotons but rather iron nuclei (atomic number A Fe = 56),the first interaction will occur at even higher altitudes becausethe cross section for iron–air interactions is correspondinglylarger.Primary high-energy photons (energy ≫ 10 MeV) inter-act via the electromagnetic process of electron–positron pairproduction. The characteristic interaction length 1 (’radiation1 The radiation length for electrons is defined in (4.7). It describesthe degrading of the electron energy by bremsstrahlungaccording to E = E 0 e −x/X 0. This ‘interaction length’ X 0 isalso characteristic for pair production by photons. The interactionlength for hadrons (protons, pions, ...) is defined through(4.3), where σ A is the total cross section. This length is sometimesalso called collision length. If the total cross section in(4.3) is replaced by its inelastic part only, the resulting lengthis called absorption length.detection of primary photonsThe primary particles carrying astrophysical information arenuclei (protons, helium nuclei, iron nuclei, ...), photons, orneutrinos. These three categories of particles are characterizedby completely different interactions. Protons and othernuclei will undergo strong interactions. They are also subjectto electromagnetic and weak interactions, however, thecorresponding cross sections are much smaller than thoseof strong interactions. Primary nuclei will therefore interactpredominantly via processes of strong interactions. Atypical interaction cross section for inelastic proton–protonscattering at energies of around 100 GeV is σ N ≈ 40 mb(1 mb = 10 −27 cm 2 ). Since high-energy primary protonsinteract in the atmosphere via proton–air interactions, thecross section for proton–air collisions is of great interest.The dependence of this cross section on the proton energy isshown in Fig. 4.1.For a typical interaction cross section of 250 mb, themean free path of protons in the atmosphere (for nitrogen:A = 14) is, see Chap. 3, (3.54),

4.2 Interaction Processes Used for Particle Detection 51length’) for electrons in air is X 0 ≈ 36 g/cm 2 . For highenergyphotons (energy ≥ 10 GeV), where pair productiondominates, the cross section is 7/9 of the cross section forelectrons ([3], Chap. 1), so the radiation length for photonsis 9/7 of that for electrons, i.e., 47 g/cm 2 . The first interactionof photon-induced electromagnetic cascades thereforealso occurs in the uppermost layers of the atmosphere.The detection of cosmic-ray neutrinos is completely different.They are only subject to weak interactions (apartfrom gravitational interactions). The cross section for neutrino–nucleoninteractions is given byσ νN = 0.7 × 10 −38 E ν [GeV] cm 2 /nucleon . (4.4)Neutrinos of 100 GeV possess a tremendously large interactionlength in the atmosphere:λ ≈ 2.4 × 10 12 g/cm 2 . (4.5)The vertex for possible neutrino–air interactions in the atmosphereshould consequently be uniformly distributed.Charged and/or neutral particles are created in the interactions,independent of the identity of the primary particle.These secondary particles will, in general, be recorded bythe experiments or telescopes. To achieve this, a large varietyof secondary processes can be used.detectionof cosmic-ray neutrinos4.2 Interaction ProcessesUsed for Particle Detection“I often say when you can measure whatyou are speaking about, and express it innumbers, you know something about it;but when you cannot measure it, whenyou cannot express it in numbers, yourknowledge is of a meagre and unsatisfyingkind.”Lord Kelvin (William Thomson)Tables 4.1 and 4.2 show the main interaction processes ofcharged particles and photons, as they are typically used inexperiments in astroparticle physics. In this overview, notonly the interaction processes are listed, but also the typicaldetectors that utilize the corresponding interaction processes.The mechanism that dominates charged-particle interactionsis the energy loss by ionization and excitation.This energy-loss process is described by the Bethe–Blochformula:interaction mechanismsenergy lossof charged particlesBethe–Bloch formula

52 4 Physics of Particle and Radiation DetectionTable 4.1Overview of interaction processesof charged particlesTable 4.2Overview of interaction processesof photonsenergy-loss dependence− dEdx∣ = Kz 2 Zion A× 1 {1β 2 2 ln 2m ec 2 β 2 γ 2 T maxI 2 − β 2 − δ 2, (4.6)whereK – 4πN A re 2m ec 2 ≈ 0.307 MeV/(g/cm 2 ),N A – Avogadro’s number,r e – classical electron radius (≈ 2.82 fm),m e c 2 – electron rest energy (≈ 511 keV),z – charge number of the incident particle,Z,A – target charge number and target mass number,β – velocity (= v/c) of the incident particle,γ – 1/ √ 1 − β 2 ,2m e p 2T max –m 2 0 + m2 e + 2m eE/c 2maximum energy transfer to an electron,m 0 – mass of the incident particle,p, E – momentum and total energyof the projectile,I – average ionization energy of the target,δ – density correction.The energy loss of charged particles, according to theBethe–Bloch relation, is illustrated in Fig. 4.2. It exhibitsa1/β 2 increase at low energies. The minimum ionizationrate occurs at around βγ ≈ 3.5. This feature is called the}

4.2 Interaction Processes Used for Particle Detection 53Fig. 4.2Energy loss of charged particles invarious targets [2]minimum of ionization, and particles with such βγ valuesare said to be minimum ionizing. For high energies, the energyloss increases logarithmically (‘relativistic rise’) andreaches a plateau (‘Fermi plateau’) owing to the density effect.The energy loss of gases in the plateau region is typically60% higher compared to the ionization minimum. Theenergy loss of singly charged minimum-ionizing particlesby ionization and excitation in air is 1.8MeV/(g/cm 2 )and2.0 MeV/(g/cm 2 ) in water (ice).Equation (4.6) only describes the average energy loss ofcharged particles. The energy loss is distributed around themost probable value by an asymmetric Landau distribution.The average energy loss is about twice as large as the mostprobable energy loss. The ionization energy loss is the basisof a large number of particle detectors.In particle astronomy, the Fly’s Eye technique takes advantageof the scintillation mechanism in air to detect particleswith energies ≥ EeV (≥ 10 18 eV). In this experiment,the atmosphere represents the target for the primary particle.The interaction products create scintillation light in the air,which is recorded by photomultipliers mounted in the focalplane of mirrors on the surface of the Earth.For high energies, the bremsstrahlung process becomessignificant. The energy loss of electrons due to this processcan be described by− dEdxZ 2∣ = 4αN AbremsA r2 eE ln183Z 1/3 = E X 0, (4.7)Landau distributionFly’s Eyebremsstrahlung

54 4 Physics of Particle and Radiation Detectionenergy loss [ MeV / (g/cm )]210 310 210where α is the fine-structure constant (α −1 ≈ 137). The def-inition of the radiation length X 0 is evident from (4.7). Theother quantities in (4.7) have the same meanings as in (4.6).Energy loss due to bremsstrahlung is of particular importancefor electrons. For heavy particles, the bremsstrahlungenergy loss is suppressed by the factor 1/m 2 . The energyloss, however, increases linearly with energy, and is thereforeimportant for all particles at high energies.In addition to bremsstrahlung, charged particles can alsolose some of their energy by direct electron–positron pairproduction, or by nuclear interactions. The energy loss dueto these two interaction processes also varies linearly withenergy. Muons as secondary particles in astroparticle physicsplay a dominant rôle in particle-detection techniques, e.g.,in neutrino astronomy. Muons are not subject to strong in-teractions and they can consequently travel relatively largedistances. This makes them important for particle detectionin astroparticle physics. The total energy loss of muons canbe described by:− dEdx ∣ = a(E) + b(E) E , (4.8)muonradiation lengthdirect pair productionnuclear interactionsenergy loss of muonsmuons in rock2110 10 2 10 3 10 4 10 5muon energy [GeV]Fig. 4.3Energy loss of muons in standardrockmuon calorimetrywhere a(E) describes the ionization energy loss, and b(E)Esummarizes the processes of muon bremsstrahlung, directelectron pair creation, and nuclear interaction. The energyloss of muons in standard rock depends on their energy. It isdisplayed in Fig. 4.3.For particles with high energies, the total energy loss isdominated by bremsstrahlung and the processes that dependlinearly on the particles’ energies. These energy-loss mechanismsare therefore used as a basis for particle calorimetry.In calorimetric techniques, the total energy of a particle isdissipated in an active detector medium. The output signal ofsuch a calorimeter is proportional to the absorbed energy. Inthis context, electrons and photons with energies exceeding100 MeV can already be considered as high-energy particlesbecause they initiate electromagnetic cascades. The mass ofthe muon is much larger than that of the electron, makingmuon calorimetry via energy-loss measurements only possiblefor energies beyond ≈ 1 TeV. This calorimetric techniqueis of particular importance in the field of TeV neutrinoastronomy.

4.3 Principles of the Atmospheric Air Cherenkov Technique 554.3 Principles of the Atmospheric AirCherenkov Technique“A great pleasure in life is doing whatpeople say you cannot do.”Walter BagehotThe atmospheric Cherenkov technique is becoming increasinglypopular for TeV γ astronomy since it allows to identifyphoton-induced electromagnetic showers, which develop inthe atmosphere. A charged particle that moves in a mediumwith refractive index n, and has a velocity v that exceeds thevelocity of light c n = c/n, emits electromagnetic radiationknown as Cherenkov radiation. There is a threshold effectfor this kind of energy loss; Cherenkov radiation only occursifCherenkov effectv ≥ c n or, equivalently, β = v c ≥ 1 n . (4.9)Cherenkov radiation is emitted at an angle ofθ C = arccos 1nβ(4.10)relative to the direction of the particle velocity. Due to thisprocess, a particle of charge number z creates a certain numberof photons in the visible spectral range (λ 1 = 400 nm upto λ 2 = 700 nm). The number of photons is calculated fromthe following equation:dNdx = λ 2παz2 2 − λ 1sin 2 θ Cλ 1 λ 2(4.11)≈ 490 z 2 sin 2 θ C cm −1 .These photons are emitted isotropically (about the axis) inazimuth. For relativistic particles (β ≈ 1), the Cherenkovangle is 42 ◦ in water, and 1.4 ◦ in air. In water, around 220photons per centimeter are produced by a singly chargedrelativistic particle. The corresponding number in air is 30photons per meter. Figure 4.4 shows the variation of theCherenkov angle and the photon yield, with the particlevelocity for water and air. The atmospheric Cherenkov techniquepermits the identification of photon-induced electromagneticshowers that develop in the atmosphere, and separatesthem from the more abundant hadronic cascades. Thisis possible because the recorded Cherenkov pattern is differentfor electromagnetic and hadronic cascades; also photonsFig. 4.4Variation of the Cherenkov angleand photon yield of singly chargedparticles in water and air

56 4 Physics of Particle and Radiation DetectionFig. 4.5Production of a Cherenkov ring ina water Cherenkov counterpoint back to their sources, while hadrons only produce anisotropic background. The axis of the Cherenkov cone followsthe direction of incidence of the primary photon. TheCherenkov cone for γ -induced cascades in air spans only±1.4 ◦ , therefore the hadronic background in such a smallangular range is relatively small.Apart from the atmospheric Cherenkov technique, theCherenkov effect is also utilized in large water Cherenkovdetectors for neutrino astronomy. The operation principle ofa water Cherenkov counter is sketched in Fig. 4.5. Cherenkovradiation is emitted along a distance x. The Cherenkovcone projects an image on the detector surface, which is ata distance d from the source. The image is a ring with anaverage radiusr = d tan θ C . (4.12)4.4 Special Aspects of Photon Detection“Are not the rays of light verysmall bodies emitted from shiningsubstances?”Sir Isaac NewtonZof the absorber material100806040photoelectriceffectdominantpairproductiondominant20 Coulomb scatteringdominant010 –2 10 –11 10 10 2photon energy [MeV]Fig. 4.6Domains, in which various photoninteractions dominate, shown intheir dependence on the photonenergy and the nuclear charge ofthe absorberThe detection of photons is more indirect compared tocharged particles. Photons first have to create charged particlesin an interaction process. These charged particles willthen be detected via the processes described, such as ionization,excitation, bremsstrahlung, and the production of Cherenkovradiation.At comparatively low energies, as in X-ray astronomy,photons can be imaged by reflections at grazing incidence.Photons are detected in the focal plane of an X-ray telescopevia the photoelectric effect. Semiconductor counters, X-rayCCDs 2 , or multiwire proportional chambers filled with anoble gas of high atomic number (e.g., krypton, xenon) canbe used for focal detectors. These types of detectors providespatial details, as well as energy information.The Compton effect dominates for photons at MeV energies(see Fig. 4.6). In Compton scattering, a photon of energyE γ transfers part of its energy E to a target electron,thereby being redshifted. Based on the reaction kinematics,2 CCD – Charge-Coupled Device (solid-state ionization chamber)

4.4 Special Aspects of Photon Detection 57the ratio of the scattered photon energy E γ ′photon energy E γ can be derived:to the incidentE γ′ 1=E γ 1 + ε(1 − cos θ γ ) . (4.13)In this equation, ε = E γ /m e c 2 is the reduced photon en- Compton telescopeergy and θ γ is the scattering angle of the photon in the γ –electron interaction. With a Compton telescope, not onlylocus of the directionof incidencethe energy, but also the direction of incidence of the photonscan be determined. In such a telescope, the energy loss of the Compton-scattered photon E = E γ − E γ ′ is determinedin the upper detector layer by measuring the energyof the Compton electron (see Fig. 4.7). The Comptonscatteredphoton of reduced energy will subsequently bedetector 1Edetected in the lower detector plane, preferentially by thephotoelectric effect. Based on the kinematics of the scatteringprocess and using (4.13), the scattering angle θ γ canbe determined. As a consequence of the isotropic emissionaround the azimuth, the reconstructed photon direction does E´ not point back to a unique position in the sky; it only definesdetector 2a circle in the sky. If, however, many photons are recorded Fig. 4.7from the source, the intercepts of these circles define the positionof the source. The detection of photons via the Comp-Schematic of a Compton telescopeton effect in such Compton telescopes is usually performedusing segmented large-area inorganic or organic scintillationcounters that are read out via photomultipliers. Alternatively,for high-resolution telescopes, semiconductor pixeldetectors can also be used. This ‘ordinary’ Compton processis taken advantage of for photon detection. In astrophysical inverse Compton scatteringsources the inverse Compton scattering plays an importantrôle. In such a process a low-energy photon might gain substantialenergy in a collision with an energetic electron, andit can be shifted into the X-ray or γ -ray domain.At high photon energies, the process of electron–positronpair creation dominates. Similarly to Compton telescopes,the electron and positron tracks enable the direction of theincident photon to be determined. The photon energy is obtainedfrom the sum of the electron and positron energy.This is normally determined in electromagnetic calorimeters,in which electrons and positrons deposit their energyto the detector medium in alternating bremsstrahlung andpair-production processes. These electromagnetic calorime-ters can be total-absorption crystal detectors such as NaI orCsI, or they can be constructed from the so-called sandwichelectron–positronpair productionelectromagnetic calorimeter

58 4 Physics of Particle and Radiation Detectionsandwich calorimeterprinciple. A sandwich calorimeter is a system where absorberand detector layers alternate. Particle multiplicationoccurs preferentially in the passive absorber sheets, whilstthe shower of particles produced is recorded in the activedetector layers. Sandwich calorimeters can be compactlyconstructed and highly segmented, however, they are inferiorto crystal calorimeters as far as the energy resolution isconcerned.4.5 Cryogenic Detection Techniques“Ice vendor to his son: ‘Stick to it, thereis a future in cryogenics.’”Anonymouscryogenic detectorsThe detection of very small energies can be performed incryogenic detectors. Cooper bonds in superconductors canalready be broken by energy deposits as low as 1 meV (=10 −3 eV). The method of classical calorimetry can also beused for particle detection at low temperatures. At low temperaturesthe specific heat of solids varies with the cube ofthe temperature (c sp ∼ T 3 ), therefore even the smallest energydeposits E provide a measurable temperature signal.These detectors are mainly used in the search for hypotheticalweakly interacting massive particles (WIMPs). To thiscategory also supersymmetric partners of ordinary particlesbelong. On the other hand, cryogenic detectors are also employedin the realm of high-resolution X-ray spectroscopywhere energy resolutions of several eV can be achieved.4.6 Propagation and Interactionsof Astroparticles in Galacticand Extragalactic Space“Space tells matter how to move ...andmatter tells space how to curve.”John A. Wheelerpropagation of astroparticlesneutrinosNow that the principles for the detection of primary andsecondary particles have been described, the interactions ofastroparticles traveling from their sources to Earth throughgalactic and extragalactic space shall be briefly discussed.Neutrinos are only subject to weak interactions withmatter, so their range is extremely large. The galactic or in-

4.6 Propagation and Interactions of Astroparticles in Galactic and Extragalactic Space 59tergalactic space does not attenuate the neutrino flux, andmagnetic fields do not affect their direction; therefore, neutrinospoint directly back to their sources.The matter density in our galaxy, and particularly in intergalacticspace, is very low. This signifies that the ionizationenergy loss of primary protons traveling from theirsources to Earth is extremely small. Protons can, however,interact with cosmic photons. Blackbody photons, in particular,represent a very-high-density target (≈ 400 photons/cm 3 ). The energy of these photons is very low, typically250 µeV, and they follow a Planck distribution (Fig. 4.8), seealso Chap. 11 on ‘The Cosmic Microwave Background’.The process of pion production by blackbody photonsinteracting with high-energy protons requires the protonenergy to exceed a certain threshold (Greisen–Zatsepin–Kuzmin cutoff). This threshold is reached if photo–pionproduction via the ∆ resonance is kinematically possible inthe photon–proton center-of-mass system (p+γ → p+π 0 ).If protons exceed this threshold energy, they quickly losetheir energy and fall below the threshold. The GZK cutofflimits the mean free path of the highest-energy cosmic rays(energy > 6 × 10 19 eV) to less than a few tens of megaparsecs,quite a small distance in comparison to typicalextragalactic scales. Of course, energetic protons lose alsoenergy by inverse Compton scattering on blackbody photons.In contrast to π 0 production via the ∆ resonance thisprocess has no threshold. Moreover, the cross section varieslike 1/s, i.e., with the inverse square of the available centerof-massenergy. Compared to the resonant π 0 productionthe cross section for inverse Compton scattering of protonson blackbody photons is small and therefore has no significantinfluence on the shape of the primary proton spectrum.A further possible process, p + γ → p + e + + e − ,eventhough it has a lower threshold than p + γ → p + π 0 , doesnot proceed through a resonance, and therefore its influenceon the propagation of energetic protons in the dense photonfield is of little importance. In addition, primary protons(charged particles) naturally interact with the galactic andextragalactic magnetic fields as well as the Earth’s magneticfield. Only the most energetic protons (energy ≫ 10 18 eV),which experience a sufficiently small magnetic deflection,can be used for particle astronomy.Photons are not influenced by magnetic fields. They do,like protons, however, interact with blackbody photons tocreate electron–positron pairs via the γγ → e + e − process.protons and nucleiFig. 4.8Blackbody spectrum of cosmicmicrowave background photonsGZK cutoffmagnetic deflection

60 4 Physics of Particle and Radiation DetectionOwing to the low electron and positron masses, the thresh-old energy for this process is only about ≈ 10 15 eV. Theattenuation of primary photons (by interactions with blackbodyphotons), as a function of the primary photon energy, isshown in Fig. 4.9 for several distances to the γ -ray sources.A potentially competing process, γγ → γγ, is connectedwith a very small cross section (it is proportional to thefourth power of the fine-stucture constant). In addition, theangular deflection of the photons due to this process is extremelysmall.galactic photon absorptionFig. 4.9Attenuation of the intensity ofenergetic primary cosmic photonsby interactions with blackbodyradiationThe mean free path for energetic photons (energy> 10 15 eV) is limited to a few tens of kiloparsecs by thisprocess. For higher energies, γγ processes with differentfinal states (µ + µ − , ...) also occur. For lower energies, photonsare attenuated by interactions with infrared or starlightphotons.4.7 Characteristic Features of Detectors“Detectors can be classified into threecategories – those that don’t work, thosethat break down, and those that getlost.”AnonymousThe secondary interaction products of astroparticles are detectedin an appropriate device, which can be a detector on

4.8 Problems 61board of a satellite, in a balloon, or at ground level, or even inan underground laboratory. The quality of the measurementdepends on the energy and position resolution of the detector.In most cases the ionization energy loss is the relevantdetection mechanism.In gaseous detectors an average of typically 30 eV is requiredto produce an electron–ion pair. The liberated chargesare collected in an external electric field and produce anelectric signal which can be further processed. In contrast,in solid-state detectors, the average energy for the creationof an electron–hole pair is only ≈ 3 eV, resulting in an improvedenergy resolution. If, instead, excitation photons producedby the process of scintillation in a crystal detectorare recorded, e.g., by photomultipliers, energy deposits ofabout 25 eV are necessary to yield a scintillation photon ininorganic materials (like NaI(Tl)), while in organic crystals≈ 100 eV are required to create a scintillation photon. Incryogenic detectors much less energy is needed to producecharge carriers. This substantial advantage which gives riseto excellent energy resolutions is only obtained at the expenseof operating the detectors at cryogenic temperatures,mostly in the milli-Kelvin range.energy andposition resolutionsolid-state detectorsscintillatorscryogenic devices4.8 Problems1. Show that (4.1) is dimensionally correct.2. The average energy required for the production ofa) a photon in a plastic scintillator is 100 eV,b) an electron–ion pair in air is 30 eV,c) an electron–hole pair in silicon is 3.65 eV,d) a quasiparticle (break-up of a Cooper pair in a superconductor)is 1 meV.What is the relative energy resolution in these countersfor a stopping 10 keV particle assuming Poisson statistics(neglecting the Fano effect 3 )?3 For any specific value of a particle energy the fluctuationsof secondary particle production (like electron–ion pairs) aresmaller than might be expected according to a Poissonian distribution.This is a simple consequence of the fact that the totalenergy loss is constrained by the fixed energy of the incidentparticle. This leads to a standard error of σ = √ FN,whereNis the number of produced secondaries and F , the Fano factor,is smaller than 1.

62 4 Physics of Particle and Radiation Detection3. The simplified energy loss of a muon is parameterizedby (4.8). Work out the range of a muon of energyE (= 100 GeV) in rock (ϱ rock = 2.5g/cm 3 ) underthe assumption that a (= 2MeV/(g/cm 2 )) and b(= 4.4 × 10 −6 cm 2 /g for rock) are energy independent.4. Show that the mass of a charged particle can be inferredfrom the Cherenkov angle θ C and momentum p bym 0 = p c√n 2 cos 2 θ C − 1 ,where n is the index of refraction.5. In a cryogenic argon calorimeter (T = 1.1 Kelvin, mass1 g) a WIMP (weakly interacting massive particle) deposits10 keV. By how much does the temperature rise?(The specific heat of argon at 1.1 K is c sp = 8 ×10 −5 J/(g K).)6. Derive (4.13) using four-momenta.7. Work out the maximum energy which can be transferredto an electron in a Compton process! As an example usethe photon transition energy of 662 keV emitted by anexcited 137 Ba nucleus after a beta decay from 137 Cs,137 Cs → 137 Ba ∗ + e − +¯ν e✲ 137 Ba + γ(662 keV) .What kind of energy does the electron get for infinitelylarge photon energies? Is there, on the other hand, a minimumenergy for the backscattered photon in this limit?8. Figure 4.2 shows the energy loss of charged particlesas given by the Bethe–Bloch formula. The abscissa isgiven as momentum and also as product of the normalizedvelocity β and the Lorentz factor γ . Show thatβγ = p/m 0 c holds.9. Equation (4.11) shows that the number of emitted Cherenkovphotons N is proportional to 1/λ 2 .Thewavelengthfor X-ray photons is shorter than that for the visiblelight region. Why then is Cherenkov light not emittedin the X-ray region?

635 Acceleration Mechanisms“Physics also solves puzzles. However,these puzzles are not posed by mankind, butrather by nature.”Maria Goeppert-MayerThe origin of cosmic rays is one of the major unsolved astrophysicalproblems. The highest-energy cosmic rays possessmacroscopic energies and their origin is likely to beassociated with the most energetic processes in the universe.When discussing cosmic-ray origin, one must in principledistinguish between the power source and the accelerationmechanism. Cosmic rays can be produced by particle interactionsat the sites of acceleration like in pulsars. Theacceleration mechanism can, of course, also be based onconventional physics using electromagnetic or gravitationalpotentials such as in supernova remnants or active galacticnuclei. One generally assumes that in most cases cosmicrayparticles are not only produced in the sources but alsoaccelerated to high energies in or near the source. Candidatesites for cosmic-ray production and acceleration are supernovaexplosions, highly magnetized spinning neutron stars,i.e., pulsars, accreting black holes, and the centers of activegalactic nuclei. However, it is also possible that cosmicrayparticles powered by some source experience accelerationduring the propagation in the interstellar or intergalacticmedium by interactions with extensive gas clouds. Thesegas clouds are created by magnetic-field irregularities andcharged particles can gain energy while they scatter off theconstituents of these ‘magnetic clouds’.In top–down scenarios energetic cosmic rays can also beproduced by the decay of topological defects, domain walls,or cosmic strings, which could be relics of the Big Bang.There is a large number of models for cosmic-ray acceleration.This appears to indicate that the actual accelerationmechanisms are not completely understood and identified.On the other hand, it is also possible that different mechanismsare at work for different energies. In the following themost plausible ideas about cosmic-ray-particle accelerationwill be presented.origin of cosmic raysacceleration mechanismssupernovaepulsarsblack holesactive galactic nucleitop–down scenario

64 5 Acceleration Mechanisms5.1 Cyclotron Mechanism“Happy is he who gets to know the reasonfor things.”Virgilsunspotscreation of magnetic fieldsRparticle trajectoryBAdsFig. 5.1Principle of particle accelerationby variable sunspotscircular accelerationEven normal stars can accelerate charged particles up to theGeV range. This acceleration can occur in time-dependentmagnetic fields. These magnetic sites appear as star spotsor sunspots, respectively. The temperature of sunspots isslightly lower compared to the surrounding regions. Theyappear darker, because part of the thermal energy has beentransformed into magnetic field energy. Sunspots in typicalstars can be associated with magnetic field strengths of upto 1000 Gauss (1 Tesla = 10 4 Gauss). The lifetime of suchsunspots can exceed several rotation periods. The spatial extensionof sunspots on the Sun can be as large as 10 9 cm.The observed Zeeman splitting of spectral lines has shownbeyond any doubt that magnetic fields are responsible forthe sunspots. Since the Zeeman splitting of spectral linesdepends on the magnetic field strength, this fact can also beused to measure the strength of the magnetic fields on stars.The magnetic fields in the Sun are generated by turbu-lent plasma motions where the plasma consists essentiallyof protons and electrons. The motions of this plasma constitutecurrents which produce magnetic fields. When thesemagnetic fields are generated and when they decay, electricfields are created in which protons and electrons can be accelerated.Figure 5.1 shows schematically a sunspot of extensionA = πR 2 with a variable magnetic field B.The time-dependent change of the magnetic flux φ producesa potential U,− dφ ∮= E · ds = U (5.1)dt(E – electrical field strength, ds – infinitesimal distancealong the particle trajectory). The magnetic flux is given by∫φ = B · dA = BπR 2 , (5.2)where dA is the infinitesimal area element. In this equationit is assumed that B is perpendicular to the area, i.e., B ‖ A,(the vector A is always perpendicular to the area). One turnof a charged particle around the time-dependent magneticfield leads to an energy gain of

5.2 Acceleration by Sunspot Pairs 65E = eU = eπR 2 dBdt. (5.3)A sunspot of an extension R = 10 9 cm and magnetic fieldB = 2000 Gauss at a lifetime of one day ( dBdt= 2000Gauss/day) leads toE = 1.6 × 10 −19 Asπ 10 14 m 2 0.2Vs86 400 s m 2= 1.16 × 10 −10 J = 0.73 GeV . (5.4)Actually, particles from the Sun with energies up to 100 GeVhave been observed. This, however, might also represent thelimit for the acceleration power of stars based on the cyclotronmechanism.The cyclotron model can explain the correct energies,however, it does not explain why charged particles propagatein circular orbits around time-dependent magnetic fields.Circular orbits are only stable in the presence of guidingforces such as they are used in earthbound accelerators.5.2 Acceleration by Sunspot Pairs“Living on Earth may be expensive, butit includes an annual free trip aroundthe Sun.”Ashleigh BrilliantSunspots often come in pairs of opposite magnetic polarity(see Fig. 5.2).The sunspots normally approach each other and mergeat a later time. Let us assume that the left sunspot is at restand the right one approaches the first sunspot with a velocityv. The moving magnetic dipole produces an electric fieldperpendicular to the direction of the dipole and perpendicularto its direction of motion v, i.e., parallel to v × B. Typicalsolar magnetic sunspots can create electrical fields of10 V/m. In spite of such a low field strength, protons can beaccelerated since the collision energy loss is smaller than theenergy gain in the low-density chromosphere. Under realisticassumptions (distance of sunspots 10 7 m, magnetic fieldstrengths 2000 Gauss, relative velocity v = 10 7 m/day) particleenergies in the GeV range are obtained. This showsthat the model of particle acceleration in approaching magneticdipoles can only explain energies which can also beFig. 5.2Sketch of a sunspot pairacceleration to GeV energies

66 5 Acceleration Mechanismsprovided by the cyclotron mechanism. The mechanism ofapproaching sunspots, however, sounds more plausible becausein this case no guiding forces (like in the cyclotronmodel) are required.5.3 Shock Acceleration“Basic research is what I am doingwhen I don’t know what I am doing.”Wernher von Braungravitational collapsesuccessive fusion processesbirth of neutron starsformation of heavy elementsneutron attachmentIf a massive star has exhausted its hydrogen, the radiationpressure can no longer withstand the gravitational pressureand the star will collapse under its own gravity. The liber-ated gravitational energy increases the central temperatureof a massive star to such an extent that helium burning canstart. If the helium reservoir is used up, the process of gravi-tational infall of matter repeats itself until the temperature isfurther increased so that the products of helium themselvescan initiate fusion processes. These successive fusion processescan lead at most to elements of the iron group (Fe,Co, Ni). For higher nuclear charges the fusion reaction isendotherm, which means that without providing additionalenergy heavier elements cannot be synthesized. When thefusion process stops at iron, the massive star will implode.In this process part of its mass will be ejected into interstellarspace. This material can be recycled for the productionof a new star generation which will contain – like the Sun –some heavy elements. As a result of the implosion a com-pact neutron star will be formed that has a density which iscomparable to the density of atomic nuclei. In the course ofa supernova explosion some elements heavier than iron areproduced if the copiously available neutrons are attached tothe elements of the iron group, which – with successive β −decays – allows elements with higher nuclear charge to beformed:56265726582657Fe + n → 26 Fe ,58Fe + n → 26 Fe ,59Fe + n →26 Fe∗ (5.5)✲ 5927 Co + e− +¯ν e ,5927 Co + n →6027 Co∗✲ 6028 Ni + e− +¯ν e .

5.3 Shock Acceleration 67The ejected envelope of a supernova represents a shock frontwith respect to the interstellar medium. Let us assume thatthe shock front moves at a velocity u 1 . Behind the shockfront the gas recedes with a velocity u 2 . This means that thegas has a velocity u 1 − u 2 in the laboratory system (see Fig.5.3).A particle of velocity v colliding with the shock frontand being reflected gains the energyshock accelerationshock frontincidentparticlegas streaming awayfrom the shock frontu 2E = 1 2 m(v + (u 1 − u 2 )) 2 − 1 2 mv2= 1 2 m(2v(u 1 − u 2 ) + (u 1 − u 2 ) 2 ). (5.6)Since the linear term dominates (v ≫ u 1 ,u 2 , u 1 >u 2 ),thissimple model provides a relative energy gain ofEE ≈ 2(u 1 − u 2 ). (5.7)vA more general, relativistic treatment of shock accelerationincluding also variable scattering angles leads toEE = 4 3u 1 − u 2c, (5.8)where it has been assumed that the particle velocity v canbe approximated by the speed of light c. Similar results areobtained if one assumes that particles are trapped betweentwo shock fronts and are reflected back and forth from thefronts.Usually the inner front will have a much higher velocity(v 2 ) compared to the outer front (v 1 ), which is deceleratedalready in interactions with the interstellar material(Fig. 5.4). The inner shock front can provide velocities up to20 000 km/s, as obtained from measurements of the Dopplershift of the ejected gas. The outer front spreads into the interstellarmedium with velocities between some 100 km/sup to 1000 km/s. For shock accelerations in active galacticnuclei even superfast shocks with v 2 = 0.9 c are discussed.A particle of velocity v being reflected at the inner shockfronts gains the energyE 1 = 1 2 m(v +v 2) 2 − 1 2 mv2 = 1 2 m(v2 2 +2vv 2). (5.9)u 1 shock-front velocityu1– u 2velocity of the gasin the laboratory systemFig. 5.3Schematics of shock-waveaccelerationinner shock frontremnantouter shock frontv 2v 1Fig. 5.4Particle acceleration by multiplereflection between two shockfrontsReflection at the outer shock front leads to an energy lossE 2 = 1 2 m(v−v 1) 2 − 1 2 mv2 = 1 2 m(v2 1 −2vv 1). (5.10)

68 5 Acceleration MechanismsOn average, however, the particle gains an energyE = 1 2 m(v2 1 + v2 2 + 2v(v 2 − v 1 )) . (5.11)Fermi mechanismof 1st orderacceleration to 100 TeVSince the quadratic terms can be neglected and because ofv 2 >v 1 , one getsE ≈ mvv ,EE≈ 2 vv . (5.12)This calculation followed similar arguments as in (5.6) and(5.7).Both presented shock acceleration mechanisms are linearin the relative velocity. Sometimes this type of shock accelerationis called Fermi mechanism of first order. Underplausible conditions using the relativistic treatment, maximumenergies of about 100 TeV can be explained in thisway.Fermi mechanism of second order (or more general Fermimechanism) describes the interaction of cosmic-ray particleswith magnetic clouds. At first sight it appears improba-ble that particles can gain energy in this way. Let us assumethat a particle (with velocity v) is reflected from a gas cloudwhich moves with a velocity u (Fig. 5.5).If v and u are antiparallel, the particle gains the energyFermi mechanismof 2nd ordercolliding magnetic clouds5.4 Fermi Mechanism“Results! Why man, I have gotten alot of results, I know several thousandthings that don’t work.”Thomas EdisonE 1 = 1 2 m(v + u)2 − 1 2 mv2 = 1 2 m(2uv + u2 ). (5.13)In case that v and u are parallel, the particle loses an energyE 2 = 1 2 m(v−u)2 − 1 2 mv2 = 1 2 m(−2uv+u2 ). (5.14)On average a net energy gain ofE = E 1 + E 2 = mu 2 (5.15)Fig. 5.5Energy gain of a particle by areflection from a magnetic cloudresults, leading to the relative energy gain ofEE= 2 u2v 2 . (5.16)

5.5 Pulsars 69Since this acceleration mechanism is quadratic in the cloudvelocity, this variant is often called Fermi mechanism of 2ndorder. The result of (5.16) remains correct even under relativistictreatment. Since the cloud velocity is rather low comparedto the particle velocities (u ≪ v ≈ c), the energy gainper collision (∼ u 2 ) is very small. Therefore, the accelerationof particles by the Fermi mechanism requires a verylong time. In this acceleration type one assumes that magneticclouds act as collision partners – and not normal gasclouds – because the gas density and thereby the interactionprobability is larger in magnetic clouds.Another important aspect is that cosmic-ray particleswill lose some of their gained energy by interactions withthe interstellar or intergalactic gas between two collisions.This is why this mechanism requires a minimum injectionenergy above which particles can only be effectively accelerated.These injection energies could be provided by theFermi mechanism of 1st order, that is, by shock acceleration.slow energy increaseinjection energy5.5 Pulsars“Rhythmically pulsating radio source,Can you not tell us what terrible forceRenders your density all so immenseTo account for your signal so sharp andintense?”Dietrick E. ThomsenJonathan EberhartSpinning magnetized neutron stars (pulsars) are remnantsof supernova explosions. While stars typically have radiiof 10 6 km, they shrink under a gravitational collapse to asize of just about 20 km. This process leads to densities of6 × 10 13 g/cm 3 comparable to nuclear densities. In this processelectrons and protons are so closely packed that in processesof weak interactions neutrons are formed:p + e − → n + ν e . (5.17)Since the Fermi energy of electrons in such a neutron staramounts to several hundred MeV, the formed neutrons cannotdecay because of the Pauli principle, since the maximumenergy of electrons in neutron beta decay is only 0.78 MeVand all energy levels in the Fermi gas of electrons up to thisenergy and even beyond are occupied.neutron star

70 5 Acceleration Mechanismsgravitational collapsepulsar periodsThe gravitational collapse of stars conserves the angularmomentum. Therefore, because of their small size, rotatingneutron stars possess extraordinary short rotational periods.Assuming orbital periods of a normal star of about onemonth like for the Sun, one obtains – if the mass loss duringcontraction can be neglected – pulsar frequencies ω Pulsar of(Θ – moment of inertia)Θ star ω star = Θ pulsar ω pulsar ,ω pulsar =R2 starRpulsar2corresponding to pulsar periods ofω star (5.18)T pulsar = T starR 2 pulsarR 2 star. (5.19)For a stellar size R star = 10 6 km, a pulsar radius R pulsar =20 km, and a rotation period of T star = 1 month one obtainsT pulsar ≈ 1ms. (5.20)The gravitational collapse amplifies the original magneticfield extraordinarily. If one assumes that the magnetic flux,e.g., through the upper hemisphere of a star, is conservedduring the contraction, the magnetic field lines will be tightlysqueezed. One obtains (see Fig. 5.6)∫∫B star · dA star = B pulsar · dA pulsar ,starpulsarFig. 5.6Increase of the magnetic fieldduring the gravitational collapse ofa starB pulsar = B starR 2 starR 2 pulsar. (5.21)For B star = 1000 Gauss magnetic pulsar fields of 2.5 ×10 12 Gauss = 2.5 × 10 8 T are obtained! These theoreticallyexpected extraordinary high magnetic field strengths havebeen experimentally confirmed by measuring quantized energylevels of free electrons in strong magnetic fields (‘Landaulevels’). The rotational axis of pulsars usually does notcoincide with the direction of the magnetic field. It is obviousthat the vector of these high magnetic fields spinningaround the non-aligned axis of rotation will produce strongelectric fields in which particles can be accelerated.For a 30 ms pulsar with rotational velocities ofv = 2πR pulsarT pulsar= 2π × 20 × 103 m3 × 10 −2 s≈ 4 × 10 6 m/s

5.6 Binaries 71one obtains, using E = v × B with v ⊥ B, electrical fieldstrengths of|E| ≈vB= 10 15 V/m . (5.22)This implies that singly charged particles can gain 1 PeV =1000 TeV per meter. However, it is not at all obvious howpulsars manage in detail to transform the rotational energyinto the acceleration of particles. Pulsars possess a rotationalenergy ofE rot = 1 2 Θ pulsar ωpulsar 2 = 1 22 5 mR2 pulsar ω2 pulsar (5.23)≈ 7 × 10 42 J ≈ 4.4 × 10 61 eV(T pulsar = 30 ms, M pulsar = 2 × 10 30 kg, R pulsar = 20 km,ω = 2π/T ). If the pulsars succeed to convert a fraction ofonly 1% of this enormous energy into the acceleration ofcosmic-ray particles, one obtains an injection rate ofdE≈ 1.4 × 10 42 eV/s , (5.24)dtif a pulsar lifetime of 10 10 years is assumed.If one considers that our galaxy contains 10 11 stars andif the supernova explosion rate (pulsar creation rate) is assumedto be 1 per century, a total number of 10 8 pulsars haveprovided energy for the acceleration of cosmic-ray particlessince the creation of our galaxy (age of the galaxy ≈ 10 10years). This leads to a total energy of 2.2 × 10 67 eV for anaverage pulsar injection time of 5 × 10 9 years. For a totalvolume of our galaxy (radius 15 kpc, average effective thicknessof the galactic disk 1 kpc) of 2 × 10 67 cm 3 this correspondsto an energy density of cosmic rays of 1.1 eV/cm 3 .One has, of course, to consider that cosmic-ray particlesstay only for a limited time in our galaxy and are furthermoresubject to energy-loss processes. Still, the above presentedcrude estimate describes the actual energy density ofcosmic rays of ≈ 1eV/cm 3 rather well.energy production rateenergy density of cosmic rays5.6 Binaries“The advantage of living near a binaryis to get a tan in half the time.”AnonymousBinaries consisting of a pulsar or neutron star and a normalstar can also be considered as a site of cosmic-ray-particleacceleration. In such a binary system matter is permanently

72 5 Acceleration Mechanismsnormal starmattertransferFig. 5.7Formation of accretion disks inbinariespulsaraccelerationin gravitational potentialsdragged from the normal star and whirled into an accretiondisk around the compact companion. Due to these enormousplasma motions very strong electromagnetic fields areproduced in the vicinity of the neutron star. In these fieldscharged particles can be accelerated to high energies (seeFig. 5.7).The energy gain of infalling protons (mass m p )inthegravitational potential of a pulsar (mass M pulsar )isE =−∫ Rpulsar∞G m p M pulsarr 2dr = G m p M pulsarR pulsar≈ 1.1 × 10 −11 J ≈ 70 MeV (5.25)(m p ≈ 1.67 × 10 −27 kg, M pulsar = 2 × 10 30 kg, R pulsar =20 km, G ≈ 6.67 × 10 −11 m 3 kg −1 s −2 gravitational constant).The matter falling into the accretion disk achieves velocitiesv which are obtained under classical treatment from12 mv2 = E = G mM pulsar(5.26)R pulsarto provide values of√2GM pulsarv =≈ 1.2 × 10 8 m/s . (5.27)R pulsar

5.7 Energy Spectra of Primary Particles 73The variable magnetic field of the neutron star whichis perpendicular to the accretion disk will produce via theLorentz force a strong electric field. Usingaccretion disksF = e(v × B) = eE , (5.28)the particle energy E is obtained, using v ⊥ B, to∫E = F · ds = evBs . (5.29)Under plausible assumptions (v ≈ c, B = 10 6 T, s =10 5 m) particle energies of 3 × 10 19 eV are possible. Even active galactic nucleimore powerful are accretion disks which form around blackholes or the compact nuclei of active galaxies. One assumesthat in these active galactic nuclei and in jets ejected fromsuch nuclei, particles can be accelerated to the highest energiesobserved in primary cosmic rays.The details of these acceleration processes are not yetfully understood. Sites in the vicinity of black holes – abillion times more massive than the Sun – could possiblyprovide the environment for the acceleration of the highestenergycosmic rays. Confined highly relativistic jets are acommon feature of such compact sources. It is assumed thatthe jets of particles accelerated near a black hole or the nucleusof a compact galaxy are injected into the radiation fieldof the source. Electrons and protons accelerated in the jetsvia shocks initiate electromagnetic and hadronic cascades. beamed jets from black holesHigh-energy γ rays are produced by inverse Compton scatteringoff accelerated electrons. High-energy neutrinos arecreated in the decays of charged pions in the development ofthe hadronic cascade. It is assumed that one detects emissionfrom these sources only if the jets are beamed into our lineof sight. A possible scenario for the acceleration of particlesin beamed jets from massive compact sources is sketched inFig. 5.8.5.7 Energy Spectra of Primary Particles“Get first your facts right and then youcan distort them as much as you please.”Mark TwainAt the present time it is not at all clear which of the presentedmechanisms contribute predominantly to the accelerationof cosmic-ray particles. There are good arguments to

74 5 Acceleration MechanismsFig. 5.8Acceleration model for relativisticjets powered by a black hole or anactive galactic nucleus (thereactions are only sketched)shock-acceleration modelassume that the majority of galactic cosmic rays is producedby shock acceleration where the particles emitted from thesource are possibly further accelerated by the Fermi mechanismof 2nd order. In contrast, it is likely that the extremelyenergetic cosmic rays are predominantly accelerated in pulsars,binaries, or in jets emitted from black holes or activegalactic nuclei. For shock acceleration in supernova explosionsthe shape of the energy spectrum of cosmic-ray particlescan be derived from the acceleration mechanism.Let E 0 be the initial energy of a particle and εE 0 theenergy gain per acceleration cycle. After the first cycle onegetsE 1 = E 0 + εE 0 = E 0 (1 + ε) (5.30)while after the nth cycle (e.g., due to multiple reflection atshock fronts) one has

5.7 Energy Spectra of Primary Particles 75E n = E 0 (1 + ε) n . (5.31)To obtain the final energy E n = E, a number ofn = ln(E/E 0)ln(1 + ε)(5.32)cycles is required. Let us assume that the escape probabilityper cycle is P . The probability that particles still take part inthe acceleration mechanism after n cycles is (1 − P) n .Thisleads to the following number of particles with energies inexcess of E:N(> E) ∼∞∑(1 − P) m . (5.33)m=ncyclic energy gainBecause ofrewritten as∞∑m=0x m = 11 − xN(> E) ∼ (1 − P) n∞ ∑m=n∑∞= (1 − P) n (1 − P) m =m=0(for x < 1), (5.33) can be(1 − P) m−n(1 − P)nP, (5.34)where m − n has been renamed m. Equations (5.32) and(5.34) can be combined to form the integral energy spectrum integral primary spectraN(> E) ∼ 1 P( EE 0) −γ∼ E −γ , (5.35)where the spectral index γ is obtained from (5.34) and (5.35)with the help of (5.32) to(1 − P) n =( EE 0) −γ,n ln(1 − P) =−γ ln(E/E 0 ),n ln(1 − P) ln(1/(1 − P))γ =− = . (5.36)ln(E/E 0 ) ln(1 + ε)This simple consideration yields a power law of primarycosmic rays in agreement with observation.The energy gain per cycle surely is rather small (ε ≪ 1).If also the escape probability P is low (e.g., at reflectionsbetween two shock fronts), (5.36) is simplified to

76 5 Acceleration Mechanismsln(1 + P)γ ≈ln(1 + ε) ≈ P ε . (5.37)Experimentally one finds that the spectral index up to energiesof 10 15 eV is γ = 1.7. For higher energies the primarycosmic-ray-particle spectrum steepens with γ = 2.5.8 Problems1. Work out the kinetic energy of electrons accelerated ina betatron for the classical (v ≪ c) and the relativisticcase (B = 1Tesla,R = 0.2 m). See also Problem 1.2.2. A star of 10 solar masses undergoes a supernova explosion.Assume that 50% of its mass is ejected and theother half ends up in a pulsar of 10 km radius. What isthe Fermi energy of the electrons in the pulsar? What isthe consequence of it?3. It is assumed that active galactic nuclei are powered byblack holes. What is the energy gain of a proton fallinginto a one-million-solar-mass black hole down to theevent horizon?4. If the Sun were to collapse to a neutron star (R NS =50 km), what would be the rotational energy of such asolar remnant (M ⊙ = 2 × 10 30 kg, R ⊙ = 7 × 10 8 m,ω ⊙ = 3 × 10 −6 s −1 )? Compare this rotational energy tothe energy which a main-sequence star like the Sun canliberate through nuclear fusion!5. In a betatron the change of the magnetic flux φ =∫B dA = πR 2 B induces an electric field,∫E ds =−˙φ ,in which particles can be accelerated,E =−˙φ2πR =−1 2 RḂ.The momentum increase is given byṗ =−eE = 1 eRḂ . (5.38)2What kind of guiding field would be required to keep thecharged particles on a stable orbit?

776 Primary Cosmic Rays“It will be found that everything dependson the composition of the forces with whichthe particles of matter act upon one another,and from these forces, as a matter of fact, allphenomena of nature take their origin.”R. J. BoscovichCosmic rays provide important information about highenergyprocesses occurring in our galaxy and beyond. Cosmicradiation produced in the sources is usually called primordialcosmic rays. This radiation is modified during itspropagation in galactic and extragalactic space. Particles ofgalactic origin pass on average through a column density of6g/cm 2 before reaching the top of the Earth’s atmosphere.Of course, the atmosphere does not really have a ‘top’ butit rather exhibits an exponential density distribution. It hasbecome common practice to understand under the top of theatmosphere an altitude of approximately 40 km. This heightcorresponds to a residual column density of 5 g/cm 2 correspondingto a pressure of 5 mbar due to the residual atmosphereabove altitudes of 40 km. Cosmic rays arriving unperturbedat the Earth’s atmosphere are usually called primarycosmic rays.Sources of cosmic rays accelerate predominantly chargedparticles such as protons and electrons. Since all elementsof the periodic table are produced during element formation,nuclei as helium, lithium, and so on can be also accelerated.Cosmic rays represent an extraterrestrial or even extragalacticmatter sample whose chemical composition exhibits certainfeatures similar to the elemental abundance in our solarsystem.Charged cosmic rays accelerated in sources can producea number of secondary particles by interactions in thesources themselves.These mostly unstable secondary particles, i.e., pionsand kaons, produce stable particles in their decay, i.e., photonsfrom π 0 → γγ and neutrinos from π + → µ + + ν µdecays. Secondary particles also emerge from the sourcesand can reach Earth. Let us first discuss the originally acceleratedcharged component of primary cosmic rays.primordial radiationtop of the atmosphereproductionof secondary particles

78 6 Primary Cosmic Rays6.1 Charged Componentof Primary Cosmic Rays“Coming out of space and incident onthe high atmosphere, there is a thin rainof charged particles known as primarycosmic rays.”C. F. PowellFig. 6.1Elemental abundance of primarycosmic rays for 1 ≤ Z ≤ 28The elemental abundance of primary cosmic rays is shownin Figs. 6.1 and 6.2 in comparison to the chemical compositionof the solar system. Protons are the dominant particlespecies (≈ 85%) followed by α particles (≈ 12%). Elementswith a nuclear charge Z ≥ 3 represent only a 3%fraction of charged primary cosmic rays. The chemical compositionof the solar system, shown in Figs. 6.1 and 6.2, hasmany features in common with that of cosmic rays. However,remarkable differences are observed for lithium, beryllium,and boron (Z = 3–5), and for the elements below theiron group (Z

6.1 Charged Component of Primary Cosmic Rays 79In the same way the fragmentation or spallation of therelatively abundant element iron populates elements belowthe iron group. The general trend of the dependence of thechemical composition of primary cosmic rays on the atomicnumber can be understood by nuclear physics arguments.In the framework of the shell model it is easily explainedthat nuclear configurations with even proton and neutronnumbers (even–even nuclei) are more abundant comparedto nuclei with odd proton and neutron numbers (odd–oddnuclei). As far as stability is concerned, even–odd and odd–even nuclei are associated with abundances between ee andoo configurations. Extremely stable nuclei occur for filledshells (‘magic nuclei’), where the magic numbers (2, 8, 20,50, 82, 126) refer separately to protons and neutrons. As aconsequence, doubly magic nuclei (like helium and oxygen)are particularly stable and correspondingly abundant. Butnuclei with a large binding energy such as iron which can beproduced in fusion processes, are also relatively abundant incharged primary cosmic rays. The energy spectra of primarynuclei of hydrogen, helium, carbon, and iron are shown inFig. 6.3.ergy so that a direct observation of the high-energy componentof cosmic rays at the top of the atmosphere with balloonsor satellites eventually runs out of statistics. Measurementsof the charged component of primary cosmic rays atenergies in excess of several hundred GeV must thereforeresort to indirect methods. The atmospheric air Cherenkovtechnique (see Sect. 6.3: Gamma Astronomy) or the measurementof extensive air showers via air fluorescence orparticle sampling (see Sect. 7.4: Extensive Air Showers) canin principle cover this part of the energy spectrum, however,a determination of the chemical composition of primary cosmicrays by this indirect technique is particularly difficult.Furthermore, the particle intensities at these high energiesare extremely low. For particles with energies in excess ofabout 10 19 eV the rate is only 1 particle per km 2 and year.shell modelmagic nucleiThe low-energy part of the primary spectrum is modi-fied by the Sun’s and the Earth’s magnetic field. The 11-yearperiod of the sunspot cycle modulates the intensity of lowenergyprimary cosmic rays (< 1GeV/nucleon). The activeSun reduces the cosmic-ray intensity because a strongermagnetic field created by the Sun prevents galactic chargedparticles from reaching Earth.In general, the intensity decreases with increasing en-Fig. 6.3Energy spectra of the maincomponents of charged primarycosmic raysmodificationof the low-energy partdecreasing intensitywith high energies

80 6 Primary Cosmic RaysThe all-particle spectrum of charged primary cosmicrays (Fig. 6.4) is relatively steep so that practically no detailsare observable. Only after multiplication of the intensitywith a power of the primary energy, structures in theprimary spectrum become visible (Fig. 6.5). The bulk ofcosmic rays up to at least an energy of 10 15 eV is believed tooriginate from within our galaxy. Above that energy which isassociated with the so-called ‘knee’ the spectrum steepens.Above the so-called ‘ankle’ at energies around 5 × 10 18 eVthe spectrum flattens again. This latter feature is often interpretedas a crossover from a steeper galactic component to aharder component of extragalactic origin.Fig. 6.4Energy spectrum of all particles ofprimary cosmic raysFig. 6.5Energy spectrum of primarycosmic rays scaled by a factor E 3 .The data from the Japaneseair-shower experiment AGASAagree well – except at very highenergies – with the air-scintillationresults of the Utah High Resolutionexperiment as far as the spectralshape is concerned, but theydisagree in absolute intensity(AGASA – Akeno Giant AirShower Array, HiRes – HighResolution Fly’s Eye)10 5Fig. 6.6Artist’s impression of the differentstructures in the primarycosmic-ray spectrum[m s sr GeV ]2.5 –2 –1 –1 1.5N(E) · E10 3 10 12 all particle spectrum10 4 = 2.7knee = 3.010 210ankle toe10 14 10 1610 18 10 20energy per nucleus [eV]

6.1 Charged Component of Primary Cosmic Rays 81Cosmic rays originate predominantly from within ourgalaxy. Galactic objects do not in general have such a combinationof size and magnetic field strength to contain particlesat very high energies.Because of the equilibrium between the centrifugal andLorentz force (v ⊥ B assumed) one hasmv 2 /ϱ = ZevB (6.1)which yields for the momentum of singly charged particlesp = eϱB(p is the particle momentum, B the magnetic field, v theparticle velocity, m the particle mass, ϱ the bending radiusor gyroradius). For a large-area galactic magnetic field ofB = 10 −10 Tesla in the galaxy (about 10 5 times weakercompared to the magnetic field on the surface of the Earth)and a gyroradius of 5 pc, from which particles start to leakfrom the galaxy, particles with momenta up top[GeV/c] =0.3 B[T] ϱ[m] ,p max = 4.6 × 10 6 GeV/c = 4.6 × 10 15 eV/c (6.2)can be contained. 1 parsec (pc) is the popular unit of distancein astronomy (1 pc = 3.26 light-years = 3.0857 ×10 16 m). Particles with energies exceeding 10 15 eV start toleak from the galaxy. This causes the spectrum to get steeperto higher energies. Since the containment radius depends onthe atomic number, see (6.1), the position of the knee shoulddepend on the charge of primary cosmic rays in this scenario,i.e., the knee for iron would be expected at higherenergies compared to the proton knee.Another possible reason for the knee in cosmic radiationcould be related to the fact that 10 15 eV is about themaximum energy which can be supplied by supernova explosions.For higher energies a different acceleration mechanismis required which might possibly lead to a steeperenergy spectrum. The knee could in principle also haveits origin in a possible change of interaction characteristicsof high-energy particles. The energy of the knee coincideswith the maximum energy presently available at accelerators( √ s = 1.8 TeV, corresponding to E lab ≈ 2 × 10 15 eV)beyond which no direct measurements are available. It isconceivable that the interaction cross section changes withenergy giving rise to features at the knee of the primarycosmic-ray spectrum. The flattening of the spectrum aboveequilibrium betweencentrifugal and Lorentz forcegalactic containment1parsec(pc)= 3.26 LYproton kneeiron kneeankle of cosmic rays

82 6 Primary Cosmic Rays10 19 eV (‘ankle’) is generally assumed to be due to an extragalacticcomponent.In 1966 it was realized by Greisen, Zatsepin, and Kuzmin(GZK) that cosmic rays above the energy of approximately6 × 10 19 eV would interact with the cosmic blackbody radiation.Protons of higher energies would rapidly lose energyby this interaction process causing the spectrum to be cut offat energies around 6 × 10 19 eV. Primary protons with theseenergies produce pions on blackbody photons via the ∆ resonanceaccording toγ + p → p + π 0 , γ + p → n + π + , (6.3)Greisen–Zatsepin–Kuzmincutoffthereby losing a large fraction of their energy.The threshold energy for the photoproduction of pionscan be determined from four-momentum conservation(q γ + q p ) 2 = (m p + m π ) 2 (6.4)(q γ , q p are four-momenta of the photon or proton, respectively;m p , m π are proton and pion masses) yieldingE p = (m 2 π + 2m pm π )/4E γ (6.5)for head-on collisions.A typical value of the Planck distribution correspondingto the blackbody radiation of temperature 2.7 Kelvinis around 1.1 meV. With this photon energy the thresholdenergy for the photoproduction of pions isE p ≈ 6 × 10 19 eV . (6.6)toe of cosmic raysγγ interactionsThe observation of several events in excess of 10 20 eV(the ‘toe’ of primary cosmic rays), therefore, represents acertain mystery. The Greisen–Zatsepin–Kuzmin cutoff limitsthe mean free path of high-energy protons to somethinglike 10 Mpc. Therefore, the experimental verification of theGZK cutoff in the spectrum of primary cosmic rays wouldbe the clearest proof that the high-energy particles beyondseveral 10 19 eV are generated at extragalactic sources. Photonsas candidates for primary particles have even shortermean free paths (≈ 10 kpc) because they produce electronpairs in gamma–gamma interactions with blackbody photons,infrared and starlight photons (γγ → e + e − ). Thehypothesis that primary neutrinos are responsible for thehighest-energy events is rather unlikely. The interactionprobability for neutrinos in the atmosphere is extremely

6.1 Charged Component of Primary Cosmic Rays 83small (< 10 −4 ). Furthermore, the observed zenith-angledistribution of energetic events and the position of primaryvertices of the cascade development in the atmosphere areinconsistent with the assumption that primary neutrinos areresponsible for these events. Because of their low interactionprobability one would expect that the primary vertices forneutrinos would be distributed uniformly in the atmosphere.In contrast, one observes that the first interaction takes placepredominantly in the 100 mbar layer which is characteristicof hadron or photon interactions. One way out would be toassume that after all protons are responsible for the eventswith energies exceeding 6 × 10 19 eV. This would supportthe idea that the sources of the highest-energy cosmic-rayevents are relatively close. A candidate source is M87, anelliptic giant galaxy in the Virgo cluster at a distance ofneutrino origin15 Mpc. From the center of M87 a jet of 1500 pc length is M87, a sourceejected that could be the source of energetic particles. M87 of cosmic rays?coincides with Virgo A (3C274), one of the strongest radiosources in the constellation Virgin.A close look at the experimental situation (see Fig.6.5) shows that the two major experiments measuring inthe ≤ EeV range do not agree in absolute intensity andalso not in the shape of the spectrum for energies in excessof ≈ 5 × 10 19 eV. Even though the HiRes experimentfinds events beyond 10 20 eV, the HiRes spectrum is not indisagreement with the expectation based on the GZK cutoff,quite in contrast to the AGASA findings. One mightargue that possibly the energy assignment for the showersfrom HiRes is superior over that from AGASA sinceHiRes records the complete longitudinal development of theshower, while AGASA only samples the shower informationin one atmospheric layer, i.e., at ground level.It is to be expected that the Auger experiment will clarifythe question concerning the GZK cutoff. In this context it isinteresting to note that the Auger-south array has recordedan event with ≈ 10 20 eV already during the constructionphase [4]. The full Auger-south detector will be completedin spring 2006.Considering the enormous rigidity of these high-energyparticles and the weakness of the intergalactic magnetic fieldone would not expect substantial deflections of these particlesover distances of 50 Mpc. This would imply that one canconsider to do astronomy with these extremely high-energycosmic rays. Since there is no correlation of the arrival di-rections of these high-energy cosmic-ray events with knownHiRes vs. AGASAAuger experimentexotic particles?

84 6 Primary Cosmic Raysantiparticlesin primary cosmic rays¯p generationastronomical sources in the immediate neighbourhood ofour galaxy, one might resort to the assumption that new andso far unknown elementary particles might be responsiblefor the events exceeding 6 × 10 19 eV, if the AGASA resultsare confirmed, e.g., by the Auger experiment.Antiparticles are extremely rare in primary cosmic rays.The measured primary antiprotons are presumably generatedin interactions of primary charged cosmic rays withthe interstellar gas. Antiprotons can be readily producedaccording top + p → p + p + p +¯p (6.7)e + generationprimary electronsstars of antimatter?(compare Example 1, Chap. 3), while positrons are mosteasily formed in pair production by energetic photons (compareExample 4, Chap. 3). The flux of primary antiprotonsfor energies > 10 GeV has been measured to beN( ¯p)N(p) ∣ ≈ 10 −4 . (6.8)>10 GeVThe fraction of primary electrons in relation to primaryprotons is only 1%. Primary positrons constitute only 10%of the electrons at energies around 10 GeV. They are presumablyalso consistent with secondary origin.One might wonder whether the continuous bombardmentof the Earth with predominantly positively chargedparticles (only 1% are negatively charged) would lead to apositive charge-up of our planet. This, however, is not true.When the rates of primary protons and electrons are compared,one normally considers only energetic particles. Thespectra of protons and electrons are very different with electronspopulating mainly low-energy regions. If all energiesare considered, there are equal numbers of protons and electronsso that there is no charge-up of our planet.To find out whether there are stars of antimatter in theuniverse, the existence of primary antinuclei (antihelium,anticarbon) must be established because secondary productionof antinuclei with Z ≥ 2 by cosmic rays is practicallyexcluded. The non-observation of primary antimatter withZ ≥ 2 is a strong hint that our universe is matter dominated.The chemical composition of high-energy primary cos-mic rays (> 10 15 eV) is to large extent an unknown territory.If the current models of nucleon–nucleon interactions areextrapolated into the range beyond 10 15 eV (correspondingto a center-of-mass energy of > ∼ 1.4 TeV in proton–protonchemical compositionof high-energy cosmic rays

6.1 Charged Component of Primary Cosmic Rays 85collisions) and if the muon content and lateral distribution ofmuons in extensive air showers are taken as a criterion forthe identity of the primary particle, then one would arriveat the conclusion that the chemical composition of primarycosmic rays cannot be very different from the compositionbelow the knee (< 10 15 eV). Some experiments, however,seem to indicate that the iron fraction of primary cosmicrays increases with energy beyond the knee. This could justbe a consequence of galactic containment: the iron knee isexpected to occur at higher energies compared to the protonknee.Even though cosmic rays have been discovered about 90years ago, their origin is still an open question. It is generallyassumed that active galactic nuclei, quasars, or supernovaexplosions are excellent source candidates for high-energycosmic rays, but there is no direct evidence for this assumption.In the energy range up to 100 TeV individual sourceshave been identified by primary gamma rays. It is conceivablethat gamma rays of these energies are decay products ofelementary particles (π 0 decay, Centaurus A?), which havebeen produced by those particles that have been originallyaccelerated in the sources. Therefore, it would be interestingto see the sources of cosmic rays in the light of theseoriginally accelerated particles.This, however, presents a serious problem: photons andneutrinos travel on straight lines in galactic and intergalacticspace, therefore pointing directly back to the sources.Charged particles, on the other hand, are subject to the influenceof homogeneous or irregular magnetic fields. Thiscauses the accelerated particles to travel along chaotic trajectoriesthereby losing all directional information beforefinally reaching Earth. Therefore, it is of very little surprisethat the sky for charged particles with energies below10 14 eV appears completely isotropic. The level of observedanisotropies lies below 0.5%. There is some hope that forenergies exceeding 10 18 eV a certain directionality could befound. It is true that also in this energy domain the galacticmagnetic fields must be taken into account, however, the deflectionradii are already rather large. The situation is evenmore complicated because of a rather uncertain topologyof galactic magnetic fields. In addition, one must in principleknow the time evolution of magnetic fields over thelast ≈ 50 million years because the sources can easily resideat distances of > 10 Mpc (̂= 32.6 million light-years).For simultaneous observation of cosmic-ray sources in theorigin of cosmic rayshadronic originof energetic γ s?isotropy of charged particlesanisotropy

86 6 Primary Cosmic RaysFig. 6.7Sketch of proton and iron-nucleustrajectories in our Milky Way at10 18 eVlight of charged particles and photons, one must take intoaccount that charged particles are delayed with respect tophotons because they travel on trajectories bent by the magneticfield.Since the magnetic deflection is proportional to thecharge of a particle, proton astronomy is more promisingthan astronomy with heavy nuclei. This idea is outlined inFig. 6.7 where the trajectories of protons and an iron nucleus(Z = 26) at an energy of 10 18 eV are sketched for ourgalaxy. This figure clearly shows that one should only use –if experimentally possible – protons for particle astronomy.Actually, there are some hints that the origins of the fewevents with energies > 10 19 eV could lie in the supergalacticplane, a cluster of relatively close-by galaxies includingour Milky Way (‘local super galaxy’). It has also been discussedthat the galactic center of our Milky Way and in particularthe Cygnus region could be responsible for a certainanisotropy at 10 18 eV. It must, however, be mentioned thatclaims for such a possible correlation are based on very lowstatistics and are therefore not unanimously supported. Theycertainly need further experimental confirmation.6.2 Neutrino Astronomy“Neutrino physics is largely an art oflearning a great deal by observing nothing.”Haim Hararilimits of classical astronomyThe disadvantage of classical astronomies like observationsin the radio, infrared, optical, ultraviolet, X-ray, or γ -rayband is related to the fact that electromagnetic radiationis quickly absorbed in matter. Therefore, with these astronomiesone can only observe the surfaces of astronomicalobjects. In addition, energetic γ rays from distant sourcesare attenuated via γγ interactions with photons of the blackbodyradiation by the processγ + γ → e + + e − .Energetic photons (> 10 15 eV), for example, from the LargeMagellanic Cloud (LMC, 52 kpc distance) are significantlyattenuated by this process (see Fig. 4.9).Charged primaries can in principle also be used in as-troparticle physics. However, the directional informationcharged primaries

6.2 Neutrino Astronomy 87is only conserved for very energetic protons (> 10 19 eV)because otherwise the irregular and partly not well-knowngalactic magnetic fields will randomize their original direction.For these high energies the Greisen–Zatsepin–Kuzmincutoff also comes into play whereby protons lose their energyvia the photoproduction of pions off blackbody photons.For protons with energies exceeding 6 × 10 19 eVthe universe is no longer transparent (attenuation lengthλ ≈ 10 Mpc). As a consequence of these facts, the requirementfor an optimal astronomy can be defined in the followingway:1. The optimal astroparticles or radiation should not be influencedby magnetic fields.2. The particles should not decay from source to Earth.This practically excludes neutrons as carriers unless neutronshave extremely high energy (τneutron 0 = 885.7s;atE = 10 19 eV one has γcτneutron 0 ≈ 300 000 light-years).3. Particles and antiparticles should be different. Thiswould in principle allow to find out whether particlesoriginate from a matter or antimatter source. This requirementexcludes photons because a photon is its ownantiparticle, γ =¯γ .4. The particles must be penetrating so that one can lookinto the central part of the sources.5. Particles should not be absorbed by interstellar or intergalacticdust or by infrared or blackbody photons.These five requirements are fulfilled by neutrinos in an idealway! One could ask oneself why neutrino astronomy hasnot been a major branch of astronomy all along. The factthat neutrinos can escape from the center of the sources isrelated to their low interaction cross section. This, unfortunately,goes along with an enormous difficulty to detectthese neutrinos on Earth.For solar neutrinos in the range of several 100 keV thecross section for neutrino–nucleon scattering isrequirementfor an optimal astronomyneutrino astronomyσ(ν e N) ≈ 10 −45 cm 2 /nucleon . (6.9)The interaction probability of these neutrinos with our planetEarth at central incidence isφ = σN A dϱ≈ 4 × 10 −12 (6.10)(N A is the Avogadro number, d the diameter of the Earth, ϱthe average density of the Earth). Out of the 7 × 10 10 neutri-

88 6 Primary Cosmic Raysneutrino telescopesneutrino detectionnos per cm 2 and s radiated by the Sun and arriving at Earthonly one at most is ‘seen’ by our planet.As a consequence of this, neutrino telescopes must havean enormous target mass, and one has to envisage long exposuretimes. However, for high energies the interaction crosssection rises with neutrino energy. Neutrinos in the energyrange of several 100 keV can be detected by radiochemicalmethods. For energies exceeding 5 MeV large-volume waterCherenkov counters are an attractive possibility.Neutrino astronomy is a very young branch of astroparticlephysics. Up to now four different sources of neutrinoshave been investigated. The physics results and implicationsof these measurements will be discussed in the followingfour sections.Therefore, the atmospheric neutrino beam contains elec-tron and muon neutrinos and one would expect a ratioN(ν µ , ¯ν µ )N(ν e , ¯ν e ) ≡ N µ≈ 2 , (6.13)N eelectronic hutneutrino productioncentraldetectorveto counterFig. 6.8The Super-Kamiokande detector inthe Kamioka mine in Japan {8}6.2.1 Atmospheric NeutrinosFor real neutrino astronomy neutrinos from atmosphericsources are an annoying background. For the particle physicsaspect of astroparticle physics atmospheric neutrinos haveturned out to be a very interesting subject. Primary cosmicrays interact in the atmosphere with the atomic nuclei of nitrogenand oxygen. In these proton–air interactions nuclearfragments and predominantly charged and neutral pions areproduced. The decay of charged pions (lifetime 26 ns) producesmuon neutrinos:π + → µ + + ν µ , π − → µ − +¯ν µ . (6.11)Muons themselves are also unstable and decay with an averagelifetime of 2.2 µs according toµ + → e + + ν e +¯ν µ , µ − → e − +¯ν e + ν µ . (6.12)as can be easily seen by counting the decay neutrinos in reexpecteddominance of ν µactions (6.11) and (6.12).The presently largest experiments measuring atmosphericneutrinos are Super-Kamiokande (see Fig. 6.8) andAMANDA II (see Sect. 6.2.4). Neutrino interactions inthe Super-Kamiokande detector are recorded in a tank ofapproximately 50 000 tons of ultrapure water. Electron neutrinostransfer part of their energy to electrons,

6.2 Neutrino Astronomy 89ν e + e − → ν e + e − , (6.14)or produce electrons in neutrino–nucleon interactionsν e + N → e − + N ′ . (6.15)Muon neutrinos are detected in neutrino–nucleon interactionsaccording toν µ + N → µ − + N ′ . (6.16)Electron antineutrinos and muon antineutrinos produce correspondinglypositrons and positive muons. The chargedleptons (e + , e − , µ + , µ − ) can be detected via the Cherenkoveffect in water. The produced Cherenkov light is measuredwith 11 200 photomultipliers of 50 cm cathode diameter.In the GeV-range electrons initiate characteristic electromagneticcascades of short range while muons producelong straight tracks. This presents a basis for distinguishingelectron from muon neutrinos. On top of that, muonscan be identified by their decay in the detector thereby givingadditional evidence concerning the identity of the initiatingneutrino species. Figures 6.9 and 6.10 show an electronand muon event in Super-Kamiokande. Muons have awell-defined range and produce a clear Cherenkov patternwith sharp edges while electrons initiate electromagneticcascades thereby creating a fuzzy ring pattern.The result of the Super-Kamiokande experiment is thatthe number of electron-neutrino events corresponds to thetheoretical expectation while there is a clear deficit of eventsinitiated by muon neutrinos. The measured electron andmuon spectra compared to expectation are shown in Figs.6.11 and 6.12.Because of the different acceptance for electrons andmuons in the water Cherenkov detector, the ratio of muonsto electrons is compared to a Monte Carlo simulation. Forthe double ratio(N µ /N e ) dataR =(6.17)(N µ /N e ) Monte Carloone would expect the value R = 1 in agreement with thestandard interaction and propagation models. However, theSuper-Kamiokande experiment obtainsFig. 6.9Cherenkov pattern of an energeticelectron in the Super-KamiokandeDetector {9}distinguishingelectron and muon neutrinosdeficit of muon neutrinosFig. 6.10Cherenkov pattern for an energeticmuon in the Super-Kamiokandedetector {9}R = 0.69 ± 0.06 , (6.18)which represents a clear deviation from expectation.

90 6 Primary Cosmic RaysFig. 6.11Momentum spectrum of single-ringelectron-like events inSuper-Kamiokande. The solid linerepresents the Monte Carloexpectation {9}Fig. 6.12Momentum spectrum of single-ringmuon-like events inSuper-Kamiokande. The solid linerepresents the Monte Carloexpectation. The cutoff arround 10GeV originates from the conditionthat the muon tracks must becontained in the detector {9}neutrino oscillationseigenstatesof the weak interactionmass eigenstatesAfter careful checks of the experimental results and investigationsof possible systematic effects the general opinionprevails that the deficit of muon neutrinos can only beexplained by neutrino oscillations.Mixed particle states are known from the quark sector(see Chap. 2). Similarly, it is conceivable that in the leptonsector the eigenstates of weak interactions ν e , ν µ ,andν τ are superpositions of mass eigenstates ν 1 ,ν 2 ,andν 3 .A muon neutrino ν µ born in a pion decay could be transformedduring the propagation from the source to the observationin the detector into a different neutrino flavour. If themuon neutrino in reality was a mixture of two different masseigenstates ν 1 and ν 2 , these two states would propagate atdifferent velocities if their masses were not identical andso the mass components get out of phase with each other.This could possibly result in a different neutrino flavour atthe detector. If, however, all neutrinos were massless, theywould all propagate precisely at the velocity of light, andthe mass eigenstates can never get out of phase with eachother.

6.2 Neutrino Astronomy 91For an assumed two-neutrino mixing of ν e and ν µ theweak eigenstates could be related to the mass eigenstates bythe following two equations:mixing angleν e = ν 1 cos θ + ν 2 sin θ,ν µ =−ν 1 sin θ + ν 2 cos θ.(6.19)The mixing angle θ determines the degree of mixing.This assumption requires that the neutrinos have non-zeromass and, in addition, m 1 ̸= m 2 must hold.In the framework of this oscillation model the probabilitythat an electron neutrino stays an electron neutrino, canbe calculated to be (see also Problem 4 in this section):(P νe →ν e(x) = 1 − sin 2 2θ sin 2 π x ), (6.20)L νwhere x is the distance from the source to the detector andL ν the oscillation lengthL ν =2.48 E ν [MeV](m 2 1 − m2 2 ) [eV2 /c 4 ] m . (6.21)The expression m 2 1 −m2 2 is usually abbreviated as δm2 . Equations(6.20) and (6.21) can be combined to give(P νe →ν e(x) = 1 − sin 2 2θ sin 2 1.27 δm 2 x )(6.22)E νwhere δm 2 is measured in eV 2 , x in km, and E ν in GeV.The idea of a two-neutrino mixing is graphically presentedin Fig. 6.13.For the general case of mixing of all three neutrinoflavours one obtains as generalization of (6.19)⎛⎝ ν ⎞ ⎛eν µ⎠ = U N⎝ ν ⎞1ν 2⎠ , (6.23)ν τ ν 3where U N is the (3 × 3) neutrino mixing matrix.The deficit of muon neutrinos can now be explained bythe assumption that some of the muon neutrinos transformthemselves during propagation from the point of productionto the detector into a different neutrino flavour, e.g., into tauneutrinos. The sketch shown in Fig. 6.13 demonstrated thatforanassumedmixingangleof45 ◦ all neutrinos of a certaintype have transformed themselves into a different neutrinoflavour after propagating half the oscillation length.Fig. 6.13Oscillation model for ν e –ν µmixing for different mixing anglesmixing matrixdeficit of muon neutrinos

92 6 Primary Cosmic Raystau production?If, however, muon neutrinos have oscillated into tau neutrinos,a deficit of muon neutrinos will be observed in thedetector because tau neutrinos would only produce taus inthe water Cherenkov counter, but not muons. Since, however,the mass of the tau is rather high (1.77 GeV/c 2 ), tauneutrinos normally would not meet the requirement to providethe necessary center-of-mass energy for tau production.Consequently, they would escape from the detector withoutinteraction. If the deficit of muon neutrinos would be interpretedby (ν µ → ν τ ) oscillations, the mixing angle and thedifference of mass squares δm 2 can be determined from theexperimental data. The measured value of the double ratioR = 0.69 leads toδm 2 ≈ 2 × 10 −3 eV 2 (6.24)neutrino mass(see also parts b,c of Problem 4 in this section). The validityof this conclusion relies on the correctly measured absolutefluxes of electron and muon neutrinos. Because of the differ-ent Cherenkov pattern of electrons and muons in the waterCherenkov detector the efficiencies for electron neutrino andmuon neutrino detections might be different. To support theoscillation hypothesis one would therefore prefer to have anadditional independent experimental result. This is providedin an impressive manner by the ratio of upward- to downward-goingmuons. Upward-coming atmospheric neutrinoshave traversed the whole Earth (≈ 12 800 km). They wouldhave a much larger probability to oscillate into tau neutrinoscompared to downward-going neutrinos which have traveledtypically only 20 km. Actually, according to the experimentalresult of the Super-Kamiokande collaboration thezenith-angle dependenceat maximal mixing (sin 2 2θ = 1, corresponding to θ =45 ◦ ). 1 If one assumes that in the neutrino sector a similarmass hierarchy as in the sector of charged leptons exists(m e ≪ m µ ≪ m τ ), then the mass of the heaviest neutrinocan be estimated from (6.24),√≈ δm 2 ≈ 0.045 eV (6.25)m ντ1 The 90% coincidence limit for δm 2 given by the Super-Kamiokande experiment is 1.3 × 10 −3 eV 2 ≤ δm 2 ≤ 3 ×10 −3 eV 2 . The accelerator experiment K2K sending muon neutrinosto the Kamioka mine gets δm 2 = 2.8 × 10 −3 eV 2 [5].K2K – from KEK to Kamioka, Long-baseline Neutrino OscillationExperiment

6.2 Neutrino Astronomy 93upward-going muon neutrinos which have traveled throughthe whole Earth are suppressed by a factor of two comparedto the downward-going muons. This is taken as a strong indicationfor the existence of oscillations (see Fig. 6.14). Forthe ratio of upward- to downward-going muon neutrinos oneobtainsS =N(ν µ, up)= 0.54 ± 0.06 , (6.26)N(ν µ , down)which presents a clear effect in favour of oscillation.Details of the observed zenith-angle dependence of atmosphericν e and ν µ fluxes also represent a particularlystrong support for the oscillation model.The comparison of electron and muon events in both thesub-GeV and multi-GeV range, in their dependence on thezenith angle, is shown in Fig. 6.15.The electron events are in perfect agreement with expectation,while the no-oscillation hypothesis for muons isclearly ruled out by the data.Since the production altitude L and energy E ν of atmosphericneutrinos are known (≈ 20 km for vertically downward-goingneutrinos), the observed zenith-angle dependenceof electron and muon neutrinos can also be convertedFig. 6.14Ratio of ν µ fluxes as a function ofzenith angle as measured in theSuper-Kamiokande experiment400300(a)(b)number of events200100number of events0200(c)(d)150100500–1 –0.5 0 0.5 1 –1 –0.5 0 0.5 1cos cos Fig. 6.15Zenith-angle distribution ofelectron-like and muon-like eventsfor the sub-GeV range(a: electrons, b: muons) and themulti-GeV range (c: electrons,d: muons) in Super-Kamiokande.The dark grey line is theexpectation for the null hypothesis(no oscillations) while the lightgrey histogram represents theexpectation for oscillations withmaximal mixing (sin 2 2θ = 1) andδm 2 = 3 × 10 −3 eV 2 {9}

94 6 Primary Cosmic RaysFig. 6.16Ratio of fully contained eventsmeasured in theSuper-Kamiokande detector as afunction of the reconstructed valueof distance over energy (L/E ν ).The lower histogram for µ-likeevents corresponds to theexpectation for ν µ ↔ ν τoscillations withδm 2 = 2.2 × 10 −3 eV andsin 2θ = 1{10}fusion reactions in the Suninto a dependence of the rate versus the reconstructed ratioof L/E ν . Figure 6.16 shows the ratio data/Monte Carlo forfully contained events as measured in the Super-Kamiokandeexperiment. The data exhibit a zenith-angle- (i.e., distance-)dependent deficit of muon neutrinos, while the electron neutrinosfollow the expectation for no oscillations. The observedbehaviour is consistent with (ν µ ↔ ν τ ) oscillations,where a best fit is obtained for δm 2 = 2.2 × 10 −3 eV formaximal mixing (sin 2θ = 1).In the Standard Model of elementary particles neutrinoshave zero mass. Therefore, neutrino oscillations representan important extension of the physics of elementary particles.In this example of neutrino oscillations the synthesisbetween astrophysics and particle physics becomes particularlyevident.6.2.2 Solar NeutrinosThe Sun is a nuclear fusion reactor. In its interior hydrogenis burned to helium. The longevity of the Sun is related tothe fact that the initial reactionp + p → d + e + + ν e (6.27)proceeds via the weak interaction. 86% of solar neutrinos areproduced in this proton–proton reaction. Deuterium madeaccording to (6.27) fuses with a further proton to producehelium 3,d + p → 3 He + γ. (6.28)In 3 He– 3 He interactions3 He + 3 He → 4 He + 2p (6.29)‘chemistry’ of nuclear fusionthe isotope helium 4 can be formed. On the other hand, theisotopes 3 He and 4 He could also produce beryllium,3 He + 4 He → 7 Be + γ. (6.30)7 Be is made of four protons and three neutrons. Light elementsprefer symmetry between the number of protons andneutrons. 7 Be can capture an electron yielding 7 Li,7 Be + e − → 7 Li + ν e , (6.31)where a proton has been transformed into a neutron. On theother hand, 7 Be can react with one of the abundant protonsto produce 8 B,

6.2 Neutrino Astronomy 957 Be + p → 8 B + γ. (6.32)7 Li produced according to (6.31) will usually interact withprotons forming helium,7 Li + p → 4 He + 4 He , (6.33)while the boron isotope 8 B will reduce its proton excess byβ + decay,8 B → 8 Be + e + + ν e , (6.34)and the resulting 8 Be will disintegrate into two helium nuclei.Apart from the dominant pp neutrinos, reaction (6.27),further 14% are generated in the electron-capture reaction(6.31), while the 8 B decay contributes only at the level of0.02% albeit yielding high-energy neutrinos. In total, the solarneutrino flux at Earth amounts to about 7×10 10 particlesper cm 2 and second.The energy spectra of different reactions which proceedin the solar interior at a temperature of 15 million Kelvinare shown in Fig. 6.17. The Sun is a pure electron–neutrinosource. It does not produce electron antineutrinos and, inparticular, no other neutrino flavours (ν µ ,ν τ ).Three radiochemical experiments and two water Cherenkovexperiments are trying to measure the flux of solar neutrinos.The historically first experiment for the search of solarneutrinos is based on the reactionFig. 6.17Neutrino spectra from solar fusionprocesses. The reaction thresholdsof the gallium, chlorine, and waterCherenkov experiments areindicated. The line fluxes ofberyllium isotopes are given inunits of cm −2 s −1ν e + 37 Cl → 37 Ar + e − , (6.35)where the produced 37 Ar has to be extracted from a hugetank filled with 380 000 liters of perchlorethylene (C 2 Cl 4 ).Because of the low capture rate of less than one neutrinoper day the experiment must be shielded against atmosphericcosmic rays. Therefore, it is operated in a gold mine at 1500meter depth under the Earth’s surface (see Fig. 6.18). Aftera run of typically one month the tank is flushed with a noblegas and the few produced 37 Ar atoms are extracted fromthe detector and subsequently counted. Counting is done bymeans of the electron-capture reaction of 37 Ar where again37 Cl is produced. Since the electron capture occurs predominantlyfrom the K shell, the produced 37 Cl atom is nowmissing one electron in the innermost shell (in the K shell).The atomic electrons of the 37 Cl atom are rearranged underemission of either characteristic X rays or by the emissionof Auger electrons. These Auger electrons and, in particu-Fig. 6.18The detector of the chlorineexperiment of R. Davis for themeasurement of solar neutrinos.The detector is installed at a depthof 1400 m in the Homestake Minein South Dakota. It is filled with380 000 liters ofperchlorethylene {11}

96 6 Primary Cosmic RaysDavis experimentGALLEX, SAGEpp fusion neutrinosFig. 6.19Arrival directions of neutrinosmeasured in theSuper-Kamiokande experimentKamiokande,Super-Kamiokandelar, the characteristic X rays are the basis for counting 37 Aratoms produced by solar neutrinos.In the course of 30 years of operation a deficit of solarneutrinos has become more and more evident. The experimentled by Davis only finds 27% of the expected solarneutrino flux. To solve this neutrino puzzle, two furtherneutrino experiments were started. The gallium experimentGALLEX in a tunnel through the Gran Sasso mountains inItaly and the Soviet–American gallium experiment (SAGE)in Caucasus measure the flux of solar neutrinos also in radiochemicalexperiments. Solar neutrinos react with gallium accordingtoν e + 71 Ga → 71 Ge + e − . (6.36)In this reaction 71 Ge is produced and extracted like in theDavis experiment and counted. The gallium experimentshave the big advantage that the reaction threshold for thereaction (6.36) is as low as 233 keV so that these experimentsare sensitive to neutrinos from the proton–proton fusionwhile the Davis experiment with a threshold of 810keV essentially only measures neutrinos from the 8 B decay.GALLEX and SAGE have also measured a deficit of solarneutrinos. They only find 52% of the expected rate whichpresents a clear discrepancy to the prediction on the basisof the standard solar model. However, the discrepancy is notso pronounced as in the Davis experiment. A strong pointfor the gallium experiments is that the neutrino capture rateand the extraction technique have been checked with neutrinosof an artificial 51 Cr source. It could be convincinglyshown that the produced 71 Ge atoms could be successfullyextracted in the expected quantities.The Kamiokande and Super-Kamiokande experiment,respectively, measure solar neutrinos via the reactionν e + e − → ν e + e − (6.37)at a threshold of 5 MeV in a water Cherenkov counter. Sincethe emission of the knock-on electron follows essentiallythe direction of the incident neutrinos, the detector can really‘see’ the Sun. This directionality gives the water Cherenkovcounter a superiority over the radiochemical experiments.Figure 6.19 shows the neutrino counting rate of theSuper-Kamiokande experiment as a function of the anglewith respect to the Sun. The Super-Kamiokande experimentalso measures a low flux of solar neutrinos representing only40% of the expectation.

6.2 Neutrino Astronomy 97A reconstructed image of the Sun in the light of neutrinosis shown in Fig. 6.20.Many proposals have been made to solve the solar neutrinoproblem. The first thing for elementary particle physicistsis to doubt the correctness of the standard solar model.The flux of 8 B neutrinos varies with the central temperatureof the Sun like ∼ T 18 . A reduction by only 5% of the centralsolar temperature would bring the Kamiokande experimentalready in agreement with the now reduced expectation.However, solar astrophysicists consider even a somewhatlower central temperature of the Sun rather improbable.The theoretical calculation of the solar neutrino flux usesthe cross sections for the reactions (6.27) up to (6.34). Anoverestimate of the reaction cross sections would also leadto a too high expectation for the neutrino flux. A variation ofthese cross sections in a range which is considered realisticby nuclear physicists is insufficient to explain the discrepancybetween the experimental data and expectation.If neutrinos had a mass, they could also possess a magneticmoment. If their spin is rotated while propagating fromthe solar interior to the detector at Earth, one would not beable to measure these neutrinos because the detectors are insensitiveto neutrinos of wrong helicity.Finally, solar neutrinos could decay on their way fromSun to Earth into particles which might be invisible to theneutrino detectors.A drastic assumption would be that the solar fire hasgone out. In the light of neutrinos this would become practicallyimmediately evident (more precisely: in 8 minutes).The energy transport from the solar interior to the surface,however, requires a time of several 100 000 years so that theSun would continue to shine for this period even though thenuclear fusion at its center has come to an end.Since all mentioned explanations are considered ratherunlikely, it is attractive to interpret a deficit of solar neutrinosalso by oscillations like in the case of atmospheric neutrinos.In addition to the vacuum oscillations described by(6.23), solar neutrinos can also be transformed by so-calledmatter oscillations. The flux of electron neutrinos and itsoscillation property can be modified by neutrino–electronscattering when the solar neutrino flux from the interior ofthe Sun encounters collisions with the abundant number ofsolar electrons. Flavour oscillations can even be magnifiedin a resonance-like fashion by matter effects so that certainFig. 6.20Reconstructed image of the Sun inthe light of solar neutrinos. Due tothe limited spatial and angularresolution of Super-Kamiokande,the image of the Sun appears largerthan it really is {12}cross sectionmagnetic momentof neutrinos?neutrino decay?extinct solar fire?neutrino oscillations?matter oscillations

98 6 Primary Cosmic RaysMSW effect ee –W –e – eFig. 6.21Feynman diagram responsible formatter oscillation (MSW effect).Given the energy of solarneutrinos and the fact that thereare only target electrons in theSun, this process can only occurfor ν e , but not for ν µ or ν τenergy ranges of the solar ν e spectrum are depleted. Thepossibility of matter oscillations has first been proposed byMikheyev, Smirnov, and Wolfenstein. The oscillation propertyof the MSW effect is different from that of vacuumoscillations. It relates to the fact that ν e e − scattering contributesa term to the mixing matrix that is not present invacuum. Due to this charged-current interaction (Fig. 6.21),which is kinematically not possible for ν µ and ν τ in the Sun,the interaction Hamiltonian for ν e is modified compared tothe other neutrino flavours. This leads to alterations for theenergy difference of the two neutrino eigenvalues in mattercompared to vacuum. Therefore, electron neutrinos aresingled out by this additional interaction process in matter.Depending on the electron density in the Sun the originallydominant mass eigenstate ν e can propagate into adifferent mass eigenstate for which the neutrino detectorsare not sensitive. One might wonder, how such matter oscillationswork in the Sun. The probability for a neutrinoto interact in matter is extremely small. The way the solarelectron density affects the propagation of solar neutrinos,however, depends on amplitudes which are square roots ofprobabilities. Therefore, even though the probabilities ofinteractions are small, the neutrino flavours can be significantlyaltered because of the amplitude dependence of theoscillation mechanism.If the three neutrino flavours ν e ,ν µ ,ν τ would completelymix, only 1/3 of the original electron neutrinos would arriveat Earth. Since the neutrino detectors, however, are blindfor MeV neutrinos of ν µ and ν τ type, the experimental resultscould be understood in a framework of oscillations.Obviously, the solar neutrino problem cannot be solved thateasily. The results of the four so far described experimentswhich measure solar neutrinos do not permit a unique solutionin the parameter space sin 2 2θ and δm 2 , compare (6.20)and (6.21). If (ν e → ν µ )or(ν e → ν τ ) oscillations are as-sumed and if it is considered that the MSW effect is responsiblefor the oscillations, a δm 2 on the order of 4 × 10 −4 –2 × 10 −5 eV 2 and a large-mixing-angle solution, althoughdisfavouring maximal mixing, is presently favoured. Assuminga mass hierarchy also in the neutrino sector, thiswould lead to a ν µ or ν τ mass of 0.02–0.004 eV. This is notnecessarily in contradiction to the results from atmosphericneutrinos since solar neutrinos could oscillate into muonneutrinos and atmospheric muon neutrinos into tau neutri-mixing anglepossible valuesfor neutrino masses

6.2 Neutrino Astronomy 99nos (or into so far undiscovered sterile neutrinos which arenot even subject to weak interactions). If this scenario werecorrect, one could have m νµ ≈ 10 meV and m ντ ≈ 50 meV.Recently the Sudbury Neutrino Observatory (SNO) hasconvincingly confirmed the oscillation picture. The SNO SNO experimentCherenkov detector installed at a depth of 2000 m undergroundin a nickel mine in Ontario, Canada, consists of a1000 ton heavy water target (D 2 O) contained in a 12 m diameteracrylic vessel. The interaction target is viewed by9600 phototubes. This central detector is immersed in a 30 mbarrel-shaped cavity containing 7000 tons of normal lightwater to suppress background reactions from cosmic raysor terrestrial radiation from radioisotopes in the surroundingrock or the mine dust. The SNO experiment can distinguishthe charged-current interaction (CC)charged currents(a) ν e + d → p + p + e − ,which can only be initiated by electron neutrinos, from theneutral-current reaction (NC)neutral currents(b) ν x + d → p + n + ν x ′ (x = e, µ, τ) ,where an incoming neutrino of any flavour interacts with adeuteron. The neutrons produced in this reaction are capturedby deuterons giving rise to the emission of 6.25 MeVphotons, which signal the NC interaction. While the ν e fluxas obtained by the CC reaction is only 1/3 of the predictedsolar neutrino flux, the total neutrino flux measured by theNC reaction is in agreement with the expectation of solarmodels, thereby providing evidence for a non-ν e component.This result solves the long-standing neutrino problem. Itdoes not, however, resolve the underlying mechanism of theoscillation process. It is not at all clear, whether the ν e oscillateinto ν µ or ν τ . It is considered very likely that matteroscillations via the MSW effect in the Sun is the most likelymechanism for the transmutation of the solar electron neutrinosinto other neutrino flavors, which unfortunately cannotdirectly be measured in a light-water Cherenkov counter.The oscillation mechanism suggested by the differentsolar experiments (ν e → ν µ ) was confirmed at the end of2002 by the KamLAND 2 reactor neutrino detector, which KamLANDremoved all doubts about possible uncertainties of the standardsolar modelreactor experimentpredictions.2 KamLAND – Kamioka Liquid-scintillator Anti-Neutrino Detector

100 6 Primary Cosmic Rays6.2.3 Supernova Neutrinosdiscovery of SN 1987AThe brightest supernova since the observation of Kepler inthe year 1604 was discovered by Ian Shelton at the Las Campanasobservatory in Chile on February 23, 1987 (see Fig.6.22). The region of the sky in the Tarantula Nebula in theLarge Magellanic Cloud (distance 170 000 light-years), inwhich the supernova exploded, was routinely photographedby Robert McNaught in Australia already 20 hours earlier.However, McNaught developed and analyzed the photographicplate only the following day. Ian Shelton was struckby the brightness of the supernova which was visible to thenaked eye. For the first time a progenitor star of the supernovaexplosion could be located. Using earlier exposures ofthe Tarantula Nebula, a bright blue supergiant, Sanduleak,was found to have exploded. Sanduleak was an inconspic-uous star of 10-fold solar mass with a surface temperatureof 15 000 K. During hydrogen burning Sanduleak increasedits brightness reaching a luminosity 70 000 higher than thesolar luminosity. After the hydrogen supply was exhausted,the star expanded to become a red supergiant. In this processits central temperature and pressure rose to such valuesthat He burning became possible. In a relatively shorttime (600 000 years) the helium supply was also exhausted.Helium burning was followed by a gravitational contractionin which the nucleus of the star reached a temperature of740 million Kelvin and a central density of 240 kg/cm 3 .These conditions enabled carbon to ignite. In a similar fashioncontraction and fusion phases occurred leading via oxygen,neon, silicon, and sulphur finally to iron, the elementwith the highest binding energy per nucleon.The pace of these successive contraction and fusionphases got faster and faster until finally iron was reached.Once the star has reached such a state, there is no way togain further energy by fusion processes. Therefore, the stabilityof Sanduleak could no longer be maintained. The starcollapsed under its own gravity. During this process the elec-trons of the star were forced into the protons and a neutronstar of approximately 20 km diameter was produced. In thecourse of this deleptonization a neutrino burst of immenseintensity was created,Fig. 6.22Supernova 1987A in the TarantulaNebula {6}fusion cycles of a stargravitational collapseproduction of a neutron stardeleptonization→ neutrino burste − + p → n + ν e . (6.38)

6.2 Neutrino Astronomy 101In the hot phase of the collapse corresponding to a temperatureof 10 MeV (≈ 10 11 K), the thermal photons producedelectron–positron pairs which, however, were immediatelyabsorbed because of the high density of the surroundingmatter. Only the weak-interaction process with a virtual Z,e + + e − → Z → ν α +¯ν α , (6.39)allowed energy to escape from the hot stellar nucleus in theform of neutrinos. In this reaction all three neutrino flavoursν e ,ν µ ,andν τ were produced ‘democratically’ in equal numbers.The total neutrino burst comprised 10 58 neutrinos andeven at Earth the neutrino flux from the supernova was comparableto that of solar neutrinos for a short period.Actually, the neutrino burst of the supernova was the firstsignal to be registered on Earth. The large water Cherenkovcounters of Kamiokande and IMB (Irvine–Michigan–Brookhaven) recorded a total of 20 out of the emitted 10 58neutrinos. The energy threshold of the Kamiokande experimentwas as low as 5 MeV. In contrast, the IMB collaborationcould only measure neutrinos with energies exceeding19 MeV. The Baksan liquid scintillator was lucky to record– even though their fiducial mass was only 200 t – five coincidentevents with energies between 10 MeV and 25 MeV.Since the neutrino energies in the range of 10 MeV areinsufficient to produce muons or taus, only electron-typeneutrinos were recorded via the reactions¯ν e + p → e + + n,¯ν e + e − →¯ν e + e − , (6.40)ν e + e − → ν e + e − .In spite of the low number of measured neutrinos onEarth some interesting astrophysical conclusions can bedrawn from this supernova explosion. If Eν i is the energy ofindividual neutrinos measured in the detector, ε 1 the probabilityfor the interaction of a neutrino in the detector, and ε 2the probability to also see this reaction, then the total energyemitted in form of neutrinos can be estimated to bemeasurement of neutrinosenergy outputE total =20∑i=1E i νε 1 (E i ν )ε 2(E i ν ) 4πr2 f(ν α , ¯ν α ), (6.41)where the correction factor f takes into account that thewater Cherenkov counters are sensitive not to all neutrino

102 6 Primary Cosmic Raysflavours. Based on the 20 recorded neutrino events a totalenergy ofE total = (6 ± 2) × 10 46 Joule (6.42)limits of neutrino massesdifferenceof propagation timeis obtained. It is hard to comprehend this enormous energy.(The world energy consumption is 10 21 Joule per year.) Duringthe 10 seconds lasting neutrino burst Sanduleak radiatedmore energy than the rest of the universe and hundred timesmore than the Sun in its total lifetime of about 10 billionyears.Measurements over the last 40 years have ever tightenedthe limits for neutrino masses. At the time of the supernovaexplosion the mass limit for the electron neutrino from measurementsof the tritium beta decay ( 3 H → 3 He + e − +¯ν e )was about 10 eV. Under the assumption that all supernovaneutrinos are emitted practically at the same time, one wouldexpect that their arrival times at Earth would be subject to acertain spread if the neutrinos had mass. Neutrinos of nonzeromass have different velocities depending on their energy.The expected difference of arrival times t of twoneutrinos with velocities v 1 and v 2 emitted at the same timefrom the supernova ist = r − r = r ( 1− 1 )= r β 2 − β 1. (6.43)v 1 v 2 c β 1 β 2 c β 1 β 2If the recorded electron neutrinos had a rest mass m 0 ,theirenergy would beE = mc 2 = γm 0 c 2 = m 0c 2√1 − β 2 , (6.44)and their velocity( ) 1/2β = 1 − m2 0 c4E 2 ≈ 1 − 1 2m 2 0 c4E 2 , (6.45)since one can safely assume that m 0 c 2 ≪ E. This meansthat the neutrino velocities are very close to the velocity oflight. Obviously, the arrival-time difference t depends onthe velocity difference of the neutrinos. Using (6.43) and(6.45) one getst ≈ r c12m 2 0 c4E 2 1− 1 2β 1 β 2m 2 0 c4E 2 2≈ 1 r E2 m2 0 c4 2 2 − E2 1. (6.46)cE 2 1 E2 2

6.2 Neutrino Astronomy 103The experimentally measured arrival-time differences andindividual neutrino energies allow in principle to work outthe electron neutrino rest mass{}2t E 2 1/21m 0 =E2 2rc 3 E2 2 − . (6.47)E2 1Since, however, not all neutrinos are really emitted simultaneously,(6.47) only allows to derive an upper limit forthe neutrino mass using pairs of particles of known energyand known arrival-time difference. Using the results of theKamiokande and IMB experiments a mass limit of the electronneutrino ofneutrino mass limitm νe ≤ 10 eV (6.48)could be established. This result was obtained in a measurementtime of approximately 10 seconds. It demonstrates thepotential superiority of astrophysical investigations over laboratoryexperiments.Similarly, a possible explanation for the deficit of solarneutrinos by assuming neutrino decay was falsified by themere observation of electron neutrinos from a distance of170 000 light-years. For an assumed neutrino mass of m 0 =10 meV the Lorentz factor of 10 MeV neutrinos would beγ =Em 0 c 2 ≈ 109 . (6.49)This would allow to derive a lower limit for the neutrinolifetime from τν 0 = τ ν/γ toneutrino lifetimeτ 0 ν = 170 000 a 1 γ≈ 5 000 s . (6.50)The supernova 1987A has turned out to be a rich astrophysicallaboratory. It has shown that the available supernovamodels can describe the spectacular death of massivestars on the whole correctly. Given the agreement of themeasured neutrinos fluxes with expectation, the supernovaneutrinos do not seem to require oscillations. On the otherhand, the precision of simulations and the statistical errorsof measurements are insufficient to draw a firm conclusionabout such a subtle effect for supernova neutrinos. The probabilitythat such a spectacle of a similarly bright supernovain our immediate vicinity will happen again in the near futureto clarify whether supernova neutrinos oscillate or notsupernova models confirmed

104 6 Primary Cosmic Raysis extremely small. It is not a surprise that the decision onthe oscillation scenario has come from observations of solarand atmospheric neutrinos and accelerator experimentswith well-defined, flavour-selected neutrino beams. Withthe experimental evidence of cosmic-ray-neutrino experiments(Davis, GALLEX, SAGE, Super-Kamiokande, SNO)and recent accelerator and reactor experiments (K2K, Kam-LAND) there is now unanimous agreement that oscillationsin the neutrino sector are an established fact.6.2.4 High-Energy Galacticand Extragalactic NeutrinosFig. 6.23Comparison of cosmic neutrinofluxes in different energy domains‘blackbody’ neutrinosThe measurement of high-energy neutrinos (≥ TeV range)represents a big experimental challenge. The arrival directionof such neutrinos, however, would directly point back tothe sources of cosmic rays. Therefore, a substantial amountof work is devoted to prototype studies for neutrino detectorsin the TeV range and the development of experimentalsetups for the measurement of galactic and extragalactichigh-energy neutrinos. The reason to restrict oneself tohigh-energy neutrinos is obvious from the inspection of Fig.6.23. The neutrino echo of the Big Bang has produced energiesbelow the meV range. About a second after the BigBang weak interactions have transformed protons into neutronsand neutrons into protons thereby producing neutrinos(p + e − → n + ν e , n → p + e − +¯ν e ). The temperature ofthese primordial neutrinos should be at 1.9 K at present time.Blackbody photons have a slightly higher temperature(2.7 K) because, in addition, electrons and positrons havetransformed their energy by annihilation into photon energy.The Big Bang neutrinos even originate from an earlier cosmologicalepoch than the blackbody photons since the universewas much earlier transparent for neutrinos. In so farthese cosmological neutrinos are very interesting as far asdetails of the creation and the development of the early universeare concerned. Unfortunately, at present it is almost inconceivablethat neutrinos of such low energies in the meVrange can be measured at all.The observation of solar (≈ MeV range) and supernovaneutrinos (≈ 10 MeV) is experimentally established.Atmospheric neutrinos represent a background for neutrinosfrom astrophysical sources. Atmospheric neutrinos originateessentially from pion and muon decays. Their productionspectra can be inferred from the measured atmospheric

6.2 Neutrino Astronomy 105muon spectra. However, they are also directly measured (seeFigs. 6.11 and 6.12). Their intensity is only known with anaccuracy of about 30%. The spectral shape and intensity ofneutrinos from extragalactic sources (AGN – Active GalacticNuclei) shown in Fig. 6.23 represents only a very roughestimate.It is generally assumed that binaries are good candidatesfor the production of energetic neutrinos. A binary consistingof a pulsar and a normal star could represent a strongneutrino source (Fig. 6.24).The pulsar and the star rotate around their common centerof mass. If the stellar mass is large compared to the pulsarmass, one can assume for illustration purposes of the neutrinoproduction mechanism that the pulsar orbits the companionstar on a circle. There are models which suggest thatthe pulsar can manage to accelerate protons to very highenergies. These accelerated protons collide with the gas ofthe atmosphere of the companion star and produce predominantlysecondary pions in the interactions. The neutral pionsdecay relatively fast (τ π 0 = 8.4 × 10 −17 s) into two energeticγ rays, which would allow to locate the astronomicalobject in the light of γ rays. The charged pions produce energeticneutrinos by their (π → µν) decay. Whether such asource radiates high-energy γ quanta or neutrinos dependscrucially on subtle parameters of the stellar atmosphere. Ifpions are produced in a proton interaction such asp + nucleus → π + + π − + π 0 + anything , (6.51)equal amounts of neutrinos and photons would be producedby the decays of charged and neutral pions (π + → µ + +ν µ ,π − → µ − +¯ν µ , π 0 → γ +γ). With increasing column densityof the stellar atmosphere, however, photons would bereabsorbed, and for densities of stellar atmospheres of ϱ ≤10 −8 g/cm 3 and column densities of more than 250 g/cm 2this source would be only visible in the light of neutrinos(Fig. 6.25).The source would shine predominantly in muon neutrinos(ν µ or ¯ν µ ). These neutrinos can be recorded in a detectorvia the weak charged current in which they produce muons(Fig. 6.26).Muons created in these interactions follow essentiallythe direction of the incident neutrinos. The energy of themuon is measured by its energy loss in the detector. For energiesexceeding the TeV range, muon bremsstrahlung andpulsar 0p + rest +starstellar atmospherelocal density (x)column density (x)dxFig. 6.24Production mechanism ofhigh-energy neutrinos in a binarysystemFig. 6.25Competition between productionand absorption of photons andneutrinos in a binary systemcompetition:neutrino, gamma emission –nW +Fig. 6.26Reaction for muon neutrinodetectionp

106 6 Primary Cosmic Raysneutrino detectorsFig. 6.27Neutrino production, propagationin intergalactic space, anddetection at Earthholes withphotomultiplierssouth poledata processingfirn layericewith bubbleshighlytransparent icelength of Cherenkov light in the detector medium. The di-rection of incidence of neutrinos can be inferred from thearrival times of the Cherenkov light at the photomultipliers.In a water Cherenkov counter in the ocean bioluminescenceand potassium-40 activity presents an annoying backgroundwhich is not present in ice. In practical applications it becameobvious that the installation of photomultiplier stringsin the antarctic ice is much less problematic compared tothe deployment in the ocean.Fig. 6.28Sketch of a neutrino detector forhigh-energy extragalactic neutrinosinstrumentationof a large volumedirect electron pair production by muons dominate. The energyloss by these two processes is proportional to the muonenergy and therefore allows a calorimetric determination ofthe muon energy (compare Sect. 7.3, Fig. 7.17).Because of the low interaction probability of neutrinosand the small neutrino fluxes, neutrino detectors must bevery large and massive. Since the whole detector volumehas to be instrumented to be able to record the interactionsof neutrinos and the energy loss of muons, it is necessary toconstruct a simple, cost-effective detector. The only practicablecandidates which meet this condition are huge wateror ice Cherenkov counters. Because of the extremely hightransparency of ice at large depths in Antarctica and therelatively simple instrumentation of the ice, ice Cherenkovcounters are presently the most favourable choice for a realisticneutrino telescope. To protect the detector against therelatively high flux of atmospheric particles, it has becomecommon practice to use the Earth as an absorber and concentrateon neutrinos which enter the detector ‘from below’.The principle of such a setup is sketched in Fig. 6.27. Protonsfrom cosmic-ray sources produce pions on a target (e.g.,stellar atmosphere, galactic medium) which provide neutrinosand γ quanta in their decay. Photons are frequentlyabsorbed in the galactic medium or disappear in γγ interactionswith blackbody photons, infrared radiation, or starlightphotons. The remaining neutrinos traverse the Earth and aredetected in an underground detector. The neutrino detectoritself consists of a large array of photomultipliers whichrecord the Cherenkov light of muons produced in ice (or inwater), Fig. 6.28. In such neutrino detectors the photomultiplierswill be mounted in a suitable distance on strings andmany of such strings will be deployed in ice (or water). Themutual distance of the photomultipliers on the strings andthe string spacing depends on the absorption and scattering

6.2 Neutrino Astronomy 107The setup of a prototype detector in the antarctic ice isshown in Fig. 6.29. In a first installation of photomultiplierstrings at depths of 810 up to 1000 meters it turned out thatthe ice still contained too many bubbles. Only for depthsbelow 1500 m the pressure is sufficiently large (≥ 150 bar)that the bubbles disappear yielding an excellent transparencywith absorption lengths of about 300 m. The working principleof the detector could be demonstrated by measuringatmospheric muons and muon neutrinos. For a ‘real’ neutrinotelescope, however, the present AMANDA detectoris still too small. To be able to record the fluxes of extragalacticneutrinos, a volume of 1 km 3 is needed. The originalAMANDA detector is continuously upgraded so thatfinally such a big volume will be instrumented (IceCube).Figure 6.30 shows a muon event in AMANDA. The followingexample will confirm that volumes of this size are reallyrequired.It is considered realistic that a point source in our galaxyproduces a neutrino spectrum according tointeraction ratein a ν detectordN= 2 × 10 −11 100dE ν Eν 2 [TeV2 ] cm−2 s −1 TeV −1 . (6.52) 60 mice surface0mThis leads to an integral flux of neutrinos ofΦ ν (E ν > 100 TeV) = 2 × 10 −11 cm −2 s −1 (6.53)(see also Fig. 6.23 for extragalactic sources).The interaction cross section of high-energy neutrinoswas measured at accelerators to beσ(ν µ N) = 6.7 × 10 −39 E ν [GeV] cm 2 /nucleon . (6.54)190 m380 m810 m1000 m1520 mFor 100 TeV neutrinos one would arrive at a cross sectionof 6.7 × 10 −34 cm 2 /nucleon. For a target thickness of onekilometer an interaction probability W per neutrino ofW = N A σdϱ= 4 × 10 −5 (6.55)is obtained (d = 1km= 10 5 cm, ϱ(ice) ≈ 1g/cm 3 ).The total interaction rate R is obtained from the integralneutrino flux Φ ν , the interaction probability W, the effectivecollection area A eff = 1km 2 , and a measurement time t.This leads to an event rate of 70 m 120 mfree of bubbles1950 mFig. 6.29Setup of the AMANDA detectorat the Sourth Pole (AMANDA –Antarctic Muon And NeutrinoDetector Array)R = Φ ν WA eff (6.56)

108 6 Primary Cosmic RaysFig. 6.30A neutrino-induced upward-goingmuon recorded in AMANDA. Thesize of the symbols is proportionalto the measured Cherenkov light{13}neutrino telescopescorresponding to 250 events per year. For large absorptionlengths of the produced Cherenkov light the effective collectionarea of the detector is even larger than the cross sectionof the instrumented volume. Assuming that there are abouthalf a dozen sources in our galaxy, the preceding estimatewould lead to a counting rate of about four events per day.In addition to this rate from point sources one would also expectto observe events from the diffuse neutrino backgroundwhich, however, carries little astrophysical information.Excellent candidates within our galaxy are the supernovaremnants of the Crab Nebula and Vela, the galactic center,and Cygnus X3. Extragalactic candidates could be representedby the Markarian galaxies Mrk 421 and Mrk 501,by M87, or by quasars (e.g., 3C273).Neutrino astronomy was pioneered with the Baikal telescopeinstalled in the lake Baikal in Siberia. The most advancedlarger telescopes are AMANDA and ANTARES 3 .AMANDA is taking data in Antarctica since several years,while the ANTARES detector which is installed in theMediterranean offshore Toulon started data taking early in2003. The NESTOR 4 detector has seen first results at theend of 2003. It is also operated in the Mediterranean offthe coast of Greece. Real neutrino astronomy, however, willrequire larger detectors like IceCube which is presently installedin Antarctica. It is expected that first results fromIceCube might become available from the year 2005 on.6.3 Gamma Astronomy“Let there be light.”The Bible; Genesis 1:36.3.1 IntroductionThe observation of stars in the optical spectral range belongsto the field of classical astronomy. Already the Chinese,Egyptians, and Greeks performed numerous systematic observationsand learned a lot about the motion of heavenlybodies. The optical range, however, covers only a minuterange of the total electromagnetic spectrum (Fig. 6.31).3 ANTARES – Abyssal Neutrino Telescope And Research Environmentof deep Sea4 NESTOR – Neutrino Extended Submarine Telescope withOceanographic Research

6.3 Gamma Astronomy 109All parts of this spectrum have been used for astronomicalobservations. From large wavelengths (radio astronomy),the sub-optical range (infrared astronomy), the classical opticalastronomy, the ultraviolet astronomy, and X-ray astronomyone arrives finally at the gamma-ray astronomy.Gamma-ray astronomers are used to characterize gammaquanta not by their wavelength λ or frequency ν, butratherby their energy,E = hν. (6.57)Planck’s constant in practical units ish = 4.136 × 10 −21 MeV s . (6.58)The frequency ν is measured in Hz = 1/s. The wavelengthλ is obtained to beλ = c/ν , (6.59)where c is the speed of light in vacuum (c = 299 792 458m/s).In atomic and nuclear physics one distinguishes gammarays from X rays by the production mechanism. X rays areemitted in transitions of electrons in the atomic shell whilegamma rays are produced in transformations of the atomicnucleus. This distinction also results naturally in a classificationof X rays and gamma rays according to their energy.X rays typically have energies below 100 keV. Electromagneticradiation with energies in excess of 100 keV is called γrays. There is no upper limit for the energy of γ rays. Evencosmic γ rays with energies of 10 15 eV = 1 PeV have beenobserved.Fig. 6.31Spectral range of electromagneticradiationdistinctionbetween γ and X rays

110 6 Primary Cosmic RaysAn important, so far unsolved problem of astroparticlephysics is the origin of cosmic rays (see also Sects. 6.1 and6.2). Investigations of charged primary cosmic rays are es-sentially unable to answer this question because chargedparticles have to pass through extended irregular magneticfields on their way from the source to Earth. This causesthem to be deflected in an uncontrolled fashion thereby‘forgetting’ their origin. Therefore, particle astronomy withcharged particles is only possible at extremely high energieswhen the particles are no longer significantly affectedby cosmic magnetic fields. This would require to go to energiesin excess of 10 19 eV which, however, creates anotherproblem because the flux of primary particles at these energiesis extremely low. Whatever the sources of cosmic raysare, they will also be able to emit energetic penetrating γrays which are not deflected by intergalactic or stellar magneticfields and therefore point back to the sources. It must,however, be kept in mind that also X and γ rays from distantsources might be subject to time dispersions. Astronomicalobjects in the line of sight of these sources can distort theirtrajectory by gravitational lensing thus making them lookblurred and causing time-of-flight dispersions in the arrivaltime also for electromagnetic radiation.charged-particle astronomy?arrival-time dispersionsmagnetic field linessources of γ rayselectronphotonFig. 6.32Production of synchrotronradiation by deflection of chargedparticles in a magnetic fieldPossible sources for cosmic rays and thereby also for γ raysare supernovae and their remnants, rapidly rotating objectslike pulsars and neutron stars, active galactic nuclei, andmatter-accreting black holes. In these sources γ rays can beproduced by different mechanisms.a) Synchrotron radiation:The deflection of charged particles in a magnetic fieldgives rise to an accelerated motion. An accelerated electricalcharge radiates electromagnetic waves (Fig. 6.32).This ‘bremsstrahlung’ of charged particles in magneticfields is called synchrotron radiation. In circular earth-bound accelerators the production of synchrotron radiationis generally considered as an undesired energy-lossmechanism. On the other hand, synchrotron radiationfrom accelerators is widely used for structure investigationsin atomic and solid state physics as well as inbiology and medicine.Synchrotron radiation produced in cosmic magnetic fieldsis predominantly emitted by the lightweight electrons.applicationsof synchrotron radiation6.3.2 Production Mechanisms for γ Rays

6.3 Gamma Astronomy 111The energy spectrum of synchrotron photons is continuous.The power P radiated by an electron of energy E ina magnetic field of strength B isatomic nucleusP ∼ E 2 B 2 . (6.60)b)Bremsstrahlung:A charged particle which is deflected in the Coulomb fieldof a charge (atomic nucleus or electron) emits bremsstrahlungphotons (Fig. 6.33). This mechanism is to a certainextent similar to synchrotron radiation, only that in thiscase the deflection of the particle occurs in the Coulombfield of a charge rather than in a magnetic field.The probability for bremsstrahlung φ varies with thesquare of the projectile charge z and also with the squareof the target charge Z, see also (4.7). φ is proportional tothe particle energy E and it is inversely proportional tothe mass squared of the deflected particle:φ ∼ z2 Z 2 Em 2 . (6.61)Because of the smallness of the electron mass bremsstrahlungis predominantly created by electrons. The energyspectrum of bremsstrahlung photons is continuous anddecreases like 1/E γ to high energies.c) Inverse Compton Scattering:In the twenties of the last century Compton discoveredthat energetic photons can transfer part of their energyto free electrons in a collision, thereby losing a certainamount of energy. In astrophysics the inverse Compton effectplays an important rôle. Electrons accelerated to highenergies in the source collide with the numerous photonsof the blackbody radiation (E γ ≈ 250 µeV, photon densityN γ ≈ 400/cm 3 ) or starlight photons (E γ ≈ 1eV,N γ ≈ 1/cm 3 ) and transfer part of their energy to the pho-tons which are ‘blueshifted’ (Fig. 6.34).d)π 0 Decay:Protons accelerated in the sources can produce chargedand neutral pions in proton–proton or proton–nucleus interactions(Fig. 6.35). A possible process iselectronphotonFig. 6.33Production of bremsstrahlung bydeflection of charged particles inthe Coulomb field of a nucleusbremsstrahlung probabilityspectrumof bremsstrahlung photonsinverse Compton scatteringFig. 6.34Collision of an energetic electronwith a low-energy photon. Theelectron transfers part of its energyto the photon and is consequentlyslowed downp+nucleus → p ′ +nucleus ′ +π + +π − +π 0 . (6.62)Charged pions decay with a lifetime of 26 ns into muonsand neutrinos, while neutral pions decay rapidly (τ =8.4 × 10 −17 s) into two γ quanta,

112 6 Primary Cosmic Rays –photonprotonnucleusof the interstellarmedium 0 +photonFig. 6.35π 0 production in protoninteractions and π 0 decay into twophotonselectronphotonphotonpositronFig. 6.36e + e − pair annihilation into twophotonsγ -ray linesπ 0 → γ + γ. (6.63)If the neutral pion decays at rest, both photons are emittedback to back. In this decay they get each half of theπ 0 rest mass (m π 0 = 135 MeV). In the π 0 decay in flightthe photons get different energies depending on their directionof emission with respect to the direction of flightof the π 0 (see Example 8, Chap. 3). Since most pionsare produced at low energies, photons from this particularsource have energies of typically 70 MeV.e) Photons from Matter–Antimatter Annihilation:In the same way as photons can produce particle pairs(pair production), charged particles can annihilate withtheir antiparticles into energy. The dominant sourcesfor this production mechanism are electron–positron andproton–antiproton annihilations,e + + e − → γ + γ. (6.64)Momentum conservation requires that at least two photonsare produced. In e + e − annihilation at rest the photonsget 511 keV each corresponding to the rest massof the electron or positron, respectively (Fig. 6.36). Anexample for a proton–antiproton annihilation reaction isp +¯p → π + + π − + π 0 , (6.65)where the neutral pion decays into two photons.f) Photons from Nuclear Transformations:Heavy elements are ‘cooked’ in supernova explosions. Inthese processes not only stable but also radioactive isotopesare produced. These radioisotopes will emit, mostlyas a consequence of a beta decay, photons in the MeVrange like, e.g.,60 Co → 60 Ni ∗∗ + e − +¯ν e✲ 60 Ni ∗ + γ(1.17 MeV) (6.66)✲ 60 Ni + γ(1.33 MeV) .neutralino annihilationg) Annihilation of Neutralinos:In somewhat more exotic scenarios, energetic γ rayscould also originate from annihilation of neutralinos, theneutral supersymmetric partners of ordinary particles, accordingtoχ +¯χ → γ + γ. (6.67)

6.3 Gamma Astronomy 1136.3.3 Measurement of γ RaysIn principle the inverse production mechanisms of γ rayscan be used for their detection (see also Chap. 4). For γ rayswith energies below several hundred keV the photoelectriceffect dominates,photoelectric effect,E γ< ∼ 100 keVγ + atom → atom + + e − . (6.68)The photoelectron can be recorded, e.g., in a scintillationcounter. For energies in the MeV range as it is typicalfor nuclear decays, Compton scattering has the largest crosssection,Compton effect, E γ< ∼ 1MeVγ + e − at rest → γ ′ + e − fast . (6.69)In this case the material of a scintillation counter can alsoact as an electron target which records at the same time thescattered electron. For higher energies (≫ 1 MeV) electron–positron pair creation dominates,γ + nucleus → e + + e − + nucleus ′ . (6.70)electron–positron pairproduction, E γ ≫ 1MeVFigure 6.37 shows the dependence of the mass attenuationcoefficient µ for the three mentioned processes in a NaI scintillationcounter.This coefficient is defined through the photon intensityattenuation in matter according toI(x) = I 0 e −µx (6.71)(I 0 – initial intensity, I(x) – photon intensity after attenuationby an absorber of thickness x).Since pair production dominates at high energies, thisprocess is used for photon detection in the GeV range. Figure6.38 shows a typical setup of a satellite experiment forthe measurement of γ rays in the GeV range.Energetic photons are converted into e + e − pairs in amodular tracking-chamber system (e.g., in a multiplate sparkchamber or a stack of semiconductor silicon counters). Theenergies E e + and E e − are measured in an electromagneticcalorimeter (mostly a crystal-scintillator calorimeter, NaI(Tl)or CsI(Tl)) so that the energy of the original photon isFig. 6.37Mass attenuation coefficient forphotons in a sodium-iodidescintillation countercrystal calorimeterE γ = E e + + E e − . (6.72)

114 6 Primary Cosmic RaysFig. 6.38Sketch of a satellite experiment forthe measurement of γ rays in theGeV rangee – e +e –e – e –3e – 1 bremsstrahlung2 pair production1 3 Compton scatteringe – 4 photoelectric effect242Fig. 6.39Schematic representation of anelectron cascadeCherenkov radiation134The direction of incidence of the photon is derived from theelectron and positron momenta where the photon momentumis determined to be p γ = p e + + p e −. For high energies(E ≫ m e c 2 ) the approximations |p e +|=E e +/c and|p e −|=E e −/c are well satisfied.The detection of electrons and positrons in the crystalcalorimeter proceeds via electromagnetic cascades. In theseshowers the produced electrons initially radiate bremsstrahlungphotons which convert into e + e − pairs. In alternatingprocesses of bremsstrahlung and pair production the initialelectrons and photons decrease their energy until absorptiveprocesses like photoelectric effect and Compton scatteringfor photons on the one hand and ionization loss for electronsand positrons on the other hand halt further particlemultiplication (Fig. 6.39).The anticoincidence counter in Fig. 6.38 serves the purposeof identifying incident charged particles and rejectingthem from the analysis.For energies in excess of 100 GeV the photon intensitiesfrom cosmic-ray sources are so small that other techniquesfor their detection must be applied, since sufficiently largesetups cannot be installed on board of satellites. In this contextthe detection of photons via the atmospheric Cherenkovtechnique plays a special rôle.When γ rays enter the atmosphere they produce – likealready described for the crystal calorimeter – a cascadeof electrons, positrons, and photons which are generally oflow energy. This shower does not only propagate longitudinallybut it also spreads somewhat laterally in the atmosphere(Fig. 6.40). For initial photon energies below 10 13 eV(= 10 TeV) the shower particles, however, do not reachsea level. Relativistic electrons and positrons of the cascadewhich follow essentially the direction of the originalincident photon emit blue light in the atmosphere which isknown as Cherenkov light. Charged particles whose velocitiesexceed the speed of light emit this characteristic electromagneticradiation (see Chap. 4). Since the speed of lightin atmospheric air isc n = c/n (6.73)(n is the index of refraction of air; n = 1.000 273 at 20 ◦ Cand 1 atm), electrons with velocitiesv ≥ c/n (6.74)will emit Cherenkov light. This threshold velocity of

6.3 Gamma Astronomy 115v = c n = 299 710 637 m/s (6.75)photoncorresponds to a kinetic electron energy ofE kin = E total − m 0 c 2 = γm 0 c 2 − m 0 c 2()= (γ − 1)m 0 c 2 1= √1 − v 2 /c − 1 m 0 c 22()1= √ − 1 m 0 c 2 (6.76)1 − 1/n 2()n= √n 2 − 1 − 1 m 0 c 2 ≈ 21.36 MeV .The production of Cherenkov radiation in an optical shockwave (Fig. 6.41) is the optical analogue to sound shockwaves which are created when aeroplanes exceed the velocityof sound.altitude [km]2520151050-5 0 +5lateral width [km]Fig. 6.40Monte Carlo simulation of anelectromagnetic shower in theatmosphere initiated by a photon ofenergy 10 14 eV. All secondarieswith energies E ≥ 3 MeV areshown {14}Fig. 6.41Emission of Cherenkov radiation inan optical shock wave by particlestraversing a medium of refractiveindex n with a velocity exceedingthe velocity of light in that medium(v >c/n)In this way energetic primary γ quanta can be recordedat ground level via the produced Cherenkov light eventhough the electromagnetic shower does not reach sea level.The Cherenkov light is emitted under a characteristic angleof( ) 1θ C = arccos . (6.77)nβFor electrons in the multi-GeV range the opening angle ofthe Cherenkov cone is only 1.4 ◦ . 55 Actually the Cherenkov angle is somewhat smaller (≈ 1 ◦ ),since the shower particles are produced at large altitudes wherethe density of air and thereby also the index of refraction issmaller.Cherenkov angle

116 6 Primary Cosmic RaysFig. 6.42Measurement of Cherenkov lightof photon-induced electromagneticcascades in the atmosphereA simple Cherenkov detector, therefore, consists of aparabolic mirror which collects the Cherenkov light and aset of photomultipliers which record the light collected at thefocal point of the mirror. Figure 6.42 shows the principle ofphoton measurements via the atmospheric Cherenkov technique.Large Cherenkov telescopes with mirror diameters≥ 10 m allow to measure comparatively low-energy photons(< 100 GeV) with correspondingly small shower size evenin the presence of light from the night sky (see Fig. 6.43).For even higher energies (> 10 15 eV) the electromagneticcascades initiated by the photons reach sea level and can berecorded with techniques like those which are used for theinvestigations and measurements of extensive air showers(particle sampling, air scintillation, cf. Sect. 7.4). At theseenergies it is anyhow impossible to explore larger regionsof the universe in the light of γ rays. The intensity of energeticprimary photons is attenuated by photon–photon interactionspredominantly with numerous ambient photons ofthe 2.7 Kelvin blackbody radiation. For the processγ + γ → e + + e − (6.78)twice the electron mass must be provided in the γγ centerof-masssystem. For a primary photon of energy E collidingwith a target photon of energy ε at an angle θ the thresholdenergy isFig. 6.43Photograph of the air Cherenkovtelescope CANGAROO(CANGAROO – Collaboration ofAustralia and Nippon (Japan) for aGAmma-Ray Observatory in theOutback) {15}γγ interactions2m 2 eE threshold =ε(1 − cos θ) . (6.79)For a central collision (θ = 180 ◦ ) and a typical blackbodyphoton energy of ε ≈ 250 µeV the threshold isE threshold ≈ 10 15 eV . (6.80)The cross section rises rapidly above threshold, reaches amaximum of 200 mb at twice the threshold energy, anddecreases thereafter. For even higher energies further absorptiveprocesses with infrared or starlight photons occur(γγ → µ + µ − ) so that distant regions of the universe(> 100 kpc) are inaccessible for energetic photons(> 100 TeV). Photon–photon interactions, therefore, causea horizon for γ astronomy, which allows us to explore thenearest neighbours of our local group of galaxies in the lightof high-energy γ quanta, but they attenuate the γ intensityfor larger distances so strongly that a meaningful observationbecomes impossible (cf. Chap. 4, Fig. 4.9).

6.3 Gamma Astronomy 117The relation between the threshold energy for absorptiveγγ interactions and the energy of target photons is shown inFig. 6.44. Distant γ -ray sources whose high-energy photonsare absorbed by blackbody and infrared photons can still beobserved in the energy range < 1TeV.6.3.4 Observation of γ -Ray Point SourcesFirst measurements of galactic γ rays were performed in theseventies with satellite experiments. The results of these investigations(Fig. 6.45) clearly show the galactic center, theCrab Nebula, the Vela X1 pulsar, Cygnus X3, and Gemingaas γ -ray point sources. Recent satellite measurements withthe Compton gamma-ray observatory (CGRO) show a largenumber of further γ -ray sources. There were four experimentson board of the CGRO satellite (BATSE, OSSE,EGRET, COMPTEL) 6 . These four telescopes cover an energyrange from 30 keV up to 30 GeV. Apart from γ -raybursts (see Sect. 6.3.5) numerous galactic pulsars and a largenumber of extragalactic sources (AGN) 7 have been discovered.It was found out that old pulsars can transform theirrotational energy more efficiently into γ rays compared toyoung pulsars. One assumes that the observed γ radiation isproduced by synchrotron radiation of energetic electrons inthe strong magnetic fields of the pulsars.Fig. 6.44Dependence of the thresholdenergy E for γγ absorption on theenergy of the target photons (BB –blackbody radiation, IR – infrared,VIS – visible spectral range, X –XraysCompton gamma-rayobservatoryOSSE – Oriented Scintillation Spectroscopy Experiment6 COMPTEL – COMpton TELescopeEGRET – Energy Gamma Ray Telescope ExperimentBATSE – Burst And Transient Source Experiment7 AGN – Active Galactic NucleiFig. 6.45Measurement of the intensity ofgalactic γ radiation for photonenergies > 100 MeV. The solidline represents the expected γ -rayintensity on the basis of thecolumn density of interstellar gasin that direction

118 6 Primary Cosmic RaysFig. 6.46All-sky survey in the light of γrays {16}possible point sourcesblack holesAmong the discovered active galaxies highly variableblazars (extremely variable objects on short time scales withstrong radio emission) were found which had their maximumof emission in the gamma range. In addition, gammaquasars at high redshifts (z >2) were observed. In this casethe gamma radiation could have been produced by inverseCompton scattering of energetic electrons off photons.Figure 6.46 shows a complete all-sky survey in the lightof γ rays in galactic coordinates. Apart from the galacticcenter and several further point sources, the galactic disk isclearly visible. Candidates for point sources of galactic γrays are pulsars, binary pulsar systems, and supernovae. Extragalacticsources are believed to be compact active galacticnuclei (AGN), quasi-stellar radio sources (quasars), blazars,and accreting black holes. According to common beliefblack holes could be the ‘powerhouses’ of quasars. Blackholes are found at the center of galaxies where the matterdensity is highest thereby providing sufficient material forthe formation of accretion disks arround black holes. Accordingto the definition of a black hole no radiation canescape from it, 8 however, the infalling matter heats up alreadybefore reaching the event horizon so that intensiveenergetic γ rays can be emitted.8 In this context the Hawking radiation, which has no importancefor γ -ray astronomy, will not be considered, since black holeshave a very low ‘temperature’.

6.3 Gamma Astronomy 119Future space-based gamma-ray telescopes like INTE-GRAL 9 (15 keV–10 MeV) and GLAST 10 (30 MeV–300GeV) will certainly supplement the results from earlier γ -ray missions.In the TeV range γ -ray point sources have been discoveredwith the air Cherenkov technique. Recently, a supernovaremnant as a source of high-energy photons has beenobserved by the HESS 11 experiment. The γ -ray spectrumof this object, SNR RX J1713.7-3946 in the galactic plane7 kpc off the galactic center, near the solar system (distance1 kpc) can best be explained by the assumption that the pho- hadron acceleratortons with energies near 10 TeV are produced by π 0 decays,i.e., this source is a good candidate for a hadron accelerator.The imaging air Cherenkov telescope MAGIC underconstruction on the Canary Island La Palma with its 17 mdiameter mirror will soon compete with the HESS telescopein Namibia. Apart from galactic sources (Crab Nebula) extragalacticobjects emitting TeV photons (Markarian 421, TeV photonsMarkarian 501, and M87) have also been unambiguouslyidentified. Markarian 421 is an elliptic galaxy with a highlyvariable galactic nucleus. The luminosity of Markarian 421in the light of TeV photons would be 10 10 times higher thanthat of the Crab Nebula if isotropic emission were assumed.One generally believes that this galaxy is powered by a massiveblack hole which emits jets of relativistic particles fromits poles. It is conceivable that this approximately 400 millionlight-years distant galaxy beams the high-energy particlejets – and thereby also the photon beam – exactly intothe direction of Earth.The highest γ energies from cosmic sources have been highest γ energiesrecorded by earthbound air-shower experiments, but also byair Cherenkov telescopes. It has become common practiceto consider the Crab Nebula, which emits photons with energiesup to 100 TeV, as a standard candle. γ -ray sourcesfound in this energy regime are mostly characterized byan extreme variability. In this context the X-ray sourceCygnus X3 plays a special rôle. In the eighties γ rays fromthis source with energies up to 10 16 eV (10 000 TeV) wereclaimed to be seen. These high-energy γ rays appeared toshow the same variability (period 4.8 hours) as the X rays9 INTEGRAL – INTErnational Gamma-Ray Astrophysics Laboratory10 GLAST – Gamma Ray Large Area Telescope11 HESS – High Energy Stereoscopic System

120 6 Primary Cosmic RaysFig. 6.47Light curve of SN 1987A. Thesolid line corresponds to completeconversion of 56 Co γ rays into theinfrared, optical, and ultravioletspectral rangeγ line emissionlight curve of SN 1987Acoming from this object. It has to be noted, however, that ahigh-energy gamma outburst of this source has never beenseen again.Apart from the investigation of cosmic sources in thelight of high-energy γ rays the sky is also searched forγ quanta of certain fixed energy. This γ -ray line emissionhints at radioactive isotopes, which are formed in the processof nucleosynthesis in supernova explosions. It could beshown beyond any doubt that the positron emitter 56 Ni wasproduced in the supernova explosion 1987A in the LargeMagellanic Cloud. This radioisotope decays into 56 Co witha half-life of 6.1 days. The light curve of this source showeda luminosity maximum followed by an exponential brightnessdecay. This could be traced back to the radioactive decayof the daughter 56 Co to the stable isotope 56 Fe with ahalf-life of 77.1 days (see Fig. 6.47).Interesting results are also expected from an all-sky surveyin the light of the 511 keV line from e + e − annihilation.This γ -ray line emission could indicate the presence of antimatterin our galaxy. The observation of the distribution ofcosmic antimatter could throw some light on the problemwhy our universe seems to be matter dominated.6.3.5 γ Bursterdiscovery of γ -ray burstsCosmic objects which emit sudden single short outbursts ofγ rays have been discovered in the early seventies by Americanreconnaissance satellites. The purpose of these satelliteswas to check the agreement on the stop of nuclear weapontests in the atmosphere. The recorded γ rays, however, didnot come from the surface of the Earth or the atmosphere,but rather from outside sources and, therefore, were not relatedto explosions of nuclear weapons which also are asource of γ rays.

6.3 Gamma Astronomy 121γ -ray bursts occur suddenly and unpredictably with arate of approximately one burst per day. The durations ofthe γ -ray bursts are very short ranging from fractions ofa second up to 100 seconds. There appear to be two distinctclasses of γ -ray bursts, one with short (≈ 0.5s), theother with longer durations (≈ 50 s), indicating the existenceof two different populations of γ -ray bursters. Figure6.48 shows the γ -ray light curve of a typical short burst.Within only one second the γ -ray intensity increases by afactor of nearly 10. The γ -ray bursters appear to be uniformlydistributed over the whole sky. Because of the shortburst duration it is very difficult to identify a γ -ray bursterwith a known object. At the beginning of 1997 researcherssucceeded for the first time to associate a γ -ray burst with arapidly fading object in the optical regime. From the spectralanalysis of the optical partner one could conclude thatthe distance of this γ -ray burster was about several billionlight-years.The angular distribution of 2000 γ -ray bursters recordeduntil the end of 1997 is shown in Fig. 6.49 in galactic coordinates.From this graph it is obvious that there is no clusteringof γ -ray bursters along the galactic plane. Therefore,the most simple assumption is that these exotic objects areat cosmological distances which means that they are extragalactic.Measurements of the intensity distributions ofbursts show that weak bursts are relatively rare. This couldimply that the weak (i.e., distant) bursts exhibit a lower spatialdensity compared to the strong (near) bursts.Even though violent supernova explosions are consideredto be excellent candidates for γ -ray bursts, it is not obviouswhether also other astrophysical objects are responsiblefor this enigmatic phenomenon. The observed spatialFig. 6.48Light curve of a typcial γ -ray burstangular distributionspatial distribution+90+180 –180–902000 eventsFig. 6.49Angular distribution of 2000 γ -raybursts in galactic coordinatesrecorded with the BATSE detector(Burst And Transient SourceExperiment) on board the CGROsatellite (Compton Gamma RayObservatory) {17}

122 6 Primary Cosmic Rayssupernova, hypernovaexplosionsWolf–Rayet starcandidates for γ -ray burstsquasi-periodic γ bursterdistribution of γ -ray bursters suggests that they are at extragalacticdistances. In this context the deficit of weak burstsin the intensity distribution could be explained by the redshiftof spectral lines associated with the expansion of theuniverse. This would also explain why weaker bursts havesofter energy spectra.A large fraction of γ -rayburstsisbelievedtobecausedby violent supernova explosions (e.g., hypernova explosionswith a collapse into a rotating black hole). This seems tohave been confirmed by the association of the γ -ray burstGRB030329 with the supernova explosion SN2003dh. Theobservation of the optical afterglow of this burst allowed tomeasure its distance to be at 800 Mpc. The afterglow of theburst, i.e., the optial luminosity of the associated supernovareached a magnitude of 12 in the first observations after theγ -ray burst. Such a bright supernova may have been visibleto the naked eye in the first minutes after the explosion.Such hypernova explosions are considered to be rare eventswhich are probably caused by stars of the ‘Wolf–Rayet’type. Wolf–Rayet stars are massive objects (M > 20 M ⊙ )which initially consist mainly of hydrogen. During theirburning phase they strip themselves off their outer layersthus consisting mainly of helium, oxygen, and heavy elements.When they run out of fuel, the core collapses andforms a black hole surrounded by an accretion disk. It is believedthat in this moment a jet of matter is ejected fromthe black hole which represents the γ -ray burst (‘collapsarmodel’).As alternative candidates for γ -ray bursts, namely forshort-duration bursts, collisions of neutron stars, collisionsof neutron stars with black holes, coalescence of two neutronstars forming a black hole, asteroid impacts on neutronstars, or exploding primordial mini black holes are discussed.From the short burst durations one can firmly concludethat the spatial extension of γ -ray bursters must bevery small. However, only the exact localization and detailedobservation of afterglow partners of γ -ray bursts will allowto clarify the problem of their origin. One important inputto this question is the observation of a γ -ray burster alsoin the optical regime by the 10 m telescope on Mauna Keain Hawaii. Its distance could be determined to be 9 billionlight-years.In the year 1986 a variant of a family of γ -ray bursterswas found. These objects emit sporadically γ bursts fromthe same source. The few so far known quasi-periodic γ -ray

6.4 X-Ray Astronomy 123bursters all reside in our galaxy or in the nearby MagellanicClouds. Most of these objects could be identified with youngsupernova remnants. These ‘soft gamma-ray repeaters’ appearto be associated with enormous magnetic fields. If sucha magnetar rearranges its magnetic field to reach a morefavourable energy state, a star quake might occasionally occurin the course of which γ bursts are emitted. The γ -rayburst of the magnetar SGR-1900+14 was recorded by sevenresearch satellites on August 27, 1998. From the observedslowing down of the rotational period of this magnetar oneconcludes that this object possesses a superstrong magneticfield of 10 11 Tesla exceeding the magnetic fields of normalneutron stars by a factor of 1000.With these properties γ bursters are also excellent candidatesas sources of cosmic rays. It is frequently discussedthat the birth or the collapse of neutron stars could be associatedwith the emission of narrowly collimated particle jets. Ifthis were true, we would only be able to see a small fractionof γ bursters. The total number of γ bursters would then besufficiently large to explain the observed particle fluxes ofcosmic rays. The enormous time-dependent magnetic fieldswould also produce strong electric fields in which cosmicrayparticles could be accelerated up to the highest energies.soft gamma-ray repeatermagnetars6.4 X-Ray Astronomy“Light brings the news of the universe.”Sir William Bragg6.4.1 IntroductionX rays differ from γ rays by their production mechanismand their energy. X rays are produced if electrons are deceleratedin the Coulomb field of atomic nuclei or in transitionsbetween atomic electron levels. Their energy rangesbetween approximately 1–100 keV. In contrast, γ rays areusually emitted in transitions between nuclear levels, nucleartransformations, or in elementary-particle processes.After the discovery of X rays in 1895 by Wilhelm ConradRöntgen, X rays were mainly used in medical applicationsbecause of their high penetration power. X rays withenergies exceeding 50 keV can easily pass through 30 cm oftissue (absorption probability ≈ 50%). The column densityproduction of X raysenergy dependenceof X-ray absorption

124 6 Primary Cosmic Raysballoon experimentsrocket flightsof the Earth’s atmosphere, however, is too large to allow extraterrestrialX rays to reach sea level. In the keV energyregion, corresponding to the brightness maximum of mostX-ray sources, the range of X rays in air is 10 cm only. To beable to observe X rays from astronomical objects one thereforehas to operate detectors at the top of the atmosphereor in space. This would imply balloon experiments, rocketflights, or satellite missions.Balloon experiments can reach a flight altitude of 35 to40 km. Their flight duration amounts to typically between 20to 40 hours. At these altitudes, however, a substantial fractionof X rays is already absorbed. Balloons, therefore, canonly observe X-ray sources at energies exceeding 50 keVwithout appreciable absorption losses. In contrast, rocketsnormally reach large altitudes. Consequently, they can measureX-ray sources unbiased by absorption effects. However,their flight time of typically several minutes, before they fallback to Earth, is extremely short. Satellites have the big ad-vantage that their orbit is permanently outside the Earth’satmosphere allowing observation times of several years.In 1962 X-ray sources were discovered by chance whenan American Aerobee rocket with a detector consisting ofthree Geiger counters searched for X rays from the Moon.No lunar X radiation was found, but instead extrasolar Xrays from the constellations Scorpio and Sagittarius wereobserved. This was a big surprise because it was known thatour Sun radiates a small fraction of its energy in the X-rayrange and – because of solid angle arguments – one did notexpect X-ray radiation from other stellar objects. This is becausethe distance of the nearest stars is more than 100 000times larger than the distance of our Sun. The brightness ofsuch distant sources must have been enormous compared tothe solar X-ray luminosity to be able to detect them with detectorsthat were in use in the sixties. The mechanism whatmade the sources Scorpio and Sagittarius shine so bright inthe X-ray range, therefore, was an interesting astrophysicalquestion.satellitesdiscoveryof cosmic X-ray sourcesbright X-ray sources6.4.2 Production Mechanisms for X RaysThe sources of X rays are similar to those of gamma rays.Since the energy spectrum of electromagnetic radiation usuallydecreases steeply with increasing energy, X-ray sourcesoutnumber gamma-ray sources. In addition to the processesalready discussed in Sect. 6.3 (gamma-ray astronomy) like

6.4 X-Ray Astronomy 125synchrotron radiation, bremsstrahlung, and inverse Comptonscattering, a further production mechanism for X rayslike thermal radiation from hot cosmic sources has to beconsidered. The Sun with its effective surface temperatureof about 6000 Kelvin emits predominantly in the eV range.Sources with a temperature of several million Kelvin wouldalso emit X rays as blackbody radiation.The measured spectra of many X-ray sources exhibit asteep intensity drop to very small energies which can be attributedto absorption by cold material in the line of sight. Athigher energies a continuum follows which can be describedeither by a power law (∼ E −γ ) or by an exponential, dependingon the type of the dominant production mechanismfor X rays. Sources in which relativistic electrons produce Xrays by synchrotron radiation or inverse Compton scatteringcan be characterized by a power-law spectrum like E −γ .Onthe other hand, one obtains an exponential decrease to highenergies if thermal processes dominate. A bremsstrahlungspectrum is usually relatively flat at low energies. In mostcases more than one production process contributes to thegeneration of X rays. According to the present understanding,the X rays of most X-ray sources appear to be of thermalorigin. For thermal X rays one has to distinguish two cases.X raysas blackbody radiationshape of spectradominanceof thermal production1. In a hot gas (≈ 10 7 K) the atoms are ionized. Electrons productionof the thermal gas produce in an optically thin medium in an optical thin medium(practically no self absorption) X rays by bremsstrahlungand by atomic level transitions. The second mechanismrequires the existence of atoms which still haveat least one bound electron. At temperatures exceeding≈ 10 7 K, however, the most abundant atoms like hydrogenand helium are completely ionized so that in thiscase bremsstrahlung is the dominant source. In this contextone understands under bremsstrahlung the emissionof X rays which are produced by interactions of electronsin the Coulomb field of positive ions of the plasmain continuum transitions (thermal bremsstrahlung). Forenergies hν > kT the spectrum decreases exponentiallylike e −hν/kT (k: Boltzmann constant). On the other hand,if hν ≪ kT , the spectrum is nearly flat. The assumptionof low optical density of the source leads to the factthat the emission spectrum and production spectrum arepractically identical.2. A hot optical dense body produces a blackbody spec- productiontrum independently of the underlying production pro- in an optical thick medium

126 6 Primary Cosmic Rayscess because both emission and absorption processes areinvolved. Therefore, an optically dense bremsstrahlungsource which absorbs its own radiation would also producea blackbody spectrum. The emission P of a black-body is given by Planck’s lawenergy spectraν 3P ∼e hν/kT − 1 . (6.81)For high energies (hν ≫ kT ) P can be described by anexponentialP ∼ e −hν/kT , (6.82)while at low energies (hν ≪ kT ), because ofe hν/kT = 1 + hν +··· , (6.83)kTthe spectrum decreases to low frequencies likeP ∼ ν 2 . (6.84)The total radiation S of a hot body is described by theStefan–Boltzmann law,S = σT 4 , (6.85)Fig. 6.50Standard X-ray spectra originatingfrom various production processeswhere σ is the Stefan–Boltzmann constant.Typical energy spectra for various production mechanismsare sketched in Fig. 6.50.6.4.3 Detection of X Rayspassive collimatorsThe observation of X-ray sources is more demanding comparedto optical astronomy. X rays cannot be imaged withlenses since the index of refraction in the keV range is veryclose to unity. If X rays are incident on a mirror they willbe absorbed rather than reflected. Therefore, the direction ofincidence of X rays has to be measured by different techniques.The most simple method for directional observationis based on the use of slit or wire collimators which aremounted in front of an X-ray detector. In this case the observationaldirection is given by the alignment of the spaceprobe. Such a geometrical system achieves resolutions onthe order of 0.5 ◦ . By combining various types of collimatorsangular resolutions of one arc minute can be obtained.

6.4 X-Ray Astronomy 127In 1952 Wolter had already proposed how to build X-ray telescopes based on total reflection. To get reflection atgrazing incidence rather than absorption or scattering, theimaging surfaces have to be polished to better than a fractionof 10 −3 of the optical wavelength. Typically, systemsof stacked assemblies of paraboloids or combinations ofparabolic and hyperbolic mirrors are used (see Fig. 6.51).Wolter telescopefromsourcediaphragmparaboloidhyperboloidreadoutX-raydetectortelemetryFig. 6.51Cross section through an X-raytelescope with parabolic andhyperbolic mirrorsTo be able to image X rays in the range between 0.5 nmand 10 nm with this technique, the angles of incidence mustbe smaller than 1.5 ◦ (Fig. 6.52). The wavelength λ[nm] isobtained from the relationλ = c ν = hchν = 1240 nm . (6.86)E[eV]The mirror system images the incident X rays onto the commonfocal point. In X-ray satellites commonly several X-raydevices are installed as focal-point detectors. They are usuallymounted on a remotely controllable device. Dependingon the particular application an appropriate detector can bemoved into the focal point. In multimirror systems angularresolutions of one arc second are obtained. The requirementof grazing incidence, however, considerably limits the acceptanceof X-ray telescopes.As detectors for X rays crystal spectrometers (Braggreflection), proportional counters, photomultipliers, singlechannelelectron multipliers (channeltrons), semiconductorcounters, or X-ray CCDs (charge-coupled devices) are inuse.In proportional counters the incident photon first createsan electron via the photoelectric effect which then producesan avalanche in the strong electrical field (Fig. 6.53).In the proportional domain gas amplifications of 10 3 upto 10 5 are achieved. Since the absorption cross section forthe photoelectric effect varies proportional to Z 5 ,aheavynoble gas (Xe, Z = 54) with a quencher should be usedas counting gas. Thin foils (≈ 1 µm) made from beryllium(Z = 4) or carbon (Z = 6) are used as entrance windows.reflection coefficient [%]1008060402000 1 2 3grazing angle [degrees]Fig. 6.52Angular-dependent reflectionpower of metal mirrorsdetector for X rays

128 6 Primary Cosmic RaysFig. 6.53Principle of operation of aproportional counterphotomultiplier, channeltronangular resolutioncharge-coupled deviceThe incident photon transfers its total energy to the photoelectron.This energy is now amplified during avalanche formationin a proportional fashion. Therefore, this techniquenot only allows to determine the direction of the incidenceof the X-ray photon but also its energy.With photomultipliers or channeltrons the incident pho-ton is also converted via the photoelectric effect into an electron.This electron is then amplified by ionizing collisions inthe discrete or continuous electrode system. The amplifiedsignal can be picked up at the anode and further processedby electronic amplifiers.The energy measurement of X-ray detectors is based onthe number of charge carriers which are produced by thephotoelectron. In gas proportional chambers typically 30 eVare required to produce an electron–ion pair. Semiconductorcounters possess the attractive feature that only approximately3 eV are needed to produce an electron–hole pair.Therefore, the energy resolution of semiconductor countersis better by a factor of approximately √ 10 compared to proportionalchambers. As solid-state materials, silicon, germanium,or gallium arsenide can be considered. Because of theeasy availability and the favourable noise properties mostlysilicon semiconductor counters are used.If a silicon counter is subdivided in a matrix-like fashioninto many quadratic elements (pixels), which are shieldedagainst each other by potential wells, the produced energydepositions can be read out line by line. Because of thecharge coupling of the pixels this type of silicon image sensoris also called charge-coupled device. Commercial CCDswith areas of 1 cm × 1 cm at a thickness of 300 µm haveabout 10 5 pixels. Even though the shifting of the charge inthe CCD is a serial process, these counters have a relativelyhigh rate capability. Presently, time resolutions of 1 ms up to100 µs have been obtained. This allows rate measurementsin the kHz range which is extremely interesting for the observationof X-ray sources with high variability.6.4.4 Observation of X-Ray SourcesThe Sun was the first star of which X rays were recorded(Friedmann et al. 1951). In the range of X rays the Sunis characterized by a strong variability. In strong flares itsintensity can exceed the X-ray brightness of the quiet Sunby a factor of 10 000.

6.4 X-Ray Astronomy 129In 1959 the first X-ray telescope was built (R. Giacconni,Nobel Prize 2002) and flown on an Aerobee rocketin 1962. During its six minutes flight time it discovered thefirst extrasolar X-ray sources in the constellation Scorpio.The observation time could be extended with the first X-raysatellite UHURU (meaning ‘freedom’ in Swahili), whichwas launched from a base in Kenya 1970. Every week inorbit it produced more results than all previous experimentscombined.In the course of time a large number of X-ray satelliteshas provided more and more precision information aboutthe X-ray sky. The satellite with the highest resolution up to1999 was a common German–British–American project: theROentgen SATellite ROSAT (see Fig. 6.54). ROSAT measuredX rays in the range of 0.1upto2.5 keV with a Woltertelescope of 83 cm diameter. As X-ray detectors, multiwireproportional chambers (PSPC) 12 with 25 arc second resolutionand a channel-plate multiplier (HRI) 13 with5arcsecond resolution were in use. One of the PSPCs was permanentlyblinded by looking by mistake into the Sun. Thesecond PSPC stopped operation after a data-taking periodof 4 years because its gas supply was exhausted. Since thenonly the channel-plate multiplier was available as an X-raydetector. Compared to earlier X-ray satellites, ROSAT hada much larger geometrical acceptance, better angular andenergy resolution, and a considerably increased signal-tonoiseratio: per angular pixel element the background ratewas only one event per day.In a sky survey ROSAT has discovered about 130 000 X-ray sources. For comparison: the earlier flown Einstein ObservatoryHEAO 14 had only found 840 sources. The mostfrequent type of X-ray sources are nuclei of active galaxies(≈ 65 000) and normal stars (≈ 50 000). About 13 000galactic clusters and 500 normal galaxies were found to emitX rays. The smallest class of X-ray sources are supernovaremnants with approximately 300 identified objects.Supernova remnants (SNR) represent the most beautifulX-ray sources in the sky. ROSAT found that the Vela pulsaralso emits X rays with a period of 89 ms known fromits optical emission. It appears that the X-ray emission ofVela X1 is partially of thermal origin. The supernova rem-12 PSPC – Position Sensitive Proportional Chamber13 HRI – High Resolution Imager14 HEAO – High Energy Astronomy Observatoryextrasolar X raysFig. 6.54Photograph of the X-ray satelliteROSAT {18}130 000 discoveredX-ray sourcesHEAO

130 6 Primary Cosmic RaysROSAT HRI2’SNR 1572 (Tycho)MPE 7/90Fig. 6.55Supernova remnant SNR 1572recorded with the HRI detector(High Resolution Instrument) onboard the ROSAT satellite {18}toroidalemissionbrightpulsarhelicaljetFig. 6.56Sketch of X-ray emission from theCrab pulsarbinariesM81 central regionPSPC 20000 sOctober 1992MPE1’M81 with SN 1993JPSPC 27000 sApril 1993Fig. 6.57Sprial galaxy M81 with thesupernova SN 1993J. The imagewas recorded with the PSPCdetector (Position SensitiveProportional Counter) of theROSAT satellite {18}nant SNR 1572 that was observed by the Danish astronomerTycho Brahe shows a nearly spherically expanding shell inthe X-ray range (Fig. 6.55). The shell expands into the interstellarmedium with a velocity of about 50 km/s and it heatsup in the course of this process to several million degrees.The topology of X-ray emission from the Crab Pulsarallows to identify different components: the pulsar itself isvery bright in X rays compared to the otherwise more diffuseemission. The main component consists of a toroidalconfiguration which is caused by synchrotron radiation ofenergetic electrons and positrons in the magnetic field of thepulsar. In addition, electrons and positrons escape along themagnetic field lines at the poles where they produce X raysin a helical wind (see Fig. 6.56).Only six days after the explosion of a supernova in thespiral galaxy M81, ROSAT has measured its X-ray emission.In the right-hand part of Fig. 6.57 the X-ray source SN1993J south of the center of M81 is visible. In an earlier exposureof the same sky region (left-hand part of the figure)this source is absent.A large number of X-ray sources are binaries. In thesebinaries mostly a compact object – a white dwarf, a neutronstar, or a black hole – accretes matter from a nearbycompanion. The matter flowing to the compact object frequentlyforms an accretion disk (see Fig. 5.7), however, thematter can also be transported along the magnetic field lineslanding directly on the neutron star. In such cataclysmicvariables a mass transfer from the companion, e.g., to awhite dwarf, can be sufficient to maintain a permanent hydrogenburning. If the ionized hydrogen lands on a neutronstar also thermonuclear X-ray flashes can occur. Initially,the incident hydrogen fuses in a thin layer at the surface ofthe neutron star to helium. If a sufficient amount of matteris accreted, the helium produced by fusion can achieve suchhigh densities and temperatures that it can be ignited in athermonuclear explosion forming carbon.The observation of thermal X rays from galactic clustersallows a mass determination of the hot plasma and the totalgravitational mass of the cluster. This method is basedon the fact that the temperature is a measure for the gravitationalattraction of the cluster. A high gas temperature –characterized by the energy of the emitted X rays – representsvia the gas pressure the counterforce to gravitation andprevents the gas from falling into the center of the cluster.

6.4 X-Ray Astronomy 131Measurements of X rays from galactic clusters have establishedthat the hot plasma between the galaxies is five timesmore massive than the galaxies themselves. The discoveryof X-ray-emitting massive hot plasmas between the galaxiesis a very important input for the understanding of thedynamics of the universe.In the present understanding of the evolution of the universeall structures are hierarchically formed from objectsof the respective earlier stages: stellar clusters combine togalaxies, galaxies form groups of galaxies which grow togalactic clusters which in turn produce superclusters. Distant,i.e., younger galactic clusters, get more massive whileclose-by galactic clusters hardly grow at all. This allows toconclude that nearby galactic clusters have essentially collectedall available matter gravitationally. The mass of theseclusters appears to be dominated by gas clouds into whichstellar systems are embedded like raisins in a cake. Therefore,the X-ray-emitting gas clouds allow to estimate thematter density in the universe. The present X-ray observationsfrom ROSAT suggest a value of approximately 30%of the critical mass density of the universe. If this were all,this would mean that the universe expands eternally (seeChap. 8).In 1999 the X-ray satellite AXAF (Advanced X-ray AstrophysicsFacility) was launched successfully. To honourthe contributions to astronomy and astrophysics the satellitewas renamed Chandra after the Indian–American astrophysicistSubrahmanyan Chandrasekhar. At the end of 1999the X-ray satellite XMM (X-ray Multi-Mirror mission) wasbrought into orbit (Fig. 6.58). It was also renamed in theyear 2000 to honour Newton and its name is now XMMNewton or Newton Observatory, respectively. Both Chandraand the Newton Observatory have better angular and energyresolutions compared to ROSAT, and therefore a better understandingof X-ray sources and the non-luminous matterin the universe is expected. From the experimental point ofview one has to take extreme care that the sensitive focalpixel detectors do not suffer radiation damage from lowenergysolar particles (p, α, e) emitted during solar eruptions.It is anticipated that Chandra and XMM Newton willdiscover ≈ 50 000 new X-ray sources per year. Of particularinterest are active galactic nuclei which are supposed to bepowered by black holes. Many or most of the black holesgalactic clustershierarchical structureof the universedensity of the universeAXAF, XMM, ASTRO-EFig. 6.58Photo of the XMM X-ray satellite{19}expected discoveriesof X-ray sources

132 6 Primary Cosmic RaysFig. 6.59Collision of two merging galacticclusters, each containing hundredsof galaxies. The site of thiscatastophic event is Abell 754 at adistance of about 9 millionlight-years. The photo shows acoded pressure map of this region,where the galaxies themselves areconfined around the white spotswhich correspond to regions ofhigh pressure, followed bydecreasing pressure as one goesaway from the centers {20}residing at the centers of distant galaxies may be difficult tofind since they are hidden deep inside vast amounts of absorbingdust so that only energetic X rays or γ rays can escape.Already now these new X-ray satellites have observedgalaxies at high redshifts emitting huge amounts of energyin the form of X rays, far more than can ever expected to beproduced by star formation. Therefore, it is conjectured thatthese galaxies must contain actively accreting supermassiveblack holes.Even though presently known classes of sources willprobably dominate the statistics of Chandra and XMM Newtondiscoveries, the possibility of finding completely newand exciting populations of X-ray sources also exists.A very recent result is the observation of the collisionof two galactic clusters in Abell 754 at a distance of about9 million light-years in the light of X rays. These clusterswith millions of galaxies merge in a catastrophic collisioninto one single very large cluster.On the other hand, the X-ray mission ASTRO-E, startedearly in the year 2000, had to be abandoned because thebooster rocket did not carry the satellite into an altituderequired for the intended orbit. The satellite presumablyburned up during reentry into the atmosphere.The diffuse X-ray background which was discovered rel-atively early consists to a large extent (75%) of resolvedextragalactic sources. It could easily be that the remainingdiffuse part of X rays consists of so far non-resolved distantX-ray sources.The largest fraction of the earlier unresolved X-raysources are constituted by active galactic nuclei and quasars.A long-term exposure of ROSAT (40 hours) revealed morethan 400 X-ray sources per square degree.diffuse background X raysFig. 6.60X-ray emission from the Moonrecorded with the PSPC detectoron board of ROSAT. The dark sideof the Moon shields the cosmicX-ray background {18}iron line in X-ray spectraX rays from the MoonIn the spectra of many X-ray sources the iron line(5.9 keV) is observed. This is a clear indication that eitheriron is synthesized directly in supernova explosions or the Xrays originate from older sources whose material has beenprocessed through several stellar generations.A surprising result of the most recent investigations wasthat practically all stars emit X rays. A spectacular observationwas also the detection of X rays from the Moon. However,the Moon does not emit these X rays itself. It is ratherreflected corona radiation from our Sun in the same way asthe Moon also does not shine in the optical range but ratherreflects the sunlight (Fig. 6.60).

6.5 Gravitational-Wave Astronomy 1336.5 Gravitational-Wave Astronomy“What would physics look like withoutgravitation?”Albert EinsteinFinally, it is appropriate to mention the new field of gravitational-waveastronomy. Gravitational waves have been predictedby Einstein als early as 1916. Apart from the observationof Taylor and Hulse concerning the energy loss of abinary pulsar (PSR 1913+16) due to the emission of gravitationalwaves over a period starting from 1974 (Nobel Prize1993) there is no direct evidence of the existence of gravitationalwaves. Nobody doubts the correctness of Einstein’sprediction, especially since the results of Taylor and Hulseon the energy loss of the binary pulsar system by emission ofgravitational radiation agree with the theoretical expectationof general relativity impressibly well (to better than 0.1%).Taylor and Hulse have observed the binary system PSR1913+16 consisting of a pulsar and a neutron star over a periodof more than 20 years. The two massive objects rotatearound their common center of mass on elliptical orbits. Theradio emission from the pulsar can be used as precise clocksignal. When the pulsar and neutron star are closest together(periastron), the orbital velocities are largest and the gravitationalfield is strongest. For high velocities and in a stronggravitational field time is slowed down. This relativistic ef-energy lossby gravitational radiationchange in periastron time [s]−−−−−−0510152025301975Taylor & HulsePSR 1913+16generalrelativityfect can be checked by looking for changes in the arrival 1995time of the pulsar signal. In this massive and compact pulsar Fig. 6.61system the periastron time changes in a single day by the Observed changes in periastrontime of the binary system PSRsame amount for which the planet Mercury needs a century1913+16 over more than 20 yearsin our solar system. Space-time in the vicinity of the binary in comparison to the expectationis greatly warped.based on Einstein’s theory ofThe theory of relativity predicts that the binary system general relativity. The agreementbetween theory and observation iswill lose energy with time as the orbital rotation energy isbetter than 0.1%converted into gravitational radiation. Fig. 6.61 shows theprediction based on Einstein’s theory of general relativity incomparison to the experimental data. The excellent agreementbetween theory and experiment presents so far the best– albeit indirect – evidence for gravitational waves.The direct detection of gravitational radiation wouldopen a new window onto violent astrophysical events andit may give a clue to processes where dark matter or darkenergy is involved.However, as far as the direct observation is concerned,the situation is to a certain extent similar to neutrino physics1985evidencefor gravitational radiation

134 6 Primary Cosmic Rayslow interaction probabilityof gravitational radiationimaging with gravity waves?quadrupole radiationaround 1950. At that time nobody really doubted the existenceof the neutrino, but there was no strong neutrino sourceavailable to test the prediction. Only the oncoming nuclearreactors were sufficiently powerful to provide a large enoughneutrino flux to be observed. The problem was related to thelow interaction cross section of neutrinos with matter.Compared to gravitational waves the neutrino interactionwith matter can be called ‘strong’. The extremely lowinteraction probability of gravitational waves ensures thatthey provide a new window to cataclysmic processes in theuniverse at the expense of a very difficult detection. For neutrinosmost astrophysical sources are almost transparent. Becauseof the feeble interaction of gravitational waves theycan propagate out of the most violent cosmological sourceseven more freely compared to neutrinos.With electromagnetic radiation in various spectral rangesastronomical objects can be imaged. This is because thewavelength of electromagnetic radiation is generally verysmall compared to the size of astrophysical objects. Thewavelength of gravitational waves is much larger so thatimaging with this radiation is almost out of question. In elec-tromagnetic radiation time-dependent electric and magneticfields propagate through space-time. For gravitational wavesone is dealing with oscillations of space-time itself.Electromagnetic radiation is emitted when electriccharges are accelerated or decelerated. In a similar waygravitational waves are created whenever there is a nonsphericalacceleration of mass–energy distributions. Thereis, however, one important difference. Electromagnetic radiationis dipole radiation, while gravitational waves presentquadrupole radiation. This is equivalent to saying that thequantum of gravitational waves, the graviton, has spin 2¯h.Electromagnetic waves are created by time-varying dipolemoments where a dipole consists of one positive andone negative charge. In contrast, gravity has no charge, thereis only positive mass. Negative mass does not exist. Even an-timatter has the same positive mass just as ordinary matter.Therefore, it is not possible to produce an oscillating massdipole. In a two-body system one mass accelerated to theleft creates, because of momentum conservation, an equaland opposite action on the second mass which moves it tothe right. For two equal masses the spacing may change butthe center of mass remains unaltered. Consequently, there isno monopole or dipole moment. Therefore, the lowest orderof oscillation generating gravitational waves is due tooscillation modes of antennae

6.5 Gravitational-Wave Astronomy 135a time-varying quadrupole moment. In the same way as adistortion of test masses creates gravitational waves where,e.g., the simplest non-spherical motion is one in which horizontalmasses move inside and vertical masses move apart,a gravitational wave will distort an antenna analogously bycompression in one direction and elongation in the other (seeFig. 6.62).The quadrupole character of gravitational radiationtherefore leads to an action like a tidal force: it squeezes theantenna along one axis while stretching it along the other.Due to the weakness of the gravitational force the relativeelongation of an antenna will be at most on the order ofh ≈ 10 −21 even for the most violent cosmic catastrophes.There is, however, one advantage of gravitational wavescompared to the measurement of electromagnetic radiation:Electromagnetic observables like the energy flux from astrophysicalsources are characterized by a 1/r 2 dependencedue to solid-angle reasons. By contrast, the direct observableof gravitational radiation (h) decreases with distanceonly like 1/r. h depends linearly on the second derivative ofthe quadrupole moment of the astrophysical object and it isinversely proportional to the distance r,h ∼ G ¨Qc 4 (G: Newton’s constant) .rConsequently, an improvement of the sensitivity of a gravitationaldetector by a factor of 2 increases the measurablevolume where sources of gravitational waves may reside bya factor of 8. The disadvantage of not being able to imagewith gravitational radiation goes along with the advantagethat gravitational-wave detectors have a nearly 4π steradiansensitivity over the sky.The most promising candidates as sources for gravitationalradiation are mergers of binary systems, accretingblack holes, collisions of neutron stars, or special binariesconsisting of two black holes orbiting around their commoncenter of mass (like in the radio galaxy 3C66B, which appearsto be the result of a merger of two galaxies).The suppression of noise in these antennae is the mostdifficult problem. There are stand-alone gravitational-wavedetectors, mostly in the form of optical interferometerswhere the elongation of one lever arm and the compressionof the other can be monitored by the technique of Michelsoninterferometry. These detectors can be operated at groundlevel or in space. A convincing signal of gravitational wavesFig. 6.62Oscillation modes of a sphericalantenna upon the impact of agravitational wave causing it toundergo quadrupole oscillationsattenuationof gravitational wavesnoise suppression

136 6 Primary Cosmic Rayswould probably require a coincidence of signals from severalindependent detectors.6.6 ProblemsProblems for Sect. 6.11. What is the reason that primary cosmic-ray nuclei likecarbon, oxygen, and neon are more abundant than theirneighbours in the periodic table of elements (nitrogen,fluorine, sodium)?2. In Sect. 6.1 it is stated that primary cosmic rays consistof protons, α particles, and heavy nuclei. Only 1%of the primary particles are electrons. Does this meanthat the planet Earth will be electrically charged in thecourse of time because of the continuous bombardmentby predominantly positively charged primary cosmicrays? What kind of positive excess charge could havebeen accumulated during the period of existence of ourplanet if this were true?3. The chemical abundance of the sub-iron elements in primarycosmic rays amounts to about 10% of the iron flux.a) Estimate the fragmentation cross section for collisionsof primary iron nuclei with interstellar/intergalacticnuclei.b) What is the chance that a primary iron nucleus willsurvive to sea level?Problems for Sect. 6.21. The Sun converts protons into helium according to thereaction4p → 4 He + 2e + + 2ν e .The solar constant describing the power of the Sun atEarth is P ≈ 1400 W/m 2 . The energy gain per reactioncorresponds to the binding energy of helium (E B ( 4 He) =28.3 MeV). How many solar neutrinos arrive at Earth?2. If solar electron neutrinos oscillate into muon or tau neutrinosthey could in principle be detected via the reactionsν µ + e − → µ − + ν e , ν τ + e − → τ − + ν e .Work out the threshold energy for these reactions to occur.

6.6 Problems 1373. Radiation exposure due to solar neutrinos.a) Use (6.9) to work out the number of interactions ofsolar neutrinos in the human body (tissue densityϱ ≈ 1gcm −3 ).b) Neutrinos interact in the human body byν e + N → e − + N ′ ,where the radiation damage is caused by the electrons.Estimate the annual dose for a human underthe assumption that on average 50% of the neutrinoenergy is transferred to the electron.c) The equivalent dose is defined asH = (E/m) w R (6.87)(m is the mass of the human body, w R the radiationweighting factor (= 1 for electrons), [H ]=1Sv =1w R Jkg −1 ), and E the energy deposit in the humanbody). Work out the annual equivalent dose dueto solar neutrinos and compare it with the normalnatural dose of H 0 ≈ 2mSv/a.4. Neutrino oscillations. 15In the most simple case neutrino oscillations can be describedin the following way (as usual ¯h and c will be setto unity): in this scenario the lepton flavour eigenstatesare superpositions|ν e 〉=cos θ|ν 1 〉+sin θ|ν 2 〉 ,|ν µ 〉=−sin θ|ν 1 〉+cos θ|ν 2 〉of mass eigenstates |ν 1 〉 and |ν 2 〉. All these states areconsidered as wave packets with well-defined momentum.In an interaction, e.g., a ν e is assumed to be generatedwith momentum p, which then propagates as freeparticle, |ν e ; t〉 = e −iHt |ν e 〉. For the mass √eigenstatesone has e −iHt |ν i 〉=e −iE ν t i |ν i 〉 with E νi = p 2 + m 2 i ,i = 1, 2. The probability to find a muon neutrino after atime t isP νe →ν µ(t) =|〈ν µ |ν e ; t〉| 2 .a) Work out P νe →ν µ(t). After a time t the particle isat x ≈ vt = pt/E νi . Show that – under the assumptionof small neutrino masses – the oscillationprobability as given in (6.22) can be derived.15 This problem is difficult and its solution is mathematically demanding.

138 6 Primary Cosmic Raysb) Estimate for (ν µ ↔ ν τ ) oscillations with maximummixing (sin 2θ = 1), E ν = 1 GeV and under theassumption of m νµ ≪ m ντ (that is, m ντ ≈ √ δm 2 )the mass of the τ neutrino, which results if the ratioof upward-to-downward-going atmospheric muonneutrinos is assumed to be 0.54.c) Does the assumption m νµ ≪ m ντ make sense here?Problems for Sect. 6.31. Estimate the detection efficiency for 1 MeV photons in aNaI(Tl) scintillation counter of 3 cm thickness.(Hint: Use the information on the mass attenuation coefficientfrom Fig. 6.37.)2. Estimate the size of the cosmological object that hasgiven rise to the γ -ray light curve shown in Fig. 6.48.3. What is the energy threshold for muons to produce Cherenkovlight in air (n = 1.000 273) and water (n =1.33)?4. The Crab Pulsar emits high-energy γ rays with a powerof P = 3 × 10 27 W at 100 GeV. How many photonsof this energy will be recorded by the planned GLASTexperiment (Gamma-ray Large Area Space Telescope)per year if isotropic emission is assumed? What is theminimum flux from the Crab that GLAST will be able todetect (in J/(cm 2 s))? (Collecting area of GLAST: A =8000 cm 2 , distance of the Crab: R = 3400 light-years.)5. The solar constant describing the solar power arriving atEarth is P S ≈ 1400 W/ m 2 .a) What is the total power radiated by the Sun?b) Which mass fraction of the Sun is emitted in 10 6years?c) What is the daily mass transport from the Sun deliveredto Earth?Problems for Sect. 6.41. The radiation power emitted by a blackbody in its dependenceon the frequency is given by (6.81). Convertthis radiation formula into a function depending on thewavelength.2. The total energy emitted per second by a star is called itsluminosity. The luminosity depends both on the radius of

6.6 Problems 139the star and its temperature. What would be the luminosityof a star ten times larger than our Sun (R = 10 R ⊙ )but at the same temperature? What would be the luminosityof a star of the size of our Sun but with ten timeshigher temperature?3. The power radiated by a relativistic electron of energy Ein a transverse magnetic field B through synchrotron radiationcan be worked out from classical electrodynamicsto beP = e2 c 32π C γ E 2 B 2 ,wherer eC γ = 4 3 π (m e c 2 ) 3 ≈ 8.85 × 10−5 mGeV −3 .Work out the energy loss due to synchrotron emissionper turn of a 1 TeV electron in a circular orbit around apulsar at a distance of 1000 km. What kind of magneticfield had the pulsar at this distance?4. Consider the energy loss by radiation of a particle movingin a transverse homogeneous time-independent magneticfield, see also Problem 3 for this section. The radiatedpower is given bye 2P = 2 3 m 2 γ 2 |ṗ| 2 .0 c3Calculate the time dependence of the particle energy andof the bending radius of the trajectory for thea) ultrarelativistic case,b) general case. 165. An X-ray detector on board of a satellite with collectionarea of 1 m 2 counts 10 keV photons from asource in the Large Magellanic Cloud with a rate of1/hour. How many 10 keV photons are emitted fromthe source if isotropic emission is assumed? The detectoris a Xe-filled proportional counter of 1 cm thickness(the attenuation coefficient for 10 keV photons isµ = 125 (g/cm 2 ) −1 , density ϱ Xe = 5.8 × 10 −3 g/cm 3 ).6. X rays can be produced by inverse Compton scatteringof energetic electrons (E i ) off blackbody photons (energyω i ). Show that the energy of the scattered photonω f is related to the scattering angles ϕ i and ϕ f by16 The solution to this problem is mathematically demanding.

140 6 Primary Cosmic Rays1 − cos ϕ iω f ≈ ω i , (6.88)1 − cos ϕ fwhere ϕ i and ϕ f are the angles between the incomingelectron and the incoming and outgoing photon. Theabove approximation holds if E i ≫ m i ≫ ω i .7. What is the temperature of a cosmic object if its maximumblackbody emission occurs at an energy of E =50 keV?(Hint: The solution of this problem leads to a transcendentalequation which needs to be solved numerically.)Problems for Sect. 6.51. A photon propagating to a celestial object of mass Mwill gain momentum and will be shifted towards theblue. Work out the relative gain of a photon approachingthe Sun’s surface from a height of H = 1km. Analogously,a photon escaping from a massive object will begravitationally redshifted.(Radius of the Sun R ⊙ = 6.9635 × 10 8 m, Mass of theSun M ⊙ = 1.993 × 10 30 kg, acceleration due to Sun’sgravity g ⊙ = 2.7398 × 10 2 m/s 2 .)2. Accelerated masses radiate gravitational waves. Theemitted energy per unit time is worked out to beP = G ...5c 2 Q 2 ,where Q is the quadrupole moment of a certain massconfiguration (e.g., the system Sun–Earth). For a rotatingsystem with periodic time dependence (∼ sin ωt)each time derivative contributes a factor ω, henceP ≈ G5c 2 ω6 Q 2 .For a system consisting of a heavy-mass object like theSun (M) and a low-mass object, like Earth (m), thequadrupole moment is on the order of mr 2 . Neglectingnumerical factors of order unity, one getsP ≈ G c 2 ω6 m 2 r 4 .Work out the power radiated from the system Sun–Earthand compare it with the gravitational power emittedfrom typical fast-rotating laboratory equipments.

1417 Secondary Cosmic Rays“There are more things in heaven and earth,Horatio, than are dreamt of in your philosophy.”Shakespeare, HamletFor the purpose of astroparticle physics the influence of theSun and the Earth’s magnetic field is a perturbation, whichcomplicates a search for the sources of cosmic rays. Thesolar activity produces an additional magnetic field whichprevents part of galactic cosmic rays from reaching Earth.Figure 7.1, however, shows that the influence of the Sunis limited to primary particles with energies below 10 GeV.The flux of low-energy primary cosmic-ray particles is anticorrelatedto the solar activity.solar modulationFig. 7.1Modulation of the primaryspectrum by the 11-year cycle ofthe SunFig. 7.2Flux densities of protons andelectrons in the radiation belts ofthe EarthOn the other hand, the solar wind, whose magnetic fieldmodulates primary cosmic rays, is a particle stream in itself,which can be measured at Earth. The particles constitutingthe solar wind (predominantly protons and electrons) are oflow energy (MeV region). These particles are captured to alarge extent by the Earth’s magnetic field in the Van Allenbelts or they are absorbed in the upper layers of the Earth’satmosphere (see Fig. 7.9). Figure 7.2 shows the flux den-solar wind

142 7 Secondary Cosmic RaysFig. 7.3(a) Relation between atmosphericdepth (column density) andpressure(b) column density of theatmosphere as a function ofaltitude up to 28 kmradiation beltssities of protons and electrons in the Van Allen belts. Theproton belt extends over altitudes from 2 000 to 15 000 km.It contains particles with intensities up to 10 8 /(cm 2 s) andenergies up to 1 GeV. The electron belt consists of two parts.The inner electron belt with flux densities of up to 10 9particles per cm 2 and s is at an altitude of approximately3 000 km, while the outer belt extends from about 15 000 kmto 25 000 km. The inner part of the radiation belts is symmetricallydistributed around the Earth while the outer partis subject to the influence of the solar wind and consequentlydeformed by it (see also Fig. 1.9 and Fig. 1.13).7.1 Propagation in the Atmosphere“Astroparticles are messengers fromdifferent worlds.”Anonymousinteraction in the atmospherecolumn densityPrimary cosmic rays are strongly modified by interactionswith atomic nuclei in the atmospheric air. The columndensity of the atmosphere amounts to approximately 1000g/cm 2 , corresponding to the atmospheric pressure of about1000 hPa. Figure 7.3 (a) shows the relation between columndensity, altitude in the atmosphere, and pressure. Figure 7.3(b) shows this relation in somewhat more detail for altitudesbelow 28 km. The residual atmosphere for flight altitudes

7.1 Propagation in the Atmosphere 143of scientific balloons (≈ 35–40 km) corresponds to approximatelyseveral g/cm 2 . For inclined directions the thicknessof the atmosphere increases strongly (approximately like1/ cos θ, with θ – zenith angle). Figure 7.4 shows the variationof atmospheric depth with zenith angle at sea level.For the interaction behaviour of primary cosmic rays thethickness of the atmosphere in units of the characteristic interactionlength for the relevant particles species in questionis important. The radiation length for photons and electronsin air is X 0 = 36.66 g/cm 2 . The atmosphere therefore correspondsto a depth of 27 radiation lengths. The relevantinteraction length for hadrons in air is λ = 90.0g/cm 2 ,correspondingto 11 interaction lengths per atmosphere. Thismeans that practically not a single particle of original primarycosmic rays arrives at sea level. Already at altitudes of15 to 20 km primary cosmic rays interact with atomic nucleiof the air and initiate – depending on energy and particlespecies – electromagnetic and/or hadronic cascades.The momentum spectrum of the singly charged componentof primary cosmic rays at the top of the atmosphereis shown in Fig. 7.5. In this diagram the particle velocityβ = v/c is shown as a function of momentum. Clearly visibleare the bands of hydrogen isotopes as well as the lowflux of primary antiprotons. Even at these altitudes severalmuons have been produced via pion decays. Since muon andpion mass are very close, it is impossible to separate themout in this scatter diagram. Also relativistic electrons andpositrons would populate the bands labeled µ + and µ − .Onegenerally assumes that the measured antiprotons are not ofprimordial origin, but are rather produced by interactions ininterstellar or interplanetary space or even in the residual atmosphereabove the balloon.The transformation of primary cosmic rays in the atmosphereis presented in Fig. 7.6. Protons with approximately85% probability constitute the largest fraction of primarycosmic rays. Since the interaction length for hadronsis 90 g/cm 2 , primary protons initiate a hadron cascade alreadyin their first interaction approximately at an altitudecorresponding to the 100 mbar layer. The secondary particlesmost copiously produced are pions. Kaons on the otherhand are only produced with a probability of 10% comparedto pions. Neutral pions initiate via their decay (π 0 → γ +γ )electromagnetic cascades, whose development is characterizedby the shorter radiation length (X 0 ≈ 1 3λ in air). Thisatmospheric depth [g/cm ]210 510 410 30°20° 40° 60° 80°zenith angle Fig. 7.4Relation between zenith angle andatmospheric depth at sea levelradiation lengthinteraction lengthelectromagneticand hadronic cascadesFig. 7.5Identification of singly chargedparticles in cosmic rays at a flightaltitude of balloons (̂= 5 g/cm 2residual atmosphere) {21}

144 7 Secondary Cosmic RaysFig. 7.6Transformation of primary cosmicrays in the atmosphereFig. 7.7Decay probabilities for chargedpions and kaons in the atmosphereas a function of their kineticenergyshower component is absorbed relatively easily and is thereforealso named a soft component. Charged pions and kaonscan either initiate further interactions or decay.The competition between decay and interaction probabilityis a function of energy. For the same Lorentz factorcharged pions (lifetime 26 ns) have a smaller decayprobability compared to charged kaons (lifetime 12.4ns).The decay probability of charged pions and kaons in theatmosphere is shown in Fig. 7.7 as a function of theirkinetic energy. The leptonic decays of pions and kaonsproduce the penetrating muon and neutrino components(π + → µ + + ν µ , π − → µ − +¯ν µ ; K + → µ + + ν µ ,K − → µ − +¯ν µ ). Muons can also decay and contributevia their decay electrons to the soft component and neutrinosto the neutrino component (µ + → e + + ν e +¯ν µ ,µ − → e − +¯ν e + ν µ ).The energy loss of relativistic muons not decaying in theatmosphere is low (≈ 1.8 GeV). They constitute with 80%of all charged particles the largest fraction of secondary particlesat sea level.Some secondary mesons and baryons can also survivedown to sea level. Most of the low-energy charged hadronsobserved at sea level are locally produced. The total fractionof hadrons at ground level, however, is very small.Apart from their longitudinal development electromagneticand hadronic cascades also spread out laterally in theatmosphere. The lateral size of an electromagnetic cascadeis caused by multiple scattering of electrons and positrons,while in hadronic cascades the transverse momenta at productionof secondary particles are responsible for the lateralwidth of the cascade. Figure 7.8 shows a comparison of theshower development of 100 TeV photons and 100 TeV protonsin the atmosphere. It is clearly visible that transversemomenta of secondary particles fan out the hadron cascade.The intensity of protons, electrons, and muons of all energiesas a function of the altitude in the atmosphere is plottedin Fig. 7.9. The absorption of protons can be approximatelydescribed by an exponential function.The electrons and positrons produced through π 0 decaywith subsequent pair production reach a maximum intensityat an altitude of approximately 15 km and soon after are relativelyquickly absorbed while, in contrast, the flux of muonsis attenuated only relatively weakly.

7.1 Propagation in the Atmosphere 145height in the atmosphere [km]35302520151050photon showerE 0 = 100 TeVθ = 0 °–6–4–2distance from the shower axis [km]035hadronsmuonsγ, e + , e − 302520151050246height in the atmosphere [km]proton showerE 0 = 100 TeVθ = 0 °–6–4–2distance from the shower axis [km]0hadronsmuonsFig. 7.8Comparison of the development ofelectromagnetic (100 TeV photon)and hadronic cascades (100 TeVproton) in the atmosphere. Only2 4 6 secondaries with E ≥ 1 GeV areshown {22}Fig. 7.9Particle composition in theatmosphere as a function ofatmospheric depthBecause of the steepness of the energy spectra the particleintensities are of course dominated by low-energy particles.These low-energy particles, however, are mostly ofsecondary origin. If only particles with energies in excess of1 GeV are counted, a different picture emerges (Fig. 7.10). sea-level compositionPrimary nucleons (protons and neutrons) with the initialhigh energies dominate over all other particle speciesdown to altitudes of 9 km, where muons take over. Becauseof the low interaction probability of neutrinos these particlesare practically not at all absorbed in the atmosphere.

146 7 Secondary Cosmic RaysFig. 7.10Intensities of cosmic-ray particleswith energies > 1 GeV in theatmosphereTheir flux increases monotonically because additional neutrinosare permanently produced by particle decays.Since the energy spectrum of primary particles is relativelysteep, the energy distribution of secondaries also hasto reflect this property.Figure 7.11 shows the proton and muon spectra for var-ious depths in the atmosphere. Clearly visible is the trendthat with increasing depth in the atmosphere muons start todominate over protons especially at high energies.proton and muon spectraFig. 7.11Momentum spectra of protons andmuons at various altitudes in theatmosphere

7.2 Cosmic Rays at Sea Level 1477.2 Cosmic Rays at Sea Level“The joy of discovery is certainly theliveliest that the mind of man can everfeel.”Claude BernardA measurement of charged particles at sea level clearlyshows that, apart from some protons, muons are the dominantcomponent (Fig. 7.12).Approximately 80% of the charged component of secondarycosmic rays at sea level are muons. Their flux througha horizontal area amounts to roughly one particle per cm 2and minute. These muons originate predominantly frompion decays, since pions as lightest mesons are producedin large numbers in hadron cascades. The muon spectrumat sea level is therefore a direct consequence of the pionsource spectrum. There are, however, several modifications.Figure 7.13 shows the parent pion spectrum at the locationof production in comparison to the observed sea-level muonspectrum. The shape of the muon spectrum agrees relativelywell with the pion spectrum for momenta between 10 and100 GeV/c. For energies below 10 GeV and above 100 GeVthe muon intensity, however, is reduced compared to thepion source spectrum. For low energies the muon decayprobability is increased. A muon of 1 GeV with a Lorentzfactor of γ = E/m µ c 2 = 9.4 has a mean decay length ofFig. 7.12Measurement and identification ofcharged particles at sea level {21}s µ ≈ γτ µ c = 6.2km. (7.1)Since pions are typically produced at altitudes of 15 kmand decay relatively fast (for γ = 10 the decay length is onlys π ≈ γτ π c = 78 m), the decay muons do not reach sea levelbut rather decay themselves or get absorbed in the atmosphere.At high energies the situation is changed. For pionsof 100 GeV (s π = 5.6 km, corresponding to a column densityof 160 g/cm 2 measured from the production altitude)the interaction probability dominates (s π > λ). Pions ofthese energies will therefore produce further, tertiary pionsin subsequent interactions, which will also decay eventuallyinto muons, but providing muons of lower energy. Therefore,the muon spectrum at high energies is always steepercompared to the parent pion spectrum.If muons from inclined horizontal directions are considered,a further aspect has to be taken into account. For largezenith angles the parent particles of muons travel relativelyFig. 7.13Sea-level muon spectrum incomparison to the pion parentsource spectrum at production

148 7 Secondary Cosmic Raysmuonsfrom inclined directionssea-level muon spectrumup to 20 TeV/clong distances in rare parts of the atmosphere. Because ofthe low area density at large altitudes for inclined directionsthe decay probability is increased compared to the interactionprobability. Therefore, for inclined directions pions willproduce predominantly high-energy muons in their decay.The result of these considerations is in agreement withobservation (Fig. 7.14). For about 170 GeV/c the muon intensityat 83 ◦ zenith angle starts to outnumber that of thevertical muon spectrum. The intensity of muons from horizontaldirections at low energies is naturally reduced becauseof muon decays and absorption effects in the thickeratmosphere at large zenith angles.The sea-level muon spectrum for inclined directions hasbeen measured with solid-iron momentum spectrometers upto momenta of approximately 20 TeV/c (Fig. 7.15). Forhigher energies the muon intensity decreases steeply.Fig. 7.14Sea-level muon momentum spectrafor vertical and inclined directionsFig. 7.15Momentum spectrum of muons atsea level for large zenith angles. Inthis figure the differential intensityis multiplied by p 3 µThe total intensity of muons, however, is dominated bylow-energy particles. Because of the increased decay probabilityand the stronger absorption of muons from inclineddirections, the total muon intensity at sea level varies likeI µ (θ) = I µ (θ = 0) cos n θ (7.2)charge ratio of muonsfor not too large zenith angles θ. The exponent of the zenithangledistribution is obtained to be n = 2. This exponentvaries very little, even at shallow depths underground, ifonly muons exceeding a fixed energy are counted.An interesting quantity is the charge ratio of muons atsea level. Since primary cosmic rays are positively charged,this positive charge excess is eventually also transferred to

7.2 Cosmic Rays at Sea Level 149muons. If one assumes that primary protons interact withprotons and neutrons of atomic nuclei in the atmospherewhere the multiplicity of produced pions is usually quitelarge, the charge ratio of muons, N(µ + )/N(µ − ), can beestimated by considering the possible charge exchange reactions:p + N → p ′ + N ′ + kπ + + kπ − + rπ 0 ,p + N → n + N ′ + (k + 1)π + + kπ − + rπ 0 (7.3).In this equation k and r are the multiplicities of the producedparticle species and N represents a target nucleon. If oneassumes that for the reactions in (7.3) the cross sections arethe same, the charge ratio of pions is obtained to beR = N(π+ )N(π − ) = 2k + 1 = 1 + 12k 2k . (7.4)For low energies k = 2 and thereby R = 1.25. Since thisratio is transferred to muons by the pion decay, one wouldexpect a similar value for muons. Experimentally one observesthat the charge ratio of muons at sea level is constantover a wide momentum range and takes on a value ofN(µ + )/N(µ − ) ≈ 1.27 . (7.5)In addition to ‘classical’ production mechanisms ofmuons by pion and kaon decays, they can also be producedin semileptonic decays of charmed mesons (for example,D 0 → K − µ + ν µ and D + → ¯K 0 µ + ν µ , D − →K 0 µ − ¯ν µ ). Since these charmed mesons are very short-lived(τ D 0 ≈ 0.4ps, τ D ± ≈ 1.1 ps), they decay practically immediatelyafter production without undergoing interactionsthemselves. Therefore, they are a source of high-energymuons. Since the production cross section of charmedmesons in proton–nucleon interactions is rather small, Ddecays contribute significantly only at very high energies.Figure 7.12 already showed that apart from muons alsosome nucleons can be observed at sea level. These nucleonsare either remnants of primary cosmic rays, which, however,are reduced in their intensity and energy by multipleinteractions, or they are produced in atmospheric hadroncascades. About one third of the nucleons at sea level areneutrons. The proton/muon ratio varies with the momentumof the particles. At low momenta (≈ 500 MeV/c) ap/µ ratio N(p)/N(µ) of about 10% is observed decreasingtolargermomenta(N(p)/N(µ)≈ 2% at 1 GeV/c,N(p)/N(µ) ≈ 0.5% at 10 GeV/c).charge exchange reactionsmuonsfrom semileptonic decaysnucleon component

150 7 Secondary Cosmic RaysIn addition to muons and protons, one also finds elec-trons, positrons, and photons at sea level as a consequenceof the electromagnetic cascades in the atmosphere. A certainfraction of electrons and positrons originates from muon decays.Electrons can also be liberated by secondary interactionsof muons (‘knock-on electrons’).The few pions and kaons observed at sea level are pre-dominantly produced in local interactions.Apart from charged particles, electron and muon neu-trinos are produced in pion, kaon, and muon decays. Theyconstitute an annoying background, in particular, for neutrinoastronomy. On the other hand, the propagation of atmosphericneutrinos has provided new insights for elementaryparticle physics, such as neutrino oscillations. A comparisonof vertical and horizontal neutrino spectra (Fig. 7.16) showsa similar tendency as for muon spectra.positrons, electrons,and photons fromelectromagnetic cascadespions and kaons at sea levelproduction of ν e and ν µFig. 7.16Energy spectra of muon andelectron neutrinos for vertical andhorizontal directionsSince the parent particles of neutrinos are dominantly pi-ons and kaons and their decay probability is increased comparedto the interaction probability at inclined directions, thehorizontal neutrino spectra are also harder in comparison tothe spectra from vertical directions. Altogether, muon neutrinoswould appear to dominate, since the (π → eν) and(K → eν) decays are strongly suppressed due to helicityconservation. Therefore, pions and kaons almost exclusivelyproduce muon neutrinos only. Only in muon decay equalnumbers of electron and muon neutrinos are produced. Athigh energies also semileptonic decays of charmed mesonsconstitute a source for neutrinos.Based on these ‘classical’ considerations the integralneutrino spectra yield a neutrino-flavour ratio ofneutrino parentsdominance of ν µ

7.3 Cosmic Rays Underground 151N(ν µ +¯ν µ )N(ν e +¯ν e ) ≈ 2 . (7.6)This ratio, however, is modified by propagation effects likeneutrino oscillations (see Sect. 6.2: Neutrino Astronomy).neutrino-flavour ratio7.3 Cosmic Rays Underground“If your experiment needs statistics,then you ought to have done a better experiment.”Ernest RutherfordParticle composition and energy spectra of secondary cosmicrays underground are of particular importance for neutrinoastronomy. Experiments in neutrino astronomy areusually set up at large depths underground to provide a sufficientshielding against the other particles from cosmic rays.Because of the rarity of neutrino events even low fluxes ofresidual cosmic rays constitute an annoying background. Inany case it is necessary to know precisely the identity andflux of secondary cosmic rays underground to be able todistinguish a possible signal from cosmic-ray sources fromstatistical fluctuations or systematical uncertainties of the atmosphericcosmic-ray background.Long-range atmospheric muons, secondary particles locallyproduced by muons, and the interaction products createdby atmospheric neutrinos represent the important backgroundsources for neutrino astronomy.Muons suffer energy losses by ionization, direct electron–positronpair production, bremsstrahlung, and nuclearinteractions. These processes have been described in ratherdetail in Chap. 4. While the ionization energy loss at highenergies is essentially constant, the cross sections for theother energy-loss processes increase linearly with the energyof the muon,− dE = a + bE. (7.7)dxThe energy loss of muons as a function of their energy isshown in Fig. 7.17 for iron as absorber material. The energyloss of muons in rock in its dependence on the muon energywas already shown earlier (Fig. 4.3).Equation (7.7) allows to work out the range R of muonsby integration,particle compositionundergroundbackground sourcesfor neutrino astrophysicsenergy loss of muonsrange of muons

152 7 Secondary Cosmic RaysFig. 7.17Contributions to the energy loss ofmuons in iron∫ 0dER =E −dE/dx = 1 b ln(1 + b E) , (7.8)aif it is assumed that the parameters a and b are energy independent.For not too large energies (E E) = AE −γ . (7.11)Using the energy–range relation (7.8), the depth–intensityrelation is obtained,

7.3 Cosmic Rays Underground 153[ a −γN(> E,R) = Ab (ebR − 1)]. (7.12)For high energies (E µ > 1TeV, bE ≫ a) the exponentialdominates and one obtains( a) −γN(> E,R) = A e −γbR . (7.13)bFor inclined directions the absorbing ground layer increaseslike 1/ cos θ = sec θ (θ – zenith angle) for a flat overburden,so that for muons from inclined directions one obtainsa depth–intensity relation ofinclined muon directions( a) −γN(>E,R,θ)= A e−γbRsec θ . (7.14)bFor shallower depths (7.12), or also (7.9), however, leads toapowerlawN(> E,R) = A(aR) −γ . (7.15)The measured depth–intensity relation for vertical directionsis plotted in Fig. 7.19. From depths of 10 km water equivalent(≈ 4000 m rock) onwards muons induced by atmosphericneutrinos dominate the muon rate. Because of thelow interaction probability of neutrinos the neutrino-inducedmuon rate does not depend on the depth. At large depths(> 10 km w.e.) a neutrino telescope with a collection areaof 100 × 100 m 2 and a solid angle of π would still measurea background rate of 10 events per day.The zenith-angle distributions of atmospheric muonsfor depths of 1500 and 7000 meter water equivalent areshown in Fig. 7.20. For large zenith angles the flux decreasessteeply, because the thickness of the overburden increasesFig. 7.19Depth–intensity relation for muonsfrom vertical directions. Thegrey-hatched band at large depthsrepresents the flux ofneutrino-induced muons withenergies above 2 GeV (upper line:horizontal, lower line: verticalupward neutrino-inducedmuons) [2]Fig. 7.20Zenith-angle distribution ofatmospheric muons at depths of1500 and 7000 m w.e.

154 7 Secondary Cosmic RaysFig. 7.21Variation of the exponent n of thezenith-angle distribution of muonswith depthFig. 7.22Ratio of stopping to penetratingmuons as a function of depth incomparison to some experimentalresults. (1) Stopping atmosphericmuons, (2) stopping muons fromnuclear interactions, (3) stoppingmuons locally produced byphotons, (4) neutrino-inducedstopping muons, and (5) sum ofall contributions(Fig. 7.21). For large depths the exponent n in this distribution,however, gets very large, so that it is preferable to use(7.14) instead.The average energy of muons at sea level is in the rangeof several GeV. Absorption processes in rock reduce predominantlythe intensity at low energies. Therefore, the averagemuon energy of the muon spectrum increases withincreasing depth. Muons of high energy can also produceother secondary particles in local interactions. Since lowenergymuons can be identified by their (µ → eνν) decaywith the characteristic decay time in the microsecond range,the measurement of stopping muons underground providesan information about local production processes. The flux ofstopping muons is normally determined for a detector thicknessof 100 g/cm 2 and the ratio P of stopping to penetratingmuons is presented (Fig. 7.22).A certain fraction of stopping muons is produced locallyby low-energy pions which decay relatively fast into muons.Since the flux of penetrating muons decreases strongly withincreasing depth, the ratio P of stopping to penetratingmuons is dominated by neutrino interactions for depthslarger than 5000 m w.e.The knowledge of the particle composition at largedepths below ground represents an important informationfor neutrino astrophysics.Also remnants of extensive air showers, which developedin the atmosphere, are measured underground. Electrons,positrons, photons, and hadrons are completely absorbedalready in relatively shallow layers of rock. Therefore,only muons and neutrinos of extensive air showerspenetrate to larger depths. The primary interaction vertexof particles which initiate the air showers is typically atan atmospheric altitude of 15 km. Since secondary particlesin hadronic cascades have transverse momenta of about300 MeV/c only, the high-energy muons essentially follateralspread of muonslike 1/ cos θ. Therefore, at large depths and from inclineddirections neutrino-induced muons dominate.For not too large zenith angles and depths the zenithangledependence of the integral muon spectrum can still berepresented byI(θ) = I(θ = 0) cos n θ (7.16)underground low the shower axis. For primaries of energy around 10 14eV lateral displacements of energetic muons (≈ 1TeV)

7.3 Cosmic Rays Underground 155Fig. 7.23Muon shower in the ALEPHexperiment. Muon tracks are seenin the central time-projectionchamber and in the surroundinghadron calorimeter. Even thoughthere is a strong 1.5 Teslamagnetic field perpendicular to theprojection shown, the muon tracksare almost straight indicating theirhigh momenta. Only a knock-onelectron produced in thetime-projection chamber by amuon is bent on a circle {23}at shallow depths underground of typically several metersexclusively caused by transferred transverse momenta areobtained. Typical multiple-scattering angles for energeticmuons (≈ 100 GeV) in thick layers of rock (50–100 m) areon the order of a few mrad.The multiplicity of produced secondary particles increaseswith energy of the initiating particle (for a 1 TeVproton the charged multiplicity of particles for proton–protoninteractions is about 15). Since the secondaries producedin these interactions decay predominantly into muons, oneobserves bundles of nearly parallel muons underground inthe cores of extensive air showers. Figure 7.23 shows sucha shower with more than 50 parallel muons observed by theALEPH experiment at a depth of 320 m w.e.High-energy muons are produced by high-energy primariesand, in particular, muon showers correlate with evenhigher primary energies. Therefore, one is tempted to localizeextraterrestrial sources of high-energy cosmic rays viathe arrival directions of single or multiple muons. SinceCygnus X3 has been claimed to emit photons with energiesup to 10 16 eV, this astrophysical source also representsan excellent candidate for the acceleration of high-energycharged primary cosmic rays. Cygnus X3 at a distance of approximately33 000 light-years is an X-ray binary consistingof a superdense pulsar and a stellar companion. The materialflowing from the companion into the direction of the pulsarforms an accretion disk around the pulsar. If apparentlyphotons of very high energy can be produced, one wouldexpect them to originate from the π 0 decay (π 0 → γγ).muon bundlesALEPHas cosmic-ray detectorCygnus X3

156 7 Secondary Cosmic RaysCygnus X3,a hadron accelerator?muon astronomyNeutral pions are usually produced in proton interactions.Therefore, the source should also be able to produce chargedpions and via their decay muons and muon neutrinos. Becauseof their short lifetime, muons would never survive the33 000 light-year distance from Cygnus X3 to Earth, so thata possible muon signal must be caused by neutrino-inducedmuons. Unfortunately, muons and multi-muons observed inthe Frejus experiment from the directions of Cygnus X3are predominantly of atmospheric origin and do not confirmthat Cygnus X3 is a strong source of high-energy particles(Fig. 7.24). The primary particles themselves acceleratedin the source could in principle point back to the sourcewhen measured on Earth. However, the arrival direction ofprimary charged particles from Cygnus X3 could also havebeen completely randomized by the irregular galactic magneticfield. Muon production by neutrinos from Cygnus X3would have been a rare event which would have required anextremely massive detector to obtain a significant rate.46°FREJUS44°Fig. 7.24Sky map of muons andmulti-muons from the direction ofCygnus X3. The cross indicates theoptically known position ofCygnus X3. The circles aroundCygnus X3 with angles of ±2 ◦ and±5 ◦ correspond to a possiblefuzziness, caused by multiplescattering of muons in rock {24}declination 42°40°38°36°34°296°300° 304° 308° 312° 316° 320°right ascension 7.4 Extensive Air Showers“Science never solves a problem withoutcreating ten more.”George Bernard ShawExtensive air showers are cascades initiated by energetic primaryparticles which develop in the atmosphere. An exten-sive air shower (EAS) has an electromagnetic, a muonic, acomponentsof an extensive air shower

7.4 Extensive Air Showers 157hadronic, and a neutrino component (see Fig. 7.6). The airshower develops a shower nucleus consisting of energetichadrons, which permanently inject energy into the electromagneticand the other shower components via interactionsand decays. Neutral pions, which are produced in nuclear interactionsand whose decay photons produce electrons andpositrons via pair production, supply the electron, positron,and photon component. Photons, electrons, and positronsinitiate electromagnetic cascades through alternating processesof pair production and bremsstrahlung. The muon andneutrino components are formed by the decay of charged pionsand kaons (see also Fig. 7.6).The inelasticity in hadron interactions is on the order of50%, i.e., 50% of the primary energy is transferred into theproduction of secondary particles. Since predominantly pionsare produced (N(π) : N(K) = 9 : 1) and all chargestates of pions (π + ,π − ,π 0 ) are produced in equal amounts,one third of the inelasticity is invested into the formation ofthe electromagnetic component. Since most of the chargedhadrons and the hadrons produced in hadron interactionsalso undergo multiple interactions, the largest fraction of theprimary energy is eventually transferred into the electromagneticcascade. Therefore, in terms of the number of particles,electrons and positrons constitute the main shower component.The particle number increases with shower depth t untilabsorptive processes like ionization for charged particlesand Compton scattering and photoelectric effect for photonsstart to dominate and cause the shower to die out.The development of electromagnetic cascades is shownin Fig. 7.25 for various primary energies. The particle intensityincreases initially in a parabolical fashion and decaysexponentially after the maximum of the shower has beenreached. The longitudinal profile of the particle number canbe parameterized byhadron, electromagnetic,muon, neutrino componentinelasticitynumber of shower particles10 510 410 310 210110 –110 2 10 3 10 410 6 = E/E 0 C0 10 20 30 40 50shower depth t[ X 0]Fig. 7.25Longitudinal shower developmentof electromagnetic cascades. (Thecritical energy in air isE c = 84 MeV)N(t) ∼ t α e −βt , (7.17)where t = x/X 0 is the shower depth in units of the radiationlength and α and β are free fit parameters. The position ofthe shower maximum varies only logarithmically with theprimary energy, while the total number of shower particlesincreases linearly with the energy. The latter can thereforebe used for the energy determination of the primary particle.One can imagine that the Earth’s atmosphere representsa combined hadronic and electromagnetic calorimeter,longitudinal particle-numberprofile

158 7 Secondary Cosmic RaysFig. 7.26Average longitudinal developmentof the various components of anextensive air shower in theatmospherein which the extensive air shower develops. The atmosphereconstitutes approximately a target of 11 interaction lengthsand 27 radiation lengths. The minimum energy for a primaryparticle to be reasonably well measured at sea levelvia the particles produced in the air shower is about 10 14 eV= 100 TeV. As a rough estimate for the particle number Nat sea level in its dependence on the primary energy E 0 , onecan use the relationN = 10 −10 E 0 [eV] . (7.18)Only about 10% of the charged particles in an extensive airshower are muons. The number of muons reaches a plateaualready at an atmospheric depth of 200 g/cm 2 (see also Fig.7.9 and Fig. 7.10). Its number is hardly reduced to sea level,since the probability for catastrophic energy-loss processes,like bremsstrahlung, is low compared to electrons because ofthe large muon mass. Muons also lose only a small fractionof their energy by ionization. Because of the relativistic timedilation the decay of energetic muons (E µ > 3GeV)intheatmosphere is strongly suppressed.Figure 7.26 shows schematically the longitudinal de-velopment of the various components of an extensive airshower in the atmosphere for a primary energy of 10 15 eV.The lateral spread of an extensive air shower is essentiallycaused by the transferred transverse momenta in hadronicinteractions and by multiple scattering of low-energy showerparticles. The muon component is relatively flat compared tothe lateral distribution of electrons and hadrons. Figure 7.27shows the lateral particle profile for the various shower components.Neutrinos essentially follow the shape of the muoncomponent.Even though an extensive air shower initiated by primaryparticles with energies below 100 TeV does not reach sealevel, it can nevertheless be recorded via the Cherenkov lightemitted by the shower particles (see Sect. 6.3 on gamma-rayastronomy). At higher energies one has the choice of variousdetection techniques.The classical technique for the measurement of extensiveair showers is the sampling of shower particles at sealevel with typically 1 m 2 large scintillators or water Cherenkovcounters. This technique is sketched in Fig. 7.28.longitudinal profileof a showerlateral distributionFig. 7.27Average lateral distribution of theshower components for N = 10 5corresponding to E ≈ 10 15 eVAuger projectIn the Auger project in Argentina 3000 sampling detectorswill be used for the measurement of the sea-level componentof extensive air showers. However, the energy assignmentfor the primary particle using this technique is not very

7.4 Extensive Air Showers 159precise. The shower develops in the atmosphere which actsas a calorimeter of 27 radiation lengths thickness. The informationon this shower is sampled in only one, the last layerof this calorimeter and the coverage of this layer is typicallyon the order of only 1%. The direction of incidence of theprimary particle can be obtained from the arrival times ofshower particles in the different sampling counters.energy measurementprimary particlethickness ~1 mdetectorsshower axiszenith angleFig. 7.28Air-shower measurement withsampling detectorsIt would be much more advantageous to measure thetotal longitudinal development of the cascade in the atmosphere.This can be achieved using the technique of the Fly’sEye (Fig. 7.29). Apart from the directional Cherenkov radiationthe shower particles also emit an isotropic scintillationlight in the atmosphere.For particles with energies exceeding 10 17 eV the fluorescencelight of nitrogen is sufficiently intense to berecorded at sea level in the presence of the diffuse backgroundof starlight. The actual detector consists of a systemof mirrors and photomultipliers, which view the whole sky.An air shower passing through the atmosphere near sucha Fly’s Eye detector activates only those photomultiplierswhose field of view is hit. The fired photomultipliers allowto reconstruct the longitudinal profile of the air shower.The total recorded light intensity is used to determine theshower energy. Such a type of detector allows much moreFly’s Eyefluorescence technique

160 7 Secondary Cosmic Raysatmosphericair showerprimaryinteractionisotropicscintillation lightFig. 7.29Principle of the measurement ofthe scintillation light of extensiveair showersFly´s EyeAir Watchprecise energy assignments, however, it has a big disadvantagecompared to the classical air-shower technique thatit can only be operated in clear moonless nights. Figure7.30 shows an arrangement of mirrors and photomultipliers,as they have been used in the original Fly’s Eye setup of theUtah group. In the Auger experiment the array of samplingdetectors is complemented by such a number of telescopeswhich measure the scintillation light produced in the atmosphere.Much larger acceptances could be provided if such aFly’s Eye detector would be installed in orbit (‘Air Watch’,Fig. 7.31).Fig. 7.30Arrangement of mirrors andphotomultipliers in the originalFly’s Eye experiment of the Utahgroup {25}radio detection of showersApart from these detection techniques it has also beentried to observe air showers via the electromagnetic radiationemitted in the radio band. It is generally believed thatthis radio signal is caused by shower electrons deflected in

7.4 Extensive Air Showers 161primary protonphotomultiplierin the focal point of a mirrorFig. 7.31Measurement of the isotropicscintillation light of extensive airshowers by Fly’s Eye detectors onboard of satellites (‘Air Watch’)the Earth’s magnetic field thereby creating synchrotron radiation.Because of the strong background in practically allwavelength ranges these attempts have not been particularlysuccessful so far. The possibility to detect large air showersvia their muon content in underground experiments has beenfollowed up in recent experiments.Apart from elementary particle physics aspects the purposeof the measurement of extensive air showers is the determinationof the chemical composition of primary cosmicrays and the search for the sites of cosmic accelerators.The arrival directions of the highest-energy particles(> 10 19 eV) which for intensity reasons can only berecorded via air-shower techniques, practically show nocorrelation to the galactic plane. This clearly indicates thattheir origin must be extragalactic. If the highest-energy primarycosmic-ray particles are protons, then their energiesmust be below 10 20 eV, if they originate from distances ofmore than 50 Mpc. Even if their original energy were muchhigher, they would lose energy by photoproduction of pionson photons of the blackbody radiation until they fallbelow the threshold of the Greisen–Zatsepin–Kuzmin cutoffmuon showers undergroundproton horizon

162 7 Secondary Cosmic Raysextensive air showersand γ -ray burstsenergies > 10 20 TeVparticle astronomy?coincidencesover large distances(≈ 6 × 10 19 eV). Protons of this energy would point backto the sources, because galactic and intergalactic magneticfields only cause angular distortions on the order of onedegree at these high energies. The irregularities of magneticfields, however, could lead to significant time delays betweenneutrinos and photons on one hand and protons, onthe other hand, from such distant sources. This comes aboutbecause the proton trajectories are somewhat longer, eventhough their magnetic deflection is rather small. Dependingon the distance from the source, time delays of months andeven years can occur. This effect is of particular importance,if γ -ray bursters are also able to accelerate the highestenergyparticles and if one wants to correlate the arrivaltimes of photons from γ -ray bursts with those of extensiveair showers initiated by charged primaries.The few measured particles with energies in excess of10 20 eV show a non-uniform distribution with a certain clusteringnear the local supergalactic plane. The fact that theattenuation length of protons with energies > 10 20 eV inthe intergalactic space is approximately 10 Mpc would makean origin in the local supercluster (maximum size 30 Mpc)plausible.Out of the six measured showers with primary energyexceeding 10 20 eV the directions of origin for two eventsare identical within the measurement accuracy. This directioncoincides with the position of a radio galaxy (3C134),whose distance unfortunately is unknown, since it lies in thedirection of the galactic plane, where optical measurementsof extragalactic objects are difficult because of interstellarabsorption. The coincidence of the radio galaxy 3C134 withthe arrival directions of the two highest-energy particles can,of course, also be an accident.Normal extensive air showers have lateral widths of atmost 10 km, even at the highest energies. However, thereare indications that correlations between arrival times of airshowers over distances of more than 100 km exist. Such coincidencescould be understood by assuming that energeticprimary cosmic particles undergo interactions or fragmentationsat large distances from Earth. The secondary particlesproduced in these interactions would initiate separateair showers in the atmosphere (Fig. 7.32).Even moderate distances of only one parsec (3×10 16 m)are sufficient to produce separations of air showers at Earthon the order of 100 km (primary energy 10 20 eV, transverse

7.5 Nature and Origin of the Highest-Energy Cosmic Rays 163primordialparticlecorrelated air showersat Earthinteractionin the galaxyFig. 7.32Possible explanation forcorrelations between distantextensive air showersmomenta ≈ 0.3GeV/c). Variations in arrival times of theseshowers could be explained by unequal energies of the fragmentswhich could cause different propagation times. Galacticor extragalactic magnetic fields could also affect the trajectoriesof the fragments in a different way thus also influencingthe arrival times.correlated showers7.5 Nature and Origin of the Highest-EnergyCosmic Rays“The universe is full of magical thingspatiently waiting for our wits to growsharper.”Eden PhillpottsAs already explained in Sect. 7.4, the highest-energy particlesof cosmic rays appear to be of extragalactic origin.The problem of the sources of these particles is closely relatedto the identity of these particles. Up to the present timeone had always assumed that the chemical composition ofprimary cosmic rays might change with energy. However,one always anticipated that the highest-energy particles wereeither protons, light, or possibly medium heavy nuclei (upto iron). For particles with energies exceeding 10 20 eV thisproblem is completely open. In the following the candidateswhich might be responsible for cosmic-ray events with energies> 10 20 eV will be critically reviewed.Up to now only a handful of events with energies exceeding10 20 eV have been observed. Due to the measurementtechnique via extensive-air-shower experiments theenergy assignments are connected with an experimental errorof typically ±30%. For the accelerated parent particlesidentityof high-energy primaries?events with E>10 20 eV

164 7 Secondary Cosmic Raysof these high-energy particles the gyroradii must be smallerthan the size of the source. Therefore, one can derive fromgalactic containmentmv 2R≤ evBa maximum value for the energy of a particle that can beaccelerated in the source,E max ≈ p max ≤ eBR (7.19)(v is a particle velocity, B is a magnetic field strength of thesource, R is the size of the source, m is the relativistic massof the particle). In units appropriate for astroparticle physicsthe maximum energy, which can be obtained by accelerationin the source, can be expressed in the following way:E max = 10 5 B RTeV3 × 10 −6 G 50 pc . (7.20)With a typical value of B = 3 µG for our Milky Way andthe very generous gyroradius of R = 5 kpc one obtainsE max = 10 7 TeV = 10 19 eV . (7.21)protonsThis equation implies that our Milky Way can hardly accelerateor store particles of these energies, so that for particleswith energies exceeding 10 20 eV one has to assume that theyare of extragalactic origin.For protons the Greisen–Zatsepin–Kuzmin cutoff (GZK)of photoproduction of pions off blackbody photons throughthe ∆ resonance takes an important influence on the propagation,γ + p → p + π 0 . (7.22)mean free path of protonsThe energy threshold for this process is at 6 × 10 19 eV (seeSect. 6.1). Protons exceeding this energy lose rapidly theirenergy by such photoproduction processes. The mean freepath for photoproduction is calculated to beλ γp = 1Nσ , (7.23)where N is the number density of blackbody photons andσ(γp → π 0 p) ≈ 100 µb the cross section at threshold.This leads toλ γp ≈ 10 Mpc . (7.24)

7.5 Nature and Origin of the Highest-Energy Cosmic Rays 165The Markarian galaxies Mrk 421 and Mrk 501, whichhave been shown to be sources of photons of the highestenergies, would be candidates for the production of highenergyprotons. Since they are residing at distances of approximately100 Mpc, the arrival probability of protons fromthese distances with energies exceeding 10 20 eV, however, isonlyMarkarian galaxiesas cosmic-ray source?≈ e −x/λ ≈ 4 × 10 −5 . (7.25)Therefore protons can initiate the high-energy air-showerevents only if they come from relatively nearby sources (i.e.,from a local GZK sphere defined by distances < 30 Mpc,i.e., several mean free paths). The giant elliptical galaxyM87 lying in the heart of the Virgo cluster (distance ≈ 20Mpc) is one of the most remarkable objects in the sky. Itmeets all of the conditions for being an excellent candidatefor a high-energy cosmic-ray source.It is, however, possible to shift the effect of the Greisen–Zatsepin–Kuzmin cutoff to higher energies by assuming thatprimary particles are nuclei. Since the threshold energy mustbe available per nucleon, the corresponding threshold energy,for example, for carbon nuclei (Z = 6, A = 12) wouldbe correspondingly higher,M87 as particle accelerator?GZK sphereE = E p cutoff A = 7.2 × 1020 eV , (7.26)so that the observed events would not be in conflict with theGreisen–Zatsepin–Kuzmin cutoff. It is, however, difficult tounderstand, how atomic nuclei can be accelerated to suchhigh energies, without being disintegrated by photon interactionsor by fragmentation or spallation processes.One remote and rather drastic assumption to explain thetrans-GZK events would be a possible violation of Lorentzinvariance. If Lorentz transformations would not only dependon the relative velocity difference of inertial frames,but also on the absolute velocities, the threshold energy forγp collisions for interactions of blackbody photons withhigh-energy protons would be washed out and different fromγp collisions when photon and proton had comparable energies,thus evading the GZK cutoff.Photons as possible candidates for the observed highenergycascades are even more problematic. Because of theprocess of pair production of electrons and positrons offblackbody photons (see Sect. 6.3.3), photons have a relativelyshort mean free path ofheavy nucleiphoto disintegrationphotonsmean free path of photons

166 7 Secondary Cosmic Raysλ γγ ≈ 10 kpc . (7.27)photonic origin?The γ -ray sources have to be relatively near to explain thehigh-energy showers. This would mean that they must beof galactic origin, which appears rather unlikely, becauseof the limited possibility for their parent particles to be acceleratedin our Milky Way up to the highest energies required.High-energy photons, furthermore, would initiateair showers at high altitudes above sea level (≈ 3000 km)due to interactions with the Earth’s magnetic field. Thereforeone would theoretically expect that they would reacha shower maximum at ≈ 1075 g/cm 2 (calculated from sealevel). The event observed by the Fly’s Eye experiment hasa shower maximum at (815 ± 40) g/cm 2 , which is typicalfor a hadron-induced cascade. Photons as candidates for thehighest-energy events can therefore be firmly excluded.Recently, neutrinos were discussed as possible candi-dates for the high-energy events. But neutrinos also encountersevere problems in explaining such events. The ratioof the interaction cross section for neutrino–air and proton–air interactions at 10 20 eV isσ(ν–air)σ(p–air) ∣ ≈ 10 −6 . (7.28)E≈10 20 eVneutrinosQuite enormous neutrino fluxes are required to explain theevents with energies > 10 20 eV. It has been argued that themeasurements of the structure function of the protons atHERA 1 have shown that protons have a rich structure of partonsat low x(x= E parton /E proton ).Eveninviewoftheseresults showing evidence for a large number of gluons in theproton, one believes that the neutrino interaction cross sec-tion with nuclei of air cannot exceed 0.3 µb. This makes interactionsof extragalactic neutrinos in the atmosphere veryimprobable, compare (3.56):rising neutrino cross section?φ = σ(ν–air) N AA d≤ 0.3 µb 6 × 1023g −1 × 1000 g/cm 214≈ 1.3 × 10 −5 (7.29)(N A is the Avogadro number, d is a column density of theatmosphere).1 HERA – Hadron Elektron Ring Anlage at the Deutsches Elektronensynchrotron(DESY) in Hamburg

7.5 Nature and Origin of the Highest-Energy Cosmic Rays 167To obtain a reasonable interaction rate only neutrino interactionsfor inclined directions of incidence or in the Earthcan be considered. The resulting expected distribution of primaryvertices due to neutrino interactions is in contrast toobservation. Therefore, neutrinos as well can very likely beexcluded as candidates for the highest-energy cosmic airshowerevents.It has been demonstrated that a large fraction of matteris in the form of dark matter. A possible way out concerningthe question of high-energy particles in cosmic rayswould be to assume that weakly interacting massive particles(WIMPs) could also be responsible for the observedshowers with energies > 10 20 eV. It has to be consideredthat all these particles have only weak or even superweakinteractions so that their interaction rate can only be on theorder of magnitude of neutrino interactions.The events with energies exceeding 10 20 eV thereforerepresent a particle physics dilemma. One tends to assumethat protons are the favoured candidates. They must comefrom relative nearby distances (< 30 Mpc), because otherwisethey would lose energy by photoproduction processesand fall below the energy of 6 × 10 19 eV. It is, however,true that up to these distances there are quite a number ofgalaxies (e.g., M87). The fact that the observed events do notclearly point back to a nearby source can be explained by thefact that the extragalactic magnetic fields are so strong thatthe directional information can be lost, even if the protonsare coming from comparably close distances. Actually, thereare hints showing that these fields are more in the µGaussrather than in the nGauss region [6].Recent measurements, however, appear to indicate thatthe GZK cutoff might have been seen at least in the dataof the HiRes experiment (see Fig. 6.5). On the other hand,this finding is in conflict with results from the large AGASAair-shower array (see also the comment on page 83).Presently one assumes that in supernova explosions particlescan only be accelerated to energies of 10 15 eV byshock-wave mechanisms. At these energies the primary spectrumgets steeper (‘knee of the primary spectrum’). As alreadyshown, our Milky Way is too small to accelerate andstore particles with energies exceeding 10 20 eV. Furthermore,the arrival directions of the high-energy particles showpractically no correlation to the galactic plane. Therefore,one has to assume that they are of extragalactic origin.vertex distributionfor neutrinosWIMPsextragalactic magnetic fieldsacceleration mechanisms

168 7 Secondary Cosmic Raysactive galactic nucleiblazarsBL-Lacertae objects, quasarsenergy conversion efficiencyparticle jets from blazarssupergalactic origin?Active galactic nuclei (AGN’s) are frequently discussedas possible sources for the highest cosmic-ray energies. Inthis group of galaxies blazars play an outstanding rôle.Blazar is a short for sources belonging to the class of BL-Lacertae objects and quasars. BL-Lacertae objects, equallyas quasars, are Milky Way-like sources, whose nuclei outshinethe whole galaxy making them to appear like stars.While the optical spectra of quasars exhibit emission andabsorption lines, the spectra of BL-Lacertae objects showno structures at all. This is interpreted in such a way thatthe galactic nuclei of quasars are surrounded by dense gas,while BL-Lacertae objects reside in low-gas-density ellipticalgalaxies.A characteristic feature of blazars is their high variability.Considerable brightness excursions have been observedon time scales as short as a few days. Therefore, these objectsmust be extremely compact, because the size of thesources can hardly be larger than the time required for lightto travel across the diameter of the source. It is generally assumedthat blazars are powered by black holes at their center.The matter falling into a black hole liberates enormousamounts of energy. While in nuclear fission only 1‰ andin nuclear fusion still only 0.7% of the mass is transformedinto energy, an object of mass m can practically liberate allits rest energy mc 2 if it is swallowed by a black hole.Many high-energy γ -ray sources which were found bythe CGRO (Compton Gamma Ray Observatory) satellite,could be correlated with blazars. This led to the conjecturethat these blazars could also be responsible for the accelerationof the highest-energy particles. The particle jetsproduced by blazars exhibit magnetic fields of more than10 Gauss and extend over 10 −2 pc and more. Therefore,according to (7.20), particles could be accelerated to energiesexceeding 10 20 eV. If protons are accelerated in suchsources they could easily escape from these galaxies, becausetheir interaction strength is smaller than that of theelectrons which must certainly be accelerated as well. Ifthese arguments are correct, blazars should also be a richsource of high-energy neutrinos. This prediction can betested with the large water (or ice) Cherenkov counters.It has already been mentioned before that for protonsto arrive at Earth the sources must not be at too large distances.The best candidates for sources should therefore liein the supergalactic plane. The local supergalaxy is a kind of

7.6 Problems 169‘Milky Way’ of galaxies whose center lies in the directionof the Virgo cluster. The local group of galaxies, of whichour Milky Way is a member, has a distance of about 20 Mpcfrom the center of this local supergalaxy and the members ofthis supergalaxy scatter around the supergalactic center onlyby about 20 Mpc.Even though the origin of the highest-energy cosmicrays is still unknown, there are some hints that the sourcesfor these high-energy events really lie in the supergalacticplane. Certainly more events are required to confirm in detailthat such a correlation really exists. The Auger experimentunder construction in Argentina should be able to solve thequestion of the origin of high-energy cosmic rays.Finally, ideas have also been put forward that the extreme-energycosmic rays are not the result of the accelerationof protons or nuclei but rather decay products of unstableprimordial objects. Candidates discussed as possiblesources are decays of massive GUT particles spread throughthe galactic halo, topological defects produced in the earlystages of the universe like domain walls, ‘necklaces’ of magneticmonopoles connected by cosmic strings, closed cosmicloops containing a superconducting circulating current,or cryptons – relic massive metastable particles born duringcosmic inflation.Virgo clusterexotic candidates7.6 Problems1. The pressure at sea level is 1013 hPa. Convert this pressureinto a column density in kg/cm 2 !2. The barometric pressure varies with altitude h in the atmosphere(assumed to be isothermal) likep = p 0 e −h/7.99 km .What is the residual pressure at 20 km altitude and whatcolumn density of residual gas does this correspond to?3. For not too large zenith angles the angular distributionof cosmic-ray muons at sea level can be parameterizedas I(θ) = I(0) cos 2 θ. Motivate the cos 2 θ dependence!4. Figure 7.22 shows the rate of stopping muons underground.Work out the rate of stopping atmosphericmuons (curve labeled 1) as a function of depth undergroundfor shallow depths!

170 7 Secondary Cosmic Rays5. Figure 7.23 shows a muon shower in the ALEPH experiment.Typical energies of muons in this shower are100 GeV. What is the r.m.s. scattering angle of muons inrock for such muons (overburden 320 m w.e., radiationlength in rock X 0 = 25 g/cm 2 ̂= 10 cm)?6. Narrow muon bundles with muons of typically 100 GeVoriginate in interactions of primary cosmic rays in theatmosphere. Estimate the typical lateral separation ofcosmic-ray muons in a bundle at a depth of 320 m w.e.underground.7. Due to the dipole character of the Earth’s magnetic fieldthe geomagnetic cutoff varies with geomagnetic latitude.The minimum energy for cosmic rays to penetrate theEarth’s magnetic field and to reach sea level can beworked out to beE min = ZeM4R 2 cos4 λ,where Z is the charge number of the incident particle,M is the moment of the Earth’s magnetic dipole, R isthe Earth radius, λ is the geomagnetic latitude (0 ◦ at theequator). For protons one gets E min = 15 GeV cos 4 λ.The Earth’s magnetic field has reversed several timesover the history of our planet. In those periods when thedipole changed polarity, the magnetic field went throughzero. In these times when the magnetic shield decayed,more cosmic-ray particles could reach the surface of theEarth causing a higher level of radiation for life developingon our planet. Whether this had a positive effecton the biological evolution or not is the object of muchdebate. The estimation of the increased radiation levelfor periods of zero field can proceed along the followinglines:• the differential energy spectrum of primary cosmicrays can be represented by a power law N(E) ∼ E −γwith γ = 2.7.• in addition to the geomagnetic cutoff there is also anatmospheric cutoff due to the energy loss of chargedparticles in the atmosphere of ≈ 2GeV.Work out the increase in the radiation level using theabove limits!8. Neutrons as candidates for the highest-energy cosmicrays have not been discussed so far. What are the problemswith neutrons?

1718 Cosmology“As far as the laws of mathematics refer toreality, they are not certain; as far as theyare certain, they do not refer to reality.”Albert EinsteinIn the following chapters the application of our knowledgeof particle physics to the very early universe in the context ofthe Hot Big Bang model of cosmology will be explored. Thebasic picture is that the universe emerged from an extremelyhot, dense phase about 14 billion years ago. The earliest timeabout which one can meaningfully speculate is about 10 −43seconds after the Big Bang (the Planck time). To go earlierrequires a quantum-mechanical theory of gravity, and this isnot yet available.At early times the particle densities and typical energieswere extremely high, and particles of all types were continuallybeing created and destroyed. For the first 10 −38 secondsor so, it appears that all of the particle interactions couldhave been ‘unified’ in a theory containing only a single couplingstrength. It was not until after this, when typical particleenergies dropped below around 10 16 GeV, that the strongand electroweak interactions became distinct. At this time,from perhaps 10 −38 to 10 −36 seconds after the Big Bang,the universe may have undergone a period of inflation, atremendous expansion where the distances between any twoelements of the primordial plasma increased by a factor ofperhaps e 100 . When the temperature of the universe droppedbelow 100 GeV, the electroweak unification broke apart intoseparate electromagnetic and weak interactions.Until around 1 microsecond after the Big Bang, quarksand gluons could exist as essentially free particles. After thispoint, energies dropped below around 1 GeV and the partonsbecame bound into hadrons, namely, protons, neutrons,and their antiparticles. Had the universe contained at thispoint equal amounts of matter and antimatter, almost all ofit would have annihilated, leaving us with photons, neutrinos,and little else. For whatever reason, nature apparentlymade one a bit more abundant than the other, so there wassome matter left over after the annihilation phase to makeHot Big Bangquantum gravityinflationannihilationof matter and antimatterannihilation phase

172 8 CosmologyBig Bang Nucleosynthesiscosmic microwavebackground radiationcritical density‘Big Crunch’vacuum energy densitydark matterneutralinosthe universe as it is now. Essentially all of the positrons hadannihilated with electrons within the first couple of seconds.Around three minutes after the Big Bang, the temperaturehad dropped to the point where protons and neutronscould fuse to form deuterons. In the course of the next fewminutes these combined to form helium, which makes up aquarter of the universe’s nuclear matter by mass, and smallerquantities of a few light elements such as deuterium, lithium,and beryllium. The model of Big Bang Nucleosynthesis(BBN) is able to correctly predict the relative abundancesof these light nuclei and this is one of the cornerstones ofthe Hot Big Bang model.As the universe continued to expand over the next severalhundred thousand years, the temperature finally droppedto the point where electrons and protons could join to formneutral atoms. After this the universe became essentiallytransparent to photons, and those which existed at that timehave been drifting along unimpeded ever since. They can bedetected today as the cosmic microwave background radia-tion. Only small variations in the temperature of the radiation,depending on the direction, at a level of one part in 10 5are observed. These are thought to be related to small densityvariations in the universe left from a much earlier period,perhaps as early as the inflationary epoch only 10 −36seconds after the Big Bang.Studies of the cosmic microwave background radiation(CMB) also lead to a determination of the total density ofthe universe, and one finds a value very close to the so-calledcritical density, above which the universe should recollapsein a ‘Big Crunch’. The same CMB data and also observationsof distant supernovae, however, show that about 70%of this is not what one would call matter at all, but rather asort of energy density associated with empty space – a vacuumenergy density.The remaining 30% appears to be gravitating matter, butof what sort? One of the indirect consequences of Big BangNucleosynthesis is that only a small fraction of the matter inthe universe appears to be composed of known particles. Theremainder of the dark matter may consist of neutralinos,particles predicted by a theory called supersymmetry.The framework in which the early universe will be studiedis based on the ‘Standard Cosmological Model’ or ‘HotBig Bang’. The basic ingredients are Einstein’s theory ofgeneral relativity and the hypothesis that the universe is

8.1 The Hubble Expansion 173isotropic and homogeneous when viewed over sufficientlylarge distances. It is in the context of this model that the lawsof particle physics will be applied in an attempt to trace theevolution of the universe at very early times. In this chapterthe important aspects of cosmology that one needs will bereviewed.Standard CosmologicalModel8.1 The Hubble Expansion“The history of astronomy is a historyof receding horizons.”Edwin Powell HubbleThe first important observation that leads to the StandardCosmological Model is Hubble’s discovery that all but thenearest galaxies are receding away from us (i.e., from theMilky Way) with a speed proportional to their distance. Thespeeds are determined from the Doppler shift of spectrallines. Suppose a galaxy receding from us (i.e., from theMilky Way) with a speed v = βc emits a photon of wavelengthλ em . When the photon is observed, its wavelengthwill be shifted to λ obs . To quantify this, the redshift z is definedasredshiftz = λ obs − λ em. (8.1)λ emFrom relativity one obtains the relation between the redshiftand the speed,√1 + βz =1 − β − 1 , (8.2)which can be approximated byz ≈ β (8.3)for β ≪ 1.To measure the distance of a galaxy one needs in ita light source of a calibrated brightness (a ‘standard candle’).The light flux from the source falls off inversely as thesquare of the distance r, so if the absolute luminosity L isknown and the source radiates isotropically, then the measuredlight flux is F = L/4πr 2 .Theluminosity distancecan therefore be determined from‘standard candle’luminosity distance

174 8 Cosmologytype-Ia supernovae√Lr =4πF . (8.4)Various standard candles can be used, such as Cepheid variablestars, used by Hubble. A plot of speed versus distancedetermined from type-Ia supernovae 1 is shown in Fig. 8.1[7].4000030000Velocity (km/sec)2000010000Fig. 8.1The speed versus distance for asample of type-Ia supernovae(from [7])00 100 200 300 400 500 600 700Distance (Mpc)1 The spectra of SN Ia are hydrogen poor. The absence of planetarynebulae allows to reconstruct the genesis of these events. Itis generally believed that the progenitor of a SN Ia is a binaryconsisting of a white dwarf and a red giant companion. Bothmembers are gravitationally bound. In white dwarves the electrondegeneracy pressure compensates the inward bound gravitationalpressure. The strong gravitational potential of the whitedwarf overcomes the weaker gravity of the red giant. At theperiphery of the red giant the gravitational force of the whitedwarf is stronger than that of the red giant causing mass fromits outer envelope to be accreted onto the white dwarf. Since forwhite dwarves the product of mass times volume is constant,it decreases in size during accretion. When the white dwarfreaches the Chandrasekhar limit (1.44 M ⊙ ), the electron degeneracypressure can no longer withstand the gravitational pressure.It will collapse under its own weight. This goes along withan increase in temperature causing hydrogen to fuse to heliumand heavier elements. This sudden burst of energy leads to athermonuclear explosion which destroys the star.Since the Chandrasekhar limit is a universal quantity, all SNIa explode in the same way. Therefore, they can be consideredas standard candles.

8.2 The Isotropic and Homogeneous Universe 175For the distances covered in the plot, the data are clearlyin good agreement with a linear relation,v = H 0 r, (8.5)which is Hubble’s Law. The parameter H 0 is Hubble’s constant,which from the data in Fig. 8.1 is determined to be64 (km/s)/Mpc. Here the subscript 0 is used to indicate thevalue of the parameter today. The relation between speedand distance is not, however, constant in time.Determinations of the Hubble constant based on differentobservations have yielded inconsistent results, althoughless so now than a decade ago. A compilation of data from2004 concludes a value ofHubble’s constantH 0 = 71 ± 5 (km/s)/Mpc , (8.6)with further systematic uncertainties of 5 to 10% [2]. In addition,one usually defines the parameter h byh parameterH 0 = h × 100 (km/s)/Mpc . (8.7)Quantities that depend on H 0 are then written with the correspondingdependence on h. To obtain a numerical value onesubstitutes for h the most accurate estimate available at thetime (h = 0.71 +0.04−0.03in 2004). For purposes of this courseone can use h ≈ 0.7 ± 0.1.8.2 The Isotropic and Homogeneous Universe“The center of the universe is everywhere,and the circumference isnowhere.”Giordano BrunoThe assumption of an isotropic and homogeneous universe,sometimes called the cosmological principle, was initiallymade by Einstein and others because it simplified the mathematicsof general relativity. Today there is ample observationalevidence in favour of this hypothesis. The cosmic microwavebackground radiation, for example, is found to beisotropic to a level of around one part in 10 5 .The isotropy and homogeneity appear today to hold atsufficiently large distances, say, greater than around 100megaparsecs. At smaller distances, galaxies are found tocosmological principleisotropy and homogeneity

176 8 CosmologyFig. 8.2Two galaxies at distances r(t) andR(t) from our owndistance scaleHubble parameterexpansion rateclump together forming clusters and voids. A typical intergalacticdistance is on the order of 1 Mpc, so a cube withsides of 100 Mpc could have a million galaxies. Therefore,one should think of the galaxies as the ‘molecules’ of a gas,which is isotropic and homogeneous when a volume largeenough to contain large numbers of them is considered.If the universe is assumed to be isotropic and homogeneous,then the only possible motion is an overall expansionor contraction. Consider, for example, two randomly chosengalaxies at distances r(t) and R(t) from ours, as shown inFig. 8.2.An isotropic and homogeneous expansion (or contraction)means that the ratioχ = r(t)/R(t) (8.8)is constant in time. Therefore, r(t) = χR(t) andṙ = χṘ = Ṙ r ≡ H(t)r , (8.9)Rwhere dots indicate derivatives with respect to time. The ratioH(t) = Ṙ/R (8.10)is called the Hubble parameter. It is the fractional change inthe distance between any pair of galaxies per unit time. H isoften called the expansion rate of the universe.Equation (8.9) is exactly Hubble’s law, where H(t) atthe present time is identified as the Hubble constant H 0 .Sothe hypothesis of an isotropic and homogeneous expansionexplains why the speed with which a galaxy moves awayfrom us is proportional to its distance.

8.3 The Friedmann Equation from Newtonian Gravity 1778.3 The Friedmann Equationfrom Newtonian Gravity“No theory is sacred.”Edwin Powell HubbleThe evolution of an isotropic and homogeneous expansion iscompletely determined by giving the time dependence of thedistance between any representative pair of galaxies. Onecan denote this distance by R(t), which is called the scalefactor. An actual numerical value for R is not important.For example, one can define R = 1 at a particular time (e.g.,now). It is the time dependence of R that gives informationabout how the universe as a whole evolves.The rigorous approach would now be to assume anisotropic and homogeneous matter distribution and to applythe laws of general relativity to determine R(t). By fortunatecoincidence, in this particular problem Newton’s theory ofgravity leads to the same answer, namely, to the Friedmannequation for the scale factor R. This approach will now bebriefly reviewed.Consider a spherical volume of the universe with a radiusR sufficiently large to be considered homogeneous, asshown in Fig. 8.3. In today’s universe this would mean takingR at least 100 Mpc. If one assumes that the universe iselectrically neutral, then the only force that is significantover these distances is gravity. As a test mass, consider agalaxy of mass m at the edge of the volume. It feels the gravitationalattraction from all of the other galaxies inside. Asa consequence of the inverse-square nature of gravity, thisFriedmann equationscale factorcosmological modelFig. 8.3A sphere of radius R containingmany galaxies, with a test galaxyof mass m at its edge

178 8 CosmologyNewtonian gravityBirkhoff’s theoremforce is the same as what one would obtain if all of the massinside the sphere were placed at the center.Another non-trivial consequence of the 1/r 2 force is thatthe galaxies outside the sphere do not matter. Their totalgravitational force on the test galaxy is zero. In Newtoniangravity these properties of isotropically distributed matterinside and outside a sphere follow from Gauss’s law for a1/r 2 force. The corresponding law holds in general relativityas well, where it is known as Birkhoff’s theorem.If one assumes that the mass of the galaxies is distributedin space with an average density ϱ, then the mass inside thesphere isM = 4 3 πR3 ϱ. (8.11)The gravitational potential energy V of the test galaxy isthereforeV =− GmMR =−4π 3 GmR2 ϱ. (8.12)The sum of the kinetic energy T and potential energy V ofthe test galaxy gives its total energy E,E = 1 2 mṘ2 − 4π 3 GmR2 ϱ = 1 2 mR2 (Ṙ2R 2 − 8π 3 Gϱ ).(8.13)curvature parameterThe curvature parameter k is now defined by()k = −2Em = 8πR2 Ṙ2Gϱ −3 R 2 . (8.14)If one were still dragging along the factors of c, k wouldhave been defined as −2E/mc 2 ; in either case k is dimensionless.Equation (8.14) can be written asFriedmann equationṘ 2R 2 + k R 2 = 8π Gϱ , (8.15)3which is called the Friedmann equation. The terms in thisequation can be identified as representingT − E =−V, (8.16)i.e., the Friedmann equation is simply an expression of con-servation of energy applied to our test galaxy. Since theenergy conservation

8.4 The Friedmann Equation from General Relativity 179sphere and test galaxy could be anywhere in the universe,the equation for R applies to any pair of galaxies sufficientlyfar apart that one can regard the intervening matter as beinghomogeneously distributed.The Friedmann equation can also be applied to the earlyuniverse, before the formation of galaxies. It will hold evenfor an ionized plasma as long as the universe is electricallyneutral overall and one averages over large enoughdistances. The scale factor R in that case represents the distancebetween any two elements of matter sufficiently farapart such that gravity is the only force that does not cancelout.early universe8.4 The Friedmann Equationfrom General Relativity 2“Since relativity is a piece of mathematics,popular accounts that try to explainit without mathematics are almost certainto fail.”Eric RogersAs mentioned above, the Friedmann equation can also beobtained from general relativity. Even though this approachis mathematically demanding, this elegant method will besketched in the following. More detailed presentations aregiven in the literature [8, 9].In general relativity the scale factor R(t) enters as a factorin the metric tensor g µν . This relates the space-time intervalds 2 to changes in coordinates x µ bymetric tensords 2 = g µν dx µ dx ν , (8.17)where µ, ν = 0, 1, 2, 3 and summation over repeated indicesis implied. The most general form of the metric tensorfor an isotropic and homogeneous universe is given by theRobertson–Walker metric,ds 2 = dt 2[dr− R 2 2 ((t)1 − kr 2 + r2 dθ 2 + sin 2 θ dφ 2)] , (8.18)2 This section is mathematically demanding and should beskipped by those readers, who are not familiar with tensors.Robertson–Walker metric

180 8 Cosmologywhere k is the curvature parameter and R the scale factor.The metric tensor is obtained as a solution to the Einsteinfield equations,R µν − 1 2 Rg µν = 8πGT µν + Λg µν , (8.19)where R µν is the Ricci tensor, R is the Ricci scalar (not tobe confused with the scale factor R), both of which are specificfunctions of g µν . On the right-hand side, Λ is the cos-mological constant and T µν is the energy–momentum tensor.An isotropic and homogeneous universe implies that T µνis of the formcosmological constantT µν = diag(ϱ, −P,−P,−P) , (8.20)energy–momentum tensorvacuum energy densitywhere ϱ is the energy density, P is the pressure, and ‘diag’means a square matrix with diagonal elements given by(8.20) and zeros everywhere else.Combining the Robertson–Walker metric and the T µνfrom (8.20) together with the field equations results in a differentialequation for the scale factor R,Ṙ 2R 2 + k R 2 = 8π 3 Gϱ + Λ 3 . (8.21)This is essentially the Friedmann equation that was foundfrom Newtonian gravity, but with three differences. First, ϱhere represents the energy density, not just the mass density.It includes all forms of energy, including, for example, photons.Second, the curvature parameter k really represents thecurvature of space, which is why it has this name. In theNewtonian case k was simply a measure of the total energy.Third, there is an additional term in the equation fromthe cosmological constant Λ. It can be absorbed into ϱ bydefining the vacuum energy density ϱ v asϱ v =Λ8πG(8.22)and then regarding the complete energy density to includeϱ v . It is called the vacuum energy because such a term ispredicted by quantum mechanics to arise from virtual particlesthat ‘fluctuate’ in and out of existence from the vacuum.

8.4 The Friedmann Equation from General Relativity 181This phenomenon is present in quantum field theories suchas the Standard Model of particle physics. In certain circumstancesit can even be observed experimentally, such as inthe Casimir effect, where virtual particles from the vacuumlead to an attractive force between two metal plates witha very small separation, see Fig. 8.4. 3 Nevertheless, naïveestimates of the magnitude of Λ from quantum field theorieslead to values that are too large by over 120 orders ofmagnitude!A more classical approach to obtain the generalized differentialequation for the scale factor is to start from (8.15)and interpret the density ϱ as the sum of the classical densityand a vacuum energy density ϱ v defined by (8.22).There is clearly much that is not understood about thecosmological constant, and Einstein famously regretted proposingit. The more modern view of physical laws, however,leads one to believe that when a term is absent froman equation, it is usually because its presence would violatesome symmetry principle. The cosmological constant isconsistent with the symmetries on which general relativity isbased. And in the last several years clear evidence from theredshifts of distant supernovae has been presented that thecosmological constant is indeed non-zero and that vacuumenergy makes a large contribution to the total energy densityof the universe. In Chaps. 12 and 13 this will be discussedin more detail.3 In the framework of quantum mechanics it is usually demonstratedthat the harmonic oscillator has a non-zero zero-pointenergy. Also in quantum field theories the vacuum is notempty. Heisenberg’s uncertainty relation suggests that thevacuum contains infinitely many virtual particle–antiparticlepairs. The evidence that such quantum fluctuations exist wasdemonstrated by the Casimir effect. Consider two parallelmetal plates at a very small separation in vacuum. Because thedistance is so small, not every possible wavelength can existin the space between the two plates, quite in contrast to thesurrounding vacuum. The effect of this limited choice of quantabetween the plates leads to a small attractive force of the plates.This pressure of the surrounding vacuum was experimentallyconfirmed. This force can be imagined in such a way that thereduced number of field quanta in the space between the platescannot resist the pressure of the unlimited number of fieldquanta in the surrounding vacuum.Casimir effectFig. 8.4Illustration of the Casimir effect:Only certain wavelengths fit intothe space between the plates. Theoutside of the plates does not limitthe number of possible frequenciessymmetry principle

182 8 Cosmology8.5 The Fluid Equation“The universe is like a safe to whichthere is a combination – but the combinationis locked in the safe.”Peter de VriesThe Friedmann equation cannot be solved yet because onedoes not know how the energy density ϱ varies with time. Instead,in the following a relation between ϱ, its time derivative˙ϱ, and the pressure P will be derived. This relation,called fluid equation, follows from the first law of thermody-namics for a system with energy U, temperature T ,entropyS, andvolumeV ,fluid equationadiabatic expansionFor the second term in (8.24) one getsdV= d dt dt R3 = 3R 2 Ṙ. (8.27)Putting (8.26) and (8.27) into (8.24) and rearranging termsgives˙ϱ + 3Ṙ (ϱ + P) = 0 , (8.28)Rwhich is the fluid equation. Unfortunately, this is still notenough to solve the problem, since an equation of state re-lating ϱ and P is needed. This can be obtained from the lawsof statistical mechanics as will be shown in Sect. 9.2. Withthese ingredients the Friedmann equation can then be usedto find R(t).equation of statedU = T dS − P dV . (8.23)The first law of thermodynamics will now be applied to avolume R 3 in our expanding universe. Since by symmetrythere is no net heat flow across the boundary of the volume,one has dQ = T dS = 0, i.e., the expansion is adiabatic.Dividing (8.23) by the time interval dt then givesdUdt+ P dVdtThe total energy U is= 0 . (8.24)U = R 3 ϱ. (8.25)The derivative dU/dt is thereforedUdt= ∂U∂RṘ + ∂U∂ϱ ˙ϱ = 3R2 ϱṘ + R 3 ˙ϱ . (8.26)

8.7 Nature of Solutions to the Friedmann Equation 1838.6 The Acceleration Equation“Observations always involve theory.”Edwin Powell HubbleIn a number of cases it can be useful to combine the Friedmannand fluid equations to obtain a third equation involvingthe second derivative ¨R. Here only the relevant steps willbe outlined and the results will be given; the derivation isstraightforward (see Problem 4 in this chapter).If the Friedmann equation (8.15) is multiplied by R 2 andthen differentiated with respect to time, one will obtain anequation involving ¨R, Ṙ, R,and˙ϱ. The fluid equation (8.28)can then be solved for ˙ϱ and substituted into the derivativeof the Friedmann equation. This gives¨RR =−4πG (ϱ + 3P) , (8.29)3which is called the acceleration equation. It does not addany new information beyond the Friedmann and fluid equationsfrom which it was derived, but in a number of problemsit will provide a more convenient path to a solution.acceleration equation8.7 Nature of Solutionsto the Friedmann Equation“In every department of physical sciencethere is only so much science,properly so-called, as there is mathematics.”Immanuel KantWithout explicitly solving the Friedmann equation one canalready make some general statements about the nature ofpossible solutions. The Friedmann equation (8.15) can bewritten asH 2 = 8πG3 ϱ − k R 2 , (8.30)where, as always, H = Ṙ/R. From the observed redshiftsof galaxies, it is known that the current expansion rate H ispositive. One expects, however, that the galaxies should beslowed by their gravitational attraction. One can thereforedecelerated expansion?

184 8 Cosmologyask whether this attraction will be sufficient to slow the expansionto a halt, i.e., whether H will ever decrease to zero.If the curvature parameter k is negative, then this cannothappen, since everything on the right-hand side of (8.30) ispositive. Recall that k =−2E/m basically gives the totalenergy of the test galaxy on the edge of the sphere of galaxies.Having k < 0 means that the total energy of the testgalaxy is positive, i.e., it is not gravitationally bound. In thiscase the universe is said to be open; it will continue to ex-pand forever.If, for example, the energy density of the universe isdominated by non-relativistic matter, then one will find thatϱ decreases as 1/R 3 . So, eventually, the term with the curvatureparameter will dominate on the right-hand side of (8.30)leading toopen universeeternal expansionṘ 2R 2 =− k R 2 , (8.31)contraction of the universeclosed universeflat universewhich means Ṙ is constant or R ∼ t.If, on the other hand, one has k>0, then as ϱ decreases,the first and second terms on the right-hand side of (8.30)will eventually cancel with the results of H = 0. That is, theexpansion will stop. At this point, all of the kinetic energyof the galaxies is converted to gravitational potential energy,just as when an object thrown vertically into the air reachesits highest point. And just as with the thrown object, the motionthen reverses and the universe begins to contract. In thiscase the universe is said to be closed.One can also ask what happens if the curvature parameterk and hence also the energy E of the test galaxy arezero, i.e., if the universe is just on the borderline betweenbeing open and closed. In this case the expansion will bedecelerated but H will approach zero asymptotically. Theuniverse is then said to be flat. This is analogous to throwinga projectile upwards with a speed exactly equal to the escapevelocity.Which of the three scenarios one obtains – open, closed,or flat – depends on what is in the universe to slow or otherwiseaffect the expansion. To see how this corresponds to theenergy density, the Friedmann equation (8.15) can be solvedfor the curvature parameter k, which gives( )8πk = R 2 3 Gϱ − H 2 . (8.32)

8.7 Nature of Solutions to the Friedmann Equation 185One can then define the critical density ϱ c byϱ c = 3H 2(8.33)8πGand express the energy density ϱ by giving the ratio Ω,Ω parameterΩ = ϱ ϱ c. (8.34)Using this with H = Ṙ/R, (8.32) becomesk = R 2 (Ω − 1)H 2 . (8.35)So, if Ω

186 8 Cosmology8.8 Experimental Evidencefor the Vacuum Energy“It happened five billion years ago. Thatwas when the Universe stopped slowingdown and began to accelerate, experiencinga cosmic jerk.”Adam Riessdeceleration parameteracceleration parameterFor a given set of contributions to the energy density of theuniverse, the Friedmann equation predicts the scale factor,R(t), as a function of time. From an observational standpoint,one would like to turn this around: from measurementsof R(t) one can make inferences about the contents ofthe universe. Naïvely one would expect the attractive forceof gravity to slow the Hubble expansion, leading to a deceleration,i.e., ¨R < 0. One of the most surprising devel-opments of recent years has been the discovery that the expansionis, in fact, accelerating, and apparently has been forseveral billion years. This can be predicted by the Friedmannequation if one assumes a contribution to the energy densitywith negative pressure, such as the vacuum energy pre-viously mentioned. Such a contribution to ϱ is sometimescalled dark energy.Because of the finite speed of light, observations ofgalaxies far away provide information about the conditionsof the universe long ago. By accurately observing the motionof very distant galaxies and comparing to that of those closerto us, one can try to discern whether the universe’s expansionis slowing down or speeding up. From each observedgalaxy, two pieces of information are required: its speed ofrecession and its distance. As type-Ia supernovae are extremelybright and therefore can be found over cosmologicaldistances, they are well suited for this type of study. Toobtain the speed, the redshift z of spectral lines can be used.As type-Ia supernovae have to good approximation a con-stant absolute luminosity, the apparent brightness providesinformation on their distance. This is essentially the sametype of analysis that was carried out to determine the Hubbleparameter H 0 as described in Sect. 8.1. Here, however,one is interested in pushing the measurement to distances sofar that the rate of expansion itself may have changed in thetime that it took for the light to arrive at Earth.The relation between the apparent brightness of a supernovaand its redshift can be obtained once one knowsthe scale factor as a function of time. This in turn is deter-negative pressuredark energytype-Ia supernovaerelation between brightnessand redshift

8.8 Experimental Evidence for the Vacuum Energy 187mined by the Friedmann equation once the contributions tothe energy density are specified. Suppose the universe contains(non-relativistic) matter and vacuum energy; the lattercould be described by a cosmological constant Λ.Theircurrentenergy densities divided by the critical density ϱ c canbe written as Ω m,0 and Ω Λ,0 , respectively. Here as usual thesubscripts 0 denote present-day values. One should also considerthe energy density of radiation, Ω r,0 , from photons andneutrinos. Such a contribution is well determined from measurementsof the cosmic microwave background and is verysmall compared to the other terms for the time period relevantto the observations. So neglecting the relativistic particles,one can show that the luminosity distance, d L ,isgivenbyd L (z) = 1 + zH 0∫ z0[Ω Λ,0 + (1 + z ′ ) 3 Ω m,0+ (1 + z ′ ) 2 (1 − Ω 0 )] −1/2dz ′ . (8.36)It is convenient to carry out the integral numerically, yieldingd L (z) for any hypothesized values of Ω m,0 and Ω Λ,0 .Recall the luminosity distance is defined by F = L/4πdL 2 ,whereF is the flux one measures at Earth and L isthe intrinsic luminosity of the source. Astronomers usuallyreplace F and L by the apparent and absolute magnitudes, mand M, which are related to the base-10 logarithm of F andL, respectively (precise definitions can be found in standardastronomy texts such as [10] and in the glossary). The observationof a supernova allows one to determine its apparentmagnitude, m. The absolute magnitude, M, is unknown apriori, but is assumed to be the same (after some correctionsand adjustments) for all type-Ia supernovae. These quantitiesare related to the luminosity distance by( )dLm = 5log 10 + 25 + M. (8.37)1MpcNotice that a higher apparent magnitude m corresponds to afainter supernova, i.e., one further away.A plot of the apparent magnitude m of a sample of distantsupernovae versus the redshift is shown in Fig. 8.6 [11].The data points at low z determine the Hubble constant H 0 .The relation at higher z, however, depends on the matter anddark-energy content of the universe. The various curves onthe plot show the predictions from (8.36) and (8.37) for differentvalues of Ω m,0 and Ω Λ,0 . The curve with no darkcosmological constantluminosity distancemagnitudesHubble diagram

188 8 Cosmology24Ω m,0 ,Ω Λ,00.25,0.750.25, 01, 022SupernovaCosmologyProjecteffective m B201816Calan/Tololo& CfAFig. 8.6Magnitudes and residuals ofsupernovae of type Ia as a functionof redshift of their host galaxies incomparison to the expectation ofvarious models. The data areconsistent with a flat universe witha fraction of about 75% of darkenergy. Shown are data from theSupernova Cosmology Project, theCalan/Tololo group, and theHarvard–Smithsonian Center forAstrophysics (CfA) {26}mag. residualfrom empty cosmology141.00.50.00.51.00.0 0.2 0.4 0.6 0.8 1.0redshift zΩ m,0 ,Ω Λ,00.25,0.750.25, 01, 0accelerating universevacuum energyenergy, Ω m,0 = 1, i.e., Ω Λ,0 = 0, is in clear disagreement.The data are well described by Ω m,0 ≈ 0.25 andΩ Λ,0 ≈ 0.75. Further measurements of the cosmic microwavebackground support this view. This picture will becompleted in Chaps. 11 and 13.Qualitatively the behaviour shown in Fig. 8.6 can be understoodin the following way. The data points at high redshiftlie above the curve for zero vacuum energy, i.e., athigher magnitudes, which means that they are dimmer thanexpected. Thus the supernovae one sees at a given z are fartheraway than one would expect, therefore the expansionmust be speeding up.The exact physical origin of the vacuum energy remainsa mystery. On the one hand, vacuum energy is expected ina quantum field theory such as the Standard Model of elementaryparticles. Naïvely, one would expect its value to beof the orderϱ v ≈ E 4 max , (8.38)

8.9 Problems 189where E max is the maximum energy at which the field theoryis valid. One expects the Standard Model of particle physicsto be incomplete, for example, at energies higher than thePlanck energy, E Pl ≈ 10 19 GeV, where quantum-gravitationaleffects come into play. So the vacuum energy mightbe roughlyϱ v ≈ EPl 4 ≈ 1076 GeV 4 . (8.39)But from the observed present-day value Ω Λ,0 ≈ 0.7, thevacuum energy density is3H02 ϱ Λ,0 = Ω Λ,0 ϱ c,0 = Ω Λ,08πG ≈ 10−46 GeV 4 . (8.40)The discrepancy between the naïve prediction and the observedvalue is 122 orders of magnitude.So something has clearly gone wrong with the prediction.One could argue that not much is understood about thephysics at the Planck scale, which is surely true, and so alower energy cutoff should be tried. Suppose one takes themaximum energy at the electroweak scale, E EW ≈ 100 GeV,roughly equal to the masses of the W and Z bosons. At theseenergies the Standard Model has been tested to high accuracy.The prediction for the vacuum energy density becomesϱ v ≈ EEW 4 ≈ 108 GeV 4 . (8.41)Now the discrepancy with the observational limit is ‘only’53 orders of magnitude. Perhaps an improvement but clearlynot enough.Finally one might argue that the entire line of reasoningwhich predicts vacuum energy might be incorrect. Butphenomena such as the Casimir effect discussed in Sect. 8.4have been observed experimentally and provide an importantconfirmation of this picture. So one cannot dismiss thevacuum energy as fiction; there must be some other reasonwhy its contribution to Ω is much less than expected. This iscurrently one of the most gaping holes in our understandingof the universe.discrepancy betweenexpectation and observationCasimir effect8.9 Problems1. Derive the relativistic relation (8.2) between redshift andvelocity of a receding galaxy.2. A gas cloud gets unstable if the gravitational energy exceedsthe thermal energy of the molecules constitutingthe cloud, i.e.,

190 8 CosmologyGM 2R > 3 2 kT M µ ,where 3 2 kT is the thermal energy of a molecule, M µis thenumber of molecules, and µ is the molecule mass. Derivefrom this condition the stability limit of a gas cloud(Jeans criterion).3. Let us assume that a large astrophysical object of constantdensity not stabilized by internal pressure is aboutto contract due to gravitation. Estimate the minimum rotationalvelocity so that this object is stabilized againstgravitational collapse! How does the rotational velocitydepend on the distance from the galactic center?4. Derive the acceleration equation (8.29) from the Friedmannequation (8.15) with the help of the fluid equation(8.28).5. Show that the gravitational redshift of light emitted froma massive star (mass M)ofradiusR isνν= GMc 2 R .6. Estimate the classical value for the deflection of starlightpassing near the Sun.7. Clocks in a gravitational potential run slow relative toclocks in empty space. Estimate the slowing-down ratefor a clock on a pulsar (R = 10 km, M = 10 30 kg)!8. Estimate the gravitational pressure at the center of theSun (average density ϱ = 1.4g/cm 3 ) and the Earth (ϱ =5.5g/cm 3 ).9. Estimate the average density ofa) a large black hole residing at the center of a galaxyof 10 11 solar masses,b) a solar-mass black hole,c) a mini black hole (m = 10 15 kg).10. The orbital velocity v of stars in our galaxy varies upto distances of 20 000 light-years as if the density werehomogeneous and constant (ϱ = 6 × 10 −21 kg/m 3 ). Forlarger distances the velocities of stars follow the expectationfrom Keplerian motion.a) Work out the dependence of v(R) for R

1919 The Early Universe“Who cares about half a second after theBig Bang; what about the half second before?”Fay Weldon quoted by Paul DaviesIn this chapter the history of the universe through the firstten microseconds of its existence will be described. First,in Sect. 9.1 the Planck scale, where quantum-mechanicaland gravitational effects both become important, will be defined.This sets the starting point for the theory to be described.In Sect. 9.2 some formulae from statistical and thermalphysics will be assembled which are needed to describethe hot dense phase out of which the universe then evolved.Then, in Sects. 9.3 and 9.4 these formulae will be used tosolve the Friedmann equation and to investigate the propertiesof the universe at very early times. Finally, in Sect. 9.5one of the outstanding puzzles of the Hot Big Bang modelwill be presented, namely, why the universe appears to consistalmost entirely of matter rather than a mixture of matterand antimatter.Planck scalematter-dominated universe9.1 The Planck Scale“A new scientific truth does not triumphby convincing its opponents and makingthem see the light, but rather becauseits opponents eventually die, anda new generation grows up that is familiarwith it.”Max PlanckThe earliest time about which one can meaningfully speculatewith our current theories is called the Planck era, around10 −43 seconds after the Big Bang. Before then, the quantummechanicalaspects of gravity are expected to be important,so one would need a quantum theory of gravity to describethis period. Although superstrings could perhaps providesuch a theory, it is not yet in such a shape that one can use itto make specific predictions.Planck timequantum theory of gravity

192 9 The Early UniverseSchwarzschild radiusevent horizonTo see how the Planck scale arises, consider the Schwarzschildradius for a mass m (see Problem 5 in this chapter),R S = 2mGc 2 , (9.1)where for the moment factors of c and ¯h will be explicitlyinserted. The distance R S gives the event horizon of a blackhole. It represents the distance at which the effects of spacetimecurvature due to the mass m become significant.Now consider the Compton wavelength of a particle ofmass m,λ C = h mc . (9.2)Planck massPlanck energyPlanck lengthPlanck timeThis represents the distance at which quantum effects becomeimportant. The Planck scale is thus defined by the conditionλ C /2π = R S /2, i.e.,¯hmc = mGc 2 . (9.3)Solving for the Planck mass gives√¯hcm Pl =G ≈ 2.2 × 10−5 g , (9.4)or the mass of a water droplet about 1/3 mm in diameter.The rest-mass energy of m Pl is the Planck energy,√¯hcE Pl =5G ≈ 1.22 × 1019 GeV , (9.5)which is about 2 GJ or 650 kg TNT equivalent. Using thePlanck mass in the reduced Compton wavelength, ¯h/mc,gives the Planck length,√¯hGl Pl =c 3 ≈ 1.6 × 10−35 m . (9.6)The time that it takes light to travel l Pl is the Planck time,t Pl = l √Pl ¯hGc = c 5 ≈ 5.4 × 10−44 s . (9.7)The Planck mass, length, time, etc. are the unique quantitieswith the appropriate dimension that can be constructed fromthe fundamental constants linking quantum mechanics andrelativity: ¯h, c, andG. Since henceforth ¯h and c will be setequal to one, and one has

9.2 Thermodynamics of the Early Universe 193m Pl = E Pl = 1/ √ G, (9.8)t Pl = l Pl = √ G. (9.9)So the Planck scale basically characterizes the strength ofgravity. As the Planck mass (or energy), 1.2 ×10 19 GeV, is anumber people tend to memorize, often 1/m 2 Plwill be usedas a convenient replacement for G.strength of gravity9.2 Thermodynamics of the Early Universe“We are startled to find a universe wedid not expect.”Walter BagehotIn this section some results from statistical and thermalphysics will be collected that will be needed to describe theearly universe. Some of the relations presented may differfrom those covered in a typical course in statistical mechanics.This is for two main reasons. First, the particles in thevery hot early universe typically have speeds comparable tothe speed of light, so the relativistic equation E 2 = p 2 + m 2must be used to relate energy and momentum. Second, thetemperatures will be so high that particles are continuallybeing created and destroyed, e.g., through reactions suchas γγ ↔ e + e − . This is in contrast to the physics of lowtemperaturesystems, where the number of particles in a systemis usually constrained to be constant. The familiar exceptionis blackbody radiation, since massless photons canbe created and destroyed at any non-zero temperature. For agas of relativistic particles expressions for ϱ, n, andP willbe found that are similar to those for blackbody radiation.Here ϱ, n, andP are the density or – more generally – theenergy density, the number density, and the pressure.The formulae in this section are derived in standard textson statistical mechanics. Some partial derivations are givenin Appendix B; here merely the results in a form appropriatefor the early universe will be quoted. In a more rigoroustreatment one would need to consider conservation ofvarious quantum numbers such as charge, baryon number,and lepton number. For each conserved quantity one has achemical potential µ, which enters into the expressions forthe energy and number densities. For most of the treatmentof the very early universe one can neglect the chemical potentials,and thus they will not appear in the formulae whichare given here.creation and annihilationof particlesgas of relativistic particlesstatistical mechanicschemical potential

194 9 The Early Universe9.2.1 Energy and Number Densitiesinternal degrees of freedomphoton spin statesStefan–Boltzmann lawnumber densitiesIn the limit where the particles are relativistic, i.e., T ≫ m, 1the energy density for a given particle type is⎧π⎪⎨230 gT 4 for bosons,ϱ =⎪⎩ 7 π 28 30 gT 4 for fermions.(9.10)Here g is the number of internal degrees of freedom forthe particle. For a particle of spin J , for example, one has2J + 1 spin degrees of freedom. In addition, the factor gincludes different colour states for quarks and gluons. Forexample, for electrons one has g e − = 2 or, for electronsand positrons considered together, g e = 4. The spin-1/2 uquark together with the ū has g u = 12, i.e., 2 from spin,3 from colour, and 2 for considering particle and antiparticletogether. Note that the photon has J = 1 but only twospin states, which correspond to the two transverse polarizationstates. The longitudinal polarization is absent as a consequenceof these particles having zero mass. For photonsthis gives ϱ γ = (π 2 /15)T 4 , the well-known formula for theenergy density of blackbody radiation (Stefan–Boltzmannlaw). Indeed, one sees that all relativistic particles have asimilar behaviour with ϱ ∼ T 4 , the only differences arisingfrom the number of degrees of freedom and from the factor7/8 if one considers fermions.In a similar manner one can show that the number densityn is given by⎧ζ(3)⎪⎨πn =2 gT 3 for bosons,(9.11)3 ζ(3) ⎪⎩4 π 2 gT 3 for fermions.Here ζ is the Riemann zeta function and ζ(3) ≈ 1.202 06 ....Notice that in particle physics units the number density hasdimension of energy cubed. To convert this to a normalnumber per unit volume, one has to divide by (¯hc) 3 ≈(0.2GeVfm) 3 .From the number and energy densities one can obtainthe average energy per particle, 〈E〉 =ϱ/n.ForT ≫ m onefinds1 Note that Boltzmann’s constant k has also been set equal to one.

9.2 Thermodynamics of the Early Universe 195⎧π 4⎪⎨ T ≈ 2.701 T for bosons,30 ζ(3)〈E〉 =7π⎪⎩4(9.12)180 ζ(3) T ≈ 3.151 T for fermions.In the non-relativistic limit one finds for the energy densityenergy densitiesϱ = mn , (9.13)where n is the number density. This is given in the nonrelativisticlimit by( ) mT 3/2n = g e −m/T , (9.14)2πwhere the same result is obtained for both the Fermi–Diracdistribution and Bose–Einstein distribution and, as above, itis assumed that chemical potentials can be neglected. Onetherefore finds that for a non-relativistic particle species, thenumber density is exponentially suppressed by the factore −m/T . In the non-relativistic limit, the average energy isthe sum of mass and kinetic terms,Fermi–Dirac distributionBose–Einstein distribution〈E〉 =m + 3 T ≈ m. (9.15)2The final approximation holds in the non-relativistic limitwhere T ≪ m.9.2.2 The Total Energy DensityThe formulae above give ϱ and n for a single particle type.What is needed for the Friedmann equation, however, is thetotal energy density from all particles. This is simply thesum of the ϱ values for all particle types, where at a giventemperature some types will be relativistic and others not.The expression that one will obtain for the total energydensity is greatly simplified by recalling that the numberdensity of a non-relativistic particle species is exponentiallysuppressed by the factor e −m/T . This will hold as long as thereduction in density is not prevented by a conserved quantumnumber, which would imply a non-zero chemical potential.Assuming this is not the case, to a good approximationthe non-relativistic matter contributes very little to the totalenergy density. At early times in the universe (t

196 9 The Early Universeyears), it will be seen that the total energy density was dominatedby relativistic particles. This is called the ‘radiationdominatedera’, and in this period one can ignore the con-tribution to ϱ from non-relativistic particles. Then, using theappropriate relativistic formulae from (9.10) for bosons andfermions, one obtains for the total energy densityradiation-dominated eraϱ =∑i=bosonsπ 230 g iT 4 + ∑j=fermions7 π 28 30 g j T 4 , (9.16)where the sums include only those particle types that arerelativistic, i.e., having m ≪ T . This can be equivalentlywritten asϱ = π 230 g ∗T 4 , (9.17)where the effective number of degrees of freedom g ∗ is de-fined aseffective numberof degrees of freedomHere as well, the sums only include particles with m ≪ T .T is always assumed to mean the photon temperature,since its value for the present era is very accurately measuredfrom the cosmic microwave background radiation tobe T ≈ 2.73 K. Some particle types may have a differenttemperature, however, since they may no longer be inthermal contact with photons. Neutrinos, for example, effec-tively decoupled from other particles at a time before mostelectrons and positrons annihilated into photons. As a result,the neutrino temperature today is around 1.95 K. In general,one can modify (9.18) to account for different temperaturesby usingdecoupling of neutrinosneutrino temperatureg ∗ =∑i=bosonsg i + 7 8∑j=fermionsg j . (9.18)g ∗ =∑i=bosons( ) 4 Tig i + 7 T 8∑j=fermions( ) 4 Tjg j .T(9.19)This more general form of g ∗ will be rarely needed exceptfor the example of neutrinos mentioned above.In order to compute g ∗ at a given temperature T , oneneeds to know what particle types have m ≪ T ,andalsothe number of degrees of freedom for these types is required.

9.2 Thermodynamics of the Early Universe 197particle mass spin colour g (particle andstates states antiparticle)photon (γ ) 0 2 1 2W + , W − 80.4 GeV 3 1 6Z 91.2 GeV 3 1 3gluon (g) 0 2 8 16Higgs > 114 GeV 1 1 1bosons 28u, ū 3MeV 2 3 12d, ¯d 6MeV 2 3 12s, ¯s 100 MeV 2 3 12c, ¯c 1.2 GeV 2 3 12b, ¯b 4.2 GeV 2 3 12t, ¯t 175 GeV 2 3 12e + ,e − 0.511 MeV 2 1 4µ + ,µ − 105.7 MeV 2 1 4τ + ,τ − 1.777 GeV 2 1 4ν e , ¯ν e < 3eV 1 1 2ν µ , ¯ν µ < 0.19 MeV 1 1 2ν τ , ¯ν τ < 18.2 MeV 1 1 2fermions 90Table 9.1Particles of the Standard Modeland their properties [2]Table 9.1 shows the masses and g values of the particles ofthe Standard Model (‘color states’ 1 means a colour-neutralparticle).Here the neutrinos will be treated as only being lefthandedand antineutrinos as only right-handed, i.e., theyonly have one spin state each, which again is related totheir being considered massless. In fact, recent evidence (seeSect. 6.2 on neutrino astronomy) indicates that neutrinos dohave non-zero mass, but the coupling to the additional spinstates is so small that their effect on g ∗ can be ignored.In addition to the particles listed, there are possible Xand Y bosons of a Grand Unified Theory and perhaps supersymmetricpartners for all particle types. It will be the taskof hadron colliders to find out whether such particles can beproduced with present-day accelerator technology. Possiblysuch new particles – if they existed in the early universe –have left imprints on astrophysical data, so that investigationson cosmoarcheology might find evidence for them. Ifone restricts oneself here to the Standard Model and considersa temperature much greater than any of the masses, e.g.,T>1 TeV, one hasneutrino masssupersymmetric particlescosmoarcheology

198 9 The Early Universeg ∗ = 28 + 7 × 90 = 106.75 . (9.20)8For the accuracy required it will usually be sufficient to takeg ∗ ≈ 10 2 for T>1TeV.A few further subtleties come into play at lower temperatures.For example, at around T ≈ 0.2 GeV, quarks and gluonsbecome confined into colour-neutral hadrons, namely,protons and neutrons. So, to obtain g ∗ at, say, T = 100 MeV,one would not have any quarks or gluons. Instead one getsa contribution of 8 × 7/8 = 7 from hadrons, i.e., two spinstates each for protons and neutrons and their antiparticles.9.2.3 Equations of Stateenergy densitypressure of relativistic matterNext the required equations of state will be recalled, that is,relations between energy density ϱ and pressure P .Theseare derived in Appendix B. They are needed in conjunctionwith the acceleration and fluid equations in order to solvethe Friedmann equation for R(t). For the pressure for a gasof relativistic particles one findsP = ϱ 3 . (9.21)pressureof non-relativistic matterThis is the well-known result from blackbody radiation, butin fact it applies for any particle type in the relativistic limitT ≫ m. In the non-relativistic limit, the pressure is given bythe ideal-gas law, P = nT . In this case, however, one has anenergy density ϱ = mn, soforT ≪ m one has P ≪ ϱ andin the acceleration and fluid equations one can approximateP ≈ 0 . (9.22)pressure of the vacuumIn addition, one can show that for the case of vacuum energydensity from a cosmological constant,P =−ϱ v . (9.23)That is, a vacuum energy density leads to a negative pressure.In general, the equation of state can be expressed asP = wϱ , (9.24)w parameterwhere the parameter w is 1/3 for relativistic particles, 0 for(non-relativistic) matter, and −1 for vacuum energy.

9.3 Solving the Friedmann Equation 1999.2.4 Relation between Temperature and Scale FactorFinally, in this section a general relation between the temperatureT and the scale factor R will be noted. All lengths,when considered over distance scales of at least 100 Mpc,increase with R. Since the de Broglie wavelength of a particle,λ = h/p, is inversely proportional to the momentum,one sees that particle momenta decrease as 1/R. For photons,one has E = p, and so their energy decreases as 1/R.Furthermore, the temperature of photons in thermal equilibriumis simply a measure of the photons’ average energy, soone gets the important relationmomentum and scale factorT ∼ R −1 . (9.25)This relation holds as long as T is interpreted as the photontemperature and as long as the Hubble expansion iswhat provides the change in T . In fact this is not exact,because there are other processes that affect the temperatureas well. For example, as electrons and positrons becomenon-relativistic and annihilate into photons, the photon temperaturereceives an extra contribution. These effects can betaken into account by thermodynamic arguments using conservationof entropy. The details of this are not critical forthe present treatment, and one will usually be able to assume(9.25) to hold.photon temperature9.3 Solving the Friedmann Equation“No one will be able to read the greatbook of the universe if he does not understandits language which is that ofmathematics.”Galileo GalileiNow enough information is available to solve the Friedmannequation. This will allow to derive the time dependence ofthe scale factor R, temperature T , and energy density ϱ. If,for the start, very early times will be considered, one cansimplify the problem by seeing that the term in the Friedmannequation (8.15) with the curvature parameter, k/R 2 ,can be neglected. To show this, recall from Sect. 8.5 the fluidequation (8.28),curvature parameter˙ϱ + 3Ṙ (ϱ + P) = 0 , (9.26)R

200 9 The Early Universewhich relates the time derivative of the energy density ϱ andthe pressure P . One can suppose that ϱ is dominated by radiation,so that the equation of state (9.21) can be used,P = ϱ 3 . (9.27)fluid equationradial dependence of ϱSubstituting this into the fluid equation (9.26) gives˙ϱ + 4ϱṘR = 0 . (9.28)The left-hand side is proportional to a total derivative, so onecan write1R 4 ddt(ϱR 4) = 0 . (9.29)This implies that ϱR 4 is constant in time, and thereforeϱ ∼ 1 R 4 . (9.30)If, instead, one would have assumed that ϱ was dominatedby non-relativistic matter, one would have used the equationof state P = 0, and in a similar way one would have found(see Problem 1 in this chapter)ϱ ∼ 1 R 3 . (9.31)In either case the dependence of ϱ on R is such that forvery early times, that is, for sufficiently small R, theterm8πGϱ/3 on the right-hand side of the Friedmann equationwill be much larger than k/R 2 . One can then ignore thecurvature parameter and effectively set k = 0; this is def-initely valid at the very early times that will be consideredin this chapter and it is still a good approximation today, 14billion years later. The Friedmann equation then becomescurvature parametermodified FriedmannequationṘ 2R 2 = 8π Gϱ . (9.32)3In the following (9.32) will be solved for the case whereϱ is radiation dominated. One can write (9.30) as( ) 4 R0ϱ = ϱ 0 , (9.33)Rwhere here ϱ 0 and R 0 represent the values of ϱ and R atsome particular (early) time. One can guess a solution of

9.3 Solving the Friedmann Equation 201the form R = At p and substitute this along with (9.33) forϱ into the Friedmann equation (9.32). With this ansatz theFriedmann equation can only be satisfied if p = 1/2, i.e.,radiation-dominated eraR ∼ t 1/2 . (9.34)The expansion rate H is thereforeH = ṘR = 1 2t . (9.35)If, instead, one assumes that ϱ is dominated by non-relativisticmatter and if one uses the equation of state P = 0, thenone finds in a similar way (see Problem 2 in this chapter)expansion ratematter-dominated eraR ∼ t 2/3 (9.36)andH = 2 3t . (9.37)First, one can combine the Friedmann equation (9.32)with the energy density (9.17) and use G = 1/m 2 Plto replacethe gravitational constant. Taking the square root then givesthe expansion rate H as a function of the temperature (seeProblem 3 in this chapter),√8πH =3 g ∗ T 2≈ 1.66 √ T 2g ∗ . (9.38)90 m Pl m PlThis can be combined with (9.35) to give a relation betweenthe temperature and the time,√t = 1 90 m Pl2 8π 3 g ∗ T 2 ≈ 0.301 m Pl√g∗ T 2 . (9.39)Remember that both of these equations use the particlephysics system of units, i.e., the expansion rate in (9.38) is inGeV and the time in (9.39) is in GeV −1 . One can also combinethe Friedmann equation with the solution H = 1/2t togive the energy density as a function of time,ϱ = 3m2 Pl32π1t 2 . (9.40)relation between temperatureand timetime dependenceof the energy density

202 9 The Early Universe9.3.1 Digression on Thermal EquilibriumHaving derived the relations for quantities such as numberand energy density as a function of temperature, it is worthasking when one expects them to apply. In order for a systemto be characterized by a temperature, there must exist interactionsbetween the particles that allow their numbers andmomentum distribution to adjust to those of thermal equilibrium.Furthermore, one has to wait long enough for equi-librium to be attained, namely, much longer than the timescale of the individual microscopic interactions.Now in any change in the temperature, the microscopicinteractions must take place quickly enough for the thermaldistribution to adjust. One can express this condition by requiringthat the rate Γ of the reaction needed to maintainequilibrium must be much greater than the fractional changein the temperature per unit time, i.e.,thermal equilibriumconditionsΓ ≫|Ṫ/T| . (9.41)But for a system of relativistic particles one has from (9.25)that T ∼ 1/R, soṪ/T =−Ṙ/R =−H . Therefore, (9.41)is equivalent to the requirementΓ ≫ H, (9.42)interaction ratescross sectionwhere the absolute value has been dropped since the expansionrate is always assumed to be positive. For a given temperatureit is straightforward to use (9.38) to determine H .It is basically proportional to T 2 , ignoring the small temperaturedependence of g ∗ .The reaction rate Γ is the number of interactions of aspecified type per unit time per particle. It is the reciprocal ofthe mean time that it will take for the particle to undergo theinteraction in question. It can be calculated as a function ofthe particle’s speed v, the number density of target particlesn, and the interaction cross section σ byΓ = n〈σv〉 , (9.43)where the brackets denote an average of σv over a thermaldistribution of velocities.If thermal equilibrium has been attained, one can findthe number density n using the appropriate formulae fromSect. 9.2. To find the cross section, one needs to bring in theknowledge of particle physics.

9.4 Thermal History of the First Ten Microseconds 2039.4 Thermal Historyof the First Ten Microseconds“An elementary particle that does notexist in particle theory should also notexist in cosmology.”AnonymousThe relations from the previous sections can now be used towork out the energy density and temperature of the universeas a function of time. As a start, one can use (9.40) to givethe energy density at the Planck time, although one needs tokeep in mind from the previous section that the assumptionof thermal equilibrium may not be valid. In any case theformula givesϱ(t Pl ) = 3m2 Pl32π1t 2 Pl= 332π m4 Pl ≈ 6×1074 GeV 4 , (9.44)where m Pl = 1/t Pl ≈ 1.2 × 10 19 GeV has been used. Onecan convert this to normal units by dividing by (¯hc) 3 ,ϱ(t Pl ) ≈ 6 × 10 74 GeV 4 ×1(0.2GeVfm) 3≈ 8 × 10 76 GeV/fm 3 . (9.45)This density corresponds to about 10 77 proton masses in thevolume of a single proton!Proceeding now more systematically, one can find thetimes and energy densities at which different temperatureswere reached. By combining this with the knowledge of particlephysics, one will see what types of particle interactionswere taking place at what time.To relate the temperature to the time, one needs the numberof degrees of freedom, g ∗ . Assuming that nature onlycontains the known particles of the Standard Model, thenfor T greater than several hundred GeV all of them can betreated as relativistic. From (9.20) one has g ∗ = 106.75.If, say, GUT bosons or supersymmetric particles also exist,then one would have a higher value. For the order-ofmagnitudevalues that one is interested in here this uncertaintyin g ∗ will not be critical.Table 9.2 shows values for the temperature and energydensity at several points within the first 10 microsecondsafter the Big Bang, where most of the values have beenrounded to the nearest order of magnitude.Planckian energy densitiescharacteristic temperaturesin the early universe

204 9 The Early UniverseTable 9.2Thermal history of the first 10microseconds‘scale’ T [GeV] ϱ [GeV 4 ] t [s]Planck 10 19 10 78 10 −45GUT 10 16 10 66 10 −39Electroweak 10 2 10 10 10 −11QCD 0.2 0.01 10 −5cosmoparticle physicsHiggs fieldvacuum expectation valueof the Higgs fieldspontaneous symmetrybreakingelectroweak scaleQCD scaleAt the Planck scale, i.e., with energies on the order of10 19 GeV, the limit of our ability to speculate about cosmologyand cosmoparticle physics has been reached. Noticefrom (9.39) that the time when the temperature is equal tothe Planck energy is not the Planck time but is in fact somewhatearlier, so the limit has even been overstepped somewhat.After 10 −39 seconds, when the universe had cooled totemperatures around 10 16 GeV (the ‘GUT scale’), the strong,electromagnetic, and weak interactions start to become distinct,each with a different coupling strength. One expectsthat around this temperature a phase transition related to theHiggs field of the Grand Unified Theory took place. At temperaturesabove the phase transition, the vacuum expectationvalue of the Higgs should be zero, and therefore all ele-mentary particles would be massless, including the X and Ybosons. During the transition, the GUT Higgs field acquiresa vacuum expectation value different from zero; this phenomenonis called spontaneous symmetry breaking (SSB).As a result, the X and Y bosons go from being massless tohaving very high masses on the order of the GUT scale. So,at lower temperatures, baryon-number-violating processesmediated by exchange of X and Y bosons are highly suppressed.After around 10 −11 seconds, the temperature is on theorder of 100 GeV; this is called the ‘electroweak scale’. Hereanother SSB phase transition is expected to occur wherebythe electroweak Higgs field acquires a non-zero vacuum expectationvalue. As a result, W and Z bosons as well as thequarks and leptons acquire their masses. At temperaturessignificantly lower than the electroweak scale, the massesM W ≈ 80 GeV and M Z ≈ 91 GeV are large compared tothe kinetic energies of other colliding particles, and the Wand Z propagators effectively suppress the strength of theweak interaction.At temperatures around 0.2 GeV (the ‘QCD scale’), theeffective coupling strength of the strong interaction, α s ,becomesvery large. At this point quarks and gluons become

9.5 The Baryon Asymmetry of the Universe 205confined into colour-neutral hadrons: protons, neutrons, andtheir antiparticles. This process, called hadronization, occursaround t ≈ 10 −5 s, where to obtain this time from(9.39) one should use a value of g ∗ somewhat smaller thanbefore, since not all of the particles are relativistic. If onetakes photons, gluons, u, d, s, e, µ, and all families of νas relativistic just before hadronization then one would getg ∗ = 61.75.It is interesting to convert the energy density at the QCDscale to normal units, which gives about 1 GeV/fm 3 .Thisis about seven times the density of ordinary nuclear matter.Experiments at the Relativistic Heavy Ion Collider (RHIC)at the Brookhaven National Laboratory near New York arecurrently underway to recreate these conditions by collidingtogether heavy ions at very high energies [12]. This will allowmore detailed studies of the ‘quark–gluon plasma’ andits transition to colour-neutral hadrons.This short sketch of the early universe has ignored atleast two major issues. First, it has not yet been explainedwhy the universe appears to be composed of matter, ratherthan a mixture of matter and antimatter. There is no definitiveanswer to this question but there are plausible scenarioswhereby the so-called baryon asymmetry of the universecould have arisen at very early times, perhaps at the GUTscale or later at the electroweak scale. This question will belooked at in greater detail in the next section.Second, one will see that the model that has been developedthus far fails to explain several observational facts,the most important of which are related to the cosmic microwavebackground radiation. A possible remedy to theseproblems will be to suppose that the energy density at somevery early time was dominated by vacuum energy. In thiscase one will not find R ∼ t 1/2 but rather an exponentialincrease, known as inflation.hadronizationenergy densityquark–gluon plasmabaryon asymmetryvacuum energyinflation9.5 The Baryon Asymmetry of the Universe“Astronomy, the oldest and one of themost juvenile of the sciences, may stillhave some surprises in store. May antimatterbe commended to its case.”Arthur SchusterFor several decades after the discovery of the positron, it appearedthat the laws of nature were completely symmetric

206 9 The Early Universebaryogenesisleptogenesisbetween matter and antimatter. The universe known to us,however, seems to consist of matter only. The relative abundanceof baryons will now be looked at in detail and thereforethis topic is called the baryon asymmetry of the universe.One has to examine how this asymmetry could evolvefrom a state which initially contained equal amounts of matterand antimatter, a process called baryogenesis. Foreveryproton there is an electron, so the universe also seems tohave a non-zero lepton number. This is a bit more difficultto pin down, however, since the lepton number could in principlebe compensated by unseen antineutrinos. In any case,models of baryogenesis generally incorporate in some waylepton production (leptogenesis) as well.Baryogenesis provides a nice example of the interplaybetween particle physics and cosmology. In the final analysisit will be seen that the Standard Model as it stands cannotexplain the observed baryon asymmetry of the universe.This is a compelling indication that the Standard Model isincomplete.9.5.1 Experimental Evidence of Baryon AsymmetryIf antiparticles were to exist in significant numbers locally,one would see evidence of this from proton–antiprotonor electron–positron annihilation. p ¯p annihilation producestypically several mesons including neutral pions, which de-cay into two photons. So, one would see γ rays in an energyrange up to around 100 MeV. No such gamma rays resultingfrom asteroid impacts on other planets are seen, man-madespace probes landing on Mars survived, and Neil Armstronglanding on the Moon did not annihilate, so one can concludethat the entire solar system is made of matter.One actually finds some antiprotons bombarding theEarth as cosmic rays at a level of around 10 −4 comparedto cosmic-ray protons (see Sect. 6.1), so one may want toleave open the possibility that more distant regions are madeof antimatter. But the observed antiproton rate is compatiblewith production in collisions of ordinary high-energy protonswith interstellar gas or dust through reactions of thetypep ¯p annihilationp + p → 3p +¯p. (9.46)antinuclei in cosmic rays?There is currently no evidence of antinuclei in cosmic rays,although the Alpha Magnetic Spectrometer based on the

9.5 The Baryon Asymmetry of the Universe 20710 0Flux [photons cm -2 s -1 MeV -1 sr -1 ]10 -110 -210 -310 -410 -510 -6COMPTELSchönfelder et al. (1980)Trombka et al. (1977)White et al. (1977)1 10Photon Energy [MeV]Fig. 9.1The measured gamma-ray flux(data points) along with the levelspredicted to arise from interactionbetween domains of matter andantimatter. The upper curvecorresponds to domain sizes of20 Mpc, the lower for1000 Mpc [14, 15]International Space Station will carry out more sensitivesearches for this in the next several years [13].If there were to exist antimatter domains of the universe,then their separation from matter regions would haveto be very complete or else one would see the γ -ray fluxfrom proton–antiproton or electron–positron annihilation.The flux that one would expect depends on the size of theseparated domains. Figure 9.1 shows the measured gammarayflux (data points) along with the predicted levels (curves)that would arise from collisions of matter and antimatter regions[14, 15]. The upper curve corresponds to domain sizesof 20 Mpc and is clearly excluded by the data. The lowercurve is for domains of 1000 Mpc and it as well is incompatiblewith the measurements. So one can conclude that ifantimatter regions of the universe exist, they must be separatedby distances on the order of a gigaparsec, which isa significant fraction of the observable universe. Given thatthere is no plausible mechanism for separating matter fromantimatter over such large distances, it is far more natural toassume that the universe is made of matter, i.e., that it has anet non-zero baryon number. Also the absence of a significantflux of 511 keV γ rays from electron–positron annihilationadds to this conclusion.If one then takes as working hypothesis that the universecontains much more matter than antimatter, one needs to askhow this could have come about. One possibility is that thenon-zero baryon number existed as an initial condition, andthat this was preserved up to the present day. This is not ane + e − annihilationabsenceof annihilation radiation

208 9 The Early Universenet baryon numberattractive idea for several reasons. First, although the asymmetrybetween baryons and antibaryons today appears to belarge, i.e., lots of the former and none of the latter, at timescloser to the Big Bang, there were large amounts of both andthe relative imbalance was very small. This will be quantifiedin Sect. 9.5.2. Going back towards the Big Bang onewould like to think that nature’s laws become in some sensemore fundamental, and one would prefer to avoid the needto impose any sort of small asymmetry by hand.Furthermore, it now appears that the laws of natureallow, or even require, that a baryon asymmetry wouldarise from a state that began with a net baryon number ofzero. The conditions needed for this will be discussed inSect. 9.5.3.9.5.2 Size of the Baryon AsymmetryAlthough the universe today seems completely dominatedby baryons and not antibaryons, the relative asymmetry wasvery much smaller at earlier times. This can be seen roughlyby considering a time when quarks and antiquarks were allhighly relativistic, at a temperature of, say, T ≈ 1TeV,and suppose that since that time there have been no baryonnumber-violatingprocesses. The net baryon number in a co-moving volume R 3 is then constant, so one has(n b − n¯b )R3 = (n b,0 − n¯b,0 )R3 0 , (9.47)where the subscript 0 on the right-hand side denotes presentvalues. Today, however, there are essentially no antibaryons,so one can approximate n¯b,0 ≈ 0. The baryon–antibaryonasymmetry A is thereforebaryon-number-violatingprocessesbaryon–antibaryonasymmetryA ≡ n b − n¯b= n b,0 R03n b n b R 3 . (9.48)One can now relate the ratio of scale factors to the ratio oftemperatures, using the relation R ∼ 1/T. Therefore, onegetsA ≈ n b,0 T 3n b T03 . (9.49)Now one can use the fact that the number densities are relatedto the temperature. Equation (9.11) had shownn b ≈ T 3 , (9.50)n γ,0 ≈ T0 3 , (9.51)

9.5 The Baryon Asymmetry of the Universe 209where these are rough approximations with the missing factorsof order unity. Using these ingredients one can expressthe asymmetry asA ≈ n b,0n γ,0. (9.52)Further, the baryon-number-to-photon ratio can be defined:baryon-to-photon ratioη = n b − n¯bn γ. (9.53)One expects this ratio to remain constant as long as there areno further baryon-number-violating processes and there areno extra influences on the photon temperature beyond theHubble expansion. So one can also assume that η refers tothe current value, (n b,0 − n¯b,0 )/n γ,0 ≈ n b,0 /n γ,0 , although– strictly speaking – one should call this η 0 . So one finallyobtains that the baryon–antibaryon asymmetry A is roughlyequal to the current baryon-to-photonratio η. A more carefulanalysis which keeps track of all the missing factors givesA ≈ 6η.The current photon density n γ,0 is well determined fromthe CMB temperature to be 410.4 cm −3 . In principle onecould determine n b,0 by adding up all of the baryons thatone finds in the universe. This in fact is expected to be anunderestimate, since some matter such as gas and dust willnot be visible and these will also obscure stars further away.A more accurate determination of η comes from the modelof Big Bang Nucleosynthesis combined with measurementsof the ratio of abundances of deuterium to hydrogen. Fromthis one finds η ≈ 5 × 10 −10 . So, finally, the baryon asymmetrycan be expressed ascurrent photon densitynucleosynthesisA ≈ 6η ≈ 3 × 10 −9 . (9.54)This means that at early times, for every billion antiquarksthere were a billion and three quarks. The matter in the universeone sees today is just the tiny amount left over afteressentially all of the antibaryons annihilated.9.5.3 The Sakharov ConditionsIn 1967 Andrei Sakharov pointed out that three conditionsmust exist in order for a universe with non-zero baryon numberto evolve from an initially baryon-symmetric state [16].Nature must provide:

210 9 The Early Universebaryon-number-violatingprocessesviolationof C and CP symmetrydeparturefrom thermal equilibriumSakharov conditionsquantum anomaliesincompletenessof the Standard Model1. baryon-number-violating processes;2. violation of C and CP symmetry;3. departure from thermal equilibrium.The first condition must clearly hold, or else a universewith B = 0 will forever have B = 0. In the second condition,C refers to charge conjugation and P to parity. C andCP symmetry roughly means that a system of particles behavesthe same as the corresponding system made of antiparticles.If all matter and antimatter reactions proceed atthe same rate, then no net baryon number develops; thus,violation of C and CP symmetry is needed. The third conditionon departure from equilibrium is necessary in order toobtain unequal occupation of particle and antiparticle states,which necessarily have the same energy levels.A given theory of the early universe that satisfies atsome level the Sakharov conditions will in principle predicta baryon density or, equivalently, a baryon-to-photonratio η.One wants the net baryon number to be consistent with themeasured baryon-to-photon ratio η = n b /n γ ≈ 5×10 −10 .Itis not entirely clear how this can be satisfied, and a detaileddiscussion goes beyond the scope of this book. Here onlysome of the currently favoured ideas will be mentioned.Baryon-number violation is predicted by Grand UnifiedTheories, but it is difficult there to understand how the resultingbaryon density could be preserved when this is combinedwith other ingredients such as inflation. Surprisingly,a non-zero baryon number is also predicted by quantumanomalies 2 in the usual Standard Model, and this is cur-rently a leading candidate for baryogenesis.CP violation is observed in decays of K and B mesonsand it is predicted by the Standard Model of particle physicsbut at a level far too small to be responsible for baryogenesis.If nature includes further CP-violating mechanisms from additionalHiggs fields, as would be present in supersymmetricmodels, then the effect could be large enough to account forthe observed baryon density. This is one of the clearest indicationsfrom cosmology that the Standard Model is incompleteand that other particles and interactions must exist. Ithas been an important motivating factor in the experimental2 Quantum anomalies can arise if a classical symmetry is brokenin the process of quantization and renormalization. The perturbativetreatment of quantum field theories requires a renormalization,and this adds non-invariant counter terms to the invariantLagrange density that one gets at the classical level.

9.6 Problems 211investigation of CP-violating decays of K and B mesons.These experiments have not, however, revealed any effectincompatible with Standard Model predictions.The departure from thermal equilibrium could beachieved simply through the expansion of the universe,i.e., when the reaction rate needed to maintain equilibriumfalls below the expansion rate: Γ ≪ H . Alternatively, itcould result from a phase transition such as those associatedwith spontaneous symmetry breaking.So, the present situation with the baryon asymmetry ofthe universe is a collection of incomplete experimental observationsand partial theories which point towards the creationof a non-zero baryon density at some point in the earlyuniverse. The details of baryogenesis are still murky and remainan active topic of research. It provides one of the closestinterfaces between particle physics and cosmology. Untilthe details are worked out one needs to take the baryon densityof the universe or, equivalently, the baryon-to-photonratio, as a free parameter that must be obtained from observation,see also Fig. 9.2.deviationfrom thermal equilibriuminterfacebetween particle physicsand cosmologyFig. 9.2The matter–antimatter symmetryobserved at microscopic scalesappears to be broken at themacroscopic level9.6 Problems1. Derive the relation between the scale factor R and theenergy density ϱ for a universe dominated by non-relativisticmatter!2. Derive the relation between the scale factor R and timefor an early universe dominated by non-relativistic matter!3. The total energy density of the early universe varies withthe temperature like

212 9 The Early Universeϱ = π 230 g ∗T 4 . (9.55)Work out the expansion rate H as a function of the temperatureand Planck mass using the Friedmann equation!4. In Sect. 9.1 the Planck length has been derived using theargument that quantum and gravitational effects becomeequally important. One can also try to combine the relevantconstants of nature (G, ¯h, c) insuchawaythatacharacteristic length results. What would be the answer?5. Estimate a value for the Schwarzschild radius assumingweak gravitational fields (which do not really apply tothe problem). What is the Schwarzschild radius of theSun and the Earth?Hint: Consider the non-relativistic expression for the escapevelocity and replace formally v by c.6. A gas cloud becomes unstable and collapses under itsown gravity if the gravitational energy exceeds the thermalenergy of its molecules. Estimate the critical densityof a hydrogen cloud at T = 1000 K which would havecollapsed into our Sun! Refer to Problem 8.2 for the stabilitycondition (Jeans criterion).7. Occasionally it has appeared that stars move at superluminalvelocities. In most cases this is due to a motion ofthe star into the direction of Earth during the observationtime. Figure out an example which would give rise to a‘superluminal’ speed!

21310 Big Bang Nucleosynthesis“In fact, it seems that present day science,with one sweeping step back across millionsof centuries, has succeeded in bearing witnessto that primordial ‘Fiat lux’ [let therebe light] uttered at the moment when, alongwith matter, there burst forth from nothinga sea of light and radiation, while the particlesof the chemical elements split andformed into millions of galaxies. Hence, creationtook place in time, therefore, there is aCreator, therefore, God exists!”Pope Pius XIIAt times from around 10 −2 seconds through the first severalminutes after the Big Bang, the temperature passed throughthe range from around 10 to below 10 −1 MeV. During thisperiod protons and neutrons combined to produce a significantamount of 4 He – one quarter of the universe’s nucleiby mass – plus smaller amounts of deuterium (D, i.e., 2 H),tritium ( 3 H), 3 He, 6 Li, 7 Li, and 7 Be. Further synthesis of nucleiin stars accounts for all of the heavier elements plus onlya relatively small additional amount of helium. The predictionsof Big Bang Nucleosynthesis (BBN) are found to agreeremarkably well with observations, and provide one of themost important pillars of the Big Bang model.The two main ingredients of BBN are the equations ofcosmology and thermal physics that have already been described,plus the rates of nuclear reactions. Although the nuclearcross sections are difficult to calculate theoretically,they have for the most part been well measured in laboratoryexperiments. Of crucial importance is the rate of thereaction ν e n ↔ e − p, which allows transformation betweenneutrons and protons. The proton is lighter than the neutronby m = m n − m p ≈ 1.3 MeV, and as long as this reactionproceeds sufficiently quickly, one finds that the neutron-to-protonratio is suppressed by the Boltzmann factore −m/T . 1 At a temperature around 0.7 MeV the reaction isno longer fast enough to keep up and the neutron-to-protonratio ‘freezes out’ at a value of around 1/6. To first approximationone can estimate the helium abundance simply byassuming that all of the available neutrons end up in 4 He.1 In the following the natural constants c, ¯h,andk are set to unity.primordial elementsBig Bang Nucleosynthesistransformation betweenneutrons and protonsfreeze-out temperature

214 10 Big Bang Nucleosynthesisbaryon densityThe one free parameter of BBN is the baryon density Ω bor, equivalently, the baryon-to-photon ratio η. By comparingthe observed abundances of the light elements with thosepredicted by BBN, the value of η can be estimated. The resultwill turn out to be of fundamental importance for thedark-matter problem, which will be dealt with in Chap. 13.10.1 Some Ingredients for BBN“The point of view of a sinner is that thechurch promises him hell in the future,but cosmology proves that the glowinghell was in the past.”Ya.B.Zel’dovichexpansion rateingredients for BBNdisappearance of antimatterneutrino familiesTo model the synthesis of light nuclei one needs the equationsof cosmology and thermal physics relevant for temperaturesin the MeV range. At this point the total energydensity of the universe is still dominated by radiation (i.e.,relativistic particles), so the pressure and energy density arerelated by P = ϱ/3. In Chap. 9 it was shown that this ledto relations for the expansion rate and time as a function oftemperature,H = 1.66 √ T 2g ∗ , (10.1)m Plt = 0.301 m Pl√g∗ T 2 . (10.2)To use these equations one needs to know the effectivenumber of degrees of freedom g ∗ . Recall that quarks andgluons have already become bound into protons and neutronsat around T = 200 MeV, and since the nucleon mass isaround m N ≈ 0.94 GeV, these are no longer relativistic. Nucleonsand antinucleons can remain in thermal equilibriumdown to temperatures of around 50 MeV, and during this periodtheir number densities are exponentially suppressed bythe factor e −mN/T . At temperatures below several tens ofMeV the antimatter has essentially disappeared and the resultingnucleon density therefore does not make a significantcontribution to the energy density.At temperatures in the MeV range, the relativistic particlesare photons, e − , ν e , ν µ , ν τ , and their antiparticles. Thephotons contribute g γ = 2ande + and e − together giveg e = 4. In general, if there are N ν families of neutrinos,

10.2 Start of the BBN Era 215these contribute together g ν = 2N ν . From (9.18) one thereforeobtainsg ∗ = 2 + 7 8 (4 + 2N ν). (10.3)For N ν = 3 one has g ∗ = 10.75. Using this value, (10.2) canbe written in a form convenient for description of the BBNera,tT 2 ≈ 0.74 s MeV 2 . (10.4)10.2 Start of the BBN Era“By the word of the Lord were the heavensmade. For he spoke, and it came tobe; he commanded, and it stood firm.”The Bible; Psalm 33:6,9From (10.4) a temperature of T = 10 MeV is reached at atime t ≈ 0.007 s. At this temperature, all of the relativisticparticles – γ , e − , ν e , ν µ , ν τ , and their antiparticles – are inthermal equilibrium through reactions of the type e + e − ↔ν ¯ν, e + e − ↔ γγ, etc. The number density of the neutrinos,for example, is given by the equilibrium formula appropriatefor relativistic fermions, see (9.11),n ν = 3 ζ(3)4 π 2 g νT 3 , (10.5)with a similar formula holding for the electron density.Already at temperatures around 20 MeV, essentially allof the antiprotons and antineutrons annihilated. The baryonto-photonratio is a number that one could, in principle, predict,if a complete theory of baryogenesis were available.Since this is not the case, however, the baryon density hasto be treated as a free parameter. Since one does not expectany more baryon-number-violating processes at temperaturesnear the BBN era, the total number of protons andneutrons in a comoving volume remains constant. That is,even though protons and neutrons are no longer relativistic,baryon-number conservation requires that the sum of theirnumber densities followsequilibriumof relativistic particlesbaryogenesisbaryon-number conservationn n + n p ∼ 1 R 3 ∼ T 3 . (10.6)At temperatures much greater than the neutron–proton massdifference, m = m n − m p ≈ 1.3 MeV, one has n n ≈ n p .

216 10 Big Bang Nucleosynthesis10.3 The Neutron-to-Proton Ratio“The most serious uncertainty affectingthe ultimate fate of the universe isthe question whether the proton is absolutelystable against decay into lighterparticles. If the proton is unstable, allmatter is transitory and must dissolveinto radiation.”Freeman J. DysonFig. 10.1Feynman diagram for the reactionnν e ↔ pe −neutron-to-proton ratioAlthough the total baryon number is conserved, protons andneutrons can be transformed through reactions like nν e ↔pe − and ne + ↔ p ¯ν e . A typical Feynman diagram is shownin Fig. 10.1.The crucial question is whether these reactions proceedfaster than the expansion rate so that thermal equilibrium ismaintained. If this is the case, then the ratio of neutron-toprotonnumber densities is given by( )n 3/2 n mn= e −(m n−m p )/T ≈ e −m/T , (10.7)n p m pwhere m p = 938.272 MeV, m n = 939.565 MeV, and m =m n − m p = 1.293 MeV. To find out whether equilibriumis maintained, one needs to compare the expansion rate Hfrom (10.1) to the reaction rate Γ . In Sect. 9.3 this rate wasfound to be given by (9.43)Γ = n〈σv〉 . (10.8)reaction ratesweak cross sectionΓ is the the reaction rate per neutron for nν e ↔ pe − ,wherethe brackets denote an average of σv over a thermal distributionof velocities. The number density n in (10.8) refers tothe target particles, i.e., neutrinos, which is therefore givenby (10.5).The cross section for the reaction ν e n ↔ e − p can bepredicted using the Standard Model of electroweak interactions.An exact calculation is difficult but to a good approximationone finds for the thermally averaged speed timescross section〈σv〉≈G 2 F T 2 . (10.9)Here G F = 1.166 × 10 −5 GeV −2 is the Fermi constant,which characterizes the strength of weak interactions.

10.3 The Neutron-to-Proton Ratio 217To find the reaction rate one has to multiply (10.9) by thenumber density n from (10.5). But since the main interesthere is to get a rough approximation, factors of order unitycan be ignored and one can take n ≈ T 3 to obtainΓ(ν e n → e − p) ≈ G 2 F T 5 . (10.10)A similar expression is found for the inverse reaction e − p →ν e n.Now the question of whether this reaction proceedsquickly enough to maintain thermal equilibrium can beaddressed. Figure 10.2 shows the expansion rate H from(10.1) and reaction rate Γ from (10.10) as a function of thetemperature.The point where Γ = H determines the decoupling orfreeze-out temperature T f .Equating the expressions for Γ and H ,G 2 F T 5 = 1.66 √ T 2g ∗ , (10.11)m Pland solving for T gives( ) 1/31.66T f =G 2 F m g∗ 1/6 . (10.12)PlEvaluating this numerically using g ∗ = 10.75 gives T f ≈1.5 MeV. It is interesting to note that (10.12) contains nothingdirectly related to the neutron–proton mass difference,and yet it gives a value very close to m n − m p ≈ 1.3MeV.This coincidence is such that the actual freeze-out temperatureis somewhat lower than that of the naïve calculation. Amore careful analysis givesFig. 10.2The reaction rate Γ(ν e n ↔ e − p)and the expansion rate H as afunction of temperaturefreeze-out temperature T fT f ≈ 0.7MeV. (10.13)At temperatures below T f the reaction ν e n ↔ e − p canno longer proceed quickly enough to maintain the equilibriumnumber densities. The neutron density is said to freezeout, i.e., the path by which neutrons could be converted toprotons is effectively closed. If neutrons were stable, thiswould mean that the number of them in a comoving volumewould be constant, i.e., their number density would follown n ∼ 1/R 3 . Actually, this is not quite true because free neutronscan still decay. But the neutron has a mean lifetimeτ n ≈ 886 s, which is relatively long, but not entirely negligible,compared to the time scale of nucleosynthesis.freeze-out conditionsneutron decay

218 10 Big Bang NucleosynthesisIgnoring for the moment the effect of neutron decay, theneutron-to-proton ratio at the freeze-out temperature isn n= e −(m n−m p )/T f≈ e −1.3/0.7 ≈ 0.16 . (10.14)n pneutron-to-proton ratioAccording to (10.4), this temperature is reached at a timet ≈ 1.5 s. By the end of the next five minutes, essentiallyall of the neutrons become bound into 4 He, and thus theneutron-to-proton ratio at the freeze-out temperature is thedominant factor in determining the amount of helium produced.Before proceeding to predict the helium abundance,a few questions will be addressed in somewhat more detail,namely, neutrino decoupling, positron annihilation, and neutrondecay.10.4 Neutrino Decoupling,Positron Annihilation, and Neutron Decay“Neutrinos, they are very small.They have no charge and have no massand do not interact at all.”John Updikeeffective number of degreesof freedomFor an estimate of the neutron freeze-out temperature ofT f ≈ 0.7 MeV the value g ∗ = 10.75 had been used for theeffective number of degrees of freedom. This corresponds tophotons, e + , e − , and three families of ν and ¯ν as relativisticparticles. The timeline of BBN is complicated slightly bythe fact that g ∗ changes from 10.75 to 3.36 as the temperaturedrops below the electron mass, m e ≈ 0.511 MeV, see(10.16). This means that somewhat more time elapses beforethe synthesis of deuterium can begin and, as a result, moreneutrons have a chance to decay than would be the case withthe higher value of g ∗ .Neutrinos are held in equilibrium with e + and e −through the reaction e + e − ↔ ν ¯ν. At a temperature around1 MeV, however, the rate of this reaction drops below the expansionrate. Muon- and tau-type neutrinos decouple com-pletely from the rest of the particles while ν e continues tointeract for a short while further through ν e n ↔ pe − .Afterthe neutron freeze-out at around 0.7 MeV, all neutrinoflavours are decoupled from other particles. As they are stable,however, their number densities continue to decreasein proportion to 1/R 3 , just like other relativistic particles,neutrino decoupling

10.4 Neutrino Decoupling, Positron Annihilation, and Neutron Decay 219and therefore they still contribute to the effective number ofdegrees of freedom.For T ≫ m e ≈ 0.5 MeV, the reaction e + e − ↔ γγproceeds at the same rate in both directions. As the temperaturedrops below the electron mass, however, the photonsno longer have enough energy to allow γγ → e + e − .Thepositrons as well as all but a small fraction of the initiallypresent electrons annihilate through e + e − → γγ.Theynolonger make a significant contribution to the total energydensity, and therefore they do not contribute towards g ∗ .The fact that the reaction e + e − → γγ produces photonsmeans that the photon temperature decreases less quicklythan it otherwise would. The neutrinos, however, are obliviousto this, and the neutrino temperature continues to scaleas T ν ∼ 1/R. Using thermodynamic arguments based onconservation of entropy, one can show that after positronannihilation the neutrino temperature is lower than that ofthe photons by a factor [17]annihilation and creationneutrino temperature( )T ν 4 1/3= ≈ 0.714 . (10.15)T γ 11In the following T is always assumed to mean the photontemperature.So, to obtain the total energy density, one needs to takeinto account that the neutrinos have a slightly lower temperature,and thus one has to use (9.19),photon temperatureg ∗ = g γ + 7 8 2N Tν4νT 4 = 2 + 7 ( ) 4 4/38 2N ν ≈ 3.36 ,11(10.16)where for the final value N ν = 3 was used.This new value of g ∗ alters the relation between expansionrate and temperature, and therefore also between timeand temperature. Using (10.2) with g ∗ = 3.36 now givestT 2 = 1.32 s MeV 2 . (10.17)In the next section it will be shown that the neutrons, presentat freeze-out, will be able to decay until deuterium productionbegins at around T = 0.085 MeV. From (10.17) thistakes place at a time t ≈ 180 s.To first approximation one can say that the neutronscan decay for around 3 minutes, after which time they arefreeze-out temperature

220 10 Big Bang Nucleosynthesisquickly absorbed to form deuterium and then helium. 2 So,from freeze-out to the start of deuterium production, the neutron-to-protonratio isFig. 10.3The ratio n n /n p as a function ofthe temperatureneutron-to-proton ration n= e −(m n−m p )/T fe −t/τ n, (10.18)n pwhere the mean neutron lifetime is τ n = 885.7 s. Figure 10.3shows n n /n p as a function of the temperature. The freezeouttemperature is at T f = 0.7 MeV, below which the ratiois almost constant, falling slightly because of neutron decay.At a time t = 180 s (T = 0.086 MeV), a value ofn n≈ 0.13 . (10.19)n pis found.10.5 Synthesis of Light Nuclei“Give me matter and I will construct aworld out of it.”Immanuel Kantdeuterium productionbaryon-to-photon ratioThe synthesis of 4 He proceeds through a chain of reactionswhich includes, for example,pn→ dγ , (10.20)dp→ 3 He γ, (10.21)d 3 He → 4 He p. (10.22)The binding energy of deuterium is E bind = 2.2MeV,so ifthe temperature is so high that there are many photons withenergies higher than this, then the deuterium will be brokenapart as soon as it is produced. One might naïvely expect thatthe reaction (10.20) would begin to be effective as soon asthe temperature drops to around 2.2 MeV. In fact this doesnot happen until a considerably lower temperature. This isbecause there are so many more photons than baryons, andthe photon energy distribution, i.e., the Planck distribution,has a long tail towards high energies.The nucleon-to-photon ratio is at this point essentiallythe same as the baryon-to-photon ratio η = n b /n γ ,which2 The calculations are based on average values of thermodynamicdistributions. The approximations therefore may showdiscontinuities which, however, would disappear if Maxwell–Boltzmann or Planck distributions, respectively, were used.

10.5 Synthesis of Light Nuclei 221is around 10 −9 . One can estimate roughly when deuteriumproduction can begin to proceed by finding the temperaturewhere the number of photons with energies greater than2.2 MeV is equal to the number of nucleons. For a nucleonto-photonratio of 10 −9 , this occurs at T = 0.086 MeV,which is reached at a time of t ≈ 3 minutes.Over the next several minutes, essentially all of the neutrons,except those that decay, are processed into 4 He. Theabundance of 4 He is usually quoted by giving its mass fraction,4 He mass fractionY P =mass of 4 Hemass of all nuclei = m Hen Hem N (n n + n p ) , (10.23)where the neutron and proton masses have both been approximatedby the nucleon mass, m N ≈ m n ≈ m p ≈0.94 GeV. There are four nucleons in 4 He, so, neglecting thebinding energy, one has m He ≈ 4m N . Furthermore, there aretwo neutrons per 4 He nucleus, so if one assumes that all ofthe neutrons end up in 4 He, one has n He = n n /2. This willturn out to be a good approximation, as the next most commonnucleus, deuterium, ends up with an abundance four tofive orders of magnitude smaller than that of hydrogen. The4 He mass fraction is thereforeY P = 4m N(n n /2)m N (n n + n p ) = 2(n n/n p )1 + n n /n p≈ 2 × 0.13 ≈ 0.23 . (10.24)1 + 0.13This rough estimate turns out to agree quite well with moredetailed calculations. 23% 4 He mass fraction means that thenumber fraction of 4 He is about 6% with respect to hydrogen.The fact that the universe finally ends up containingaround one quarter 4 He by mass can be seen as the result ofa number of rather remarkable coincidences. For example,mean lifetimes for weak decays can vary over many ordersof magnitude. The fact that the mean neutron lifetime turnsout to be 885.7 s is a complicated consequence of the ratherclose neutron and proton masses combined with strong andweak interaction physics. If it had turned out that τ n were,say, only a few seconds or less, then essentially all of theneutrons would decay before they could be bound up intodeuterium, and the chain of nuclear reactions could not havestarted.4 He number fractiontuning of parameters

222 10 Big Bang Nucleosynthesisfine-tuning of parametersastrochemistry, astrobiologyThe exact value of the decoupling temperature is alsoa complicated mixture of effects, being sensitive, for example,to g ∗ , which depends on the number of relativisticparticle species in thermal equilibrium. If, for example, thedecoupling temperature T f had been not 0.7 MeV but, say,0.1 MeV, then the neutron-to-proton ratio would have beene −1.3/0.1 ≈ 2 × 10 −6 , and essentially no helium would haveformed. On the other hand, if it had been at a temperaturemuch higher than m n − m p , then there would have beenequal amounts of protons and neutrons. Then the entire universewould have been made of helium. Usual hydrogen-burning stars would be impossible, and the universe wouldcertainly be a very different place (see also Chap. 14: Astrobiology).10.6 Detailed BBN“Little by little, time brings out eachseveral thing into view, and reasonraises it up into the shores of light.”LucretiusA detailed modeling of nucleosynthesis uses a system of differentialequations involving all of the abundances and reac-tion rates. The rates of nuclear reactions are parameterizationsof experimental data. Several computer programs thatnumerically solve the system of rate equations are publiclyavailable [18]. An example of predicted mass fractions versustemperature and time is shown in Fig. 10.4.Around t ≈ 1000 s, the temperature has dropped to T ≈0.03 MeV. At this point the kinetic energies of nuclei aretoo low to overcome the Coulomb barriers and the fusionprocesses stop.In order to compare the predictions of Big Bang Nucleosynthesiswith observations, one needs to measure the primordialabundances of the light elements, i.e., as they werejust after the BBN era. This is complicated by the fact thatthe abundances change as a result of stellar nucleosynthesis.For example, helium is produced in stars and deuteriumis broken apart.To obtain the most accurate measurement of the 4 Hemass fraction, for example, one tries to find regions of hotionized gas from ‘metal-poor’ galaxies, i.e., those where rel-mass fractionsstellar nucleosynthesisprimordial 4 He abundance

10.6 Detailed BBN 223Fig. 10.4Evolution of the mass and numberfractions of primordial elements.4 He is given as a mass fractionwhile the other elements arepresented as number fractionsatively small amounts of heavier elements have been producedthrough stellar burning of hydrogen. A recent surveyof data [2] concludes for the primordial 4 He mass fractionY P = 0.238 ± 0.002 ± 0.005 , (10.25)where the first error is statistical and the second reflects systematicuncertainties. In contrast to the helium-4 content ofthe universe which is traditionally given as a mass fraction,the abundances of the other primordial elements are presentedas number fractions, e.g., n7 Li /n p ≡ n7 Li /n H for 7 Li.The best determinations of the 7 Li abundance come fromhot metal-poor stars from the galactic halo. As with 4 He, oneextrapolates to zero metallicity to find the primordial value.Recent data [2] give a lithium-to-hydrogen ratio of7 Li abundancemetal-poor starsn7 Li /n H = 1.23 × 10 −10 . (10.26)The systematic uncertainty on this value is quite large, however,corresponding to the range from about 1 to 2 × 10 −10 .Although deuterium is produced by the first reaction inhydrogen-burning stars through pp → de + ν e , it is quicklyprocessed further into heavier nuclei. Essentially, no netdeuterium production takes place in stars and any presentwould be quickly fused into helium. So to measure the primordialdeuterium abundance, one needs to find gas cloudsat high redshift, hence far away and far back in time, thatprimordial deuterium

224 10 Big Bang Nucleosynthesisdeuterium spectroscopyhave never been part of stars. These produce absorptionspectra in light from even more distant quasars. The hydrogenLyman-α line at λ = 121.6 nm appears at veryhigh redshift (z ≥ 3) in the visible part of the spectrum.The corresponding line from deuterium has a small isotopicshift to shorter wavelengths. Comparison of the two componentsgives an estimate of the deuterium-to-hydrogen ratioD/H ̂= n d /n p . A recent measurement finds [19]n d /n p = (3.40 ± 0.25) × 10 −5 . (10.27)primordial elementabundancesMeasuring of the primordial abundance of 3 He turns outto be more difficult. There do not yet exist sufficiently reliablevalues for 3 He to test or constrain Big Bang Nucleosynthesis.Finally, the predicted abundances of light nuclei are confrontedwith measurements. The predictions depend, however,on the baryon density n b or, equivalently, η = n b /n γ .The predicted mass fraction of 4 He as well as the numbersrelative to hydrogen for D, 3 He, and 7 Li are shown as a functionof η in Fig. 10.5 [2].Fig. 10.5Predictions for the abundances of4 He,D,and 7 Li as a function ofthe baryon-to-photon ratio η. Y P isthe primordial 4 He mass fraction.Traditionally, the 4 He content ofthe universe is given as massfraction, while the other primordialelements are presented as numberfraction (see also the brokenvertical scale). The larger box for7 Li/H includes the systematicalerror added in quadrature to thestatistical errorThe deuterium fraction D/H decreases for increasing ηbecause a higher baryon density means that deuterium isprocessed more completely into helium. Since the resulting

10.6 Detailed BBN 225prediction for D/H depends quite sensitively on the baryondensity, the measured D/H provides the most accurate determinationof η. The measured value from (10.27), namely,n d /n p = (3.40 ± 0.25) × 10 −5 , gives the range of allowedη values, which are shown by the vertical band in Fig. 10.5.This corresponds tosensitive D/H ratioη = (5.1 ± 0.5) × 10 −10 . (10.28)The boxes in Fig. 10.5 indicate the measured abundancesof 4 He and 7 Li (from slightly different analyses than thosementioned above). The size of the boxes shows the measurementuncertainty. These measurements agree remarkablywell with the predictions, especially when one considersthat the values span almost 10 orders of magnitude.The value of η = n b /n γ determines the baryon density,since the photon density is well-known from the measuredCMB temperature, T = 2.725 K, through (9.11) to be n γ =2ζ(3)T 3 /π 2 . Therefore, the value of η can be converted intoa prediction for the energy density of baryons divided by thecritical density,baryon-to-photon ratioΩ b = ϱ bϱ c. (10.29)The critical density is given by (8.33) as ϱ c = 3H 2 0 /8πG.The baryons today are non-relativistic, so their energy densityis simply the number of nucleons per unit volume timesthe mass of a nucleon, i.e., ϱ b = n b m N ,wherem N ≈0.94 GeV. Putting these ingredients together gives Ω b determinationΩ b = 3.67 × 10 7 × ηh −2 , (10.30)where h is defined by H 0 = 100 h km s −1 Mpc −1 .Usingh = 0.71 +0.04−0.03and η from (10.28) gives baryon fractionof the universeΩ b = 0.038 ± 0.005 , (10.31)where the uncertainty originates both from that of the Hubbleconstant and also from that of η. Recently η and Ω b havebeen measured to higher acuracy using the temperature variationsin the cosmic microwave background radiation, leadingto Ω b = 0.044 ± 0.004; this will be followed up inChap. 11 (see also Table 11.1). The values from the BBNand CMB studies are consistent with each other and, takentogether, provide a convincing confirmation of the Big Bangmodel.

226 10 Big Bang Nucleosynthesis10.7 Constraints on the Numberof Neutrino Families“If there were many neutrino generations,we would not be here to countthem, because the whole universe wouldbe made of helium and no life could develop.”AnonymousN ν from cosmologyhow many neutrino families?In this section it will be shown how the comparison of themeasured and predicted 4 He mass fractions can result inconstraints on the particle content of the universe at BBNtemperatures. For example, the Standard Model has N ν = 3,but one can ask whether additional families exist. It will beseen that BBN was able to constrain N ν to be quite closeto three – a number of years earlier than accelerator experimentswere able to determine the same quantity to high precisionusing electron–positron collisions at energies near theZ resonance.Once the parameter η has been determined, the predicted4 He mass fraction is fixed to a narrow range of values closeto Y P = 0.24. As was noted earlier, this prediction is ingood agreement with the measured abundance. The predictiondepended, however, on the effective number of degreesof freedom,g ∗ = 2 + 7 8 (4 + 2N ν), (10.32)where N ν is the number of neutrino families. Earlier theStandard Model value N ν = 3 was used, which gave g ∗ =10.75. This then determines the expansion rate throughFig. 10.6The reaction rate Γ(ν e n ↔ e − p)and the expansion rate H forN ν = 2, 3, and 4 as a function oftemperatureH = 1.66 √ T 2g ∗ . (10.33)m PlThe effective number of degrees of freedom therefore has animpact on the freeze-out temperature, where one has H =Γ(nν e ↔ pe − ), see (10.11). This can be seen in Fig. 10.6,which shows the reaction rate Γ(nν e ↔ pe + ) ≈ G 2 F T 5 andthe expansion rate H versus temperature. The expansion rateis shown using three different values of g ∗ , corresponding toN ν = 2, 3, and 4.From Fig. 10.6 one can see that the freeze-out temperatureT f is higher for larger values of g ∗ , i.e., for higher N ν .At T f the neutron-to-proton ratio freezes out to n n /n p =

10.7 Constraints on the Number of Neutrino Families 227Fig. 10.7The predicted 4 He mass fraction asa function of η for different valuesof N νe −(m n−m p )/T f . If this occurs at a higher temperature, then theratio is higher, i.e., there are more neutrons available to makehelium, and the helium abundance will come out higher.This can be seen in Fig. 10.7, which shows the predictedhelium abundance as a function of the baryon density. Thethree diagonal bands show the predicted Y P for different valuesof an equivalent number of neutrino families N ν = 3.0,3.2, and 3.4. Of course, this no longer represents the (integer)number of neutrino flavours but rather an effective parameterthat simply gives g ∗ . The data are consistent withN ν = 3 and are clearly incompatible with values muchhigher than this [19].The equivalent number of light (i.e., with masses ≤m Z /2) neutrino families has also been determined at theLarge Electron–Positron (LEP) collider from the total widthof the Z resonance, as was shown in Fig. 2.1 (see also Problem3 in this chapter). From a combination of data from theLEP experiments one finds [20]N ν and helium abundanceaccelerator data on N νN ν = 2.9835 ± 0.0083 . (10.34)Although this is 1.7 times the quoted error bar below 3, itis clear that N ν = 3 fits reasonably well and that any otherinteger value is excluded.Before around 1990, when N ν was determined to highprecision in accelerator experiments, BBN measurements

228 10 Big Bang Nucleosynthesishad already found that there could be at most one additionalneutrino family [21] (see also Chap. 2). The interplay betweenthese two different determinations of N ν played animportant rôle in alerting particle physicists to the relevanceof cosmology. Although the example that has just been dis-cussed is for the number of neutrino families, the same argumentsapply to any particles that would contribute to g ∗ suchas to affect the neutron freeze-out temperature. Thus theabundances of light elements provide important constraintsfor any theory involving new particles that would contributesignificantly to the energy density during the BBN era.impact of cosmologyon particle physics10.8 Problems1. Equation (10.4) shows thattT 2 ≈ 0.74 s MeV 2 .How can this numerical result be derived from the previousequations?2. In the determination of the neutron-to-proton ratio thecross section for the (n ↔ p) conversion reactionν e + n → e − + pwas estimated from〈σ v〉≈G 2 F T 2 .How can the T 2 dependence of the cross section be motivated?3. The measurement of the total width Γ Z of the Z resonanceat LEP led to a precise determination of the numberof light neutrinos N ν . How can N ν be obtained fromthe Z width?The solution of this problem requires some intricate detailsof elementary particle physics.

22911 The Cosmic MicrowaveBackground“God created two acts of folly. First, He createdthe universe in a Big Bang. Second, Hewas negligent enough to leave behind evidencefor this act, in the form of the microwaveradiation.”Paul ErdösIn this chapter the description of the early universe to coverthe first several hundred thousand years of its existence willbe presented. This leads to one of the most important pillarsof the Big Bang model: the cosmic microwave backgroundradiation or CMB. It will be seen how and when theCMB was formed and what its properties are. Most importantamong these are its blackbody energy spectrum characterizedby an average temperature of T = 2.725 K, andthe fact that one sees very nearly the same temperature independentof direction. Recent measurements of the slightdependence of the temperature on direction have been usedto determine a number of cosmological parameters to a precisionof several percent.overviewof cosmic microwaveand blackbody radiationprecision determinationof cosmological parameters11.1 Prelude: Transition to aMatter-Dominated Universe“We know too much about matter todayto be materialists any longer.”Harides ChaudhuriPicking up the timeline from the last chapter, Big Bang Nucleosynthesiswas completed by around t ≈ 10 3 s, i.e., whenthe temperature had fallen to several hundredths of an MeV.Any neutrons that by this time had not become bound intoheavier nuclei soon decayed.The break-neck pace of the early universe now shiftsgears noticeably. The next interesting event takes place whenthe energy density of radiation, i.e., relativistic particles(photons and neutrinos), drops below that of the matter (nonrelativisticnuclei and electrons), called the time of ‘matter–radiation equality’. In order to trace the time evolution of thenucleosynthesismatter–radiation equality

230 11 The Cosmic Microwave Backgrounduniverse, one needs to determine when this occurred, sincethe composition of the energy density influences the timedependence of the scale factor R(t).Estimates of the time of matter–radiation equality dependon what is assumed for the contents of the universe.For matter it will be seen that estimates based on the CMBproperties as well as on the motion of galaxies in clusterspresent-day energy densities give Ω m,0 = ϱ m,0 /ϱ c,0 ≈ 0.3, where as usual the sub-script 0 indicates a present-day value. For photons it willbe found from the CMB temperature: Ω γ,0 = 5.0 × 10 −5 .Taking into account neutrinos brings the total for radiationto Ω r,0 = 8.4 × 10 −5 , so currently, matter contributes some3 600 times more to the total energy density than does radiation.In Chap. 8 it has been derived by solving the Friedmannequation how to predict the time dependence of theR dependence different components of the energy density. For radiationof energy densities ϱ r ∼ 1/R 4 was obtained, whereas for matter ϱ m ∼ 1/R 3was found. Therefore, the ratio follows ϱ m /ϱ r ∼ R, anditwas thus equal to unity when the scale factor R was 3 600times smaller than its current value.In order to pin down when this occurred, one needs totime dependence of R know how the scale factor varies in time. If it is assumedthat the universe has been matter dominated from the timeof matter–radiation equality up to now, then one has R ∼t 2/3 , and this leads to a time of matter–radiation equality, t mr ,of around 66 000 years. In fact, there is now overwhelmingevidence that vacuum energy makes up a significant portionof the universe, with Ω Λ ≈ 0.7. Taking this into accountleads to a somewhat earlier time of matter–radiation equalityof around t mr ≈ 50 000 years.When the dominant component of the energy densitytransition changes from radiation to matter, this alters the relation be-to a matter-dominated tween the temperature and time. For non-relativistic particleenergy density types with mass m i and number density n i , one now gets forthe energy densityϱ ≈ ∑ im i n i , (11.1)where the sum includes at least baryons and electrons, andperhaps also ‘dark-matter’ particles. The Friedmann equation(neglecting the curvature term) then readsH 2 = 8πG ∑m i n i . (11.2)3i

11.2 Discovery and Basic Properties of the CMB 231Assuming the particles that contribute to the energy densityare stable, one obtainsn i ∼ 1/R 3 ∼ T 3 . (11.3)Furthermore, since R ∼ t 2/3 , the expansion rate is nowgiven bytime dependence of R and Tunder matter dominationH = ṘR = 2 3t . (11.4)Combining (11.3) with (11.4) or R ∼ t 2/3 then leads toT 3 ∼ 1t 2 . (11.5)This is to be contrasted with the relation T 2 ∼ t −1 valid forthe era when the energy density was dominated by relativisticparticles.11.2 Discovery and Basic Propertiesof the CMB“What we have found is evidence for thebirth of the universe. It’s like looking atGod.”George SmootThe existence of the CMB was predicted by Gamow [22]in connection with Big Bang Nucleosynthesis. It was shownin Chap. 10 that BBN requires temperatures around T ≈0.08 MeV, which are reached at a time t ≈ 200 s. By knowingthe cross section for the first reaction, p + n → d + γ ,and the number density n of neutrons and protons, one canpredict the reaction rate Γ = n〈σv〉.In order for BBN to produce the observed amount of helium,one needs a sufficiently high rate for the deuteriumfusion reaction over the relevant time scale. This correspondsto requiring Γt to be at least on the order of unityat t ≈ 200 s, when the temperature passes through the relevantrange. This assumption determines the nucleon densityduring the BBN phase.Since the BBN era, the nucleon and photon densitieshave both followed n ∼ 1/R 3 ∼ T 3 . So, by comparing thenucleon density in the BBN era to what one finds today, thecurrent temperature of the photons can be predicted. Reasoningalong these lines, Alpher and Herman [23] estimateddeuterium fusionphoton temperatureestimation

232 11 The Cosmic Microwave Backgroundphoton contributionto the cosmic energy densityprediction and observationof the CMBearth-based vs.satellite-supportedCMB measurementsa CMB temperature of around 5 K, which turned out to benot far off.Even without invoking BBN one can argue that the con-tribution of photons to the current energy density cannotexceed by much the critical density. If one assumes, say,Ω γ ≤ 1, then this implies T ≤ 32 K.Gamow’s prediction of the CMB was not pursued for anumber of years. In the 1960s, a team at Princeton (Dicke,Peebles, Roll, and Wilkinson) did take the prediction seriouslyand set about building an experiment to look forthe CMB. Unknown to them, a pair of radio astronomers,A. Penzias and R. Wilson at Bell Labs in New Jersey, werecalibrating a radio antenna in preparation for studies unrelatedto the CMB. They reported finding an “effective zenithnoise temperature . . . about 3.5 ◦ K higher than expected.This excess temperature is, within the limits of our observations,isotropic, unpolarized, and free from seasonal variations...” [24]. The Princeton team soon found out aboutPenzias’ and Wilson’s observation and immediately suppliedthe accepted interpretation [25].Although the initial observations of the CMB were consistentwith a blackbody spectrum, the earth-based observationswere only able to measure accurately the radiation atwavelengths of several cm; shorter wavelengths are stronglyabsorbed by the water in the atmosphere. The peak of theblackbody spectrum for a temperature of 3 K, however, isaround 2 mm. It was not until 1992 that the COBE satellitemade accurate measurements of the CMB from space. Thisshowed that the form of the energy distribution is extremelyclose to that of blackbody radiation, i.e., to a Planck distribution,as shown in Fig. 11.1.Wavelength [mm]2 1 0.67 0.5Fig. 11.1The spectrum of the CMBmeasured by the COBE satellitetogether with the blackbody curvefor T = 2.725 K. The error barshave been enlarged by a factor of400; any deviations from thePlanck curve are less than 0.005%(from [26])Intensity [MJy/sr]400300200100FIRAS data with 400σ errorbars2.725 K Blackbody00 5 10 15 20ν [1/cm]

11.3 Formation of the CMB 23311.3 Formation of the CMB“The old dream of wireless communicationthrough space has now been realizedin an entirely different manner thanmany had expected. The cosmos’ shortwaves bring us neither the stock marketnor jazz from different worlds. With softnoises they rather tell the physicists ofthe endless love play between electronsand protons.”Albrecht UnsöldAt very early times, any protons and electrons that managedto bind together into neutral hydrogen would be dissociatedvery quickly by collision with a high-energy photon. As thetemperature decreased, the formation of hydrogen eventuallybecame possible and the universe transformed from anionized plasma to a gas of neutral atoms; this process iscalled recombination. The reduction of the free electron densityto almost zero meant that the mean free path of a photonsoon became so long that most photons have not scatteredsince. This is called the decoupling of photons from matter.By means of some simple calculations one can estimatewhen recombination and decoupling took place.Neutral hydrogen has an electron binding energy of13.6 eV and is formed through the reactionrecombination: creationof hydrogen atomsdecoupling of photonsfrom matterp + e − → H + γ. (11.6)Naïvely one would expect the fraction of neutral hydrogento become significant when the temperature drops below13.6 eV. But because the baryon-to-photon ratio η ≈5 × 10 −10 is very small, the temperature must be significantlylower than this before the number of photons withE > 13.6 eV is comparable to the number of baryons.(This is the same basic argument for why deuterium productionbegan not around T = 2.2 MeV, the binding energyof deuterium, but rather much lower.) One finds thatthe numbers of neutral and ionized atoms become equal ata recombination temperature of T rec ≈ 0.3 eV (3500 K). Atthis point the universe transforms from an ionized plasma toan essentially neutral gas of hydrogen and helium.It can be estimated when recombination took place bycomparing the temperature of the CMB one observes today,T 0 ≈ 2.73 K, to the value of the recombination temperatureT rec ≈ 0.3 eV. Recall from Sect. 9.2.4 that the wavelengthrecombination temperatureestimationof the recombination era

234 11 The Cosmic Microwave Backgroundof a photon follows λ ∼ R. Therefore, the ratio of the scalefactor R at a previous time to its value R 0 now is related tothe redshift z throughR 0R = λ 0= 1 + z. (11.7)λFurthermore, today’s CMB temperature is measured to beT 0 ≈ 2.73 K and it is known that T ∼ 1/R. Therefore onegets1+z = T ≈ 0.3eVT 0 2.73 K × 18.617 × 10 −5 ≈ 1300 ,eV K−1 (11.8)recombination time≈ 300 000 yearsdecoupling of photonsfrom matterdecoupling temperaturewhere Boltzmann’s constant was inserted to convert temperaturefrom units of eV to K. If one assumes that the scalefactor follows R ∼ t 2/3 from this point up to the present,then it is found that recombination was occurring at( ) R 3/2 ( ) 3/2 T0t 0t rec = t 0 = t 0 =R 0 T rec (1 + z rec ) 3/2≈ 1.4 × 1010 years(1300) 3/2 ≈ 300 000 years . (11.9)Shortly after recombination, the mean free path for aphoton became so long that photons effectively decoupledfrom matter. While the universe was an ionized plasma, thephoton scattering cross section was dominated by Thomsonscattering, i.e., elastic scattering of a photon by an electron.The mean free path of a photon is determined by the numberdensity of electrons, which can be predicted as a functionof time, and by the Thomson scattering cross section,which can be calculated. As the universe expands, the electrondensity decreases leading to a longer mean free pathfor the photons. This path length becomes longer than thehorizon distance (the size of the observable universe at agiven time) at a decoupling temperature of T dec ≈ 0.26 eV(3000 K) corresponding to a redshift of 1 + z ≈ 1100. Thiscondition defines decoupling of photons from matter. Thedecoupling time is( ) 3/2 T0t 0t dec = t 0 =≈ 380 000 years .T dec (1 + z dec ) 3/2(11.10)

11.4 CMB Anisotropies 235Once the photons and matter decoupled, the photonssimply continue unimpeded to the present day. One can definea surface of last scattering as the sphere centered aboutus with a radius equal to the mean distance to the last placewhere the CMB photons scattered. To a good approximationthis is equal to the distance to where decoupling took placeand the time of last scattering is essentially the same as t dec .So, when the CMB is detected, one is probing the conditionsin the universe at a time of approximately 380 000 years afterthe Big Bang.surface of last scatteringtransparent universe11.4 CMB Anisotropies“As physics advances farther and fartherevery day and develops new axioms,it will require fresh assistancefrom mathematics.”Francis BaconThe initial measurements of the CMB temperature by Penziasand Wilson indicated that the temperature was independentof direction, i.e., that the radiation was isotropic, towithin an accuracy of around 10%. More precise measurementseventually revealed that the temperature is about onepart in one thousand hotter in one particular direction of thesky than in the opposite. This is called the dipole anisotropyand is interpreted as being caused by the motion of the Earththrough the CMB. Then in 1992 the COBE satellite foundanisotropies at smaller angular separations at a level of onepart in 10 5 . These small variations in temperature have recentlybeen measured down to angles of several tenths of adegree by several groups, including the WMAP 1 satellite,from which one can extract a wealth of information aboutthe early universe.In order to study the CMB anisotropies one begins witha measurement of the CMB temperature as a function of direction,i.e., T(θ,φ),whereθ and φ are spherical coordinates,i.e., polar and azimuthal angle, respectively. As withany function of direction, it can be expanded in sphericalharmonic functions Y lm (θ, φ) (a Laplace series),dipole anisotropydiscoveryof small-angle anisotropiesLaplace series∞∑ l∑T(θ,φ)= a lm Y lm (θ, φ) . (11.11)l=0 m=−l1 WMAP – Wilkinson Microwave Anisotropy Probe

236 11 The Cosmic Microwave BackgroundSome of the mathematical formalism of the Laplace seriesis given in Appendix A.2. This expansion is analogous toa Fourier series, where the higher-order terms correspondto higher frequencies. Here, terms at higher l correspond tostructures at smaller angular scales. The same mathematicaltechnique is used in the multipole expansion of the poten-tial from an electric charge distribution. The terminology isborrowed from this example and the terms in the series arereferred to as multipole moments. The l = 0termisthemonopole, l = 1 the dipole, etc.Once one has estimates for the coefficients a lm ,theamplitudeof regular variation with angle can be summarizedby definingmultipole expansionangular power spectrumC l = 12l + 1l∑m=−l|a lm | 2 . (11.12)The set of numbers C l is called the angular power spectrum.The value of C l represents the level of structure found at anangular separationθ ≈ 180◦ . (11.13)lThe measuring device will in general only be able to resolveangles down to some minimum value; this determines themaximum measurable l.11.5 The Monopole and Dipole Terms“The universe contains the record of itspast the way that sedimentary layers ofrock contain the geological record of theEarth’s past.”Heinz R. PagelsThe l = 0 term in the expansion of T(θ,φ)gives the temperatureaveraged over all directions. The most accurate de-termination of this value comes from the COBE satellite,monopole termIn the 1970s it was discovered that the temperature ofthe CMB in a particular direction was around 0.1% hotterthan in the opposite direction. This corresponds to a nonzerovalue of the l = 1ordipole term in the Laplace expan-sion. The dipole anisotropy has recently been remeasureddipole term〈T 〉=2.725 ± 0.001 K . (11.14)

11.5 The Monopole and Dipole Terms 237by the WMAP experiment, which finds a temperature differenceofTT = 1.23 × 10−3 . (11.15)The temperature appears highest in the direction (right ascensionand declination, see Appendix C for the definition ofthese coordinates) (α, δ) = (11.20 h , −7.22 ◦ ). This temperaturevariation has a simple interpretation, namely, the movementof the Earth through the (local) unique reference framein which the CMB has no dipole anisotropy. This frame is insome sense the (local) ‘rest frame’ of the universe. The solarsystem and with it the Earth are moving through it with aspeed of v = 371 km/s towards the constellation Crater (betweenVirgo and Hydra). The CMB is blueshifted to slightlyhigher temperature in the direction of motion and redshiftedin the opposite direction. This dipole pattern in the map ofthe CMB temperature is shown in Fig. 11.2. The map is anequal-area projection in galactic coordinates with the planeof the Milky Way running horizontally through the plot.dipole anisotropylocal rest frameFig. 11.2Map of the CMB temperaturemeasured by the COBE satellite.The dipole pattern is due to themotion of the Earth through theCMB (from [27]) {27}11.5.1 Small-Angle AnisotropyIf small density fluctuations were to exist in the early universe,then one would expect these to be amplified by gravity,with more dense regions attracting even more matter, untilthe matter of the universe was separated into clumps. Thisis how galaxy formation is expected to have taken place.Given that one sees a certain amount of clumpiness today,one can predict what density variations must have existedat the time of last scattering. These variations would correspondto regions of different temperature, and so from thefluctuationsand galaxy formation

238 11 The Cosmic Microwave Backgroundobserved large-scale structure of the universe one expectedto see anisotropies in the CMB temperature at a level ofaround one part in 10 5 .These small-angle anisotropies were finally observed bythe COBE satellite in 1992. COBE had an angular resolu-tion of around 7 ◦ and could therefore determine the powerspectrum up to a multipole number of around l = 20. In thefollowing years, balloon experiments were able to resolvemuch smaller angles but with limited sensitivity. Finally, in2003 the WMAP project made very accurate measurementsof the CMB temperature variations with an angular resolutionof 0.2 ◦ . One of the maps (with the dipole term subtracted)is shown in Fig. 11.3.angular resolutionof COBE and WMAPFig. 11.3Cosmographic map of the CMBtemperature measured by theWMAP satellite with the dipolecomponent subtracted (from [28]){28}Because of the better angular resolution of WMAP comparedto COBE, it was possible to make an accurate measurementof the angular power spectrum up to l ≈ 1000, asshown in Fig. 11.4.11.6 Determinationof Cosmological Parameters“There are probably few features of theoreticalcosmology that could not becompletely upset and rendered uselessby new observational discoveries.”Sir Hermann Bondiconclusions from the CMBangular power spectrumThe angular power spectrum of the CMB can be used tomake accurate determinations of many of the most importantcosmological parameters, including the Hubble constant H ,the baryon-to-photon ratio η, the total energy density over

11.6 Determination of Cosmological Parameters 239Fig. 11.4CMB power spectrum. The set ofmeasurements with the smallererror bars is from WMAP; thosewith the larger errors represent anaverage of measurements prior toWMAP (from [29])the critical density, Ω, as well as the components of the energydensity from baryons, Ω b , and from all non-relativisticmatter Ω m .As an example, in the following a rough idea will begiven of how the angular power spectrum is sensitive to Ω.Consider the largest region that could be in causal contactat the time of last scattering t ls ≈ t dec ≈ 380 000 years.This distance is called the particle horizon d H . Naïvely onewould expect this to be d H = t (i.e., ct, butc = 1 has beenassumed). This is not quite right because the universe is expanding.The correct formula for the particle-horizon distanceat a time t in an isotropic and homogeneous universeis (see, e.g., [30]),∫ tdt ′d H (t) = R(t)0 R(t ′ ) . (11.16)If the time before matter–radiation equality is considered(t mr ≈ 50 000 years), then R ∼ t 1/2 and d H (t) = 2t. Forthe matter-dominated era one has R ∼ t 2/3 and d H = 3t. Ifa sudden switch from R ∼ t 1/2 to R ∼ t 2/3 is assumed att mr , then one finds from integrating (11.16) a particle horizonat t ls ofd H (t ls ) = 3t ls − t 2/3lstmr 1/3 ≈ 950 000 (light-)years .(11.17)As most of the time up to t ls is matter dominated, the resultis in fact close to 3t ls .Ω determinationparticle horizondistancesin an expanding universe

240 11 The Cosmic Microwave Backgrounddensity fluctuationsas sound wavesproper distanceangular diameter distanceA detailed modeling of the density fluctuations in theearly universe predicts a large level of structure on distancescales roughly up to the horizon distance. These fluctuationsare essentially sound waves in the primordial plasma, i.e.,regular pressure variations resulting from the infalling ofmatter into small initial density perturbations. These initialperturbations may have been created at a much earlier time,e.g., at the end of the inflationary epoch.By looking at the angular separation of the temperaturefluctuations, in effect one measures the distance betweenthe density perturbations at the time when the photons wereemitted.To relate the angles to distances, one needs to reviewbriefly the proper distance and angular diameter distance.The proper distance d p at a time t is the length one wouldmeasure if one could somehow stop the Hubble expansionand lay meter sticks end to end between two points. In anexpanding universe one finds that the current proper distance(i.e., at t 0 ) to the surface of last scattering is given by∫ t0dtd p (t ls ) = R(t 0 )t lsR(t) , (11.18)angular variationsof temperature variationsNote that this is the current proper distance to the positionof the photon’s emission, assuming that that place hasbeen carried along with the Hubble expansion. (The particle-horizondistance used above is simply the proper distanceto the source of a photon emitted at t = 0.) If matterdomination is assumed, i.e., R ∼ t 2/3 since t ls , then onegets d p (t ls ) = 3(t 0 − t ls ), which can be approximated byd p (t ls ) ≈ 3t 0 .Now, what one wants to know is the angle subtended bya temperature variation which was separated by a distanceperpendicular to our line of sight of δ = 3t ls when the photonswere emitted. To obtain this one needs to divide δ notby the current proper distance from us to the surface of lastscattering, but rather by the distance that it was to us at thetime when the photons started their journey. This locationhas been carried along with the Hubble expansion and isnow further away by a factor equal to the ratio of the scalefactors, R(t 0 )/R(t ls ). Using (11.7), therefore, one findsθ =δ R(t 0 )d p (t ls ) R(t ls ) = δ (1 + z) , (11.19)d p (t ls )

11.6 Determination of Cosmological Parameters 241where z ≈ 1100 is the redshift of the surface of last scattering.Thus, if a region is considered whose size was equal tothe particle-horizon distance at the time of last scattering asgiven by (11.17) as viewed from today, then it will subtendan angleθ ≈ 3t ls − t 2/3lstmr1/3(1 + z) (11.20)3t 0950 000 a180◦≈3 × 1.4 × 10 10 × 1100 ×a π ≈ 1.4◦ .The structure at or just below this angular scale correspondsto the ‘acoustic peaks’ visible in the power spectrum startingat around l ≈ 200.The naming of the structures in the power spectrumas ‘acoustic peaks’ comes about for the following reason:as already mentioned, the density fluctuations in the earlyuniverse caused gravitational instabilities. When the matterfell into these gravitational potential wells, this matterwas compressed, thereby getting heated up. This hot matterradiated photons causing the plasma of baryons to expand,thereby cooling down and producing less radiation as a consequence.With decreasing radiation pressure the irregularitiesreach a point where gravity again took over initiatinganother compression phase. The competition between gravitationalaccretion and radiation pressure caused longitudinalacoustic oscillations in the baryon fluid. After decouplingof matter from radiation the pattern of acoustic oscillationsbecame frozen into the CMB. CMB anisotropies thereforeare a consequence of sound waves in the primordial protonfluid.The angle subtended by the horizon distance at the timeof last scattering depends, however, on the geometry of theuniverse, and this is determined by Ω, the ratio of the energydensity to the critical density. This is illustrated schematicallyin Fig. 11.5. In Fig. 11.5(a), Ω = 1isassumedandtherefore the universe is described by a flat geometry. Theangles of the triangle sum to 180 ◦ and the angle subtendedby the acoustic horizon has a value a bit less than 1 ◦ . If,however, one has Ω

242 11 The Cosmic Microwave BackgroundFig. 11.5The horizon distance at the time oflast scattering as viewed by ustoday in (a) a flat universe (Ω = 1)and (b) an open universe (Ω

11.6 Determination of Cosmological Parameters 243are shown in Table 11.1. Among the most important of these,one sees that the Hubble constant is now finally known toan accuracy of several percent, and the value obtained is ingood agreement with the previous average (h = 0.7 ± 0.1).Furthermore, the universe is flat, i.e., Ω is so close to unitythat surely this cannot be a coincidence.value andparameterexperimentalerrorHubble constant h 0.71 +0.04−0.03ratio of total energy density to ϱ c , Ω 1.02 ± 0.02baryon-to-photon ratio η 6.1 +0.3−0.2 × 10−10baryon energy density over ϱ c , Ω b 0.044 ± 0.004matter energy density over ϱ c , Ω m 0.27 ± 0.04vacuum energy density over ϱ c , Ω Λ 0.73 ± 0.04Table 11.1Some of the cosmologicalparameters determined by theWMAP experiment throughmeasurement of the CMB angularpower spectrumData from the Cosmic Background Imager (CBI) [33]– an array of antennae operating at frequencies from 26 to36 GHz in the Atacama Desert – confirm the picture of a flatuniverse, giving Ω = 0.99 ± 0.12 in agreement with resultsfrom the Boomerang 2 and Maxima 3 experiments, and, ofcourse, with WMAP.In addition, WMAP has determined the baryon-to-photonratio η to an accuracy of several percent. The value foundis a bit higher than that obtained from the deuterium abundance,η = (5.1 ± 0.5) × 10 −10 , but the latter measurement’serror did not represent the entire systematic uncertaintyand, in fact, the agreement between the two is quitereasonable. The fact that two completely independent measurementsyield such close values is surely a good sign. Thedensities from non-relativistic matter Ω m and from vacuumenergy Ω Λ are similarly well determined, and these measurementsconfirm earlier values based on completely differentobservables.Further improvement in the determination of theseand other parameters is expected from the PLANCKproject (formerly called COBRAS/SAMBA), scheduled tobe launched by the European Space Agency in 2007 [34].2 Boomerang – Balloon Observations of Millimetric ExtragalacticRadiation and Geophysics3 Maxima – Millimeter Anisotropy Experiment Imaging Arraybaryon-to-photon ratiomeasured by WMAPimproved measurementsin future

244 11 The Cosmic Microwave Backgroundparticle physicsin the early universeThese experiments provide another close bridge betweencosmology and the particle physics of the early universe.In addition to helping pin down important cosmological parameters,the CMB may provide the most important cluesneeded to understand particle interactions at ultrahigh energyscales – energies that will never be attained by manmadeparticle accelerators. This theme will be further exploredin the next chapter on inflation.11.7 Problems1. What is the probability that a photon from the Big Bangundergoes a scattering after having passed the surface oflast scattering?2. Estimate a limit for the cosmological constant!3. The Friedmann equation extended by the cosmologicalconstant is nothing but a relation between differentforms of energy:m2 Ṙ2} {{ }kinetic(+− GmMR − 1 6 mΛc2 R 2 )} {{ }potential=−kc 2 m 2} {{ }total energy(m is the mass of a galaxy at the edge of a galactic clusterof mass M).Work out the pressure as a function of R related to theclassical potential energy in comparison to the pressurecaused by the term containing the cosmological constant.4. Work out the average energy of blackbody microwavephotons! For the integration of the Planck distributionrefer to Chap. 27 in [35] or Formulae 3.411 in [36] orcheck with the web page:http://jove.prohosting.com/~skripty/page_998.htm.For the Riemann zeta function look athttp://mathworld.wolfram.com/RiemannZetaFunction.html.5. Estimate the energy density of the cosmic microwavebackground radiation at present and at the time of lastscattering.6. Why is the decoupling temperature for photons at lastscattering (0.3 eV) much lower compared to the ionizationenergy of hydrogen (13.6 eV)?

24512 Inflation“Inflation hasn’t won the race, but so far it’sthe only horse.”Andrei LindeThe Standard Cosmological Model appears to be very successfulin describing observational data, such as the abundancesof light nuclei, the isotropic and homogeneous expansionof the universe, and the existence of the cosmicmicrowave background. The CMB’s very high degree ofisotropy and the fact that the total energy density is closeto the critical density, however, pose problems in that theyrequire a very specific and seemingly arbitrary choice of initialconditions for the universe. Furthermore, it turns outthat Grand Unified Theories (GUTs) as well as other possibleparticle physics theories predict the existence of stableparticles such as magnetic monopoles, which no one hasyet succeeded in observing. These problems can be solvedby assuming that at some very early time, the total energydensity of the universe was dominated by vacuum energy.This leads to a rapid, accelerating increase in the scale factorcalled inflation.This chapter takes a closer look at the problems mentionedabove, how inflation solves them, what else inflationpredicts, and how these predictions stand in comparison toobservations. In doing this it will be necessary to make predictionsfor the expansion of the universe from very earlytimes up to the present. For these purposes it will be sufficientto treat the universe as being matter dominated after atime of around 50 000 years, preceded by an era of radiationdomination (except for the period of inflation itself). In factit is now believed this picture is not quite true, and the currentenergy density is dominated again by a sort of vacuumenergy. This fact will not alter the arguments relevant for thepresent chapter and it will be ignored here; the topic of vacuumenergy in the present universe will be taken up again inthe next chapter.initial conditionsfor the universemonopoles?predictions of inflationeras of the universevacuum energy

246 12 Inflation12.1 The Horizon Problem“The existing universe is bounded innone of its dimensions; for then it musthave an outside.”Lucretiuscosmic microwavebackgroundregions in causal contacthorizon problemIn the previous chapter it was shown that, after correcting forthe dipole anisotropy, the CMB has the same temperature towithin one part in 10 5 coming from all directions. This radiationwas emitted from the surface of last scattering at a timeof around t ls ≈ 380 000 years. In Sect. 11.6 it was calculatedthat two places separated by the particle-horizon distance att ls are separated by an angle of around 1.4 ◦ as viewed today.This calculation assumed R ∼ t 1/2 for the first 50 000 yearsduring the radiation-dominated era, followed by a matterdominatedphase with R ∼ t 2/3 fromthenuptot ls .Foranymixture of radiation and matter domination one would findvalues in the range from one to two degrees.So, if regions of the sky separated in angle by more thanaround 2 ◦ are considered, one would expect them not to havebeen in causal contact at the time of last scattering. Further-more, the projected entire sky can be divided into more than10 4 patches which should not have been in causal contact,and yet, they are all at almost the same temperature.The unexplained uniform temperature in regions that appearto be causally disconnected is called the ‘horizon problem’.It is not a problem in the sense that this model makesa prediction that is in contradiction with observation. Thedifferent temperatures could have perhaps all had the sametemperature ‘by chance’. This option is not taken seriously.The way that systems come to the same temperature is by interacting,and it is hard to believe that any mechanism otherthan this was responsible.12.2 The Flatness Problem“This type of universe, however, seemsto require a degree of fine-tuning ofthe initial conditions that is in apparentconflict with ‘common wisdom’.”Idit Zehavi and Avishai DekelΩ ≈ 1It was found in Sect. 11.6 that the total energy density ofthe universe is very close to the critical density or Ω ≈ 1.

12.2 The Flatness Problem 247Although this is now known to hold to within around 2%, ithas been clear for many years that Ω is at least constrainedto roughly 0.2

248 12 InflationΩ at the Planck timeGoing further back in time makes the problem moreacute. Suppose that the universe was radiation dominatedfor times less than t mr going all the way back to the Plancktime, t Pl ≈ 10 −43 s. Then one should take R ∼ t 1/2 ,whichmeans Ṙ ∼ t −1/2 and therefore 1Ω − 1 ∼ kR 2 ∼ kt . (12.6)flatness problem≈ 10 −59 . (12.7)Thatis,tobeabletofindΩ of order unity today, the modelrequires it to be within 10 −59 of unity at the Planck time.As with the horizon problem, the issue is not one ofa prediction that stands in contradiction with observation.There is nothing to prevent Ω from being arbitrarily closeto unity at early times, but within the context of the cosmo-logical model that has been described so far, it could havejust as easily had some other value. And if other values areapriorijust as likely, then it seems ridiculous to believe thatnature would pick Ω ‘by chance’ to begin so close to unity.One feels that there must be some reason why Ω came outthe way it did.fine-tuning of ΩUsing this dependence for times earlier than t mr , one findsfor Ω − 1 at the Planck time relative to the value now:( )Ω(t Pl ) − 1Ω(t 0 ) − 1 = R(tPl ) 2R(t mr )R(t mr )) R(t 0 ) = t ( ) 2/3Pl tmrt mr t 010 −43 ( )s50 000 a 2/3≈50 000 a × 3.2 × 10 7 s/a 1.4 × 10 10 a12.3 The Monopole Problem“From the theoretical point of view onewould think that monopoles should exist,because of the prettiness of the mathematics.”Paul Adrian Maurice Diracphase transitionsin the early universeIn order to understand the final ‘problem’ with the modelthat has been presented up to now, one needs to recallsomething about phase transitions in the early universe. InChap. 9 it was remarked that a sort of phase transition took1 The Boltzmann constant and the curvature parameter are traditionallydenoted with the same letter k. From the context itshould always be clear which parameter is meant.

12.3 The Monopole Problem 249place at a critical temperature around the GUT energy scale,T c ≈ E GUT ≈ 10 16 GeV. 2 As the temperature dropped belowT c , the Higgs field, which is responsible for the massesof the X and Y bosons, acquired a non-zero vacuum expectationvalue. There is an analogy between this process andthe cooling of a ferromagnet, whereby the magnetic dipolessuddenly line up parallel to their neighbours. Any directionis equally likely, but once a few of the dipoles have randomlychosen a particular direction, their neighbours follow along.The same is true for the more common transition of waterwhen it freezes to snowflakes. In water the molecules moverandomly in all directions. When the water is cooled belowthe freezing point, ice crystals or snowflakes form wherethe water molecules are arranged in a regular pattern whichbreaks the symmetry of the phase existing at temperaturesabove zero degree Celsius. In the case of the Higgs, the analogueof the dipole’s direction or crystal orientation is nota direction in physical space, but rather in an abstract spacewhere the axes correspond to components of the Higgs field.As with the ferromagnet or the orientation axis of the icecrystals, the components of the field tend to give the sameconfiguration the same way in a given local region.If one considers two regions which are far enough apartso they are not in causal contact, then the configuration acquiredby the Higgs field in each will not in general be thesame. At the boundary between the regions there will bewhat is called a ‘topological defect’, analogous to a dislocationin a ferromagnetic crystal. The simplest type of defectis the analogue of a point dislocation, and in typical GrandUnified Theories, these carry a magnetic charge: they aremagnetic monopoles. Magnetic monopoles behave as particleswith masses of roughlyHiggs fieldferromagnetsymmetry-breakingmechanismstopological defectmagnetic monopolesm mon ≈ M Xα U≈ 10 17 GeV , (12.8)where the X boson’s mass, M X ≈ 10 16 GeV, is roughly thesame as the GUT scale and the effective coupling strength isaround α U ≈ 1/40.A further crucial prediction is that these monopoles arestable. Owing to their high mass they contribute essentiallyfrom the moment of their creation to the non-relativisticmatter component of the universe’s energy density. One expectsmonopoles to be produced with a number density of2 As usual, here and in the following the standard notation c = 1,¯h = 1, and k = 1 (Boltzmann constant) is used.stable monopoles

250 12 Inflationcausally isolated regionmonopole number densitymonopole energy densityroughly one in every causally isolated region. The size ofsuch a region is determined by the distance which light cantravel from the beginning of the Big Bang up to the timeof the phase transition at t c . This distance is simply theparticle horizon from (11.16) at the time t c . If a radiationdominatedphase for times earlier than t c is assumed, thenone gets R ∼ t 1/2 and therefore a particle horizon of 2t c .The monopole number density is therefore predicted to ben mon ≈ 1(2t c ) 3 . (12.9)The critical temperature T c is expected to be on the orderof the GUT scale, M X ≈ 10 16 GeV, from which one findsthe time of the phase transition to be t c ≈ 10 −39 s. As themonopoles are non-relativistic, their energy density is givenbyϱ mon = n mon m mon ≈ M Xα U1(2t c ) 3 ≈ 2 × 1057 GeV 4 .(12.10)photon energy densityplethora of monopolesThis can compared with the energy density of photons at thesame time, which isϱ γ = π 215 T GUT 4 ≈ 2 × 1063 GeV 4 . (12.11)So, initially, the energy of photons still dominates overmonopoles by a factor ϱ γ /ϱ mon ≈ 10 6 . But as photonsare always relativistic, one has ϱ γ ∼ 1/R 4 , whereas forthe monopoles, which are non-relativistic, ϱ mon ∼ 1/R 3holds. The two energy densities would become equal afterR increased by 10 6 , which is to say, after the temperaturedropped by a factor of 10 6 , because of R ∼ 1/T. Sincethe time follows t ∼ T −2 , equality of ϱ γ and ϱ mon occursafter the time increases by a factor of 10 12 .So,startingat the GUT scale around T GUT ≈ 10 16 GeV or at a timet GUT ≈ 10 −39 s, one would predict ϱ γ = ϱ mon at a temperatureT γ mon ≈ 10 10 GeV or at a time t γ mon ≈ 10 −27 s.This is clearly incompatible with what one observes today.Searches for magnetic monopoles have been carriedout, and in a controversial experiment by Cabrera in 1982,evidence for a single magnetic monopole was reported [37].This appears, however, to have been a one-time glitch, asno further monopoles were found in more sensitive experiments.Indeed, far more serious problems would arise from

12.4 How Inflation Works 251the predicted monopoles. Foremost among these is that theenergy density would be so high that the universe wouldhave recollapsed long ago. Given the currently observed expansionrate and photon density, the predicted contributionfrom monopoles would lead to a recollapse in a matter ofdays.So, here is not merely a failure to explain the universe’sinitial conditions, but a real contradiction between a predictionand what one observes. One could always argue, ofcourse, that Grand Unified Theories are not correct, and indeedthere is no direct evidence that requires such a pictureto be true. One can also try to arrange for types of GUTswhich do not produce monopoles, but these sorts of the theoryare usually disfavoured for other reasons. Historically,the monopole problem was one of the factors that motivatedAlan Guth to propose the mechanism of inflation, an acceleratingphase in the early universe. In the next sections itwill first be defined more formally what inflation is and howit could come about, and then it will be shown how it solvesnot only the monopole problem but also the horizon and flatnessproblems as well.recollapsedue to magnetic monopolesmonopole probleminflation solves the puzzles12.4 How Inflation Works“No point is more central than this, thatempty space is not empty. It is the seatof the most violent physics.”John Archibald WheelerInflation is defined as meaning a period of accelerating expansion,i.e., where ¨R >0. In this section it will be investigatedhow this can arise. Recall first the Friedmann equation,which can be written asṘ 2R 2 + k R 2 = 8πG ϱ, (12.12)3where here the term ϱ is understood to include all formsof energy including that of the vacuum, ϱ v .Itwasshownin Chap. 8 that this arises if one considers a cosmologicalconstant Λ. This gives rise to a constant contribution to thevacuum which can be interpreted as vacuum energy densityof the formϱ v =Λ8πG . (12.13)vacuum energycosmological constant

252 12 InflationOne can now ask what will happen if the vacuum energyor, in general, if any constant term dominates the totalenergy density. Suppose this is the case, i.e., ϱ ≈ ϱ v ,andassumeas well that one can neglect the k/R 2 term; this shouldalways be a good approximation for the early universe. TheFriedmann equation then readsṘ 2R 2 = 8πG3 ϱ v . (12.14)Thus, one finds that the expansion rate H is a constant,exponential increaseof the scale factoracceleration equationH = Ṙ√8πGR = 3 ϱ v . (12.15)The solution to (12.15) for t>t i is[√ ]8πGR(t) = R(t i ) e H(t−ti) = R(t i ) exp3 ϱ v (t − t i )[√ ]Λ= R(t i ) exp3 (t − t i) . (12.16)That is, the scale factor increases exponentially in time.More generally, the condition for a period of acceleratingexpansion can be seen by recalling the accelerationequation from Sect. 8.6,¨RR =−4πG (ϱ + 3P) . (12.17)3This shows that one will have an accelerating expansion, i.e.,¨R >0, as long as the energy density and pressure satisfyϱ + 3P

12.5 Mechanisms for Inflation 253One now needs to ask several further questions: “Whatcould cause such a vacuum energy density?”, “How and whydid inflation stop?”, “How does this solve any of the previouslymentioned problems?”, and “What are the observationalconsequences of inflation?”.12.5 Mechanisms for Inflation“Using the forces we know now, youcan’t make the universe we know now.”George SmootTo predict an accelerating scale factor, an equation of staterelating pressure P and energy density ϱ of the form P =wϱ with w

254 12 Inflationexcitation of this field corresponds to a spin-0 particle. Theenergy density ϱ and pressure P associated with φ are (inunits with ¯h = c = 1) given by [38]ϱ = 1 2 ˙φ 2 + 1 2 (∇φ)2 + V(φ), (12.20)P = 1 2 ˙φ 2 − 1 6 (∇φ)2 − V(φ). (12.21)Higgs fieldThe term V corresponds to a potential which could emergefrom some particle physics theory. A particular form for Vis, for example, predicted for a Higgs field (see below). Fornow V(φ)will be treated as a function which one can choosefreely.Now, for a field φ(x,t), which is almost constant in timeand space such that the ˙φ 2 and (∇φ) 2 terms can be neglectedrelative to V(φ), one then hasϱ ≈−P ≈ V(φ), (12.22)Fig. 12.1Schematic illustration of thepotential V(φ)associated with theHiggs field φ in the StandardModel of particle physicsHiggs potentialFig. 12.2Schematic illustration of thepotential V(φ)first proposed toprovide inflationmodified Higgs potentialand therefore the equation-of-state parameter w is approximately−1. Since this fulfills the relation w0.Now, in constructing a quantum field theory there is afairly wide degree of freedom in writing down the potentialterm V(φ). For example, Fig. 12.1 shows the potential associatedwith the Higgs field. This is a scalar field in the StandardModel of particle physics needed to explain the massesof the known particles.In 1981, Alan Guth [39] proposed that a scalar Higgsfield associated with a Grand Unified Theory could be responsiblefor inflation. The potential of this field should havea dip around φ = 0, as shown in Fig. 12.2. As a consequenceof the local minimum at φ = 0, there should be a classicallystable configuration of the field at this position. In this state,the energy density of the field is given by the height of thepotential at φ = 0. Suppose the field goes into this state at atime t i . The scale factor then follows an exponential expansion,(√ )8πGϱR(t) = R(t i ) exp (t − t i ) , (12.23)3with ϱ = V(0).In a classical field theory, if the field were to settle intothe local minimum at φ = 0, then it would stay there forever.In a quantum-mechanical theory, however, it can tunnel from

12.5 Mechanisms for Inflation 255this ‘false vacuum’ to the true vacuum with V = 0. Whenthis happens, the vacuum energy will no longer dominateand the expansion will be driven by other contributions to ϱsuch as radiation.In the quantum field theories such as the Standard Modelof particle physics, different fields interact. That is, energyfrom one field can be transformed into excitations in another.This is how reactions are described in which particles arecreated and destroyed. So, when the inflaton field (see belowfor more details) moves from the false to the true vacuum,one expects its energy to be transformed into other ‘normal’particles such as photons, electrons, etc. Thus, the end of inflationsimply matches onto the hot expanding universe ofthe existing Big Bang model. It may then appear that inflationhas not made any predictions that one can verify byobservation, and as described so far no verification has beengiven. Later in this chapter, however, it will be shown howinflation provides explanations for the initial conditions ofthe Big Bang, which otherwise would have to be imposedby hand.Shortly after the original inflaton potential was proposed,it was realized by Guth and others that the modelhad important flaws. As the quantum-mechanical tunnelingis a random process, inflation should end at slightly differenttimes in different places. Therefore, some places will undergoinflation for longer than others. And because these regionscontinue to inflate, their contribution to the total volumeof the universe remains significant. Effectively, inflationwould never end. This has been called inflation’s ‘gracefulexit problem’.It was pointed out by Linde [40] and Albrecht and Steinhardt[41] that by a suitable modification of the potentialV(φ), one could achieve a graceful ending to inflation. Thepotential needed for this is shown schematically in Fig. 12.3.The field it describes is called an inflaton, as its potential isnot derived from particle physics considerations such as theHiggs mechanism, but rather its sole motivation is to providefor inflation. The local minimum in V at φ = 0isreplacedby an almost flat plateau. The field can settle downinto a metastable state near φ ≈ 0, and then effectively ‘roll’down the plateau to the true vacuum. One can show that inthis scenario, called new inflation, the exponential expansionends gracefully everywhere.inflaton fieldhow inflation endsquantum-mechanicaltunneling problemgraceful exit probleminflatonnew inflation

256 12 InflationFig. 12.3Schematic illustration of thepotential V(φ)for ‘new inflation’quantum fluctuationsend of inflationdensity fluctuationsAs in Guth’s original theory, however, new inflation doesnot end everywhere at the same time. But now this featureis turned to an advantage. It is used to explain the structureor clumpiness of the universe currently visible on distancescales less than around 100 Mpc. Because of quantum fluctuations,the value of the field φ at the start of the inflationaryphase will not be exactly the same at all places. As with therandomness of quantum-mechanical tunneling, these quantumfluctuations lead to differences, depending on position,in the time needed to move to the true vacuum state.When a volume of space is undergoing inflation, the energydensity ϱ ≈ V(0) is essentially constant. After inflationends, the energy is transferred to particles such as photons,electrons, etc. Their energy density then decreases as theuniverse continues to expand, e.g., ϱ ∼ R −4 for relativisticparticles. The onset of this decrease is therefore delayed inregions where inflation goes on longer. Thus, the variation inthe time of the end of inflation provides a natural mechanismto explain spatial variations in energy density. These densityfluctuations are then amplified by gravity and finally resultin the structures that one sees today, e.g., galaxies, clusters,and superclusters.12.6 Solution to the Flatness Problem“I have just invented an anti-gravitymachine. It’s called a chair.”Richard P. Feynmanwhy is ϱ = ϱ c ?It will now be shown that an early period of inflationary expansioncan explain the flatness problem, i.e., why the energydensity today is so close to the critical density. Supposethat inflation starts at some initial time t i and continues until

12.6 Solution to the Flatness Problem 257a final time t f , and suppose that during this time the energydensity is dominated by vacuum energy ϱ v , which could resultfrom some inflaton field. The expansion rate H duringthis period is given by√8πGH =3 ϱ v . (12.24)So from t i to t f , the scale factor increases by a ratioR(t f )R(t i ) = eH(t f−t i ) ≡ e N , (12.25)where N = H(t f − t i ) represents the number of e foldingsof expansion during inflation.Referring back to (12.3), it was shown that the Friedmannequation could be written asnumber of e foldingsof the scale factorΩ − 1 = k Ṙ 2 . (12.26)Now, during inflation, one has R ∼ e Ht , with H = Ṙ/Rconstant. Therefore, one findsΩ during inflationΩ − 1 ∼ e −2Ht (12.27)during the inflationary phase. That is, during inflation, Ω isdriven exponentially towards unity.What one would like is to have this exponential decreaseof |Ω − 1| to offset the divergence of Ω from unity duringthe radiation- and matter-dominated phases. To do this onecan work out how far one expects Ω to differ from unityfor a given number of e foldings of inflation. It has beenshown that during periods of matter domination one has|Ω − 1| ∼t 2/3 , see (12.4), and during radiation domination|Ω − 1| ∼t, see (12.6). Let us assume inflation starts at t i ,ends at t f , then radiation dominates until t mr ≈ 50 000 a, followedby matter domination to the present, t 0 ≈ 14 × 10 9 a.The current difference between Ω and unity is thereforeΩ is driven to unityduring inflation( )( ) 2/3|Ω(t 0 ) − 1| =|Ω(t i ) − 1| e −2H(t f−t i ) tmr t0.t f t mr(12.28)In Chap. 11 it was seen that the WMAP data indicate|Ω(t 0 ) − 1| < 0.04. If, for example, one assumes that theinflaton field is connected to the physics of a Grand UnifiedTheory, then one expects inflation to be taking place

258 12 Inflationinflationary periodbeginning of inflationin time scales around 10 −38 to 10 −36 s. Suppose one takest f = 10 −36 s and uses the WMAP limit |Ω(t 0 )−1| < 0.04. Ifone assumes that before inflation Ω(t i ) − 1 is of order unity,then this implies that the number of e foldings of inflationmust be at least[N> 1 ( )( ) ]2 ln |Ω(t i ) − 1|2/3 tmr t0≈ 61 .|Ω(t 0 ) − 1| t f t mr(12.29)Suppose that before inflation began, the universe was radiationdominated. This means that the expansion rate followedH = 1/2t. By the ‘beginning of inflation’ at time t i , the timeis meant when the vacuum energy began to dominate, so oneshould have roughly H ≈ 1/t i during inflation. So, the numberof e foldings is basically determined by the duration ofthe inflationary period,N = H(t f − t i ) ≈ t f − t it i. (12.30)N ≈ 100?So, if t f ≈ 10 −36 s, then t i could not have been much earlierthan 10 −38 s. Of course, inflation might have gone onlonger. Equation (12.29) only gives the smallest number ofe foldings needed to explain why Ω is within 0.04 of unitytoday.12.7 Solution to the Horizon Problem“There was no ‘before’ the beginning ofour universe, because once upon a timethere was no time.”John D. Barrowhorizon problemIt can also be shown that an early period of inflationary expansioncan explain the horizon problem, i.e., why the entiresky appears at the same temperature. To be specific, supposethat inflation starts at t i = 10 −38 s, ends at t f = 10 −36 s,and has an expansion rate during inflation of H = 1/t i .Assumingthe universe was radiation dominated up to t i ,theparticle-horizon distance d H at this point wasd H = 2ct i ≈ 2 × 3 × 10 8 m/s × 10 −38 s = 6 × 10 −30 m .(12.31)

12.8 Solution to the Monopole Problem 259This is the largest region where one would expect to find thesame temperature, since any region further away would notbe in causal contact. Now, during inflation, a region of sized expands in proportion to the scale factor R by a factor e N ,where the number of e foldings is from (12.30) N ≈ 100.After inflation ends, the region expands following R ∼ t 1/2up to the time of matter–radiation equality (50 000 a) andthen in proportion to t 2/3 from then until the present, assumingmatter domination. So, the current size of the regionwhich would have been in causal contact before inflation is( ) 1/2 ( ) 2/3d(t 0 ) = d(t i ) e N tmr t0≈ 10 38 m .t f t mr(12.32)This distance can be compared to the size of the currentHubble distance, c/H 0 ≈ 10 26 m. So, the currently visibleuniverse, including the entire surface of last scattering, fitseasily into the much larger region which one can expect tobe at the same temperature. With inflation, it is not true thatopposite directions of the sky were never in causal contact.So, the very high degree of isotropy of the CMB can be understood.region in causal contactHubble distance12.8 Solution to the Monopole Problem“If you can’t find them, dilute them.”AnonymousThe solution to the monopole problem is equally straightforward.One simply has to arrange for the monopoles tobe produced before or during the inflationary period. Thisarises naturally in models where inflation is related to theHiggs fields of a Grand Unified Theory and works, of course,equally well if inflation takes place after the GUT scale. Themonopole density is then reduced by the inflationary expansion,leaving it with so few monopoles that one would notexpect to see any of them.To see this in numbers, let us suppose the monopoles areformed at a critical time t c = 10 −39 s. Suppose, as in theprevious example, the start and end times for inflation aret i = 10 −38 sandt f = 10 −36 s, and let us assume an expansionrate during inflation of H = 1/t i .ThisgivesN ≈ 100e foldings of exponential expansion. The volume containinga given number of monopoles increases in proportion tomonopole densitydilution during inflation

260 12 Inflationdilutionof magnetic monopolespresent-daymonopole densityR 3 , so that during inflation, the density is reduced by a factore 3N ≈ 10 130 . Putting together the entire time evolutionof the monopole density, one takes R ∼ t 1/2 during radiationdomination from t c to t i ,thenR ∼ e Ht during inflation,followed again by a period of radiation domination up tothe time of matter–radiation equality at t mr = 50 000 a, followedby matter domination with R ∼ t 2/3 up to the present,t 0 = 14 × 10 9 a. Inserting explicitly the necessary factorsof c, one would therefore expect today a monopole numberdensity of( )1 −3/2 tin m (t 0 ) ≈(2ct c ) 3 e −3(t f−t i )/t it c( ) −3/2 ( ) −2 tmr t0×. (12.33)t f t mrUsing the relevant numbers one would getn m (t 0 ) ≈ 10 −114 m −3 ≈ 3 × 10 −38 Gpc −3 . (12.34)So, the number of monopoles one would see today would besuppressed by such a huge factor that one would not evenexpect a single monopole in the observable part of the universe.12.9 Inflation and the Growth of Structure“The universe is not made, but is beingmade continuously. It is growing, perhapsindefinitely.”Henri Bergsonspecific initial conditionsThe primary success of inflationary models is that they providea dynamical explanation for specific initial conditions,which otherwise would need to be imposed by hand. Furthermore,inflation provides a natural mechanism to explainthe density fluctuations which grew into the structures suchas galaxies and clusters as are seen today. But beyond thisqualitative statement, one can ask whether the properties ofthe predicted structure indeed match what one observes.The level of structure in the universe is usually quantifiedby considering the relative difference between the densityat a given position, ϱ(x), and the average density 〈ϱ〉,δ(x) =ϱ(x) −〈ϱ〉〈ϱ〉. (12.35)

12.9 Inflation and the Growth of Structure 261The quantity δ(x) is called the density contrast. Suppose acube of size L is considered, i.e., volume V = L 3 ,andthedensity contrast inside V is expanded into a Fourier series.Assuming periodic boundary conditions, this givesdensity contrastδ(x) = ∑ δ(k) e i k·x , (12.36)where the sum is taken over all values of k = (k x ,k y ,k z )which fit into the box, e.g., k x = 2πn x /L, with n x =0, ±1, ±2,..., and similarly for k y and k z . Averaging overall directions provides the average magnitude of the Fouriercoefficients as a function of k =|k|. Nowthepower spectrumis defined aspower spectrumP(k) =〈|δ(k)| 2 〉 . (12.37)Its interpretation is simply a measure of the level of structurepresent at a wavelength λ = 2π/k. Different observationscan provide information about the power spectrum at differentdistance scales. For example, at distances up to around100 Mpc, surveys of galaxies such as the Sloan Digital Sky Sloan Digital Sky SurveySurvey (SDSS) can be used to measure directly the galaxydensity. At larger distance scales, i.e., smaller values of k,the temperature variations in the CMB provide the most accurateinformation. Some recent measurements of P(k) areshown in Fig. 12.4 [42].In most cosmological theories of structure formation,one finds a power law valid for large distances (small k) ofthe formP(k) ∼ k n . (12.38)Using the value n = 1forthescalar spectral index giveswhat is called the scale-invariant Harrison–Zel’dovich spectrum.Most inflationary models predict n ≈ 1 to withinaround 10%. The exact value for a specific model is relatedto the form of the potential V(φ), i.e., how long and flat itsplateau is (see, e.g., [43] and Fig. 12.3).The curve shown in Fig. 12.4 is based on a model withn = 1 for large distances, and this agrees well with the data.Depending on the specific model assumptions, the best determinedvalues of the spectral index are equal to unity towithin around 10%. This is currently an area with a close interplaybetween theory and experiment, and future improvementsin the measurement of the power spectrum shouldlead to increasingly tight constraints on models of inflation.scalar spectral indexmodels of inflation

262 12 InflationFig. 12.4Measurements of the powerspectrum P(k)from several typesof observations. The curve showsthe prediction of a model with thespectral index n equal to unity. Theparameter h isassumedtobe0.72(from [42])12.10 Outlook on Inflation“The argument seemed sound enough,but when a theory collides with a fact,the result is tragedy.”Louis Nizertests of inflationdark energy nowSo far, inflation seems to have passed several important tests.Its most generic predictions, namely an energy density equalto the critical density, a uniform temperature for the observableuniverse, and a lack of relic particles created at any preinflationarytime, are well confirmed by observation. Fur-thermore, there are no serious alternative theories that comeclose to this level of success. But inflation is not a singletheory but rather a class of models that include a period ofaccelerating expansion. There are still many aspects of thesemodels that remain poorly constrained, such as the nature ofthe energy density driving inflation and the time when it existed.Concerning the nature of the energy density, it will beseen in Chap. 13 that around 70% of the current energy densityin the universe is in fact ‘dark energy’, i.e., somethingwith properties similar to vacuum energy. Data from type-Ia supernovae and also information from CMB experimentssuch as WMAP indicate that for the last several billion

12.10 Outlook on Inflation 263years, the universe has been undergoing a renewed quasiexponentialexpansion compatible with a value of Ω Λ,0 =ϱ v,0 /ϱ c,0 ≈ 0.7. (The subscript Λ refers to the relation betweenvacuum energy and a cosmological constant; the subscript0 denotes as usual present-day values.) The criticaldensity is ϱ c,0 = 3H0 2/8πG,whereH 0 ≈ 70 km s −1 Mpc −1is the Hubble constant. This gives a current vacuum energydensity of around (¯hc ≈ 0.2GeVfm)current vacuum energydensityϱ v,0 ≈ 10 −46 GeV 4 . (12.39)In the example of inflation considered above, however, themagnitude of the vacuum energy density is related to the expansionrate during inflation by (12.15). Furthermore, it wasargued that the expansion rate is approximately related to thestart time of inflation by H ≈ 1/t i . So, if one believes thatinflation occurred at the GUT scale with, say, t i ≈ 10 −38 s,then one needs a vacuum energy density of, see (9.40),dark energy at inflationϱ v = 3m2 Pl32πt 2 i≈ 10 64 GeV 4 . (12.40)The vacuum energies now and those which existed duringan earlier inflationary period may well have a common explanation,but given their vast difference, it is by no meansobvious what their relationship is.One could try to argue that inflation took place at a latertime, and therefore had a smaller expansion rate and correspondinglylower ϱ v . The latest possible time for inflationwould be just before the Big Bang Nucleosynthesis era,around t ≈ 1 s. With inflation any later than this, the predictionsof BBN for abundances of light nuclei would be alteredand would no longer stand in such good agreement with observation.Even at an inflation time of t i = 1 s, one wouldneed a vacuum energy density of ϱ v ≈ 10 −12 GeV 4 , still 34orders of magnitude greater than what is observed today.In addition to those observable features mentioned, inflationpredicts the existence of gravitational waves. Thesewould present a fossilized record of the first moments oftime. In particular, the energy density of gravity waves isexpected to beϱ gravity waves = h2 ω 2which gives32πG(12.41)inflation timegravitational wavesfrom inflation

264 12 InflationΩ gravity waves = ϱ gravity wavesϱ c= h2 ω 212H 2 . (12.42)gravity-wave detectionnew experimentalpossibilities?This would result in distortions of gravity-wave antennaeon the order of h = 10 −27 for kHz gravity waves. Eventhough present gravitational-wave antennae do not reach thiskind of sensitivity, a detection of the predicted gravitationalwavebackground at the level expected in inflationary modelswould result in a credibility comparable to that whichwas achieved for the Big Bang by the observation of the2.7 K blackbody radiation.Even though the detection of gravity waves has not beenestablished experimentally, it might still be possible in thefuture to find means of testing these predictions of inflationarymodels with new techniques or new ideas. At the time ofthe discovery of the blackbody radiation in 1965 by Penziasand Wilson it seemed unrealistic to assume that one mightbe able to measure the spectrum at the level at which it isknown now. With future satellites, like with the EuropeanPlanck mission, even further improvements are envisaged.12.11 Problems1. A value of the cosmological constant Λ can be estimatedfrom the Friedmann equation extended by the cosmologicalterm. It is often said that this value disagrees byabout 120 orders of magnitude with the result from theexpected vacuum energy in a unified supergravity theory.How can this factor be illustrated?2. Work out the time dependence of the size of the universefor a flat universe with a cosmological constant Λ.3. Work out the time evolution of the universe if Λ werea dynamical constant (Λ = Λ 0 (1 + αt)) for a Λ-dominated flat universe.4. Estimate the size of the universe at the end of the inflationperiod (≈ 10 −36 s). Consider that for a matterdominateduniverse its size scales as t 2 3 , while for aradiation-dominated universe one has R ∼ t 1 2 .

26513 Dark Matter“There is a theory, which states that if everanyone discovers exactly what the universeis for and why it is here, it will instantly disappearand be replaced by something evenmore bizarre and inexplicable. There is anothertheory, which states that this has alreadyhappened.”Douglas AdamsRecent observations have shown that the universe is flat, i.e.,Ω = 1. The baryonic matter constitutes only a small fractionon the order of 4%. The main ingredients of the universeare presented by material and/or energy that one can onlyspeculate about. 23% of the matter in the universe is darkand 73% is given by dark energy (see Fig. 13.1). There isno clear idea what kind of material dominates over the wellknownbaryonic matter. Even the visible matter constitutesonly a small fraction of the total baryonic matter.The following sections discuss some candidates for theunseen baryonic matter and offer some proposals for themissing non-baryonic matter and dark energy.energy contentof the universe13.1 Large-Scale Structure of the Universe“There are grounds for cautious optimismthat we may now be near the endof the search for the ultimate laws of nature.”Stephen W. HawkingOriginally it has been assumed that the universe is homogeneousand isotropic. However, all evidence speaks to thecontrary. Nearly on all scales one observes inhomogeneities:stars form galaxies, galaxies form galactic clusters, andthere are superclusters, filaments of galaxies, large voids,and great walls, just to name a few of them. The largescalestructure has been investigated up to distances of about100 Mpc. On these large scales a surprising lumpiness wasfound. One has, however, to keep in mind that the spatialdistribution of galaxies must not necessarily coincide withthe distribution of matter in the universe.Fig. 13.1Illustration of the relative fractionsof dark matter, dark energy, andbaryonic matterinhomogeneitiesof the universe

266 13 Dark Matterearly universedevelopmentof large-scale structuresThe measurement of the COBE and WMAP satellites onthe inhomogeneities of the 2.7 Kelvin blackbody radiationhas shown that the early universe was much more homoge-neous. On the other hand, small temperature variations ofthe blackbody radiation have served as seeds for structures,which one now observes in the universe.It is generally assumed that the large-scale structureshave evolved from gravitational instabilities, which can betraced back to small primordial fluctuations in the energydensity of the early universe. Small perturbations in the energydensity are amplified due to the associated gravitationalforces. In the course of time these gravitational aggregationscollect more and more mass and thus lead to structure formation.The reason for the original microscopic small inhomogeneitiesis presumably to be found in quantum fluctua-tions. Cosmic inflation and the subsequent slow expansionhave stretched these inhomogeneities to the presently observedsize. One consequence of the idea of cosmic inflationis that the exponential growth has led to a smooth and flatuniverse which means that the density parameter Ω shouldbe very close to unity. To understand the formation of thelarge-scale structure of the universe and its dynamics in detail,a sufficient amount of mass is required because otherwisethe original fluctuations could never have been transformedinto distinct mass aggregations. It is, however, truethat the amount of visible matter only does not support thecritical density of Ω = 1 as required by inflation.To get a flat universe seems to require a second Copernicanrevolution. Copernicus had noticed that the Earth wasnot the center of the universe. Cosmologists now conjecturethat the kind of matter, of which man and Earth are made,only plays a minor rôle compared to the dark non-baryonicmatter, which is – in addition to the dark energy – neededto understand the dynamics of the universe and to reach thecritical mass density.quantum fluctuationsinflation → Ω = 1major rôleof non-baryonic dark matter13.2 Motivation for Dark Matter“If it’s not DARK, it doesn’t MATTER.”AnonymousThe idea that our universe contains dark matter is not entirelynew. Already in the thirties of the last century Zwicky

13.2 Motivation for Dark Matter 267[44] had argued that clusters of galaxies would not be gravitationallystable without additional invisible dark matter.Recent observations of high-z supernovae and detailed measurementsof the cosmic microwave background radiationhave now clearly demonstrated that large quantities of darkmatter must exist, which fill up the universe. An argumentfor the existence of invisible dark matter can already be inferredfrom the Keplerian motion of stars in galaxies. Keplerhad formulated his famous laws, based on the precision measurementsof Tycho Brahe. The stability of orbits of planetsin our solar system is obtained from the balance of centrifugaland the attractive gravitational force:circumstantial evidencefor dark mattermv 2r= G mMr 2 (13.1)(m is the mass of the planet, M is the mass of the Sun, r isthe radius of the planet’s orbit assumed to be circular). Theresulting orbital velocity is calculated to bev = √ GM/r . (13.2)The radial dependence of the orbital velocity of v ∼ r −1/2is perfectly verified in our solar system (Fig. 13.2).The rotational curves of stars in galaxies, however, showa completely different pattern (Fig. 13.3). Since one assumesthat the majority of the mass is concentrated at the centerof a galaxy, one would at least expect for somewhat largerdistances a Keplerian-like orbital velocity of v ∼ r −1/2 .Instead,the rotational velocities of stars are almost constanteven for large distances from the galactic center.The flat rotational curves led to the conclusion that thegalactic halo must contain nearly 90% of the mass of thegalaxy. To obtain a constant orbital velocity, the mass ofthe galactic nucleus in (13.1) has to be replaced by the nowdominant mass of invisible matter in the halo. This requirementleads to a radial dependence of the density of this massofϱ ∼ r −2 , (13.3)becausemv 2= G m ∫ r0 ϱ dVrr 2 ∼ G mϱVr 2 ∼ G mr−2 r 3r 2⇒ v 2 = const . (13.4)Fig. 13.2Rotational curves of planets in oursolar system, 1 Astronomical Unit(AU) = distance Earth to Sunorbital velocity [km/s]150100500radius [kpc]NGC 6503halogalacticdiskgas0 10 20Fig. 13.3Rotational curves of the spiralgalaxy NGC 6503. Thecontributions of the galactic disk,the gas, and the halo are separatelyshown

268 13 Dark Mattermass density54321parameterizationof the mass densityof a galaxy(π a 2 /M) σ(r)(π a 3 /M) ρ(r)00 0.2 0.4 0.6 0.8 1radius [a (galaxy radius)]Fig. 13.4Mass density of a galaxy for a twoandthree-dimensional model of themass densityorbital velocity [(GM/a) 1/2 ]1.41.21see Fig. 13.4, also leads to a flat rotational curve in the galac-tic disk [45]. In contrast to the three-dimensional distributionthe surface density has a 1/r singularity for r → 0andvanishes at the radius of the galaxy. The radial force on amass m in the plane and the orbital velocity readflat galaxies?2-dim.0.83-dim.0.60 0.5 1 1.5 2radius [a (galaxy radius)]Fig. 13.5Flat rotational curves of stars in agalactic disk for two- andthree-dimensional rotationallysymmetric mass distributionscontributionof baryonic matterFrequently the proportionality of (13.3) for larger r is parameterizedin the following form:R0 2 ϱ(r) = ϱ + a20r 2 + a 2 , (13.5)where r is the galactocentric distance, R 0 = 8.5 kpc (forthe Milky Way) is the galactocentric radius of the Sun, anda = 5 kpc is the radius of the halo nucleus. ϱ 0 is the localenergy density in the solar system,ϱ 0 = 0.3GeV/cm 3 . (13.6)If the mass density, like in (13.3) for flat rotational curvesrequired, decreases like r −2 , the integral mass of a galaxygrows like M(r) ∼ r, since the volume increases like r 3 .So far, three-dimensional rotationally symmetric massdistributions of galaxies have been considered. In anotherlimiting case, one can study purely two-dimensional distributions,since the luminous matter is practically confined toa disk in a galactic plane with the exception of the bulk. IfM denotes the mass and a the radius of the galaxy, the twodimensionalmass densityσ(r) ={ M2πararccos ( ra),r

13.2 Motivation for Dark Matter 269density ϱ c = 3H 2 /8πG and Ω b = ϱ b /ϱ c – is obtained tobeΩ b ≈ 0.04 . (13.9)This corresponds to just 0.3 baryons per m 3 . The visibleluminous matter only provides a density ofvisible luminous matter0.003 ≤ Ω lum ≤ 0.007 , (13.10)that is, Ω lum < 1%. From the fact that the rotational curvesof all galaxies are essentially flat, the contribution of galactichalos to the mass density of the universe can be estimated tobecontribution of galactic halos0.2 ≤ Ω gal ≤ 0.4. (13.11)Since galaxies also have the tendency to form clusters,an estimation of the cosmic mass density can be obtainedfrom the dynamics of galaxies in galactic clusters. The massobtained from these considerations also surmounts the visibleluminous matter by a large margin.The existence of large amounts of invisible non-luminousdark matter seems to be established in a convincingfashion. Consequently, the question arises what this matteris and how it is distributed.dynamics of galaxiesin clusters13.2.1 Dark StarsThe rotational curves of stars in galaxies require a considerablefraction of gravitating matter which is obviouslyinvisible. The idea of primordial nucleosynthesis suggeststhat the amount of baryonic matter is larger than the visiblematter, (13.9) and (13.10). Therefore, it is obvious toassume that part of the galactic halos consists of baryonicmatter. Because of the experimental result of Ω lum < 0.007,this matter cannot exist in form of luminous stars. Also hotand therefore luminous gas clouds, galactic dust, and coldgalactic gas clouds are presumably unlikely candidates, too,because they would reveal themselves by their absorption.These arguments only leave room for special stellar objectswhich are too small to shine sufficiently bright or celestialobjects which just did not manage to initiate hydrogenburning. In addition, burnt-up stars in form of neutron stars,black holes, or dwarf stars could also be considered. Neutronstars and black holes are, however, unlikely candidates,gas cloudsgalactic dustbaryonic matterin dark stellar objectsneutron starsblack holes

270 13 Dark Matterbrown dwarvesFig. 13.6Image of a distant backgroundgalaxy as Einstein ring, where theforeground galaxy in the center ofthe figure acts as gravitational lens{29}microlensingFig. 13.7Apparent light curve of a brightstar produced by microlensing,when a brown dwarf star passes theline of sight between source andobserver. The brightness excursionis given in terms of magnitudesgenerally used in astronomy {30}because they would have emerged from supernova explosions.From the observed chemical composition of galaxies(dominance of hydrogen and helium) one can exclude thatmany supernova explosions have occurred, since these are asource of heavier elements.Therefore, it is considered likely that galactic baryonicmatter could be hidden in brown dwarves. Since the massspectrum of stars increases to small masses, one would expecta significant fraction of small brown stars in our galaxy.The question is, whether one is able to find such massive,compact, non-luminous objects in galaxies (MACHO) 1 .Ashas already been shown in the introduction (Chap. 1), a darkstar can reveal itself by its gravitational effect on light. Apoint-like invisible deflector between a bright star and theobserver produces two star images (see Fig. 1.7). If the deflectingbrown star is directly on the line of sight between thestar and the observer, a ring will be produced, the Einsteinring (Fig. 13.6). The radius r E of this ring depends on themass M d of the deflector like r E ∼ √ M d . Such phenomenahave frequently been observed. If, however, the mass of thebrown star is too small, the two star images or the Einsteinring cannot be resolved experimentally. Instead, one wouldobserve an increase in brightness of the star, if the deflectorpasses through the line of sight between star and observer(microlensing). This brightness excursion is related to thefact that the light originally emitted into a larger solid angle(without the deflecting star) is focused by the deflectoronto the observer. The brightness increase now depends onhow close the dark star comes to the line of sight betweenobserver and the bright star (‘impact parameter’ b). The expectedapparent light curve is shown in Fig. 13.7 for variousparameters. One assumes that b is the minimum distance ofthe dark star with respect to the line of sight between sourceand observer and that it passes this line of sight with thevelocity v. A characteristic time for the brightness excursionis given by the time required to pass the Einstein ring(t = r E /v).To be able to find non-luminous halo objects by gravitationallensing, a large number of stars has to be observedover a long period of time to search for a brightness excursionof individual stars. Excellent candidates for such asearch are stars in the Large Magellanic Cloud (LMC). The1 MACHO – MAssive Compact Halo Object, sometimes alsocalled ‘Massive Astrophysical Compact Halo Objects’

13.2 Motivation for Dark Matter 271Large Magellanic Cloud is sufficiently distant so that thelight of its stars has to pass through a large region of thegalactic halo and therefore has a chance to interact gravitationallywith many brown non-luminous star candidates.Furthermore, the Large Magellanic Cloud is just above thegalactic disk so that the light of its stars really has to passthrough the halo. From considerations of the mass spectrumof brown stars (‘MACHOs’) and the size of the Einstein ringone can conclude that a minimum of 10 6 stars has to be observedto have a fair chance to find some MACHOs.The experiments MACHO, EROS 2 , and OGLE 3 havefound approximately a dozen MACHOs in the halo of ourMilky Way. Figure 13.8 shows the light curve of the firstcandidate found by the MACHO experiment. The observedwidth of the brightness excursion allows to determine themass of the brown objects.If the deflector has a mass corresponding to one solarmass (M ⊙ ), one would expect an average brightness excur-sion of three months while for 10 −6 M ⊙ one would obtainonly two hours. The measured brightness curves all leadto masses of approximately 0.5 M ⊙ . The non-observationof short brightness signals already excludes a large massrange of MACHOs as candidates for dark halo matter. Ifthe few seen MACHOs in a limited solid angle are extrapolatedto the whole galactic halo, one arrives at the conclusionthat possibly 20% of the halo mass, which takes an influenceon the dynamics of the Milky Way, could be hiddenin dark stars. Because of the low number of observed MA-CHOs, this result, however, has a considerable uncertainty((20 +30−12 )%).A remote possibility for additional non-luminous baryonicdark matter could be the existence of massive quarkstars (several hundred solar masses). Because of their anticipatedsubstantial mass the duration of brightness excursionswould be so large that it would have escaped detection.The Andromeda Galaxy with many target stars wouldbe an ideal candidate for microlensing experiments. Thisgalaxy is right above the galactic plane. Unfortunately, it istoo distant that individual stars can be resolved. Still onecould employ the ‘pixel-lensing’ technique by observing theapparent brightness excursions of individual pixels with aCCD camera. In such an experiment one pixel would cover2 EROS – Expérience pour la Recherche d’Objets Sombres3 OGLE – Optical Gravitational Lens Experimentamplification A10864200 50100time [d]Fig. 13.8Light curve of a distant star causedby gravitational lensing. Shown isthe first brown object found by theMACHO experiment in thegalactic halo {30}expected durationsof brightness excursionsestimated mass fractionof dark stars in the halomassive quark starspixel-lensing technique

272 13 Dark MatterNACHOsultracold gas cloudsweak gravitational lensing:image distortionsseveral non-resolved star images. If, however, one of thesestars would increase in brightness due to microlensing, thiswould be noticed by the change in brightness of the wholepixel.One generally assumes that MACHOs constitute a certainfraction of dark matter. However, it is not clear, whichobjects are concerned and where these gravitational lensesreside. To understand the dynamics of galaxies there mustbe also some NACHOs (Not Astrophysical Compact HaloObjects).A certain contribution to baryonic dark matter could alsobe provided by ultracold gas clouds (temperatures < 10 K),which are very difficult to detect.Recently a promising new technique of weak gravitationallensing has been developed for the determination ofthe density of dark matter in the universe. Weak gravitationallensing is based on the fact that images of distant galaxieswill be distorted by dark matter between the observer andthe galaxy. The particular pattern of the distortions mirrorsthe mass and its spatial distribution along the line of sight tothe distant galaxy. First investigations on 145 000 galaxiesin three different directions of observation have shown thatΩ ≤ 1 and that the cosmological constant very likely playsa dominant rôle in our universe.13.2.2 Neutrinos as Dark MatterFor a long time neutrinos were considered a good candidatefor dark matter. A purely baryonic universe is in contradictionwith the primordial nucleosynthesis of the Big Bang.Furthermore, baryonic matter is insufficient to explain thelarge-scale structure of the universe. The number of neutrinosapproximately equals the number of blackbody photons.If, however, they had a small mass, they could provide a significantcontribution for dark matter.From direct mass determinations only limits for the neu-trino masses can be obtained (m νe < 3eV,m νµ < 190 keV,m ντ < 18 MeV). The deficit of atmospheric muon neutrinosbeing interpreted as ν µ –ν τ oscillations, leads to a mass ofthe ν τ neutrino of approximately 0.05 eV.Under the assumption of Ω = 1 the expected numberdensity of primordial neutrinos allows to derive an upperlimit for the total mass that could be hidden in the threeneutrino flavours. One expects that there are approximatelyupper neutrino mass limitsfrom direct measurements

13.2 Motivation for Dark Matter 273equal numbers of blackbody neutrinos and blackbody photons.With N ≈ 300 neutrinos/cm 3 and Ω = 1 (correspondingto the critical density of ϱ c ≈ 1 × 10 −29 g/cm 3 at an ageof the universe of approximately 1.4 × 10 10 years) one obtains4N ∑ ∑m ν ≤ ϱ c , mν ≤ 20 eV . (13.12)The sum extends over all sequential neutrinos includingtheir antiparticles. For the three known neutrino generationsone has ∑ m ν = 2(m νe + m νµ + m ντ ). The consequenceof (13.12) is that for each individual neutrino flavour a masslimit can be derived:m ν ≤ 10 eV . (13.13)It is interesting to see that on the basis of these simplecosmological arguments the mass limit for the τ neutrino asobtained from accelerator experiments can be improved byabout 6 orders of magnitude.If the contribution of neutrino masses to dark matter isassumedtobeΩ ν > 0.1, a similar argument as before providesalso a lower limit for neutrino masses. Under the assumptionof Ω ν > 0.1 (13.12) yields for the sum of massesof all neutrino flavours ∑ m ν > 2 eV. If one assumes thatin the neutrino sector the same mass hierarchy holds as withcharged leptons (m e ≪ m µ ≪ m τ → m νe ≪ m νµ ≪ m ντ ),the mass of the τ neutrino can be limited to the rangeupper summative neutrinomass limit from flat universeupper individual neutrinomass limits from flat universelower neutrino mass limitfrom minimumenergy density1eV≤ m ντ ≤ 10 eV . (13.14)This conclusion, however, rests on the assumption ofΩ ν > 0.1. Recent estimates of Ω ν indicate much smallervalues (Ω ν < 1.5%) and hence the ν τ mass can be substantiallysmaller. Neutrinos with low masses are relativistic andwould constitute in the early universe in thermal equilibriumto the so-called ‘hot dark matter’. With hot dark matter, however,it is extremely difficult to understand the structures inthe universe on small scales (size of galaxies). Therefore,neutrinos are not considered a good candidate for dark matter.There is a possibility to further constrain the margin forneutrino masses. To contribute directly to the dark matterof a galaxy, neutrinos must be gravitationally bound to the4 For simplification c = 1 has been generally used. If numbers,however, have to be worked out, the correct numerical value forthe velocity of light, c ≈ 3 × 10 8 m/s, must be used.hot dark mattergravitational bindingto a galaxy

274 13 Dark Matterneutrinosas free relativistic fermion gasneutrino mass densitylower neutrino mass limitfrom galaxy dynamicsgalaxy, i.e., their velocity must be smaller than the escapevelocity v f . This allows to calculate a limit for the maximummomentum p max = m ν v f . If the neutrinos in a galaxy aretreated as a free relativistic fermion gas in the lowest energystate (at T = 0), one can derive from the Fermi energyE F = ¯hc(3π 2 n max ) 1/3 = p max c (13.15)an estimation for the neutrino mass density (n max – numberdensity):n max m ν = m4 ν v3 f3π 2 ¯h 3 . (13.16)Since n max m ν must be at least on the order of magnitude ofthe typical density of dark matter in a galaxy, if one wantsto explain its dynamics with neutrino masses, these argumentslead to a lower limit for the neutrino mass. Usingv f = √ 2GM/r, whereM and r are galactic mass and radius,one obtains under plausible assumptions about the neutrinomass density and the size and structure of the galaxym ν > 1eV. (13.17)Again, this argument is based on the assumption that neutrinomasses might contribute substantially to the matterdensity of the universe. These cosmological arguments leaveonly a relatively narrow window for neutrino masses.These considerations are not necessarily in contradictionto the interpretation of results on neutrino oscillations, becausein that case one does not directly measure neutrinomasses but rather the difference of their masses squared.From the deficit of atmospheric muon neutrinos one obtainsδm 2 = m 2 1 − m2 2 = 3 × 10−3 eV 2 . (13.18)If ν µ –ν τ oscillations are responsible for this effect, muonand tau neutrino masses could still be very close withoutneutrino masses getting into conflict with the cosmological mass limits. Onlyfrom oscillations if the known mass hierarchy (m e ≪ m µ ≪ m τ ) fromthe sector of charged leptons is transferred to the neutrinosector and if one further assumes m νµ ≪ m ντ ,theresult(m ντ ≈ 0.05 eV) would be in conflict with the cosmologicallimits, but then the cosmological limits have been derivedunder the assumption that neutrino masses actually play animportant rôle for the matter density in the universe, whichis now known not to be the case.

13.2 Motivation for Dark Matter 275If light neutrinos would fill up the galactic halo, onewould expect a narrow absorption line in the spectrum ofhigh-energy neutrinos arriving at Earth. The observation ofsuch an absorption line would be a direct proof of the existenceof neutrino halos. Furthermore, one could directly inferthe neutrino mass from the energetic position of this line.For a neutrino mass of 10 eV the position of the absorptionline can be calculated from, see (3.16), ν +¯ν → Z 0 → tohadrons or leptons,to be2m ν E ν = M 2 Z (13.19)E ν = M2 Z= 4.2 × 10 20 eV . (13.20)2m νThe verification of such an absorption line, which wouldresult in a burst of hadrons or leptons from Z decay (‘Zbursts’) represents a substantial experimental challenge. Thefact that recent fits to cosmological data indicate that thecontribution of neutrino masses to the total matter density inthe universe is rather small (Ω ν ≤ 1.5%), makes the observationof such an absorption line rather unlikely.neutrino absorption lineneutrino halosZ bursts13.2.3 Weakly Interacting Massive Particles (WIMPs)“In questions like this, truth is only to behad by laying together many variationsof error.”Virginia WolfBaryonic matter and neutrino masses are insufficient to closethe universe. A search for further candidates of dark mattermust concentrate on particles, which are only subjectto weak interactions apart from gravitation, otherwise onewould have found them already. There are various scenarioswhich allow the existence of weakly interacting massiveparticles (WIMPs). In principle, one could consider afourth generation of leptons with heavy neutrinos. However,the LEP 5 measurements of the Z width have shown that thenumber of light neutrinos is exactly three (see Chap. 2, Fig.2.1, and Sect. 10.7) so that for a possible fourth generationthe mass limitm νx ≥ m(Z)/2 ≈ 46 GeV (13.21)fourth generation?heavy neutrinos5 LEP – Large Electron–Positron Collider at CERN in Geneva

276 13 Dark Mattersupersymmetric extensionsof the Standard Model(a)e +e +~ e +photino Z 0photino e – ~ e –e –(b)e +invisiblephotino~ e +e + e –~ e –e – invisiblephotinoFig. 13.9Production and decay ofsupersymmetric particles asFeynman diagram (a) and in thedetector (b)WIMP, LSPdetection possibilitiesholds. Such a mass, however, is considered to be too largeto expect that a sizable amount of so heavy particles couldhave been created in the Big Bang.An alternative to heavy neutrinos is given by WIMPs,which would couple even weaker to the Z than sequentialneutrinos. Candidates for such particles are provided in supersymmetricextensions of the Standard Model. Supersym-metry is a symmetry between fundamental fermions (leptonsand quarks) and gauge bosons (γ , W + , W − , Z, gluons,Higgs bosons, gravitons). In supersymmetric models allparticles are arranged in supermultiplets and each fermionis associated with a bosonic partner and each boson gets afermionic partner.Bosonic quarks and leptons (called squarks and sleptons)are associated to the normal quarks and leptons. Thecounterparts of the usual gauge bosons are supersymmetricgauginos, where one has to distinguish between charginos(supersymmetric partners of charged gauge bosons: winos( ˜W + , ˜W − ) and charged higgsinos ( ˜H + , ˜H − )) and neutralinos(photino ˜γ ,zino˜Z, neutral higgsinos ( ˜H 0 ,...), gluinos˜g, and gravitinos). The non-observation of supersymmetricparticles at accelerators means that supersymmetry mustbe broken and the superpartners obviously are heavier thanknown particles and not in the reach of present-day accelerators.The theory of supersymmetry appears to be at leastfor the theoreticians so aesthetical and simple that they expectit to be true. A new quantum number, the R parity, distinguishesnormal particles from their supersymmetric partners.If the R parity is a conserved quantity – and this is assumedto be the case in the most simple supersymmetric theories– the lightest supersymmetric particle (LSP) must bestable (it could only decay under violation of R parity). Thislightest supersymmetric particle represents an ideal candidatefor dark matter (Fig. 13.9). It is generally assumedthat the Large Hadron Collider (LHC) under construction atCERN has a fair chance to produce and to be able to successfullyreconstruct the creation of supersymmetric particles.The first collisions in LHC are expected for the year 2007.The low interaction strength of the LSP, however, is atthe same time an obstacle for its detection. If, for example,supersymmetric particles would be created at an acceleratorin proton–proton or electron–positron interactions, the finalstate would manifest itself by missing energy, because in decaysof supersymmetric particles one lightest supersymmetricparticle would always be created, which would leave the

13.2 Motivation for Dark Matter 277detector without significant interaction. Primordially producedsupersymmetric particles would have decayed alreadya long time ago, apart from the lightest supersymmetric particles,which are expected to be stable if R parity is conserved.These would lose a certain amount of energy in collisionswith normal matter, that could be used for their detection.Unfortunately, the recoil energy transferred to a targetnucleus in a WIMP interaction (mass 10–100 GeV) is rathersmall, namely in the range of about 10 keV. Still one tries tomeasure the ionization or scintillation produced by the recoilingnucleus. Also a direct calorimetric measurement ofthe energy deposited in a bolometer is conceivable. Becauseof the low energy Q to be measured and the related minutetemperature riseT = Q/c sp m (13.22)(c sp is the specific heat and m the mass of the calorimeter),these measurements can only be performed in ultrapurecrystals (e.g., in sapphire) at extremely low temperatures(milli-Kelvin range, c sp ∼ T 3 ). It has also been consideredto use superconducting strips for the detection, which wouldchange to a normal-conducting state upon energy absorption,thereby producing a usable signal.Based on general assumptions on the number density ofWIMPs one would expect a counting rate of at most oneevent per kilogram target per day. The main problem in theseexperiments is the background due to natural radioactivityand cosmic rays.Due to their high anticipated mass WIMPs could also begravitationally trapped by the Sun or the Earth. They wouldoccasionally interact with the material of the Sun or Earth,lose thereby energy, and eventually obtain a velocity belowthe escape velocity of the Sun or the Earth, respectively.Since WIMPs and in the same way their antiparticles wouldbe trapped, they could annihilate with each other and provideproton–antiproton or neutrino pairs. One would expectto obtain an equilibrium between trapping and annihilationrate.The WIMP annihilation signal could be recorded in largeexisting neutrino detectors originally designed for neutrinoastronomy. Also large neutrino detectors under construction(ANTARES 6 or IceCube) would have a chance to pick up aWIMP signal.6 ANTARES – Astronomy with a Neutrino Telescope and Abyssenvironmental RESearchrecoil energyionization and scintillationof recoiling nucleuscalorimetric measurementbolometerexpected WIMP rategravitational bindingto the solar systemWIMP annihilationpossible recordingin neutrino detectors

278 13 Dark MatterIt is not totally inconceivable that particularly heavyWIMP particles could be represented by primordial blackholes, which could have been formed in the Big Bang be-fore the era of nucleosynthesis. They would provide an idealcandidate for cold dark matter. However, it is very difficult toimagine a mechanism by which such primordial black holescould have been produced in the Big Bang.Recently the Italian–Chinese DAMA 7 collaboration haspublished a result, which could hint at the existence ofWIMPs. Similarly to our Sun, the Milky Way as a wholecould also trap WIMPs gravitationally. During the rotationof the solar system around the galactic center the velocityof the Earth relative to the hypothetical WIMP halo wouldchange depending on the season. In this way the Earth whileorbiting the Sun would encounter different WIMP fluxes dependingon the season. In the month of June the Earth movesagainst the halo (→ large WIMP collision rate) and in Decemberit moves parallel to the halo (→ low WIMP collisionrate).The DAMA collaboration has measured a seasonal vari-ation of the interaction rate in their 100 kg heavy sodiumiodidecrystal. The modulation of the interaction rate withan amplitude of 3% is interpreted as evidence for WIMPs.The results obtained in the Gran Sasso laboratory wouldfavour a WIMP mass of 60 GeV. This claim, however, isin contradiction to the results of an American collaboration(CDMS) 8 , which only observes a seasonal-independentbackground due to neutrons in their highly sensitive lowtemperaturecalorimeter.black holeas WIMP candidategravitational bindingto the Milky Way:possible seasonal-dependentWIMP fluxresults from DAMAand CDMS collaborations13.2.4 Axions“For every complex natural phenomenonthere is a simple, elegant,compelling, but wrong explanation.”Thomas GoldC, P, CP violationWeak interactions not only violate parity P and charge conjugationC, but also the combined symmetry CP. TheCPviolation is impressively demonstrated by the decays of neutralkaons and B mesons. In the framework of quantumchromodynamics (QCD) describing strong interactions, CPviolatingterms also originate in the theory. However, experimentallyCP violation is not observed in strong interactions.7 DAMA – DArk MAtter search8 CDMS – Cryogenic Dark Matter Search

13.2 Motivation for Dark Matter 279Based on theoretical considerations in the framework ofQCD the electric dipole moment of the neutron should be onthe same order of magnitude as its magnetic dipole moment.Experimentally one finds, however, that it is much smallerand even consistent with zero. This contradiction has beenknown as the so-called strong CP problem. The solution tothis enigma presumably lies outside the Standard Model ofelementary particles. A possible solution is offered by the introductionof additional fields and symmetries, which eventuallyrequire the existence of a pseudoscalar particle, calledaxion. The axion is supposed to have similar properties asthe neutral pion. In the same way as the π 0 it would havea two-photon coupling and could be observed by its twophotondecay or via its conversion in an external electromagneticfield (Fig. 13.10).Theoretical considerations appear to indicate that the axionmass should be somewhere between the µeV and meVrange. To reach the critical density of the universe with axionsonly, the axion density – assuming a mass of 1 µeV –should be enormously high (> 10 10 cm −3 ). Since the conjecturedmasses of axions are very small, they must possessnon-relativistic velocities to be gravitationally bound to agalaxy, because otherwise they would simply escape fromit. For this reason axions are considered as cold dark matter.As a consequence of the low masses the photons generatedin the axion decay are generally of low energy. Foraxions in the preferred µeV range the photons produced byaxion interactions in a magnetic field would lie in the microwaverange. A possible detection of cosmological axionswould therefore involve the measurement of a signal in a microwavecavity, which would significantly stand out abovethe thermal noise. Even though axions appear to be necessaryfor the solution of the strong CP problem and thereforeare considered a good candidate for dark matter, all experimentson their search up to date gave negative results.strong CP problem(a)(b)axionfexternalfieldfFig. 13.10Coupling of an axion to twophotons via a fermion loop (a).Photons could also be provided byan electromagnetic field for axionconversion (b)gravitational bindingto the galaxycold dark matteraxion decayfff13.2.5 The Rôle of the Vacuum Energy DensityTo obtain a flat universe a non-zero cosmological constantis required, which drives the exponential expansion of theuniverse. The cosmological constant is a consequence of thefinite vacuum energy. This energy could have been storedoriginally in a false vacuum, i.e., a vacuum, which is notat the lowest-energy state. The energy of the false vacuumflat universeand cosmological constantfalse vacuum

280 13 Dark Mattertrue vacuumcosmological constantand static universecosmological constantas dominant energycould be liberated in a transition to the true vacuum (seeSect. 12.5).Paradoxically, a non-zero cosmological constant was introducedby Einstein as a parameter in the field equations ofthe theory of general relativity to describe a static universe,which was popular at that time and to prevent a dynamicbehaviour, which followed from his theory. Now it appearsthat the dominant energy, which determines the dynamics ofthe universe, is stored in the empty space itself. The questionnow arises, whether the cosmological constant is a practicablesupplement to the required dark matter. This questionwas controversial just as the question of the existence ofsupersymmetric particles or axions over the last years. Recently,however, several experiments have found compellingevidence for a substantial amount of dark energy, which canbe interpreted, e.g., in terms of a non-zero cosmological constant,see Chap. 12 on inflation.There is a fundamental difference between classical darkmatter in form of MACHOs, WIMPs, or axions and the effectof the cosmological constant Λ.The potential energy on a test mass m created by matterand the vacuum energy density can easily be written afterNewton to beE pot =−G mM matter− G mM vacuum energyRR∼− ϱ matterR 3− ϱ vacuumR 3. (13.23)R RThere is a fundamental difference between the matter den-sity and the vacuum energy density during the expansionof the universe. For the vacuum-energy term in (13.23) thevacuum energy density remains constant, since this energydensity is a property of the vacuum. In contrast to this thematter density does not remain constant during expansion,since only the mass is conserved leading to the dilution ofthe matter density. Therefore, the spatial dependence of thepotential energy is given bymatter densityvs. vacuum energy densityE pot ∼− M matterR− ϱ vacuum R 2 . (13.24)Since ϱ vacuum ∼ Λ, (13.24) shows that the radial dependenceof the mass term is fundamentally different fromthe term containing the cosmological constant. Therefore,the question of the existence of dark matter (M matter )anda

13.2 Motivation for Dark Matter 281finite vacuum energy density (Λ ̸= 0) are not trivially coupled.Furthermore, Λ could be a dynamical constant, whichis not only of importance for the development of the earlyuniverse. Since the Λ term in the field equations appears todominate today, one would expect an accelerated expansion.The experimental determination of the acceleration parametercould provide evidence for the present effect of Λ. Todo this one would have to compare the expansion rate ofthe universe in earlier cosmological times with that of thepresent.Such measurements have been performed with the SupernovaCosmology Project at Berkeley and with the HighzSupernova Search Team at Australia’s Mount StromloSpring Observatory. The surprising results of these investigationswas that the universe is actually expanding at ahigher pace than expected (see also Sect. 8.8 and, in particular,Fig. 8.6). It is important to make sure that SN Ia uponwhich the conclusions depend explode the same way nowand at much earlier times so that these supernovae can beconsidered as dependable standard candles. This will be investigatedby looking at older and more recent supernovaeof type Ia. Involved in these projects surveying distant galaxiesare the Cerro Tololo Interamerican Observatory (CTIO)in the Chilean Andes, the Keck Telescope in Hawaii, andfor the more distant supernovae the Hubble Space Telescope(HST).cosmological constantand expanding universeSupernova CosmologyProjectHigh-z Supernova Search13.2.6 Galaxy FormationAs already mentioned in the introduction to this chapter, thequestion of galaxy formation is closely related to the problemof dark matter. Already in the 18th century philosopherslike Immanuel Kant and Thomas Wright have speculatedabout the nature and the origin of galaxies. Today it seems tobe clear that galaxies originated from quantum fluctuations,which have been formed right after the Big Bang.With the Hubble telescope one can observe galaxies upto redshifts of z = 3.5 (λ observed = (1 + z)λ emitted ); thiscorresponds to 85% of the total universe. The idea of cosmicinflation predicts that the universe is flat and expandsforever, i.e., the Ω parameter is equal to unity.The dynamics of stars in galaxies and of galaxies ingalactic clusters suggests that less than ≈ 5% of matter isin form of baryons. Apart from the vacuum energy the mainpart of matter leading to Ω = 1 has to be non-baryonic,Kant, Wrightquantum fluctuationsHubble telescopeΩ parameter

282 13 Dark Matterpropertiesof non-baryonic matter‘cold’ and ‘hot’ particlesmodels of galaxy formationquantum fluctuationsand gravitational instabilities‘protogalaxies’‘top–down’ scenariowhich means it must consist of particles that do not occurin the Standard Model of elementary particle physics.The behaviour of this matter is completely different fromnormal matter. This dark matter interacts with other matterpredominantly via gravitation. Therefore, collisions of darkmatterparticles with known matter particles must be veryrare. As a consequence of this, dark-matter particles loseonly a small fraction of their energy when moving throughthe universe. This is an important fact when one discussesmodels of galaxy formation.Candidates for dark matter are subdivided into ‘cold’and ‘hot’ particles. The prototype of hot dark matter is theneutrino (m ν ̸= 0), which comes in at least in three differentflavour states. Low-mass neutrinos are certainly insufficientto close the universe. Under cold dark matter one normallysubsumes heavy weakly interacting massive particles(WIMPs) or axions.The models of galaxy formation depend very sensitivelyon whether the universe is dominated by hot or cold darkmatter. Since in all models one assumes that galaxies haveoriginated from quantum fluctuations which have developedto larger gravitational instabilities, two different cases canbe distinguished.If the universe would have been dominated by the lowmassneutrinos, fluctuations below a certain critical masswould not have grown to galaxies because fast relativisticneutrinos could easily escape from these mass aggregations,thereby dispersing the ‘protogalaxies’. For a neutrino massof 20 eV a critical mass of about 10 16 solar masses is required,so that structure formation can really set in. Withsuch large masses one lies in the range of the size of superclusters.Neutrinos as candidates for dark matter thereforewould favour a scenario, in which first the large structures(superclusters), later clusters, and eventually galaxies wouldhave been formed (‘top–down’ scenario). This would implythat galaxies have formed only for z ≤ 1. However, fromHubble observations one already knows that even for z ≥ 3large populations of galaxies existed. This is also an argumentto exclude a neutrino-dominated universe.Massive, weakly interacting and mostly non-relativistic(i.e., slow) particles, however, will be bound gravitationallyalready to mass fluctuations of smaller size. If cold dark matterwould dominate, initially small mass aggregations wouldcollapse and grow by further mass attractions to form galaxies.These galaxies would then develop galactic clusters and

13.2 Motivation for Dark Matter 283later superclusters. Cold dark matter therefore favours a scenarioin which smaller structures would be formed first andonly later develop into larger structures (‘bottom–up’ scenario).In particular, the COBE and WMAP observations ofthe inhomogeneities of the 2.7 Kelvin radiation confirm theidea of structure formation by gravitational amplification ofsmall primordial fluctuations. These observations thereforesupport a cosmogony driven by cold dark matter, in whichsmaller structures (galaxies) are formed first and later developinto galactic clusters.The dominance of cold dark matter, however, does notexclude non-zero neutrino masses. The values favoured bythe observed neutrino anomaly in the Super-Kamiokandeand SNO experiments are compatible with a scenario ofdominating cold dark matter. In this case one would have acocktail – apart from baryonic matter – consisting predominantlyof cold dark matter with a pinch of light neutrinos.‘bottom–up’ scenarioinhomogeneitiesof blackbody radiationneutrino anomaly13.2.7 Resumé on Dark MatterIt is undisputed that large quantities of dark matter must behidden somewhere in the universe. The dynamics of galaxiesand the dynamics of galactic clusters can only be understood,if the dominating part of gravitation is caused by nonluminousmatter. In theories of structure formation in theuniverse in addition to baryonic matter (Ω b ) also other formsof matter or energy are required. These could be representedby hot dark matter (e.g., light relativistic neutrinos: Ω hot )or cold dark matter (e.g., WIMPs: Ω cold ). At the presenttime cosmologists prefer a mixture of these three components.Recent measurements of distant supernovae and preciseobservations of inhomogeneities of the blackbody radiationlead to the conclusion that the dark energy, mostly interpretedin terms of the cosmological constant Λ, alsohasa very important impact on the structure of the universe. Ingeneral, the density parameter Ω can be presented as a sumof four contributions,non-luminous matterbaryonic matterhot dark mattercold dark mattercosmological constantΩ parameterΩ = Ω b + Ω hot + Ω cold + Ω Λ . (13.25)The present state of the art in cosmology gives Ω b ≈ 0.04,Ω hot ≤ 1%, Ω cold ≈ 0.23, and Ω Λ ≈ 0.73.As demonstrated by observations of distant supernovae,the presently observed expansion is accelerated. Therefore,accelerated expansion

284 13 Dark MatterquintessenceBoomerangMaximasurface of last scatteringrecombinationBoomerang, Maxima,WMAPtemperature clustersit is clear that the cosmological constant plays a dominantrôle even today. Even if the repulsive force of the vacuumis small compared to the action of gravity, it will eventuallywin over gravity if there is enough empty space, see also(13.24). The present measurements (status 2005) indicate avalue for Ω Λ of ≈ 0.7. This means that the vacuum is filledwith a weakly interacting hypothetical scalar field, which essentiallymanifests itself only via a repulsive gravitation andnegative pressure, which consequently corresponds to a tension.In this model the field (e.g., ‘quintessence’, see glossary)produces a dynamical energy density of the vacuumin such a way that the repulsive gravity plays an importantrôle also today. This idea is supported by the findings of theAmerican Boomerang 9 and Maxima 10 experiments. Thesetwo balloon experiments measure the blackbody radiationover a limited region of the sky with a much higher precisionthan the COBE experiment. The observed temperatureanisotropies are considered as a seed for galaxy formation.These temperature variations throw light on the structureof the universe about 380 000 years after the Big Bang whenthe universe became transparent. The universe before thattime was too hot for atoms to form and photons were trappedin a sea of electrically charged particles. Pressure waves pulsatedin this sea of particles like sound waves in water. Theresponse of this sea to the gravitational potential fluctuationsallows to measure the properties of this fluid in an expandinguniverse consisting of ordinary and dark matter. Whenthe universe cooled down, protons and electrons recombinedto form electrically neutral hydrogen atoms, thereby freeingthe photons. These hot photons have cooled since then to atemperature of around 2.7 K at present. Temperature variationsin this radiation are fingerprints of the pattern of soundwaves in the early universe, when it became transparent. Thesize of the spots in the thermal map is related to the curvatureor the geometry of the early universe and gives informationsabout the energy density. Both experiments, Boomerang andMaxima, and the satellite experiment WMAP observed tem-perature clusters of a size of about one degree across. Thisinformation is experimentally obtained from the power spectrumof the primordial sound waves in the dense fluid ofparticles (see Chap. 11). The observed power spectrum correspondingto characteristic cluster sizes of the temperature9 Boomerang – Balloon Observations of Millimetric ExtragalacticRadiation and Geophysics10 Maxima – Millimeter Anisotropy Experiment Imaging Array

13.3 Problems 285map of about 1 ◦ indicates – according to theory – that theuniverse is flat. If the results of the supernova project (seeSect. 8.3), the WMAP results, and the Boomerang and Maximadata are combined, a value for the cosmological constantcorresponding to Ω Λ ≈ 0.7 is favoured. Such a significantcontribution would mean that the vacuum is filledwith an incredibly weakly interaction substance, which revealsitself only via its repulsive gravitation (e.g., in termsof a quintessence model). The suggested large contributionof the vacuum energy to the Ω parameter would naturallyraise the question, whether appreciable amounts of exoticdark matter (e.g., in the form of WIMPs) are required at all.However, it appears that for the understanding of the dynamicsof galaxies and the interpretation of the fluctuations inthe blackbody temperature a contribution of classical, nonbaryonicdark matter corresponding to Ω dm ≈ 0.23 is indispensable.In this scenario the long-term fate of the universe is characterizedby eternal accelerated expansion. A Big Crunchevent as anticipated for Ω larger than unity appears to becompletely ruled out. Since the nature of the dark energyis essentially unknown, also its long-term properties are notunderstood. Therefore, one must be prepared for surprises.For example, it is possible that the dark energy becomes sopowerful that the known forces are insufficient to preservethe universe. Galactic clusters, galaxies, planetary systems,and eventually atoms would be torn apart. In such theoriesone speculates about an apocalyptic Big Rip, which wouldshred the whole physical structure of the universe at the endof time.flat universevacuum energyBig CrunchBig Rip13.3 Problems1. In the vicinity of the galactic center of the Milky Waycelestial objects have been identified in the infrared andradio band which appear to rotate around an invisiblecenter with high orbital velocities. One of these objectscircles the galactic center with an orbital velocity of v ≈110 km/s at a distance of approximately 2.5 pc. Workout the mass of the galactic center assuming Kepleriankinematics for a circular orbit!2. What is the maximum energy that a WIMP (m W = 100GeV) of 1 GeV kinetic energy can transfer to an electronat rest?

286 13 Dark Matter3. Derive the Fermi energy of a classical and relativisticgas of massive neutrinos. What is the Fermi energy ofthe cosmological neutrino gas?4. If axions exist, they are believed to act as cold dark matter.To accomplish the formation of galaxies they mustbe gravitationally bound; i.e., their velocities should notbe too high.Work out the escape velocity of a 1 µeV-mass axion froma protogalaxis with a nucleus of 10 10 solar masses ofradius 3 kpc!5. Consider a (not highly singular) spherically symmetricmass density ϱ(r), the total mass of which is assumed tobe finite.a) Determine the potential energy of a test mass min the gravitational field originating from ϱ(r). Theforce is given by Newton’s formula in which onlythe mass M(r) inside that sphere enters, whose radiusis given by the position of m.b) With the previous result calculate the potential energyof the mass density, where the test mass is replacedby dM = M ′ (r) dr and the r integration isperformed to cover the whole space. Show with anintegration by parts in the outer integral that the totalpotential energy can be written as∫ ∞M 2 (r)E pot =−Gr 2 dr.0c) Determine the potential energy of a massive sphericalshell of radius R and mass M, i.e., M(r) = 0forrR.d) Work out the potential energy of a sphere of radiusR and mass M with homogeneous mass density.(This problem is a little difficult and tricky.)6. Motivate the lower mass limit of neutrinos based on theirmaximum escape velocity from a galaxy (13.17), if it isassumed that they are responsible for the dynamics ofthe galaxy.7. In the discussion on the search for MACHOs there is astatement that the radius of the Einstein ring varies asthe square root of the mass of the deflector. Work outthis dependence and determine the ring radius for• distance star–Earth = 55 kpc (LMC),• distance deflector–Earth = 10 kpc (halo),• Schwarzschild radius of the deflector = 3 km (correspondingto the Schwarzschild radius of the Sun).

28714 Astrobiology“In the beginning the universe was created.This has made a lot of people very angryand been widely regarded as a bad move.”Douglas AdamsApart from considering the physical development of our universe,it is interesting to speculate about alternatives to ouruniverse, i.e., whether the evolution in a different universemight have led to different physical laws, other manifestationsof matter, or to other forms of life. Einstein had alreadypondered about such ideas and had raised the questionwhether the necessity of logical simplicity leaves anyfreedom for the construction of the universe at all.The numerous free parameters in the Standard Model ofelementary particles appear to have values such that stablenuclei and atoms, in particular, carbon, which is vital forthe development of life, could be produced. 1 Also the coincidenceof time scales for stellar evolution and evolution oflife on planets is strange. In the early universe, hydrogen andhelium were mainly created, whereas biology, as we knowit, requires all elements of the periodic table. These other elementsare provided in the course of stellar evolution and insupernova explosions. The long-lived, primordial, radioactiveelements ( 238 U, 232 Th, 40 K) that have half-lives on theorder of giga-years, appear to be particularly important forthe development of life. Furthermore, a sufficient amount ofiron must be created in supernovae explosions such that habitableplanets can develop a liquid iron core. This is essentialin the generation of magnetic fields which shield life on theplanets against lethal radiation from solar and stellar windsand cosmic radiation.1 α particles are formed in pp fusion processes. In αα collisions8 Be could be formed, but 8 Be is highly unstable and does notexist in nature. To reach 12 C, a triple collision of three α particleshas to happen to get on with the production of elements.This appears at first sight very unlikely. Due to a curious moodof nature, however, this reaction has a resonance-like large crosssection (see also Problem 2 in this chapter).free parametersearly universedevelopment of lifecosmic radiation

288 14 Astrobiologyage of life on Earthbioastronomymasses of quarksThe oldest sedimentary rocks on Earth (3.9 × 10 9 yearsold) contain fossiles of cells. It is known that bacteria evenexisted 3.5 × 10 9 years ago. If the biological evolution hadrequired extremely long time scales, it is conceivable thatthe development of the higher forms of life would have beenimpossible, since the fuel of stars typically only lasts for10 10 years.The origin of life is a question of much debate. In additionto the theory of a terrestrial origin also the notion ofextra-terrestrial delivery of organic matter to the early Earthhas gained much recognition. These problems are investigatedin the framework of bioastronomy.The many free parameters in the Standard Model of electroweakand strong interactions can actually be reduced to asmall number of parameters, whose values are of eminentimportance for the astrobiological development of the universe.Among these important parameters are the masses ofthe u and d quarks and their mass difference. Experimentsin elementary particle physics have established the fact thatthe mass of the u quark (≈ 5 MeV) is smaller than that ofthe d quark (≈ 10 MeV). The neutron, which has a quarkcontent udd, is therefore heavier than the proton (uud), andcan decay according ton → p + e − +¯ν e . (14.1)unstable proton?If, on the other hand, m u were larger than m d , the protonwould be heavier than the neutron, and it would decay accordingtop → n + e + + ν e . (14.2)stability of deuteriumStable elements would not exist, and there would be nochance for the development of life as we know it. On theother hand, if the d quark were much heavier than 10 MeV,deuterium (d = 2 H) would be unstable and heavy elementscould not be created. This is because the synthesis of elementsin stars starts with the fusion of hydrogen to deuterium,p + p → d + e + + ν e , (14.3)and the formation of heavier elements requires stable deuterium.Life could also not develop under these circumstances.Also the value of the lifetime of the neutron takesan important influence on primordial chemistry and thereby

14 Astrobiology 289on the chances to create life (see also Sect. 10.5). There aremany other parameters which appear to be fine-tuned to thedevelopment of life in the form that we know.One of the problems of the unification of forces in anall-embracing Theory of Everything is the ‘weakness’ of thegravitational force. If, however, gravitation would be muchstronger, we would live in a short-lived miniature universe.No creatures could grow larger than a few centimeters, andthere would be hardly time for biological evolution.The fact that we live in a flat universe with Ω = 1appearsalso to be important for us. If Ω were much larger thanunity, the universe would have recollapsed long ago, alsowith the impossibility for the development of life. In thiscontext the exact value of the cosmological constant whichdescribes a repulsive gravity, plays an important rôle for thedevelopment of the universe.It is remarkable that the effect of Λ on microscopicalscales, just as that of attractive gravity, is negigible. The discrepancybetween the measured small value of Λ in cosmologyand the extremely large values of the vacuum energy inquantum field theories tells us that important ingredients inthe understanding of the universe are still missing.Another crucial parameter is the number of dimensions.It is true that superstring theories can be formulated ineleven dimensions, out of which seven are compactified, sothat we live in three spatial and one time dimension. But lifewould be impossible, e.g., in two spatial dimensions. Whathas singled out our case?Also the efficiency of energy generation in stars is importantfor the production of different chemical elements.If this efficiency would be much larger than the value thatwe know (0.7%), stars would exhaust their fuel in a muchshorter time compared to that needed for biological evolution.It has become clear that the fine-tuning of parametersin the Standard Model of elementary particle physics andcosmology is very important for the development of stars,galaxies, and life. If some of the parameters which describeour world were not finely tuned, then our universe wouldhave grossly different properties. These different propertieswould have made unlikely that life – in the form we knowit – would have developed. Consequently, physicists wouldnot be around to ask the question why the parameters haveexactly the values which they have. Our universe could havefine-tuned parametersgravitationΩ parametercosmological constantnumber of dimensionsefficiencyof energy generationfine-tuning of parameters

290 14 Astrobiologybeen the result of a selection effect on the plethora of uni-verses in a multiverse. It is also conceivable that there couldexist even a diversity of physical laws in other universes.Only in universes where the conditons allow life to develop,questions about the specialness of the parameters can beposed. As a consequence of this anthropic principle thereis nothing mysterious about our universe being special. Itmight just be that we are living in a most probable universewhich allows life to develop.Nevertheless, it is the hope of particle theorists and cosmologiststhat a Theory of Everything can be found suchthat all sensitive parameters are uniquely fixed to the valuesthat have been found experimentally. Such a theory mighteventually also require a deeper understanding of time. Tofind such a theory and also experimental verifications of it,is the ultimate goal of cosmoparticle physics.This is exactly what Einstein meant when he said: “WhatI am really interested in, is whether God could have madethe world in a different way; that is, whether the necessity oflogical simplicity leaves any freedom at all.”plethora of universesmultiverseanthropic principlecosmoparticle physics14.1 Problems1. Estimate the maximum stable size of a human in theEarth’s gravitational field. What would happen to thisresult if the acceleration due to gravity would double?Hint: Consider that the weight of a human is proportionalto the cube of a typical dimension, while itsstrength only varies as its cross section.2. What are the conditions to synthesize carbon in stars?Hint: The detailed, quantitative solution of this problemis difficult. To achieve carbon fusion one has to havethree alpha particles almost at the same time in the sameplace. Consequently high alpha-particle densities are required,and high temperatures to overcome the Coulombbarriers. Also the cross section must be large. Checkwith the web pageshttp://www.campusprogram.com/reference/en/wikipedia/t/tr/triple_alpha_process.html,http://en.wikipedia.org/wiki/Triple-alpha_process,and nucl-th/0010052 onhttp://xxx.lanl.gov/.

14.1 Problems 2913. What determines whether a planet has an atmosphere?Hint: Gas atoms or molecules will be lost from an atmosphereif their average velocity exceeds the escapevelocity from the planet.4. Rotationally invariant long-range radial forces: (nonrelativistic)two-body problem in n dimensions. 2a) How does such a force F between two bodies dependon the distance between two point-like bodies?Recall from three dimensions how the force dependson the surface or the solid angle of a sphere, respectively(‘force flow’). The (hyper)surface s n (r)of the n-dimensional (hyper)sphere can be writtenas s n (r) = s n r g(n) ,whereg(n) needs to be fixed.b) What is the corresponding potential energy and howdoes the effective potential (which includes the centrifugalbarier) look like? Find conditions for stableorbits with attractive forces.c) Consider the introduction of a field-energy densityw ∼ F 2 (mediated by a field strength ∼ F )andintegrate w within the range of the two radii λ andΛ. Then-dimensional volume element is given bydV n = s n (r) dr, see also part (a). At what n doesthis expression diverge for λ → 0orΛ → ∞?If quantum corrections are supposed to compensatethe divergences, at what limit should they becomerelevant?2 This problem discusses unconventional dimensions. It is quitetricky and mathematically demanding.

29315 Outlook“My goal is simple. It is complete understandingof the universe: why it is as it isand why it exists at all.”Stephen HawkingAstroparticle physics has developed from the field of cosmicrays. As far as the particle physics aspect is concerned,accelerator experiments have taken over and played a leadingrôle in this field since the sixties. However, physicistshave realized that the energies in reach of terrestrial acceleratorsrepresent only a tiny fraction of nature’s wide window,and it has become obvious that accelerator experiments willnever match the energies required for the solution of importantastrophysical and cosmological problems. Actually,recently cosmic ray physics has again taken the lead overaccelerator physics by discovering physics beyond the StandardModel, namely, by finding evidence for neutrino oscillationsin cosmic-ray experiments, which require non-zeroneutrino masses.The electromagnetic and weak interactions unify to acommon electroweak force at center-of-mass energies ofaround 100 GeV, the scale of W ± and Z masses. Alreadythe next unification of strong and electroweak interactionsat the GUT scale of 10 16 GeV is beyond any possibility toreach in accelerator experiments even in the future. This iseven more true for the unification of all interactions whichis supposed to happen at the Planck scale, where quantumeffects of gravitation become important (≈ 10 19 GeV).These energies will never be reached, not even in theform of energies of cosmic-ray particles. The highest energyobserved so far of cosmic-ray particles, which has beenmeasured in extensive-air-shower experiments (3×10 20 eV),corresponds to a center-of-mass energy of about 800 TeV inproton–proton collisions, which is 60-fold the energy thatis going to be reached at the Large Hadron Collider LHCin the year 2007. It has, however, to be considered that therate of cosmic-ray particles with these high energies is extremelylow. In the early universe (< 10 −35 s), on the otherhand, conditions have prevailed corresponding to GUT- andcosmic raysphysicsbeyond the Standard Modelelectroweak scaleGUT scalePlanck scaleLHC

294 15 OutlookTheory of Everythingorigin of cosmic rayscosmic acceleratorsneutrino astronomybeyond the Standard ModelPlanck-scale energies. The search for stable remnants ofthe GUT or Planck time probably allows to get informationabout models of the all-embracing theory (TOE – Theory ofEverything). These relics could manifest themselves in exoticobjects like heavy supersymmetric particles, magneticmonopoles, axions, or primordial black holes.Apart from the unification of all interactions the problemof the origin of cosmic rays has still not been solved. Thereis a number of known cosmic accelerators (supernova explosions,pulsars, active galactic nuclei, M87(?), ...), but itis completely unclear how the highest energies (> 10 20 eV)are produced. Even the question of the identity of these par-ticles (protons?, heavy nuclei?, photons?, neutrinos?, newparticles?) has not been answered. It is conjectured that activegalactic nuclei, in particular those of blazars, are ableto accelerate particles to such high energies. If, however,protons or gamma rays are produced in these sources, ourfield of view into the cosmos is rather limited due to theshort mean free path of these particles (λ γp ≈ 10 Mpc,λ γγ ≈ 10 kpc). Therefore, the community of astroparticlephysicists is optimistic to be able to explore the universewith neutrino astronomy where cosmic neutrinos are pro-duced in a similar way to γ rays in cosmic beam-dump experimentsvia pion decays. Neutrinos directly point back tothe sources, they are not subject to deflections by magneticfields and they are not attenuated or even absorbed by interactions.The enigmatic neutrinos could also give a small contributionto the dark matter, which obviously dominates theuniverse. The investigations on the flavour composition ofatmospheric neutrinos have shown that a deficit of muonneutrinos exists, which obviously can only be explained byoscillations. Such neutrino oscillations require a non-zeroneutrino mass, which carries elementary particle physics alreadybeyond the well-established Standard Model. It wasclear already for a long time that the Standard Model of elementaryparticles with its 26 free parameters cannot be thefinal answer, however, first hints for a possible extension ofthe Standard Model do not originate from accelerator experimentsbut rather from investigations of cosmic rays.Neutrinos alone are by far unable to solve the problemof dark matter. To which extent exotic particles (WIMPs,SUSY particles, axions, quark nuggets, ...) contribute tothe invisible mass remains to be seen. In addition, there is

15 Outlook 295also the cosmological constant, not really beloved by Einstein,which provides an important contribution to the structureand to the expansion of the universe via its associatedvacuum energy density. In the standard scenario of the classicalBig Bang model the presently observed expansion ofthe universe is expected to be decelerated due to the pull ofgravity. Quite on the contrary, the most recent observationsof distant supernovae explosions indicate that the expansionhas changed gear. It has turned out that the cosmologicalconstant – even at present times – plays a dominant rôle. Aprecise time-dependent measurement of the acceleration parameter– via the investigation of the expansion velocity indifferent cosmological epochs, i.e., distances – has shownthat the universe expands presently at a faster rate than atearlier times.Another input to cosmology which does not come fromparticle physics experiments is due to precise observationsof the cosmic microwave background radiation. The resultsof these experiments in conjunction with the findings of thedistant supernova studies have shown that the universe isflat, i.e., the Ω parameter is equal to unity. Given the lowamount of baryonic mass (about 4%) and that dark matterconstitutes only about 23%, there is large room for darkenergy (73%) to speculate about. These results from nonacceleratorexperiments have led to a major breakthrough incosmology.Eventually the problem of the existence of cosmic antimatterremains to be solved. From accelerator experimentsand investigations in cosmic rays it is well-known that thebaryon number is a sacred conserved quantity. If a baryonis produced, an antibaryon is always created at the sametime. In the same way with each lepton an antilepton is produced.The few antiparticles ( ¯p, e + ) measured in astroparticlephysics are presumably of secondary origin. Since onehas to assume that equal amounts of quarks and antiquarkshad been produced in the Big Bang, a mechanism is urgentlyrequired, which acts in an asymmetric fashion on particlesand antiparticles. Since it is known that weak interactionsviolate not only parity P and charge conjugation C but alsoCP, aCP-violating effect could produce an asymmetry inthe decay of the parent particles of protons and antiprotonsso that the numbers of protons and antiprotons developed ina slightly different way. In subsequent p ¯p annihilations veryfew particles, which we now call protons, would be left over.cosmological constantacceleration parameterΩ parameterbaryonic massdark energycosmic antimatterCP violationp ¯p annihilation

296 15 Outlookmatter dominanceprimordial antimatterannihilation radiationWhether a small CP violation is sufficient to accomplish thisremains to be seen. Since in p ¯p annihilations a substantialamount of energy is transformed into photons, the observedγ/p ratio (̂= n γ /n p ) of about 10 9 indicates that even aminute difference in the decay properties of the parent particlesof protons and antiprotons is sufficient to explain thematter dominance in the universe. Such an asymmetry couldhave its origin in Grand Unified Theories of electroweak andstrong interactions. The matter–antimatter asymmetry couldthen be explained by different decay properties of heavy Xand ¯X particles which are supposed to have existed in theearly universe. Equal numbers of X and ¯X particles decayinginto quarks, antiquarks, and leptons could lead to differentnumbers of quarks and antiquarks if baryon-number andlepton-number conservation were violated, thereby leadingto the observed matter dominance of the universe.The non-observation of primordial antimatter is still nota conclusive proof that the universe is matter dominated. Ifgalaxies made of matter and ‘antigalaxies’ made of antimatterwould exist, one would expect that they would attracteach other occasionally by gravitation. This should lead to aspectacular annihilation event with strong radiation. A clearsignal for such a catastrophe would be the emission of the511 keV line due to e + e − annihilation. It is true that such a511 keV γ -ray line has been observed – also in our galaxy– but its intensity is insufficient to be able to understand itas an annihilation of large amounts of matter and antimatter.On the other hand, it is conceivable that the annihilationradiation produced in interactions of tails of galaxies withtails of antigalaxies would establish such a radiation pressurethat the galaxies would be driven apart and the reallygiant spectacular radiation outburst would never happen.Questions about a possible dominance of matter are unlikelyto be answered in accelerator experiments alone. Onthe other hand, the early universe provides a laboratory,which might contain – at least in principle – the answers.However, the step to bridge the gap from the Planck erato present-day theories is still hidden in the mist of spaceand time.The investigations in the framework of astroparticlephysics represent an important tool for a deeper understandingof the universe, from its creation to the present and futurestate.

15.1 Problems 29715.1 Problems1. It is generally believed that the cosmological constantrepresents the energy of the vacuum. On the other hand,quantum field theories lead to a vacuum energy whichis larger by about 120 orders of magnitude. This dicrepancycan be solved by an appropriate quantizationof Einstein’s theory of general relativity. Start from theStandard Model of particle physics (see Chap. 2) anduse the Friedmann equation (see Sect. 8.4) extended bya term for the vacuum energy ϱ v to solve the discrepancybetween the value of the cosmological constant obtainedfrom astroparticle physics measurements and thevacuum energy as obtained from quantum field theories!

29916 Glossary“A good notation has a subtlety and suggestiveness,which at times makes it almostseem like a live teacher.”Bertrand RussellAThe total luminosity emitted from an astrophysical object.Lowest possible temperature, 0 Kelvin, at which all motion comesto rest except for quantum effects (0 Kelvin =−273.15 ◦ Celsius).Percentage of an element occurring on Earth, in the solar system,or in primary cosmic rays in a stable isotopic form.It describes the acceleration of the universe in its dependence on theenergy density and negative pressure (i.e., due to the cosmologicalconstant).A measurement for the change of the expansion rate with time. Foran increased expansion rate the acceleration parameter is positive.It was generally expected that the present expansion rate is reducedby the attractive gravitation (negative acceleration parameter = deceleration).The most recent measurements of distant supernovae,however, contradict this expectation by finding an accelerated expansion.A machine used to accelerate charged particles to high speeds.There are, of course, also cosmic accelerators.Accumulation of dust and gas into a disk, normally rotating arounda compact object.After decoupling of matter from radiation the pattern of acousticoscillations of the primordial baryon fluid became frozen as structuresin the power spectrum of the CMB.Galaxy with a bright central region, an active galactic nucleus.Seyfert galaxies, quasars, and blazars are active galaxies, whichare presumably powered by a black hole.In the Big Bang model the inverse of the Hubble constant is identifiedwith the age of the universe. The present estimate of the ageof the universe is about 14 billion years.absolute brightnessabsolute zeroabundanceacceleration equationacceleration parameteracceleratoraccretion diskacoustic peaksactive galaxyage of the universe

300 16 GlossaryMeasurement of extensive air showers via their produced Cheren-kov light in the atmosphere.AGNair-shower Cherenkov techniquealpha decayAMANDAAndromeda NebulaÅngströmannihilationanthropic principleantigravitationantimatterantiparticlesActive Galactic Nucleus. If one assumes that black holes poweredby infalling matter reside at centers of AGNs and if these AGNsemit polar jets, then the different AGN types (Seyfert galaxies, BL-Lacertae objects, radioquasars, radiogalaxies, quasars) can be understoodas a consequence of the random direction of view fromEarth.Nuclear decay consisting of the emission of an α particle (helium-4nucleus).Antarctic Muon And Neutrino Detector Array for the measurementof high-energy cosmic neutrinos.M31, one of the main galaxies of the local group; distance from theMilky Way: 700 kpc.The unit of length equal to 10 −10 m.A process, in which particles and antiparticles desintegrate, e.g.,e + e − → γγ.One might ask why the universe is as it is. If some of the physicsconstants which describe our universe were not finely tuned, thenit would have grossly different properties. It is considered highlylikely that life forms as we know them would not develop underan even slightly different set of basic parameters. Therefore, if theuniverse were different, nobody would be around to ask why theuniverseisasitis.Repulsive gravitation caused by a negative pressure as a conse-quence of a finite non-zero cosmological constant.Matter consisting only of antiparticles, like positrons, antiprotons,or antineutrinos. When antimatter particles and ordinary matterparticles meet, they annihilate mostly into γ rays (e.g., e + e − →γγ).For each particle there exists a different particle type of exactlythe same mass but opposite values for all other quantum numbers.This state is called antiparticle. For example, the antiparticle of anelectron is a particle with positive electric unit charge, which iscalled positron. Also bosons have antiparticles apart from those forwhich all charge quantum numbers vanish. An example for this isthe photon or a composite boson consisting of a quark and its correspondingantiquark. In this particular case there is no distinctionbetween particle and antiparticle, they are one and the same object.

16 Glossary 301The antiparticle of a quark.The point of greatest separation of two stars as in a binary star orbit.The point in the orbit of a planet, which is farthest from the Sun.The point in the orbit of an Earth satellite, which is farthest fromEarth.The brightness of a star as it appears to the observer. It is measuredin units of magnitude. See magnitude.Considerations on the formation and evolution of life in universeswith different fundamental parameters (different physical laws,quark masses, etc.)The average distance from Earth to Sun. 1 AU ≈ 149 597 870 km.At large momenta quarks will be eventually deconfined and becomeasymptotically free.The mass of a neutral atom or a nuclide. The atomic weight of anatom is the weight of the atom based on the scale where the massof the carbon-12 nucleus is equal to 12. For natural elements withmore than one isotope, the atomic weight is the weighted averagefor the mixture of isotopes.The number of protons in the nucleus.The northern (aurora borealis) or southern (aurora australis) lightsare bright emissions of atoms and molecules in the polar upperatmosphere around the south and north geomagnetic poles. Theaurora is associated with luminous processes caused by energeticsolar cosmic-ray particles incident on the upper atmosphere of theEarth.Advanced X-ray Astrophysics Facility; now named Chandra.Hypothetical pseudoscalar particle, which was introduced as quantumof a field to explain the non-observation of CP violation instrong interactions.See cosmic background radiation.Formation of baryons out of the primordial quark soup in the veryearly universe.antiquarkapastronaphelionapogeeapparent magnitudeastrobiologyastronomical unit (AU)asymptotic freedomatomic massatomic numberauroraAXAFaxionBbackground radiationbaryogenesis

302 16 GlossaryAn asymmetry created by some, so far unknown baryon-number-violating process in the early universe (see Sakharov conditions).baryon–antibaryon asymmetrybaryon numberbaryonsBATSEbeam-dump experimentbeta decayBethe–Bloch formulaBethe–Weizsäcker cycleBethe–Weizsäcker formulaBig BangBig Bang theoryBig CrunchThe baryon number B is unity for all strongly interacting particlesconsisting of three quarks. Quarks themselves carry baryon number1/3. For all other particles B = 0.Elementary particles consisting of three quarks like, e.g., protonsand neutrons.Burst And Transient Source Experiment on board of the CGROsatellite.If a high-energy particle beam is stopped in a sufficiently thicktarget, all strongly and electromagnetically interacting particles areabsorbed. Only neutral weakly interacting particles like neutrinoscan escape.In nuclear β decay a neutron of the nucleus is transformed into aproton under the emission of an electron and an electron antineutrino.The transition of the proton in the nucleus into a neutronunder emission of a positron and an electron neutrino is called β +decay, contrary to the aforementioned β − decay. The electron capturewhich is the reaction of proton and electron into neutron andan electron neutrino (p + e − → n + ν e ) is also considered as betadecay.Describes the energy loss of charged particles by ionization andexcitation when passing through matter.Carbon–nitrogen–oxygen cycle: nuclear fusion process in massivestars, in which hydrogen is burnt to helium with carbon as catalyzer(CNO cycle).Describes the nuclear binding energy in the framework of theliquid-drop model.Beginning of the universe when all matter and radiation emergedfrom a singularity.The theory of an expanding universe that begins from an infinitelydense and hot primordial soup. The initial instant is called the BigBang. The theory says nothing at all about time zero itself.If the matter density in the universe is larger than the critical density,the presently observed expansion phase will turn over into acontraction, which eventually will end in a singularity.

16 Glossary 303The Big Rip is a cosmological hypothesis about the ultimate fateof the universe. The universe has a large amount of dark energycorresponding to a repulsive gravity, and in the long run it will winover the classical attractive gravity anyhow. Therefore, all matterwill be finally pulled apart. Galaxies in galactic clusters will beseparated, the gravity in galaxies will become too weak to hold thestars together, and also the solar system will become gravitationallyunbound. Eventually also the Earth and the atoms themselves willbe destroyed with the consequence that the Big Rip will shred thewhole physical structure of the universe.Binary stars are two stars that orbit around their common center ofmass. An X-ray binary is the special case where one of the starsis a collapsed object such as a neutron star or black hole. Matteris stripped from the normal star and falls onto the collapsed starproducing X rays or also gamma rays.The energy that has to be invested to desintegrate a nucleus into itssingle constituents (protons and neutrons, nuclear binding energy).Scientific branch of astronomy that deals with the question of theorigin of life, e.g., whether organic material has been delivered toEarth from exraterrestrial sources. The techniques are to look forbiomolecules with spectroscopic methods used in astronomy.A hypothetical body that absorbs all the radiation falling on it. Theemissivity of a blackbody depends only on its temperature.The radiation produced by a blackbody. The blackbody is a perfectradiator and absorber of heat or radiation, see also Planck distribution.A massive star that has used up all its hydrogen can collapse underits own gravity to a mathematical singularity. The size of a blackhole is characterized by the event horizon. The event horizon is theradius of that region, in which gravity is so strong that even lightcannot escape (see also Schwarzschild radius).Short for variable active galactic nuclei, which are similar to BL-Lacertae objects and quasars except that the optical spectrum isalmost featureless.Reduction of the wavelength of electromagnetic radiation by theDoppler effect as observed, e.g., during the contraction of the universe.Variable extragalactic objects in the nuclei of some galaxies, whichoutshine the whole galaxy. BL stands for radio loud emission in theBband(≈ 100 MHz); lacerta = lizard.Big Ripbinary starsbinding energybioastronomyblackbodyblackbody radiationblack holeblazarsblueshiftBL-Lacertae objects

304 16 GlossarySensitive resistance thermometer for the measurement of small en-ergy depositions.bolometerBoltzmann constantBose–Einstein distributionbosonbottom quarkBrahebranebremsstrahlungbrown dwarvesCCalabi–Yau spacecarbon cyclecascadeCasimir effectA constant of nature which describes the relation between the temperatureand kinetic energy for atoms or molecules in an ideal gas.The energy distribution of bosons in its dependence on the temperature.A particle with integer spin. The spin is measured in units of ¯h (spins = 0, 1, 2,...). All particles are either fermions or bosons. Gaugebosons are associated with fundamental interactions or forces,which act between the fermions. Composite particles with an evennumber of fermion constituents (e.g., consisting of a quark and antiquark)are also bosons.b; the fifth quark flavour (if the quarks are ordered with increasingmass). The b quark has the electric charge − 1 3 e.Tycho Brahe, a Danish astronomer, whose accurate astronomicalobservations form the basis for Johannes Kepler’s laws of planetarymotion. The supernova remnant SNR 1572 (Tycho) is named afterBrahe.Branes are higher-dimensional generalisations of strings. A p braneis a space-time object with p spatial dimensions.Emission of electromagnetic radiation when a charged particle isdecelerated in the Coulomb field of a nucleus. Bremsstrahlung canalso be emitted during the deceleration of a charged particle in theCoulomb field of an electron or any other charge.Low-mass stars (< 0.08 solar masses), in which thermonuclearreactions do not ignite, which, however, still shine because gravitationalenergy is transformed into electromagnetic radiation duringa slow shrinking process of the objects.Complex space of higher dimension, which has become popular forthe compactification of extraspatial dimensions in the frameworkof string theories.See Bethe–Weizsäcker cycle.See shower.A reduction in the number of virtual particles between two flat parallelmetal plates in vacuum compared to the surrounding leads toa measurable attractive force between the plates.

16 Glossary 305Stars which have rapid and unpredictable changes in brightness,mostly associated with a binary star system.See cosmic background radiation.Charge-Coupled Device, a solid-state camera.Strong galactic radio source in the constellation Centaurus.Type of pulsating variable stars where the period of oscillationis proportional to the average absolute magnitude; often used as‘standard’ candles.Centre Européen pour la Recherche Nucléaire. The major Europeancenter for particle physics located near Geneva in Switzerland.Compton Gamma-Ray Observatory. Satellite with four experimentson board for the measurement of galactic and extragalactic gammarays.X-ray satellite of the NASA (original name: AXAF) started in July1999; named after Subrahmanyan Chandrasekhar.Limiting mass for white dwarves (1.4 solar masses). If a star exceedsthis mass, gravity will eventually defeat the pressure of thedegenerate electron gas leading to a compact neutron star.Quantum number carried by a particle. The charge quantum numberdetermines whether a particle can participate in a special interactionprocess. Particles with electric charge participate in electromagneticinteractions, such with strong charge undergo stronginteractions, and those with weak charge are subject to weak interactions.The principle of charge invariance claims that all processes againconstitute a physical reality if particles are exchanged by their antiparticles.This principle is violated in weak interactions.The observation that the charge of a system of particles remains unchangedin an interaction or transformation. In this context chargestands for electric charge, strong charge, or also weak charge.Interaction process mediated by the exchange of a virtual chargedgauge boson.cataclysmic variablesCBRCCDCentaurus ACepheid variablesCERNCGROChandraChandrasekhar masschargecharge conjugationcharge conservationcharged currentc; the fourth quark flavour (if quarks are ordered with increasing charm quarkmass). The c quark has the electric charge + 2 3 e.For a given component in a particle mixture the chemical potentialdescribes the change in free energy with respect to a change in theamount of the component at fixed pressure and temperature.chemical potential

306 16 GlossaryA Friedmann–Lemaître model of the universe with positive curva-ture of space. Such a universe will eventually contract leading to aBig Crunch.Cherenkov effectclosed universecluster of galaxiesCMBCOBEcold dark mattercollidercolourcolour chargecompactificationCOMPTELCompton effectconfinementCherenkov radiation occurs, if the velocity of a charged particle ina medium with index of refraction n exceeds the velocity of lightc/n in that medium.A set of galaxies containing from a few tens to several thousandmember galaxies, which are all gravitationally bound together. TheVirgo cluster includes 2500 galaxies.Cosmic microwave background, see cosmic background radiation.Cosmic Background Explorer. Satellite with which the minute temperatureinhomogeneities ( TT ≈ 10 −5 ) of the cosmic backgroundradiation were first discovered.Type of dark matter that was moving at much less than the ve-locity of light some time after the Big Bang. It could consist ofWeakly Interacting Massive Particles (WIMPs, such as supersymmetricparticles) or axions.An accelerator in which two counter-rotating beams are steeredtogether to provide a high-energy collision between the particlesfrom one beam with those of the other.The strong ‘charge’ of quarks and gluons is called colour.The quantum number that determines the participation of hadronsin strong interactions. Quarks and gluons carry non-zero colourcharges.The universe may have extra dimensions. In string theories thereare 10 or more dimensions. Since our world appears to have onlythree plus one dimension, it is assumed that the extra dimensionsare curled up into sizes so small that one can hardly detect themdirectly, which means they are compactified.Compton telescope on board the CGRO satellite.Compton effect is the scattering of a photon off a free electron. Thescattering off atomic electrons can be considered as pure Comptoneffect, if the binding energy of the electrons is small compared tothe energy of the incident photon.The property of strong interactions, which says that quarks andgluons can never be observed as single free objects, but only insidecolour-neutral objects.

16 Glossary 307A quantity is conserved, if it does not change in the course of areaction between particles. Conserved quantities are, for example,electric charge, energy, and momentum.A group of stars that produce in projection a shape often namedafter mythological characters or animals.An alternative process to X-ray emission during the de-excitationof an excited atom or an excited nucleus. If the excitation energyof the atomic shell is transferred to an atomic electron and if thiselectron is liberated from the atom, this electron is called an Augerelectron. If the excitation energy of a nucleus is transferred to anatomic electron of the K, L, orM shell, this process is called internalconversion. Auger electron emission or emission of characteristicX rays is often a consequence of internal conversion.A very hot outer layer of the Sun’s atmosphere consisting of highlydiffused ionized gas and extending into interplanetary space. Thehot gas in the solar corona forms the solar wind.European gamma-ray satellite started in 1975.Standard composition of elements in the universe as determinedfrom terrestrial, solar, and extrasolar matter.CBR; nearly isotropic blackbody radiation originating from the BigBang (‘echo of the Big Bang’). The cosmic background radiationhas now a temperature of 2.7 K.Nuclear and subatomic particles moving through space at velocitiesclose to the speed of light. Cosmic rays originate from stars and,in particular, from supernova explosions. The origin of the highestenergycosmic rays is an unsolved problem.One-dimensional defects, which might have been created in theearly universe.Topological defect which involves a kind of twisting of the fabricof space-time.The branch of particle physics and astronomy that tries to dig outresults on elementary particles and cosmology from experimentalevidence obtained from data about the early universe.The science of the origins of galaxies, stars, planets, and satellites,and, in particular, of the universe as a whole.Two-dimensinonal representation of the universe in the light of the2.7 K blackbody radiation showing temperature variations as seedfor galaxy formation.conserved quantityconstellationconversion electroncoronaCOS-Bcosmic abundancecosmic background radiationcosmic rayscosmic stringscosmic texturescosmoarcheologycosmogonycosmographic map

308 16 GlossaryThe cosmological constant was introduced by hand into the Ein-stein field equations with the intention to allow an aesthetic cosmologicalsolution, i.e., a steady-state universe which was believed torepresent the correct picture of the universe at the time. The cosmologicalconstant is time independent and spatially homogeneous.It is physically equivalent to a non-zero vacuum energy. In an expandinguniverse the repulsive gravity due to Λ will eventually winover the classical attractive gravity.cosmological constant (Λ)cosmological principlecosmological redshiftcosmologycosmoparticle physicsCP invarianceCPT invarianceCrab Nebulacritical densitycross sectioncurvaturecyclotron mechanismHypothesis that the universe at large scales is homogeneous andisotropic.Light from a distant galaxy appears redshifted by the expansion ofthe universe.Science of the structure and development of the universe.The study of elementary particle physics using informations fromthe early universe.Nearly all interactions are invariant under simultaneous interchangeof particles with antiparticles and space inversion. The CP invariance,however, is violated in weak interactions (e.g., in the decayof the neutral kaon and B meson).All interactions lead again to a physically real process, if particlesare replaced by antiparticles and if space and time are reversed. It isgenerally assumed that CPT invariance is an absolutely conservedsymmetry.Crab; supernova explosion of a star in our Milky Way (observedby Chinese astronomers in 1054). The masses ejected from the starform the Crab Nebula in whose center the supernova remnant resides.Cosmic matter density ϱ c leading to a flat universe. In a flat uni-verse without cosmological constant (Λ = 0) the presently observedexpansion rate will asymptotically tend to zero. For ϱ>ϱ cthe expansion will turn into a contraction (see Big Crunch); forϱ

16 Glossary 309X-ray binary consisting of a blue supergiant and a compact object,which is considered to be a black hole.X-ray and gamma-ray binary system consisting of a pulsar anda companion. The pulsar appears to be able to emit occasionallygamma rays with energies up to 10 16 eV.A special form of energy that creates a negative pressure andis gravitationally repulsive. It appears to be a property of emptyspace. Dark energy appears to contribute about 70% to the energydensity of the universe. The vacuum energy, the cosmological constant,and light scalar fields (like quintessence) are particular formsof dark-energy candidates.Unobserved non-luminous matter, whose existence has been inferredfrom the dynamics of the universe. The nature of the darkmatter is an unsolved problem.Historically first experiment for the measurement of solar neutrinos.Quantum-mechanical wavelength λ of a particle: λ = h/p (p –momentum, h – Planck’s constant).A process in which a particle disappears and in its place differentparticles appear. The sum of the masses of the produced particlesis always less than the mass of the original decaying particle.A measure of how far an object is above or below the celestialequator (in degrees); similar to latitude on Earth.The pressure in a degenerate gas of fermions caused by the Pauliexclusion principle and the Heisenberg uncertainty principle.Fermi gas (electrons, neutrons), whose stability is guaranteed bythe Pauli principle. In a gas of degenerate particles quantum effectsbecome important.The number of values of a system that are free to vary.Process during supernova explosions where electrons and protonsmerge to form neutrons and neutrinos (e − + p → n + ν e ).Local increase or decrease of the mass or radiation density in theearly universe leading to galaxy formation.Isotope of hydrogen with one additional neutron in the nucleus.Cygnus X1Cygnus X3Ddark energydark matterDavis experimentde Broglie wavelengthdecaydeclinationdegeneracy pressuredegenerate matterdegrees of freedomdeleptonizationdensity fluctuationsdeuterium

310 16 GlossaryThe apparent change in the cosmic background radiation tempera-ture caused by the motion of the Earth through the CBR.differential gravitationdipole anisotropydiskdistance ladderdomain wallDoppler effectdouble pulsardown quarkEGeomagnetically caused asymmetry in the arrival direction of pri-mary cosmic rays, which is related to the fact that most primarycosmic-ray particles carry positive charge.east–west effectEGRETelectromagnetic interactionselectronelectron captureDifferent forces of gravity acting on two different points of a bodyin a strong gravitational field lead to a stretching of the body.The visible surface of any heavenly body projected against the sky.Anumber oftechniques (e.g.,redshift,standardcandles,...)usedby astronomers to obtain the distances to progressively more distantastronomical objects.Topological defect that might have been created in the early uni-verse. Domain walls are two-dimensional defects.Change of the wavelength of light caused by a relative motion betweensource and observer.Predictions of the general theory of relativity about the perihelionrotation and the energy loss by emission of gravitational waveshave been confirmed for the binary pulsar system PSR1913+16with extreme precision.d; the second quark flavour, (if quarks are ordered according toincreasing mass). The d quark has the electric charge − 1 3 e.Energy Gamma Ray Telescope Experiment on board the CGROsatellite.The interactions of particles due to their electric charge. This typeincludes also magnetic interactions.e; the lightest electrically charged particle. As a consequence ofthis it is absolutely stable, because there are no lighter electricallycharged particles in which it could decay. The electron is the mostfrequent lepton with the electric charge −1 in units of the elementarycharge.Nuclear decay by capture of an atomic electron. If the decay energyis larger than twice the electron mass, positron emission can alsooccur in competition with electron capture.

16 Glossary 311The electron and its associated electron neutrino are assigned theelectronic lepton number +1, its associated antiparticles the electroniclepton number −1. All other elementary particles have theelectron number 0. The electron number is generally a conservedquantity. Only in neutrino oscillations it is violated.eV; unit of energy and also of mass of particles. One eV is the energythat an electron (or, more generally, a singly charged particle)gets, if it is accelerated in a potential difference of one volt.Standard Model of elementary particles in which the electromagneticand weak interactions are unified.Galaxy of smooth elliptical structure without spiral arms.Radiation density in Joule/cm 3 or eV/cm 3 .Expérience pour la Recherche d’Objets Sombres. Experimentsearching for dark objects; see also MACHO.The minimum velocity of a body to escape a gravitational fieldcaused√by a mass M from a distance R from its center is v =2GMR . For the Earth the escape velocity is v E = 11.2km/s.Surface of a black hole. Particles or light originating from insidethe event horizon cannot escape from the black hole.Theory in which all galaxies fly apart from each other. The expansionbehaviour looks the same from every galaxy.Our universe has three spatial and one time dimension. In stringtheories and supersymmetric theories there are in general p extendedspatial dimensions, some of which may be compactified.Large particle cascade in the atmosphere initiated by a primarycosmic-ray particle of high energy.Radiation originating from outside our galaxy.A metastable state describing a quantum field which is zero, eventhough its corresponding vacuum energy density does not vanish.A decaying false vacuum can in principle liberate the stored energy.Organization of matter particles into three groups with each groupbeing known as a family. There is an electron, a muon, and a taufamily on the lepton side and equivalently there are three familiesof quarks. LEP has shown that there are only three families (generations)of leptons with light neutrinos.electron numberelectron voltelectroweak theoryelliptical galaxyenergy densityEROSescape velocityevent horizonexpansion modelextended dimensionextensiveairshower(EAS)extragalactic radiationFfalse vacuumfamilies

312 16 GlossaryFermi–Dirac distributionFermi energyFermi mechanismfermionFeynman diagramsfissionfixed-target experimentflareflatness problemflavourfluid equationThe energy distribution of fermions in its dependence on the temperature.Energy of the highest occupied electron level at absolute zero for afree Fermi gas (e.g., electron, neutron gas).Acceleration mechanism for charged particles by shock waves (the1.-order process) or extended magnetic clouds (the 2.-order process).A particle with half integer spin ( 1 2 , 3 2, etc.) if the spin is measuredin units of ¯h. An important consequence of this particular angularmomentum is that fermions are subject to the Pauli principle. ThePauli principle says that no two fermions can exist in the samequantum-mechanical state at the same time. Many properties ofordinary matter originate from this principle. Electrons, protons,and neutrons – all of them are fermions – just as the fundamentalconstituents of matter, i.e., quarks and leptons.Feynman diagrams are pictorial shorthands to describe the interactionprocesses in space and time. With the necessary theoreticaltools Feynman diagrams can be translated into cross sections. Inthis book the agreement is such that time is plotted on the horizontaland spatial coordinates on the vertical axis. Particles moveforward in space-time, while antiparticles move backwards.The splitting of heavier atomic nuclei into lighter ones. Fission isthe way how nuclear power plants produce energy.An experiment in which the beam of particles from an acceleratoris aimed at a stationary target.Short duration outburst from stars in various spectral ranges.In classical cosmology any value of Ω in the early universe evenslightly different from one would be amplified with time and drivenaway from unity. In contrast, in inflationary cosmology Ω is approachingexponentially a value of one, thus explaining naturallythe presently observed value of Ω = 1.02 ± 0.02. In classical cosmologysuch a value would have required an unbelievably carefulfine-tuning.Characterizes the assignment of a fermion (lepton or quark) to aparticle family or a particle generation.It describes the evolution of the energy density in its dependenceon the expansion rate, pressure, and density. It can be derived usingclassical thermodynamics.

16 Glossary 313Particle detector for the measurement of large extensive air showersbased on the measurement of the emitted scintillation light in theatmosphere.Vector with four components comprising energy and the three momentumcomponents.Vector with four components mostly comprising time in the firstcompoment and three additional spatial components.Break-up of a heavy nucleus into a number of lighter nuclei in thecourse of a collision.A differential equation that expresses the evolution of the universedepending on its energy density. This equation can be derived fromEinstein’s field equations of general relativity. Surprisingly, a classicaltreatment leads to the same result.Standard Big Bang models with negative (Ω 1), or flat (Ω = 1) space curvature.A particle with no observable inner structure. In the StandardModel of elementary particles quarks, leptons, photons, gluons,W + , W − bosons, and Z bosons are fundamental. All other objectsare composites of fundamental particles.In fusion lighter elements are combined into heavier ones. Fusionis the way how stars produce energy.Aggregation of galaxies in a spatially limited region.A spherical region mainly consisting of old stars surrounding thecenter of a galaxy.Magnetic field in our Milky Way with an average strength of3 µG = 3 × 10 −10 Tesla.Coordinate transformation for frames of reference in which forcefreebodies move in straight lines with constant speed without considerationof the fact that there exists a limiting velocity (velocity oflight). This theorem of the addition of velocities leads to conflictswith experience, in particular, for velocities close to the velocity oflight.Gallium experiment in the Gran Sasso laboratory for the detectionof solar neutrinos from the pp cycle.Fly’s Eyefour-momentumfour-vectorfragmentationFriedmann equationFriedmann–Lemaître universesfundamental particlefusionGgalactic clustergalactic halogalactic magnetic fieldGalilei transformationGALLEX experiment

314 16 GlossaryExtragalactic gamma-ray sources flaring up only once. Spectacu-lar supernova explosions (hypernova) or colliding neutron stars arepossible candidates for producing gamma-ray bursts.gamma astronomygamma burstergamma raysgeneral principle of relativitygeneral relativitygeneral theory of relativitygenerationGLASTgluinogluongrand unificationgravitational collapsegravitational instabilityAstronomy in the gamma energy range (> 0.1MeV).Short-wavelength electromagnetic radiation corresponding to en-ergies ≥ 0.1MeV.The principle of relativity argues that accelerated motion and beingexposed to acceleration by a graviational field is equivalent.Einstein’s general formulation of gravity which shows that spaceand time communicate the gravitational force through their curvature.Generalization of Newton’s theory of gravity to systems with rela-tive acceleration with respect to each other. In this theory gravitationalmass (caused by gravitation) and inertial mass (mass resistingacceleration) are equivalent.A set of two quarks and two leptons which form a family. Theorder parameter for families is the mass of the family members.The first generation (family) contains the up and down quark, theelectron and the electron neutrino. The second generation containsthe charm and strange quark as well as the muon and its associatedneutrino. The third generation comprises the top and bottom quarkand the tau with its neutrino.Gamma-ray Large Area Space Telescope. GLAST will measure γrays in the energy range 10 keV–300 GeV. The space experiment isexpected to be launched in 2005.Supersymmetric partner of the gluon.g; the gluon is the carrier of strong interactions. There are alto-gether eight different gluons, which differ from each other by theircolour quantum numbers.A theory which combines the strong, electromagnetic, and weakinteractions into one unified theory (GUT).If the gas and radiation pressure of a star can no longer resist the in-wardly directed gravitational pressure, the star will collapse underits own gravity.Process by which density fluctuations of a certain size grow byself-gravitation.

16 Glossary 315The interaction of particles due to their mass or energy. Also particleswithout rest mass are subject to gravitational interactions ifthey have energy, since energy corresponds to mass according tom = E/c 2 .A large mass causes a strong curvature of space thereby deflectingthe light of a distant radiation source. Depending on the relativeposition of source, deflector, and observer, multiple images, arcs,or rings are produced as images of the source.Increase of the wavelength of electromagnetic radiation in thecourse of emission against a gravitational field.Supersymmetric partner of the graviton.Massless boson with spin 2¯h mediating the interaction betweenmasses.Attractive force between two bodies caused by their mass. Thegravitation between two elementary particles is negligibly smalldue to their low masses.In the same way as an accelerated electrical charge emits electromagneticradiation (photons), accelerated masses emit gravitationalwaves. The quantum of the gravity wave is called the graviton.A 100 Mpc structure seen in the distribution of galaxies.Threshold energy of energetic protons for pion production off photonsof the cosmic background radiation via the ∆ resonance.Unified theory of strong, electromagnetic, and weak interactions.The very early universe when the strong, weak, and electromagneticforces were unified and of equal importance.Energy scale at which strong and electroweak interactions mergeinto one common force, E ≈ 10 16 GeV.A charged particle moving in a magnetic field will orbit around themagnetic field lines. The radius of this orbit is called the gyroradiusor sometimes also the Lamor radius.A particle consisting of strongly interacting constituents (quarksor gluons). Mesons and baryons are hadrons. All these particlesparticipate in strong interactions.gravitational interactiongravitational lensgravitational redshiftgravitinogravitongravitygravity wavesGreat WallGreisen–Zatsepin–Kuzmin cutoff(GZK)GUT (Grand Unified Theory)GUT epochGUT scalegyroradiusHhadron

316 16 GlossaryHawking radiationHawking temperatureHEAOHeisenberg’s uncertaintyprincipleheliocentricheliosphereHertzsprung–Russell diagramHiggs bosonHiggs fieldhorizonhorizon problemhot dark matterThe gravitational field energy of a black hole allows to createparticle–antiparticle pairs of which, for example, one can be absorbedby the black hole while the other can escape if the processoccurs outside the event horizon. By this quantum process blackholes can evaporate because the escaping particle reduces the energyof the system. The time scales for the evaporation of largeblack holes exceeds, however, the age of the universe considerably.The temperature of a black hole manifested by the emission ofHawking radiation.High Energy Astronomy Observatory, X-ray satellite.Position and momentum of a particle cannot be determined simultaneouslywith arbitrary precision, x p ≥ ¯h/2. This uncertaintyrelation refers to all complementary quantities, e.g., also to the energyand time uncertainty.Centered on the Sun.The large region starting at the Sun’s surface and extending to thelimits of the solar system.Representation of stars in a colour–luminosity diagram. Stars withlarge luminosity are characterized by a high colour temperature,i.e., shine strongly in the blue spectral range.Hypothetical particle as a quantum of the Higgs field predicted inthe framework of the Standard Model to give masses to fermionsvia the mechanism of spontaneous symmetry breaking. Named afterthe Scottish physicist Peter Higgs.A hypothetical scalar field which is assumed to be responsible forthe generation of masses in a process called spontaneous symmetrybreaking.The observable range of our universe. The radius of the observableuniverse corresponds to the distance that light has travelled sincethe Big Bang.In classical cosmology regions in the sky separated by more than ≈2 ◦ are causally disconnected. Experimentally it is found that theystill have the same temperature. This can naturally be explained byan exponential expansion in the early universe which has smoothedout existing temperature fluctuations.Relativistic dark matter. Neutrinos are hot-dark-matter candidates.

16 Glossary 317High Resolution Imager, focal detector of the X-ray satelliteROSAT.Constant of proportionality between the receding velocity v ofgalaxies and their distance r, v = Hr. According to recentmeasurements the value of the Hubble constant H is about70 (km/s)/Mpc. The inverse Hubble constant corresponds to theage of the universe.The receding velocity of galaxies is proportional to their distance.Space telescope with a mirror diameter of 2.2 m (HST – HubbleSpace Telescope).Hydrogen burning is the fusion of four hydrogen nuclei into a singlehelium nucleus. It starts with the fusion of two protons into adeuterium, positron, and electron neutrino (p +p → d +e + +ν e ).Irvine–Michigan–Brookhaven collaboration for the search of nucleondecay and for neutrino astronomy.A violent inward-bound collapse.Property of matter that requires a force to act on it to change theway it is moving.Hypothetical exponential phase of expansion in the early universestarting at 10 −38 s after the Big Bang and extending up to 10 −36 s.The presently observed isotropy and uniformity of the cosmicbackground radiation can be understood by cosmic inflation.At low momenta quarks are confined in hadrons. They cannot escapetheir hadronic prison.Astronomy in the light of infrared radiation emitted from celestialobjects.A process in which a particle decays or responses to a force due tothe presence of another particle as in a collision.Characteristic collision length for strongly interacting particles inmatter.The gas and dust which exist in the space between the stars.Energy transfer by an energetic electron to a low-energy photon.HRIHubble constant HHubble lawHubble telescopehydrogen burningIIMBimplosioninertiainflationinfrared slaveryinfrared astronomyinteractioninteraction length λinterstellar mediuminverse Compton scattering

318 16 GlossaryNuclei of fixed charge representing the same chemical element al-beit with different masses. The chemical element is characterizedby the atomic number. Therefore, isotopes are nuclei of fixed protonnumber, but variable neutron number.ionizationisomersisospinisotopeJJeans massjetsKK; a meson consisting of a strange quark and an anti-up or anti-down quark or, correspondingly, an anti-strange quark and an upor down quark.KamiokandekaonkpcLlarge attractorlatitude effectLiberation of atomic electrons by photons or charged particles,namely photo ionization and ionization by collision.Long-lived excited states of a nucleus.Hadrons of equal mass (apart from electromagnetic effects) aregrouped into isospin multipletts, in analogy to the spin. Individualmembers of the multipletts are considered to be presented by thethird component of the isospin. Nucleons form an isospin doublet(I = 1 2 , I 3(p) =+ 1 2 , I 3(n) =− 1 2 ) and pions an isospin triplet(I = 1, I 3 (π + ) =+1, I 3 (π − ) =−1, I 3 (π 0 ) = 0).An inhomogeneity in a matter distribution will grow due to gravitationalattraction and contribute to the formation of structure inthe universe (e.g., galaxy formation), if it exceeds a certain criticalsize (Jeans mass).Long narrow streams of matter emerging from radiogalaxies andquasars, or bundles of particles in particle–antiparticle annihilations.Nucleon Decay Experiment in the Japanese Kamioka mine for themeasurement of cosmic and terrestrial neutrinos.Kilo parsec (see parsec).Hypothetical supercluster complex with a mass of > 10 4 galaxiesat a distance of several 100 million light-years which impresses anoriented proper motion on the local group of galaxies.Increase of the cosmic-ray intensity to the geomagnetic polescaused by the interaction of charged primary cosmic rays withthe Earth’s magnetic field.

16 Glossary 319Short for Large Electron–Positron Collider, the e + e − storage ringat CERN with a circumference of 27 km.The production of leptons in the early universe.A fundamental fermion which does not participate in strong interactions.The electrically charged leptons are the electron (e), themuon (µ),andthetau(τ), and their antiparticles. Electrically neutralleptons are the associated neutrinos (ν e , ν µ , ν τ ).Quantum number describing the lepton flavour. The three generationsof charged leptons (e − , µ − , τ − ) have different individuallyconserved lepton numbers, while their neutrinos are allowed to oscillate,thereby violating lepton-number conservation.Large Hadron Collider at CERN in Geneva. In the LHC protonswill collide with a center-of-mass energy of approximately 14 TeV.LHC is expected to start operation in 2007. LHC will be the acceleratorwith the highest center-of-mass energy in the world. It isgenerally hoped that the physics of the Large Hadron Collider willhelp to clarify some open questions of particle physics, for example,the question whether the Higgs boson responsible for the generationof masses of fundamental fermions exists and which massit has.The distance traveled by light in vacuum during one year; 1 ly =9.46 × 10 15 m = 0.307 pc.The model that explains pulsars as flashes of light from a rotatingpulsar.Irregular galaxy in the southern sky at a distance of 170 000 lightyears(̂= 52 kpc).A system of galaxies comprising among others our Milky Way, theAndromeda Nebula, and the Magellanic Clouds. The diameter ofthe local group amounts to approximately 5 million light-years.Virgo supercluster; ‘Milky Way’ of galaxies to which also the localgroup and the Virgo cluster belong. Its extension is approximately30 Mpc.Feature emerging from special relativity in which a moving objectappears shortened along its direction of motion.Transformation of kinematical variables for frames of referencewhich move at linear uniform velocity with respect to each otherunder the consideration that the speed of light in vacuum is amaximum velocity.LEPleptogenesisleptonlepton numberLHClight-yearlighthouse modelLMC – Large Magellanic Cloudlocal grouplocal superclusterLorentz contractionLorentz transformation

320 16 GlossaryTotal light emission of a star or a galaxy in all ranges of wave-lengths. The luminosity therefore corresponds to the absolutebrightness.luminosityluminosity distanceMA measure of the relative brightness of a star or some other ce-lestial object. The brightness differences between two stars withintensities I 1 and I 2 are fixed through the magnitude m 1 − m 2 =−2.5log 10 (I 1 /I 2 ). The zero of this scale is defined in such a waythat the apparent brightness of the polestar comes out to be 2 m , 12.An apparent brightness difference of m = 2.5 corresponds to anintensity ratio of 10 : 1. On this scale the planet Venus has the magnitudem =−4.4. With the naked eye one can see stars down tothe magnitude m = 6. The strongest telescopes are able to observestars down to the 28th magnitude.M87MACHOMagellanic Cloudmagic numbersmagnetarmagnetic monopolemagnitudemain-sequence starThe distance to an astronomical object derived by comparing itsapparent brightness to its total known or presumed luminosity byassuming isotropic emission. If the luminosity is known, the sourcecan be used as standard candle.Galaxy in the Virgo cluster at a distance of 11 Mpc. M87 is consideredan excellent candidate for a source of high-energy cosmicrays.Massive Compact Halo Object. Experiment for the search for darkcompact objects in the halo of our Milky Way, see also EROS.Nearest galaxy neighbour to our Milky Way consisting of the Small(SMC) and the Large Magellanic Cloud (LMC); distance 52 kpc.Nuclei whose atomic number Z or neutron number N is one ofthe magic numbers 2, 8, 20, 28, 50, or 126. In the framework ofthe shell model of the nucleus these magic nucleon numbers formclosed shells. Nuclei whose proton and also neutron number aremagic are named doubly magic.Special class of neutron stars having a superstrong magnetic field(upto10 11 Tesla). Magnetars emit sporadic γ -ray bursts.Hypothetical particle which constitutes separate sources andcharges of the magnetic field. Magnetic monopoles are predictedby Grand Unified Theories.Star of average age lying on the main sequence in the Hertzsprung–Russell diagram.

16 Glossary 321Distant galaxies with a bright active galactic nucleus emitting inthe blue and UV range. The Markarian galaxies are also sources of≥ TeV gamma rays.Rest mass; the rest mass m 0 of a particle is that mass, which oneobtains, if the energy of an isolated free particle in the state ofrest is divided by the square of the velocity of light. When particlephysicists use the name mass, they always refer to the rest mass ofa particle.The sum (A) of the number of neutrons (N) and protons (Z) inanucleus.An asymmetry created by some, so far unknown baryon- and lepton-number-violatingprocess in the early universe (see Sakharovconditions).Flavour oscillations which can be amplified by matter effects in aresonance-like fashion; e.g., suppression of solar neutrinos in ν e e −interactions in the Sun.Moment when the energy density of radiation dropped below thatof matter at around 50 000 years after the Big Bang.A hadron consisting of a quark and an antiquark.A small dark object in the line of sight to a bright background starcan give rise to a brightness excursion of the background star dueto a bending of the light rays by the dark body.Extremely small (≈ µg) black holes could have been formed in theearly universe. These primordial black holes are not final states ofcollapsing or evaporating stars.Mass or energy in cosmology which must be present because ofits gravitational force it exerts on other matter. The missing massdoes not emit detectable electromagnetic radiation (see dark matter).Missing mass is also encountered in relativistic kinematics,where in many experiments, in which total energy is fully constrainedby the center-of-mass energy, the mass, energy, and momentumof particles escaping from the detector without interaction(like neutrinos or SUSY particles), can be reconstructed due to thedetection of all other particles.See Friedmann–Lemaître universes.All grand unified theories predict large numbers of massive magneticmonopoles in contrast to observation. Inflation during the eraof monopole production would have diluted the monopole densityto a negligible level, thereby solving the problem.Markarian galaxiesmassmass numbermatter–antimatter asymmetrymatter oscillationmatter–radiation equalitymesonmicrolensingmini black holesmissing massmodels of the universemonopole problem

322 16 GlossaryMpcMSW effectMtheorymultiplicitymultiversemuonmuon numberNInteraction mechanism mediated by the exchange of a virtual neu-tral gauge boson.negative pressureneutral currentneutralinoneutrinoMegaparsec (see parsec).Matter oscillations first proposed by Mikheyev, Smirnov, andWolfenstein; see matter oscillation.Supersymmetric string theory (‘superstring theory’) unifying allinteractions. In particular, the M theory also contains a quantumtheory of gravitation. The smallest objects of the M theory are p-dimensional membranes. It appears now that all string theories canbe embedded into an 11-dimensional theory, called M theory. Outof the 10 spatial dimensions 7 are compactified into a Calabi–Yauspace.Number of secondary particles produced in an interaction.Hypothetical enlargement of the universe in which our universe isonly one of an enormous number of separate and distinct universes.µ; the muon belongs to the second family of charged leptons. It hasthe electric charge −1.The muon µ − and its associated muon neutrino have the muoniclepton number +1, their antiparticles the muonic lepton number−1. All other elementary particles have the muon number 0. Themuon number – except for neutrino oscillations – is a conservedquantity.In the same way as a positive pressure increases gravitation throughits field, negative pressure (like, for example, in a spring) leads toa repulsive gravity. Dark energy represents a form of energy that isgravitationally repulsive, due to its negative effective pressure.Candidate for a non-baryonic dark-matter particle. In the frame-work of supersymmetry fermions and bosons come in supermultiplets.The neutralino is the bosonic partner of the neutrino. Thelightest supersymmetric partner is expected to be stable and wouldtherefore be a good dark-matter candidate.A lepton without electric charge. Neutrinos only participate inweak and gravitational interactions and therefore are very difficultto detect. There are three known types of neutrinos, which are allvery light. These leptons are ν e , ν µ ,andν τ for the electron, muon,and tau family. At the LEP experiments it could be shown that apartfrom the three already known neutrino generations there is no furthergeneration with light neutrinos (m ν < 45 GeV/c 2 ).

16 Glossary 323Transmutation of a neutrino flavour into another by vacuum or matteroscillations.Neutrino flavour oscillation in vacuum which can occur if neutrinoshave mass and if the weak eigenstates are superpositions ofdifferent mass eigenstates.neutrino oscillationsneutrino vacuum oscillationn; a baryon with electric charge zero. A neutron is a fermion with neutronan internal structure consisting of two down quarks and one upquark, which are bound together by gluons. Neutrons constitutethe neutral component of atomic nuclei. Different isotopes of thesame chemical element differ from each other only by a differentnumber of neutrons in the nucleus.Neutron decay process into a proton, an electron, and an electronantineutrino.Nuclear decay by emission of a neutron.Star of extremely high density consisting predominantly of neutrons.Neutron stars are remnants of supernova explosions wherethe gravitational pressure in the remnant star is so large that theelectrons and protons are merged to neutrons and neutrinos (e − +p → n + ν e ). Neutron stars have a diameter of typically 20 km.If the gravitational pressure surmounts the degeneracy pressure ofneutrons, the neutron star will collapse to a black hole.For an external observer the black hole has only three properties:mass, electric charge, and angular momentum. If two black holesagree in these properties they are indistinguishable.Star with a sudden increase in luminosity (≈ 10 6 fold). The starexplosion is not as violent as in supernova explosions. A nova canalso occur several times on the same star.The energy, which is required to desintegrate an atomic nucleusinto its constituents; ≈ 8MeV/nucleon.Fission of a nucleus in two fragments of approximately equal size.In most cases fission leads to asymmetric fragments. It can occurspontaneously or be induced by nuclear reactions.Fusion of light elements to heavier elements. In a fusion reactorprotons are combined to produce helium via deuterium. Our Sun isa fusion reactor.neutron decayneutron evaporationneutron star‘no hair’ theoremnovanuclear binding energynuclear fissionnuclear fusion

324 16 GlossaryOA Friedmann–Lemaître model of an infinite, permanently expand-ing universe.OGLEOlbert’s paradoxopen universeorbitOSSEPpair productionparallaxparityparsecparticle horizonOptical Gravitational Lens Experiment looking for dark stars in thegalactic halo of the Milky Way.If the universe were infinite, uniform, and unchanging, the sky atnight would be bright since in whatever direction one looked, onewould eventually see a star. The number of stars would increasein proportion to the square of the distance from Earth while itsintensity is inversely proportional to the square of the distance.Consequently, the whole sky should be about as bright as the Sun.The paradox is resolved by the fact that the universe is not infinite,not uniform, and not unchanging. Also light from distant stars andgalaxies displays an extreme redshift and sometimes ceases to bevisible.The path of an object (e.g., satellite) that is moving around a secondobject or point.Oriented Scintillation Spectroscopy Experiment on board theCGRO satellite.Creation of fermion–antifermion pairs by photons in the Coulombfield of atomic nuclei. Pair production – in most cases electron–positron pair production – can also occur in the Coulomb field ofan electron or any other charged particle.An apparent displacement of a distant object with respect to a moredistant background when viewed from two different positions.The property of a wave function that determines its behaviour,when all its spatial coordinates are reversed. If a wave function ψsatisfies the equation ψ(r) =+ψ(−r), it is said to have even parity.If, however, ψ(r) =−ψ(−r), the parity of the wave functionis odd. Experimentally only the square of the absolute value of thewave function is observable. If parity were conserved, there wouldbe no fundamental way of distinguishing between left and right.Parity is conserved in electromagnetic and strong interactions, butit is violated in weak interactions.Parallax second; the unit of length used to express astronomicaldistances. It is the distance at which the mean radius of the Earth’sorbit around the Sun appears under an angle of 1 arc second. 1 pc =3.086 × 10 16 m = 3.26 light-years.Largest region that can be in causal contact.

16 Glossary 325See fermion.The point in an orbit of a star in a double-star system in which thebody describing the orbit is nearest to the star.The point in the orbit of the Moon or an artificial Earth satellite, atwhich it is closest to Earth.The point in the orbit of a planet, comet, or artificial satellite in asolar orbit, at which it is nearest to the Sun.Positron Electron Tandem Ring Accelerator, an electron–positronstorageringatDESYinHamburg.Intensity maximum of secondary cosmic rays at an altitude of approximately15 km produced by interactions of primary cosmicrays in the atmosphere.Supersymmetric partner of the photon.Liberation of atomic electrons by photons.The gauge boson of electromagnetic interactions.Pauli principleperiastronperigeeperihelionPETRAPfotzer maximumphotinophotoelectric effectphotonπ; the lightest meson; pions constitute an isospin triplet the mem- pionbers of which have electric charges of +1, −1, or 0.Intensity distribution of blackbody radiation of a blackbody of temperatureT as a function of the wavelength following Planck’s radiationlaw.Scale, at which the quantum nature of gravitation becomes visible,L Pl = √ G¯h/c 3 ≈ 1.62 × 10 −35 m.Energy scale, at which all forces including gravity can be describedby a unified theory; m Pl = √¯hc/G ≈ 1.22 × 10 19 GeV/c 2 .A law giving the distribution of energy radiated by a blackbody.The frequency-dependent radiation density depends on the temperatureof the blackbody. See also Planck distribution.The tension of a typical string in string theories.The time taken for a photon to pass the distance equal to the Plancklength: t Pl = √ G¯h/c 5 ≈ 5.39 × 10 −44 s.A shell of gas ejected from and expanding about a certain kind ofextremely hot star.Planck distributionPlanck lengthPlanck massPlanck’s radiation lawPlanck tensionPlanck timeplanetary nebula

326 16 GlossaryRotating neutron star with characteristic, pulsed emission in dif-ferent spectral ranges (radio, optical, X-ray, gamma-ray emission;‘pulsating radiostar’).positronpositron annihilationpower spectrumprimary cosmic raysprimordial black holesprimordial nucleosynthesisprimordial particlesprotonproton–proton chainprotostarProxima CentauripseudoscalarPSPCpulsarQA quantum is the minimum discrete amount by which certain prop-erties such as energy or angular momentum of a system can change.Planck’s constant is the smallest quantity of a physical action. Theelementary charge is the smallest charge of freely observable particles.quantume + ; antiparticle of the electron.Positron decay in matter by annihilation with an electron (e + e − →γγ).Describes a measure of the level of structure related to the densitydifference with respect to the average density in the early universewhich became frozen during the growth of structure.Radiation of particles originating from our galaxy and beyond. Itconsists mainly of protons and α particles, but also elements up touranium are present in primary cosmic rays.See Mini Black Holes.Production of atomic nuclei occurring during the first three minutesafter the Big Bang.Particles from the sources.p; the most commonly known hadron, a baryon of electric charge+1. Protons are made up of two up quarks and one down quarkbound together by gluons. The nucleus of a hydrogen atom is a proton.A nucleus with electric charge Z contains Z protons. Therefore,the number of protons determines the chemical properties ofthe elements.Nuclear reaction, in which hydrogen is eventually fused to helium.The pp cycle is the main energy source of our Sun.Very dense regions or aggregations of gas clouds, from which starsare formed.Nearest neighbour of our Sun at a distance of 4.27 light-years.Scalar quantity, which changes sign under spatial inversion.Position Sensitive Proportional Chamber on board of ROSAT.

16 Glossary 327Quantum anomalies can arise if a classical symmetry is broken inthe process of quantization and renormalization.QCD; theory of strong interactions of quarks and gluons.Quantum-mechanical theory applied to systems that have an infinitenumber of degrees of freedom. Quantum field theory also describesprocesses, in which particles can be created or annihilated.Frothy character of the space-time fabrique on ultramicroscopicscales.Quantum theory of gravitation aiming at the unification with theother types of interactions.Quantum mechanics describes the laws of physics which hold atvery small distances. The main feature of quantum mechanics isthat, for example, the electric charge, the energies, and the angularmomenta come in discrete amounts, which are called quanta.quantum anomaliesquantum chromodynamicsquantum field theoryquantum foamquantum gravitationquantum mechanicsq; a fundamental fermion subject to strong interactions. Quarks quarkcarry the electric charge of either + 2 3 (up, charm, top) or − 1 3(down, strange, bottom) in units, in which the electric charge ofaprotonis+1.Quasistellar radio sources; galaxies at large redshifts with an activenucleus which outshines the whole galaxy and therefore makes thequasar appear like a bright star.A scalar field model for the dark energy. In contrast to the cosmologicalconstant the energy density in the quintessence field representsa time-varying inhomogeneous component with a negativepressure which satisfies −1

328 16 GlossaryThe number of neutral and ionized atoms become equal at the re-combination temperature (3500 K).radio astronomyradio galaxyrecombinationrecombination temperaturered giantredshiftrelativity principleresidual interactionrest massright ascensionAstronomy in the radioband; it led to the discovery of the cosmicblackbody radiation.A galaxy with high luminosity in the radioband.Capture of an electron by a positive ion, frequently in connectionwith radiation emission.If a main-sequence star has used up its hydrogen supply, its nu-cleus will contract leading to a temperature increase so that heliumburning can set in. This causes the star to expand associated withan increase in luminosity. The diameter of a red giant is large comparedto the size of the original star.Increase of the wavelength of electromagnetic radiation by theDoppler effect, the expansion of the universe, or by strong gravitationalfields:z = λ = λ − λ 0λ 0 λ 0(λ 0 – emitted wavelength, λ – observed wavelength),⎧⎪⎨ v/c , classicalz = √ .⎪⎩ c+vc−v − 1 , relativisticEquivalence of observation of the same event from different framesof reference having different velocities and acceleration. There isno frame of reference that is better or qualitatively different fromany other.Interaction between objects that do not carry a charge but do con-tain constituents that have that charge. The residual strong interactionbetween protons and neutrons due to the strong charges oftheir quark constituents is responsible for the binding of the nucleus.In the thirties it was believed that the binding of protons andneutrons in a nucleus was mediated by the exchange of pions.The rest mass of a particle is the mass defined by the energy of anisolated free particle at rest divided by the speed of light squared.A coordinate which along with declination may be used to locateany object in the sky. Right ascension is the angular distance measuredeastwards along the celestial equator from the vernal equinoxto the intersection of the hour circle passing through the body. It isthe celestial equivalent to longitude.

16 Glossary 329The metric describing an isotropic and homogeneous space-time ofthe universe.German–British–American Roentgen satellite (launched 1990).Quantum number which distinguishes supersymmetric particlesfrom normal particles.Soviet–American Gallium Experiment for the measurement of solarneutrinos from the pp cycle.Necessary conditions first formulated by Sakharov which are requiredto create a matter-dominated universe: a baryon-numberviolatingprocess, violation of C or CP symmetry, and departurefrom thermal equilibrium.Small Astronomy Satellite; gamma-ray satellites launched byNASA, 1972 (SAS-2) resp. 1975 (SAS-3).The scale factor denotes an arbitrary distance which can be usedto describe the expansion of the universe. The ratio of scale factorsbetween two different epochs indicates by how much the size ofthe universe has grown.Event horizon of a spherical black hole, R = 2GM/c 2 .Excitation of atoms and molecules by the energy loss of chargedparticles with subsequent light emission.Secondary cosmic rays are a complex mixture of elementary particles,which are produced in interactions of primary cosmic rayswith the atomic nuclei of the atmosphere.Supersymmetric partner of the electron.Member of a small class of galaxies with brilliant nuclei and inconspicuousspiral arms. Seyfert galaxies are strong emitters in theinfrared and are also detectable as radio and X-ray sources. Thenuclei of Seyfert galaxies possess many features of quasars.Soft Gamma-Ray Repeaters are objects with repeated emission ofγ bursts. Presumably soft gamma-ray repeaters are neutron starswith extraordinary strong magnetic fields (see magnetar).A sudden pressure, density, and temperature gradient.Robertson–Walker MetricROSATR paritySSAGE experimentSakharov conditionsSAS-2, SAS-3scale factorSchwarzschild radiusscintillationsecondary cosmic raysselectronSeyfert galaxySGR objectsshock front

330 16 GlossaryOr cascade. High-energy elementary particles can generate numer-ous new particles in interactions, which in turn can produce particlesin further interactions. The particle cascade generated in thisway can be absorbed in matter. The energy of the primary initiatingparticle can be derived from the number of particles observedin the particle cascade. One distinguishes electromagnetic (initiatedby electrons and photons) and hadronic showers (initiated bystrongly interacting particles, e.g., p, α,Fe,π ± ,...).shock waveshowersingularitysleptonSMCSN 1987ASNAPSNOSNRsolar flaresolar windspaghettificationspallationA very narrow region of high pressure and temperature. Particlespassing through a shock front can be effectively accelerated, if thevelocity of the shock front and that of the particle have oppositedirection.A space-time region with infinitely large curvature – a space-timepoint.Supersymmetric partner of a lepton.Small Magellanic Cloud. Galaxy in the immediate vicinity of theLarge Magellanic Cloud (LMC).Supernova explosion in the Large Magellanic Cloud in 1987. Theprogenitor star was Sanduleak.SuperNova/Acceleration Probe. A space-based experiment to measurethe properties of the accelerating universe, which depend onthe amounts of dark energy and dark matter. The proposed SNAPsatellite is expected to be launched in ≈ 2014.Sudbury Neutrino Observatory.Remnant after a supernova explosion; mostly a rotating neutronstar (pulsar).Violent eruption of gas from the Sun’s surface.Flux of solar particles (electrons and protons) streaming away fromthe Sun.A body falling into a black hole will be stretched because of thedifferential gravitation.Nuclear transmutation by high-energy particles in which – contraryto fission – a large number of nuclear fragments, α particles, andneutrons are produced.

16 Glossary 331This theory refers to inertial non-accelerated frames of reference. Itassumes that physical laws are identical in all frames of referenceand that the speed of light in vacuum c is constant throughout theuniverse and independent of the speed of the observer. The theoremof the addition of velocities is modified to account for the deviationfrom the Galilei transformation.Intrinsic angular momentum of a particle quantized in units of ¯h,where ¯h =2π h and h is Planck’s constant.Emergence of different properties of a system at low energies (e.g.,weak and electromagnetic interaction) which do not exist at highenergies where these interactions are described by the Unified Theory.Supersymmetric partner of a quark.An astronomical object with well-known absolute luminosity whichcan be used to determine distances.Theory of fundamental particles and their interactions. Originallythe Standard Model described the unification of weak and electromagneticinteractions, the electroweak interaction. In a more generalsense it is used for the common description of weak, electromagnetic,and strong interactions.Galaxies with a high star-formation rate.A bunch of stars which are bound to each other by their mutualgravitational attraction.A quake in the crust of a neutron star.Older model of the universe in which matter is continuously producedto fill up the empty space created by expansion, therebymaintaining a constant density (‘steady-state universe’).This law states that the amount of energy radiated by a blackbodyis proportional to the fourth power of its temperature.The ejection of gas off the surface of a star.Synchrotron in which counter-rotating particles and antiparticlesare stored in a vacuum pipe. The particles usually stored in bunchescollide in interaction points, where the center of mass energy isequal to twice the beam energy for a head-on collision. Storagerings are used in particle physics and for experiments with synchrotronradiation.special theory of relativityspinspontaneous symmetry breakingsquarkstandard candleStandard Modelstarburst galaxiesstar clusterstar quakesteady-state universeStefan–Boltzmann lawstellar windstorage ring

332 16 GlossaryPhenomenon of apparent superluminal speed caused by a geomet-rical effect related to the finite propagation time of light.strangenessstrange quarkstringstring theorystrong interactionsunspot cyclesunspotssuperclustersupergalactic planeSuper-Kamiokande detectorsuperluminal speedsupermassive black holesupernovasupernova remnantThe strangeness of an s quark is −1. Strangeness is conserved instrong and electromagnetic interactions, however, violated in weakinteractions. In weak decays or weak interactions the strangenesschanges by one unit.s; the third quark flavour (if quarks are ordered with increasingmass). The strange quark has the electric charge − 1 3 e.In the framework of string theories the known elementary particlesare different excitations of elementary strings. The length of stringsis given by the Planck scale.Theory, which unifies the general theory of relativity with quantummechanics by introducing a microscopic theory of gravitation.Interaction responsible for the binding between quarks, antiquarks,and gluons, the constituents of hadrons. The residual interaction ofstrong interactions is responsible for nuclear binding.The recurring 11-year rise and fall in the number of sunspots.A disturbance of the solar surface which appears as a relativelydark center surrounded by less dark area. Sunspots appear darkbecause part of the thermal energy is transformed into magneticfield energy.See galactic cluster.Concentration of many galaxies around the Virgo cluster into aplane of diameter of about 30 Mpc.Successor experiment of the Kamiokande detector (see Kamiokande).Black hole at the center of a galaxy containing about 10 9 solarmasses.Star explosion initiated by a gravitational collapse, if a star hasexhausted its hydrogen and helium supply and collapses under itsown gravity. In a supernova explosion the luminosity of a star isincreased by a factor of 10 9 for a short period of time. The remnantstar of a supernove explosion is a neutron star or pulsar.Remnant of a supernova explosion; mostly a neutron star or pulsar.

16 Glossary 333Particles whose spin differs by 1/2 unit from normal particles. Superpartnersare paired by supersymmetry.In supersymmetric theories each fermion is associated with a bosonicpartner and each boson has a fermionic partner. In this waythe number of elementary particles is doubled. The bosonic partnersof leptons and quarks are sleptons and squarks. The fermionicpartners, for example, of the photon, gluon, Z,andW are photino,gluino, zino, and wino. Up to now no supersymmetric particleshave been found.A reduction in the amount of symmetry of a system, usually associatedwith a phase transition.Circular accelerator in which charged particles travel in bunchessynchronized with external and magnetic fields at a fixed radius.The orbit is stabilized by synchronizing an external magnetic guidingfield with the increasing momentum of the accelerated particles.Electromagnetic radiation emitted by an accelerated charged particlein a magnetic field.Particle that moves faster than the speed of light. Its mass squared isnegative. Its presence in a theory generally yields inconsistencies.superpartnerssupersymmetry (SUSY)symmetry breakingsynchrotronsynchrotron radiationTtachyonτ; the third flavour of charged leptons (if the leptons are arranged tauaccording to increasing mass). The tau carries the electric charge−1.τ − and its associated tau neutrino have the tau-lepton number +1,the antiparticles τ + and ¯ν τ the tau-lepton number −1. All otherelementary particles have the tau-lepton number 0. The tau-leptonnumber is a conserved quantity except for neutrino oscillations.The ultimate theory in which the different phenomena of electroweak,strong, and gravitational interactions are unified.Nuclear fusion of light elements to heavier elements at high temperatures(e.g., pp fusion at T ≈ 10 7 K).Stretching of time explained by special relativity; also called timedilation.tau-lepton numberTheory of Everything (TOE)thermonuclear reactiontime dilatation

334 16 Glossarytime-reversal invarianceTopological defects like magnetic monopoles, cosmic strings, do-main walls, or cosmic textures might have been created in the earlyuniverse.topological defectt; the sixth quark flavour (if quarks are arranged according to in-creasing mass). The top quark carries the electric charge + 2 3 e.Themass of the top quarks is comparable to the mass of a gold nucleus(≈ 175 GeV/c 2 ).top quarktriple-alpha processtritiumUuncertainty relationUnified Theoryup quarkVvacuum energy densityVan Allen beltsVela pulsarT invariance or simply time reversal. The operation of replacingtime t by time −t. As with CP violation, T violation occurs inweak interactions of kaon decays. The CPT operation which is asuccession of charge conjugation, parity transformation, and timereversal is regarded as a conserved quantity.Reaction in which three α particles are fused to carbon. Such aprocess requires high α-particle densities and a large resonancelikecross section.Hydrogen isotope with two additional neutrons in the nucleus.The quantum principle first formulated by Heisenberg which statesthat it is not possible to know exactly both the position x and themomentum p of an object at the same time. The same is true forthe complementary quantities energy and time.Any theory that describes all four forces and all of matter within asingle all-encompassing framework (see also Grand Unified Theory,GUT).u; the lightest quark flavour with electric charge + 2 3 e.Quantum fields in the lowest energy state describing the vacuummust not necessarily have the energy zero.Low-energy particles of the solar wind are trapped in certain regionsof the Earth’s magnetic field and stored.Vela X1, supernova remnant in the constellation Vela at a distanceof about 1500 light-years; the Vela supernova explosion was observedby the Sumerians 6000 years ago.

16 Glossary 335(c) The value of the velocity of light forms the basis for the definitionof the length unit meter. It takes 1/299 792 458 seconds forlight in vacuum to pass 1 meter. The velocity of light in vacuum isthe same in all frames of reference.Concentration of galaxies in the direction of the constellation Virgin.A particle that exists only for an extremely brief instant of time inan intermediate process. For virtual particles the Lorentz-invariantmass does not coincide with the rest mass. Virtual particles withnegative mass squared are called space-like, those with a positivemass squared time-like. Virtual particles can exist for timesallowed by the Heisenberg’s uncertainty principle.Charged gauge quanta of weak interactions. These quanta participatein so-called charged-current processes. These are interactionsin which the electric charge of the participating particles changes.Probability waves upon which quantum mechanics is founded.The weak interaction acts on all fermions, e.g., in the decay ofhadrons. It is responsible for the beta decay of particles and nuclei.In charged-current interactions the quark flavour is changed, whilein neutral-current interactions the quark flavour remains the same.A star that has exhausted most or all of its nuclear fuel and hascollapsed to a very small size. The stability of a white dwarf is notmaintained by the radiation or gas pressure as with normal stars butrather by the pressure of the degenerate electron gas. If the whitedwarf has a mass larger than 1.4 times the solar mass, also thispressure is overcome and the star will collapse to a neutron star.Weakly Interacting Massive Particles are candidates for dark matter.Wilkinson Microwave Anisotropy Probe launched in 2001 to measurethe fine structure of the cosmic background radiation.A proposed channel of space-time connecting distant regions ofthe universe. It is not totally inconceivable that wormholes mightprovide the possibility of time travel.The w parameter is defined as the ratio of pressure over energydensity (w ≡ P/ϱ). In models with a cosmological constant thepressure of the vacuum density P equals exactly the negative ofthe energy density ϱ; i.e., w = −1, in contrast to quintessencemodels, where −1

336 16 GlossaryXX bosonXMM-NewtonX-ray astronomyX-ray bursterYY bosonylemYukawa particleZThe Heisenberg uncertainty relation does not allow a finite quan-tum-mechanical system to have a definite position and definite momentumat the same time. Thus any system even in the lowest energystate must have a non-zero energy. Zero energy would lead tozero momentum and thereby to an infinite position uncertainty.Z bosonzero-point energySee Y boson.European X-ray space observatory mission, with its X-ray Multi-Mirror design using three telescopes. Named in honour of Sir IsaacNewton.Astronomy in the X-ray range (0.1 keV–100 keV).X-ray source with irregular sudden outbursts of X rays.The observed baryon–antibaryon asymmetry in the universe requires,among others, a baryon-number-violating process. Hypotheticalheavy X and Y bosons, existing at around the GUT scalewith masses comparable to the GUT energy scale (≈ 10 16 GeV),could have produced this matter–antimatter asymmetry in their decayto quarks and antiquarks, resp. baryons and antibaryons. Thecouplings of the X and Y bosons to fermion species is presently thesimplest idea for which baryon-number violation appears possible.Name for the state of matter before the Big Bang. Gamow proposedthat the matter of the universe originally existed in a promordialstate, which he coined ‘ylem’ (from the Greek υ´λη ↩ which standsfor ‘matter’, ‘wood’ via the medieval latin hylem meaning the primordialelements of life). According to his idea all elements wereformed shortly after the Big Bang from this primary substance.Yukawa predicted a particle which should mediate the binding betweenprotons and neutrons. After its discovery the muon was initiallymistaken to be this particle. The situation was resolved withthe discovery of the pion as Yukawa particle.Neutral boson of weak interactions. It mediates weak interactionsin all those processes, where electric charge and flavour do notchange.

33717 Solutions“The precise statement of any problem is the most important stepin its solution.”Edwin Bliss17.1 Chapter 11. a) The centrifugal force is balanced by the gravitational force between the Earth andthe satellite:mv 2= G mM ♁R ♁ R♁2 ≈ mg ,where it has been taken into account that the altitude is low and R ≈ R ♁ .Asaresult,√v ≈ gR ♁ ≈ 7.9km/s .b) The escape velocity is found from the condition that at infinity the total energyequals zero. SinceE tot = 1 2 mv2 + E potand the potential energy is given byE pot =−G mM ♁R ♁,one obtains12 mv2 = G mM √♁⇒ v = 2MR ♁ G/R ♁ ≈ 11.2km/s .♁c) The centrifugal force is balanced by the gravitational force between Earth and thesatellite. From the equality of the magnitudes of these two forces one gets:v 2r = GM ♁r 2 .From the relation between the velocity v and revolution frequency ω one obtains:

338 17 Solutionsv 2 = r 2 ω 2 = G M ♁⇒ r 3 = GM ♁rω 2 .Taking into account that for the geostationary satellite the revolution period T ♁ =1day = 86 400 s and ω ♁ = 2π/T ♁ , one obtains√r = 3 GM ♁ T♁24π 2 ≈ 42 241 km .The altitude above ground level therefore is H = r − R ♁ = 35 871 km.Geostationary satellites can only be positioned above the equator because only therethe direction of the centrifugal force can be balanced by the direction of the gravitationalforce. In other words, the centrifugal force for a geostationary object pointsoutward perpendicular to the Earth’s axis and only in the equator plane the gravitationalforce is collinear to the former.2. The centrifugal force is balanced by the Lorentz force, somv 2ϱ= evB ⇒ p = eBϱ ⇒ ϱ =peB .Since the kinetic energy of the proton (1 MeV) is small compared to its mass (≈938 MeV), a classical treatment for the energy–momentum relation, E = p 2 /2m 0 ,isappropriate from which one obtains p = √ 2m 0 E kin .Then the bending radius is given byϱ = p √eB = 2m0 E kineBDimensional analysis:{ } kg mp = eBϱ, psor ϱ ≈ 2888 m .{ m }c = (pc) {J} =e {As} B {T} ϱ {m} c {ms −1 } ,s(pc) [J]×1.6 × 10 −19 J/eV = 1.6 × 10 −19 × B [T]×ϱ [m]×3 × 10 8 ,(pc)[eV] =3 × 10 8 B[T] ϱ[m] =300 B [Gauss] ϱ [cm](pc) [eV]ϱ =300 B [Gauss] cm = 2.888 × 105 cm .3. From Fig. 4.2 one can read the average energy loss of muons in air-like materials to be≈ 2MeV/(g/cm 2 ). This number can also be obtained from (4.6). The column densityof the atmosphere from the production altitude can be read from Fig. 7.3. It is approximately940 g/cm 2 . Finally, the average energy loss in the atmosphere is− dEdx x ≈ 2MeV/(g/cm2 ) × 940 g/cm 2 = 1.88 GeV .

17.1 Chapter 1 3394. By definition (see the Glossary) the following relation holds between the ratio of intensitiesI 1 ,I 2 and the difference of two star magnitudes m 1 ,m 2 :so thatm 1 − m 2 =−2.5log 10 (I 1 /I 2 ),I 1 /I 2 = 10 −0.4(m 1−m 2 ) ,and from m = 1 one gets I 1 /I 2 ≈ 0.398 or I 2 ≈ 2.512 I 1 .5. Let N be the number of atoms making up the celestial body, µ the mass of the nucleon,and A the average atomic number of the elements constituting the celestial object. Gravitationalbinding dominates ifGM 2R>Nε,where ε is a typical binding energy for solid material (≈ 1 eV per atom). Here numericalfactors of order unity are neglected; for a uniform mass distribution the numerical factorwould be 6/5, see Problem 13.5. The mass of the object is M = NµA, so that thecondition above can be written asGM 2R = GM R NµA>Nε or MR > εµAG .M/R can be rewritten asMR = 4 3 πR2 ϱ = (4 3 π) 1/3 (43 π) 2/3 R 2 ϱ 2/3} {{ }M 2/3 ϱ 1/3 = (4 3 π) 1/3 M 2/3 ϱ 1/3 ,which leads to( 43 π) 1/3 M 2/3 ϱ 1/3 > εµAGε ) 3/2 1or M>( √ .µAG 43 πϱThe average density can be estimated to be (again neglecting numerical factors of orderunity; for spherical molecules the optimal arrangement of spheres without overlappingleads to a packing fraction of 74%)ϱ = µ ,43 πr3 Bwhere r B = r e /α 2 is the Bohr radius (r B = 0.529 × 10 −10 m), r e the classical electronradius, and α the fine-structure constant. This leads toM> ( ε ) 3/2 1√= ( ε ) 3/2 r 3/2B√ = 1 (εr B ) 3/2 .µAG 43 πµ/(4 3 πr3 B ) µAG µ µ 2 AG

340 17 SolutionsWithµ = 1.67 × 10 −27 kg , ε = 1eV= 1.6 × 10 −19 J ,G = 6.67 × 10 −11 m 3 kg −1 s −2 , A = 50one gets the conditionM>4.58 × 10 22 kg .This means that our moon is gravitationally bound, while the moons of Mars (Phobosand Deimos) being much smaller are bound by solid-state effects, i.e., essentially byelectromagnetic forces.6. In special relativity the redshift is given byz = λ − λ 0λ 0=√1 + β1 − β − 1 .Let us assume that this relation can be applied even for the young universe, where atreatment on the basis of general relativity would be more appropriate.Solve z(β) for β or v:β = (z + 1)2 − 1(z + 1) 2 + 1⇒ v = (z + 1)2 − 1(z + 1) 2 + 1 c = Hd,t = d c = 1 (z + 1) 2 − 1H (z + 1) 2 + 1 ≈ 0.967H .SinceH 1 is the age of the universe, we see the distant quasar when it was ≈ 3.3% of itspresent age.17.2 Chapter 21. a) lepton-number conservation is violated: not allowed,b) possible,c) possible, a so-called Dalitz decay of the π 0 ,d) both charge and lepton-number conservation violated, not allowed,e) kinematically not allowed (m K − + m p >m Λ ),f) possible,g) possible,h) possible.

17.2 Chapter 2 3412. For any unstable elementary particle like, e.g., a muon, the quantity ‘lifetime’ shouldbe considered in the average sense only. In other words, it does not mean that a particlewith lifetime τ will decay exactly the time τ after it was produced. Its actual lifetime tis a random number distributed with a probability density function,f(t; τ)dt = 1 τ e− t τ dt,giving a probability that the lifetime t lies between t and t + dt. It can easily be checkedthat the mean value of t equals τ. For an unstable relativistic particle, a mean rangebefore it decays is given by the product of its velocity βc and lifetime γτ to βγcτ.Formuons cτ = 658.653 m. Therefore, to survive to sea level from an altitude of 20 km, theaverage√range should equal l = 20 km or βγcτ µ = l = 20 km and βγ = l/cτ µ .Fromβγ = γ 2 − 1 one gets γ 2 = (l/cτ µ ) 2 + 1.Since l/cτ ≫ 1, one finally obtainsγ ≈ l/cτ µ = 20 × 103 m658.653 m ≈ 30.4 .Then the total energy isE µ = γm µ c 2 ≈ 3.2GeVand the kinetic energy isE kinµ = E µ − m µ c 2 ≈ 3.1GeV.3. The Coulomb force isF Coulomb = 1 q 1 q 21(1.602 × 10 −19 As) 24πε 0 r 2 ≈4π × 8.854 × 10 −12 Fm −1 (10 −15 m) 2≈ 230.7Nand the gravitational force isF gravitation = G m 1m 2r 2−11 m3 (2.176 × 10 −8 kg) 2≈ 6.674 × 10kg s 2 (10 −15 m) 2 ≈ 31 600 N .4. Energy–momentum conservation requiresq e + + q e − = q f ,where q e +,q e −,andq f are the four-momenta of the positron, electron, and final state,respectively. To produce a Z, the invariant mass of the initial state squared should be notless than the invariant mass of the required final state squared or(q e + + q e −) 2 ≥ m 2 Z .Since q e + = (E e +, p e +) and q e − = (m e , 0), one obtains

342 17 Solutionsor2m e (E e + + m e ) ≥ m 2 ZE e + ≥ m2 Z2m e− m e ≈ 8.1 × 10 15 eV = 8.1PeV.17.3 Chapter 31. Similarly to Problem 2.4, for the reaction γ + γ → µ + + µ − the threshold condition is(q γ1 + q γ2 ) 2 ≥ (2m µ ) 2 .For photons qγ 2 1= qγ 2 2= 0. Then(q γ1 + q γ2 ) 2 = 2(E γ1 E γ2 − p γ1 · p γ2 ) = 4E γ1 E γ2for the angle π between the photon 3-momenta. Finally,E γ1 ≥ (m µc 2 ) 2≈ 1.1 × 10 19 eV .E γ22. The number of collisions is obtained by subtracting the number of unaffected particlesfrom the initial number of particles:N = N 0 − N = N 0 (1 − e −x/λ ).For thin targets x/λ ≪ 1 and the expansion in the Taylor series givesN = N 0 (1 − (1 − x/λ +···)) = N 0xλ = N 0N A σ N x.For the numerical example one getsN = 10 8 × 6.022 × 10 23 g −1 × 10 −24 cm 2 × 0.1g/cm 2 ≈ 6 × 10 6 .3. The threshold condition is(q¯νe + q p ) 2 ≥ (m n + m e ) 2 .Taking into account that q 2¯ν e= 0andq p = (m p , 0), one obtainsandm 2 p + 2E¯ν em p ≥ (m n + m e ) 2E¯νe ≥ (m n + m e ) 2 − m 2 p2m p≈ 1.8MeV,where the following values of the particles involved were used: m n = 939.565 MeV,m p = 938.272 MeV, m e = 0.511 MeV.

17.4 Chapter 4 3434.p b❅■ ✒❅✻ target (mass A,chargeZ)rz b✲ particle track pForce F = zeZe rr ,p b =∫ +∞−∞= zZe2βcr 2∫ +∞zZe 2 b dx|F b | dt =−∞ r 2 r βc∫ +∞b dx−∞ ( √ x 2 + b 2 ) = zZe23 βcbmomentum transferperpendicular to pd(x/b)(√1 + ( )) 3x 2b∫ +∞−∞= 2zZe2βcb= 2r em e cbβ zZ ,} {{ }=2where the classical electron radius is r e =e2m e c 2 .5. Previously the transverse momentum transfer was obtained to be (z = 1)p b = 2 Z r em e cbβ= 2p r eZbβ 2 ,where p = m e v was assumed (classical treatment). The transverse momentum transferis given by p b = p sin ϑ. Sincesin 2γ = 2sinγ cos γ = 2tanγ cos 2 γ =2tanγ1 + tan 2 γ ,one gets, using Rutherford’s scattering formula:r e Z/bβ 2p b = 2p1 + (r e Z/bβ 2 ) 2 .17.4 Chapter 41. φ = N AA σ A = N A σ N ,[N A ]=mol −1 ⎫⎪⎬[A] = gmol −1 ⇒ [φ] =(g/cm 2 ) −1 .[σ A ] = cm 2 ⎪ ⎭

344 17 Solutions2. The relative energy resolution is determined by the fluctuations of the number N ofproduced particles. If W is the energy required for the production of a particle (pair),the relative energy resolution isEE= NN= √NN = 1 √N=√WE ,since N = E/W. HereE and E are the energy of the particle and its uncertainty,⎧⎪ a) 10%E ⎨E = b) 5.5%.⎪ c) 1.9% ⎩d) 3.2 × 10 −4∫ 0∫dE E3. R =E dE/dx = dE0 a + bE = 1 b ln(1 + b a E) .The numerical calculation givesR ≈()14.4 × 10 −6 ln 4.4 × 10−61 + 10 52g≈ 45 193gcm2 or R = 181 m of rock taking into account that ϱ rock = 2.5g/cm 3 .4. The Cherenkov angle θ C is given by the relationcos θ C = 1nβ .From the relation between the momentum p and β:p = γm 0 βc ⇒ β = pγm 0 candcos θ C = γm 0cnp .cm 2Thennp cos θ Cm 0 c√= Em 0 c 2 = c p 2 + m 2 0 c2m 0 c 2and(np cos θ C ) 2 = p 2 + m 2 0 c2 ⇒ p 2 (n 2 cos 2 θ C − 1) = m 2 0 c2 .Finally,m 0 = p c√n 2 cos 2 θ C − 1 .

17.4 Chapter 4 3455. From the expression for the change of the thermal energy, Q = c sp mT , one can findthe temperature rise:T = Qc sp m = 104 eV × 1.6 × 10 −19 J/eV8 × 10 −5 = 2 × 10 −11 K .J/(gK) × 1g6. q γ + q e = q ′ γ + q′ e ⇒ q γ − q ′ γ = q′ e − q e ⇒q 2 γ + q2 γ ′ − 2q γ q ′ γ =−2(E γ E ′ γ − p γ p ′ γ ) = q′2 e + q2 e − 2q′ e q e ⇒−2E γ E ′ γ (1 − cos θ) = 2m2 e − 2E′ e m e ; (p e = 0)=−2m e (E ′ e − m e) =−2m e E kine =−2m e (E γ − E ′ γ ) ⇒E γ − E ′ γE ′ γ= E γE ′ γ− 1 = E γm e(1 − cos θ)⇒E γ′ 1=E γ 1 + E γm e(1 − cos θ) = 11 + ε(1 − cos θ) .7. Start from (4.13). The maximum energy is transferred for backscattering, θ = π;E γ′ = 1E γ 1 + 2ε ,E maxe = E γ − E ′ γ = E γ − E γ1 + 2ε = E γwith numbers: Eemax = 478 keV (‘Compton edge’).For ε →∞(∗) yields Ee max = E γ .For θ γ = π one hasE ′ γ = E γ11 + 2 εand consequentlyE ′ γ = m ec 2 ε1 + 2 ε = m ec 22 + 1/ε .For ε ≫ 1 this fraction approaches m e c 2 /2.2ε1 + 2ε = 2ε21 + 2ε m ec 2 ; (∗)8. Momentum: p = mv; m = γm 0 ,wherem 0 is the rest mass and γ =p = γm 0 βc ⇒ γβ = p/m 0 c.1√1 − β 2 :9. In the X-ray region the index of refraction is n = 1, therefore there is no dispersion andconsequently no Cherenkov radiation.

346 17 Solutions17.5 Chapter 51. a) In the classical non-relativistic case (v ≪ c) the kinetic energy is justE kin = 1 2 m 0v 2 = 1 2 m 0( ) eRB 2= 1 e 2 R 2 B 2,m 0 2 m 0where the velocity v was found from the usual requirement that the centrifugal forcebe balanced by the Lorentz force:m 0 v 2R= evB ⇒ v =eBRm 0.b) In the relativistic case⎛⎞√E kin = γm 0 c 2 − m 0 c 2 = m 0 c 2 1⎝√− 1⎠ = c p 2 + m 2 0 c2 − m 0 c 21 − v2c 2 √√= c e 2 R 2 B 2 + m 2 0 c2 − m 0 c 2 = ecRB 1 + m2 0 c2e 2 R 2 B 2 − m 0c 2≈ 5.95 × 10 7 eV .Alternatively, with an early relativistic approximation√E kin = c p 2 + m 2 0 c2 − m 0 c 2 ≈ cp = ecRB ≈ 6 × 10 7 eV .2. Let us first find the number of electrons N e in the star:N e = M star /m p = 10 M ⊙ /m p ≈2 × 1034 g1.67 × 10 −24 g ≈ 1.2 × 1058 .Here it is assumed that the star consists mainly of hydrogen (m p ≈ 1.67 × 10 −24 g) andis electrically neutral, so that N e = N p .The pulsar volume isV = 4 3 πr3 ≈ 4.19 × 10 18 cm 3and the electron density is n = 0.5 N e /V ≈ 1.43 × 10 39 /cm 3 . Then the Fermi energyisE F = ¯hc(3π 2 n) 1/3≈ 6.582 × 10 −22 MeV s × 3 × 10 10 cm s(3 × 3.14162 × 1.43 × 10 39 ) 1/3 cm −1≈ 688 MeV .

17.5 Chapter 5 347Consequences:The electrons are pressed into the protons,e − + p → n + ν e .The neutrons cannot decay, since in free neutron decay the maximum energy transfer tothe electron is only ≈ 780 keV, and all energy levels in the Fermi gas are occupied, evenif only 1% of the electrons are left: (n ∗ = 0.01 n ⇒ E F ≈ 148 MeV).3. Event horizon of a black hole with mass M = 10 6 M ⊙ :R S = 2GMc 2 ≈ 2.96 × 10 9 m ,∫ RSE =− G m pM∞ r 2 dr = G m pM= 1 R S 2 m pc 2 ≈ 469 MeV ≈ 7.5 × 10 −8 J .The result is independent of the mass of the black hole. This classical calculation, however,is not suitable for this problem and just yields an idea of the energy gain. In addition,the value of the energy depends on the frame of reference. In the vicinity of blackholes only formulae should be used which hold under general relativity.4. a) Conservation of angular momentumL = r × p ⇒ L ≈ mrv = mr 2 ωrequires r 2 ω to be constant (no mass loss):(R⊙ 2 ω ⊙ = RNS 2 ω R⊙) 2ω⊙NS ⇒ ω NS =≈ 588 Hz .R NSThen the rotational energy isE rot = 1 22 5 M NSRNS 2 ω2 NS ≈ 0.4 × 1030 kg × (5 × 10 4 ) 2 m 2 × 588 2 s −2≈ 0.35 × 10 45 J .b) Assume that the Sun consists of protons only (plus electrons, of course). Four protonseach are fused to He with an energy release of 26 MeV corresponding to amass–energy conversion efficiency of ≈ 0.7%:E = M ⊙ c 2 × 7 × 10 −3 = 2 × 10 30 kg × (3 × 10 8 ) 2 m 2 /s 2 × 7 × 10 −3= 1.26 × 10 45 J ,which is comparable to the rotational energy of the neutron star.5. A dipole field is needed to compensate the centrifugal forcemv 2R = evB guide . (∗)Equation (5.38) yields p = mv = 1 2eRB. Comparison with (∗)givesB guide = 1 B (Wideroe condition) .2

348 17 Solutions17.6 Chapter 6Sect. 6.11. C, O, and Ne are even–even nuclei, oxygen is even doubly magic, while F as even–odd and N, Na as odd–odd configurations are less tightly bound. The pairing term δ inthe Bethe–Weizsäcker formula gives the difference in the nuclear binding energies foreven–even, even–odd, and odd–odd nuclei:m(Z, A) = Zm p + (A − Z)m n − a v A + a s A 2/3 Z 2+ a CA 1/3 + a (Z − A/2) 2a+ δAa v A – volume term,a s A 2/3 – surface term,Za 2C A 1/3(Z−A/2)a 2a A– Coulomb repulsion,– asymmetry term,⎧⎨ −a p A −3/4 for even–evenδ = 0 for even–odd/odd–even⎩+a p A −3/4 for odd–odd, δ – pairing term .2. From Fig. 7.9 the rate of primary particles can be estimated asΦ ≈ 0.2 (cm 2 ssr) −1 .The surface of the Earth isS = 4πR 2 ♁ ≈ 4π(6370 × 105 ) 2 cm 2 ≈ 5.10 × 10 18 cm 2 .The age of the Earth is T = 4.5 × 10 9 years or 1.42 × 10 17 s. The total charge accumulatedin the solid angle of 2π during the time T is∫Φ(x,t)dx dt ≈ 1.45 × 10 35 × 2π≈ 9.1 × 10 35 equivalent protons ̂= 1.5 × 10 17 Coulomb .Still, there is no charge-up. Primary particles mentioned in Sect. 6.1 are mainly those ofhigh energy (typically > 1 GeV). In this high-energy domain positively charged particlesactually dominate. If all energies are considered, there are equal amounts of positiveand negative particles. Our Sun is also a source of large numbers of protons, electrons,and α particles. In total, there is no positive charge excess if particles of all energiesare considered. This is not in contrast to the observation of a positive charge excessof sea-level muons because they are the result of cascade processes in the atmosphereinitialized by energetic (i.e., mainly positively charged) primaries.

17.6 Chapter 6 3493. a) The average column density traversed by primary cosmic rays is ≈ 6g/cm 2 (seeSect. 6.1). The interaction rate Φ is related to the cross section by, see (3.57),Φ = σN A .Because of Φ[(6g/cm 2 ) −1 ]≈0.1 and assuming that the collision partners are nucleons,one getsσ ≈ 0.1 × (6g/cm 2 ) −1 /N A ≈ 1.66 × 10 −25 cm 2 = 166 mbarn .b) If the iron–proton fragmentation cross section is ≈ 170 mb, the cross sectionfor iron–air collisions can be estimated to be σ frag (iron–air) ≈ 170 mb × A α ,where A ≈ 0.8 A N + 0.2 A O = 11.2 + 3.2 = 14.4. With α ≈ 0.75 onegets σ frag (iron–air) ≈ 1.26 b. The probability to survive to sea level is P =exp (−σN A d/A) = 1.3 × 10 −23 , where it has been used that the thickness of theatmosphere is d ≈ 1000 g/cm 2 .Sect. 6.21. The neutrino flux φ ν is given by the number of fusion processes 4p → 4 He + 2e + + 2ν etimes 2 neutrinos per reaction chain:φ ν =solar constantenergy gain per reaction chain × 21400 W/m 2≈28.3MeV× 1.6 × 10 −13 J/MeV × 2 ≈ 6.2 × 1010 cm −2 s −1 .2. (q να + q e −) 2 = (m α + m νe ) 2 , α = µ, τ ;assuming m να to be small (≪ m e ,m µ ,m τ ) one gets2E να m e + m 2 e = m2 α ⇒ E να = m2 α − m2 e⇒2m eα = µ : E νµ = 10.92 GeV , α = τ : E ντ = 3.09 TeV ;since solar neutrinos cannot convert into such high-energy neutrinos, the proposed reactionscannot be induced.3. a) The interaction rate isR = σ N N A dAφ νwhere σ N is the cross section per nucleon, N A = 6.022 × 10 23 g −1 is the Avogadroconstant, d the area density of the target, A the target area, and φ ν the solar neutrinoflux. With d ≈ 15gcm −2 , A = 180 × 30 cm 2 , φ ν ≈ 7 × 10 10 cm −2 s −1 ,andσ N = 10 −45 cm 2 one gets R = 3.41 × 10 −6 s −1 = 107 a −1 .b) A typical energy of solar neutrinos is 100 keV, i.e., 50 keV are transferred to theelectron. Consequently, the total annual energy transfer to the electrons isE = 107 × 50 keV = 5.35 MeV = 0.86 × 10 −12 J .

350 17 Solutionsc) With the numbers used so far the mass of the human is 81 kg. Therefore, the equivalentannual dose comes out to beH ν = Em w R = 1.06 × 10 −14 Sv ,actually independent of the assumed human mass. The contribution of solar neutrinosto the normal natural dose rate is negligible, sinceH = H νH 0= 5.3 × 10 −12 .4. a) The time evolution of the electron neutrino is|ν e ; t〉 =cos θ e −iE ν 1 t |ν 1 〉+sin θ e −iE ν 2 t |ν 2 〉 .This leads to〈ν µ |ν e ; t〉 =sin θ cos θbecause(e −iE ν 2 t − e −iE ν 1 t ) ,(− sin θ|ν 1 〉+cos θ|ν 2 〉)(cos θ e −iE ν 1 t |ν 1 〉+sin θ e −iE ν 2 t |ν 2 〉)=−sin θ cos θ e −iE ν 1 t + sin θ cos θ e −iE ν 2 tsince |ν 1 〉; |ν 2 〉 are orthogonal states.Squaring the time-dependent part of this relation, i.e., multiplying the expression byits complex conjugate, yields as an intermediate step(e −iE ν 2 t − e −iE ν 1 t )( e +iE ν 2 t − e +iE ν 1 t ) = 1 − e i(E ν 2 −E ν1 )t − e i(E ν 1 −E ν2 )t + 1= 2 − (e i(E ν 2 −E ν1 )t + e −i(E ν 2 −E ν1 )t ) = 2 − 2cos[(E ν2 − E ν1 )t]= 2 (1 − cos [(E ν2 − E ν1 )t]) = 4sin 2 [(E ν2 − E ν1 )t/2] .Using 2 sin θ cos θ = sin 2θ one finally getsP νe →ν µ(t) =|〈ν µ |ν e ; t〉| 2 = sin 2 2θ sin 2 [(E ν1 − E ν2 )t/2] .Since the states |ν e 〉 and |ν µ 〉 are orthogonal one has P νe →ν e(t) = 1 − P νe →ν µ(t).In the approximation of small masses, E νi = p + m 2 i /2p + O(m4 i ), one gets E ν 1−E ν2 ≈ (m 2 1 − m2 2 )/2p.Since t = E νi x/p and because of m i ≪ E νi , the momentum p can be identifiedwith the neutrino energy E ν . If, finally, also the correct powers of ¯h and c areintroduced, one obtains( 1P νe →ν e(x) = 1 − sin 2 2θ sin 2 x )4¯hc δm2 E νwith 1/4¯hc = 1.27 × 10 9 eV −1 km −1 , which is the desired relation.

17.6 Chapter 6 351b) On both sides of the Earth a number of N muon neutrinos are created. The ones fromabove, being produced at an altitude of 20 km almost have no chance to oscillate,while the ones crossing the whole diameter of the Earth have passed a distance ofmore than 12 000 km, so that from below only NP νµ →ν µ(2R E ) muon neutrinos willarrive, leading to a ratio ofS = 0.54 = NP ν µ →ν µ(2R E )= 1 − sin 2 (1.27 × 12 700 × δm 2 ).NSolving for δm 2 one obtains δm 2 = 4.6 × 10 −5 eV 2 and m ντ ≈ 6.8 × 10 −3 eV.c) No. The masses of the lepton flavour eigenstates are quantum-mechanical expectationvalues of the mass operator M = √ H 2 − p 2 . For an assumed (ν µ ↔ ν τ )mixing with a similar definition of the mixing angle as for the (ν e ↔ ν µ ) mixing,one getsm νµ =〈ν µ |M|ν µ 〉=m 1 cos 2 θ + m 2 sin 2 θ,m ντ =〈ν τ |M|ν τ 〉=m 1 sin 2 θ + m 2 cos 2 θ.For maximum mixing (θ = 45 ◦ ,cos 2 θ = sin 2 θ = 1/2) one obtainsm νµ = m ντ = (m 1 + m 2 )/2 .Sect. 6.31. I = I 0 e −µx photons survive, I 0 − I = I 0 (1 − e −µx ) get absorbed;from Fig. 6.37 one reads µ = 0.2cm −1 ;detection efficiency: η = I 0(1 − e −µx )= 1 − e −µx ≈ 0.45 = 45% .I 02. The duration of the brightness excursion cannot be shorter than the time span for thelight to cross the cosmological object. Figure 6.48 shows t = 1s⇒ size ≈ ct =300 000 km.3. cos ϑ = 1nβ ; threshold at β>1 ⇒ v> c nn ;E µ = γm 0 c 2 =1√1 − β 2 m 0c 2 =1√1 − 1 n 2 m 0 c 2 =n√n 2 − 1 m 0c 2 ,{ {4.5 GeV in air4.4 GeV in airE µ ≈160.3 MeV in water , Ekin µ ≈ 54.6 MeV in water .4. Number of emitted photons:N E = P hν ≈3 × 10 27 W10 11 eV × 1.602 × 10 −19 J/eV ≈ 1.873 × 1035 s −1 .

352 17 SolutionsSolid angle: Ω =A4πR 2 ≈ 6.16 × 10−41 .Number of recorded photons: N R = ΩN E ≈ 1.15 × 10 −5 /s ≈ 364/a .Minimum flux: assumed 10/a, P min ≈ P Ω 10A 364 ≈ 6.35 × 10−19 J/(cm 2 s).5. (a) Assume isotropic emission, which leads to a total power of P = 4πr 2 P S ,wherer = 150 × 10 6 km is the astronomical unit (distance Sun–Earth). One getsP ≈ 3.96 × 10 26 W .(b) In a period of 10 6 years the emitted energy is E = P × t ≈ 1.25 × 10 40 J, whichcorresponds to a mass of m = E/c 2 ≈ 1.39 × 10 23 kg, which represents a relativefraction of the solar mass off = m M ⊙≈ 6.9 × 10 −8 .(c) The effective area of the Earth is A = πR 2 , so that the daily energy transport toEarth is worked out to be E = πR 2 P S t. This corresponds to a mass ofm = E c 2 ≈ 1.71 × 105 kg = 171 tons .Sect. 6.41. The power emitted in the frequency interval [ν, ν + dν] corresponds to the one emittedin the wavelength interval [λ, λ + dλ], λ = c/ν, i.e., P(ν)dν = P(λ)dλ, orP(λ) = P(ν) dνdλ ∼ ν 3 dνe hν/kT − 1 dλ ∼ 1λ 5 (e hc/λkT − 1) ,since dν/dλ =−c/λ 2 .2. The luminosity of a star is proportional to the integral over Planck’s radiation formula:L ∼∫ ∞0ϱ(ν, T ) dν =∫ ∞08πhν 3c 3 1e hν/kT − 1 dν ;use x =kThν ⇒L ∼ 8π ( ) kT 3 ∫ ∞c 3 h x 3 1 kT 8π k 4 T 4 ∫ ∞x 3 dxh 0 e x dx =− 1 h c 3 h 3 0 e x − 1= 8π k 4 π 4c 3 h 3 15 T 4 ∼ T 4 .In addition, the luminosity varies with the size of the surface (∼ R 2 ).

17.6 Chapter 6 353This gives the scaling law( )L R 2 ( ) T 4=;L ⊙ R ⊙ T ⊙a) R = 10 R ⊙ , T = T ⊙ ⇒ L = 100 L ⊙ ;b) R = R ⊙ , T = 10 T ⊙ ⇒ L = 10 000 L ⊙ .3. The motion of an electron in a transverse magnetic field is described bymv 2ϱ = evB ⇒ p ϱ = eB .Since at high energies cp ≈ E, one getscpϱ = ceB ⇒ B = Eceϱ ,where ϱ is the bending radius. This leads toP = e2 c 32π C γ E 2 E 2c 2 e 2 ϱ 2 = cC γ2πE 4ϱ 2 .The energy loss for one revolution around the pulsar isE =∫ T0P dτ = cC γ2π= 8.85 × 10 −5 × 1012 GeV 4The magnetic field isB =E 4ϱ 2 2πϱ/c = C γE 4ϱ10 6 m mGeV−3 = 88.5GeV.Eceϱ = 10 12 eV × 1.6 × 10 −19 J/eV3 × 10 8 m/s × 1.6 × 10 −19 As× 10 6 = 0.0033 T = 33 Gauss .m4. The Lorentz force is related to the absolute value of the momentum change by |ṗ| =F = evB. The total radiated power is taken from the kinetic (or total) energy of theparticle, i.e., Ė kin = Ė =−P . From the centrifugal force the bending radius results toϱ = p/eB, see also Problem 3 in this section. As differential equation for the energyone getsĖ =−P =− 2 e 4 B 23 m 2 γ 2 v 2 .0 c3a) In general E = γm 0 c 2 holds. The ultrarelativistic limit is given by v → c, henceĖ =− 2 e 4 B 2 c 33 (m 0 c 2 ) 4 E2 =−αE 2 , α = 2 e 4 B 2 c 33 (m 0 c 2 ) 4 .

354 17 SolutionsThe solution of this differential equation is obtained by separation of variables,∫ EdE ′ ∫ tE 0E ′2 =−α 0dt ′ ⇒ 1 E 0− 1 E =−αt ⇒ E = E 01 + αE 0 t .It is only valid for E ≫ m 0 c 2 . (This limit is not respected for longer times.)The bending radius follows with E → pc toϱ =EceB = 1 E 0ceB 1 + αE 0 t = ϱ 01 + αceBϱ 0 t .b) In the general (relativistic) case there is p 2 = γ 2 m 2 0 v2 , i.e., γ 2 v 2 = p 2 /m 2 0 =(E 2 − m 2 0 c4 )/m 2 0 c2 , thusĖ =− 2 e 4 B 2 c 33 (m 0 c 2 ) 4 (E2 − m 2 0 c4 ) =−α(E 2 − m 2 0 c4 ), αas in (a) .The solution is calculated as in (a):∫ EdE ′ ∫ tE 0 E ′2 − m 2 =− dt ′ =−αt ⇒0 c4 0∫1 E()∣12m 0 c 2 E 0E ′ − m 0 c 2 − 1E ′ + m 0 c 2 dE ′ 1=2m 0 c 2 ln E′ − m 0 c 2 ∣∣∣∣ EE ′ + m 0 c 2⎛⎞=− 1 ⎝ 1m 0 c 2 2 ln 1 + m 0c 2E− 11 − m 0c 2 2 ln 1 + m 0c 2E 0 ⎠E1 − m 0c 2E 0()=− 1m 0 c 2 artanh m 0c 2E − artanh m 0c 2=−αt .E 0E ′ =E 0Solving this equation for E leads to()m 0 c 2E = () = m 0 c 2 coth αm 0 c 2 t + artanh m 0c 2.tanh αm 0 c 2 t + artanh m 0c 2E 0E 0For larger times (in the non-relativistic regime) the rest energy is appoached exponentially.From this the bending radius results in√ϱ = peB = E 2 − m 2 0 c4ceBm 0 c/eB=() =sinh αm 0 c 2 t + artanh m 0c 2E 0= m √0c ( )coth 2 αm 0 c 2 t + artanh m 0c 2eBE 0− 1m 0 c/eB(sinh αm 0 c 2 t + artanh) .√m 0 cϱ0 2e2 B 2 +m 2 0 c2

17.6 Chapter 6 3555. Measured flux R = source flux R ∗ × efficiency ε × solid angle Ω;ε = 1 − e −µx = 1 − e −125×5.8×10−3 ≈ 51.6% , Ω = 104 cm 24π(55 kpc) 2 ≈ 2.76 × 10−44 ,R ∗ =RεΩ ≈ 1.95 × 1040 /s .6. In contrast to Compton scattering of photons on an electron target at rest, all threemomentaare different from zero in this case. For the four-momenta k i , k f (photon) andq i , q f (electron) one has k i − k f = q f − q i . Squaring this equation gives k i k f = q i q f − m 2 e .On the other hand, rewriting the four-momentum conservation as q f = q i + k i − k f andmultiplying with q i leads to q i q f − m 2 e = q i(k i − k f ), yieldingk i k f = q i (k i − k f ).Let ϑ be the angle between k i and k f , which givesω i ω f (1 − cos ϑ) = E i (ω i − ω f ) −|q i |(ω i cos ϕ i − ω f cos ϕ f ).Solving for ω f leads to1 − √ 1 − (m e /E i )ω f = ω 2 cos ϕ ii1 − √ .1 − (m e /E i ) 2 cos ϕ f + ω i (1 − cos ϑ)/E iThis expression is still exact. If the terms ω i /E i and m e /E i are neglected, the relationquoted in the problem is obtained.7. Starting from the derivative dP/dν,dP/dν ∼ 3ν2 (e hν/kT − 1) − ν 3 kT h ehν/kT(e hν/kT − 1) 2 = ν 2 3 (ex − 1) − x e x(e x − 1) 2 ,and the condition dP/dν = 0 one obtains the equatione −x = 1 − x 3 , x = hνkT .An approximated solution to this transcendental equation gives x ≈ 2.8, leading to afrequency, resp. energy of the maximum of the Planck distribution ofhν M ≈ 2.8 kT .This linear relation between the frequency in the maximum and the temperature is calledWien’s displacement law.Forhν M = 50 keV a temperature ofT ≈ 2 × 10 8 K .is obtained.

356 17 SolutionsSect. 6.51. E = GMm γREE= GMc 2for H ≪ R.2. a) P ≈ G c 2 ω6 m 2 r 4 .− GMm γR + H = GMm HγR(H+ R) ,HR(H+ R) = GMR 2 c 2HRH + R = g ⊙c 2For ω = 1/year, m = 5.97 × 10 24 kg, r = 1 AU one getsP ≈ 1.34 × 10 22 W .HRH + R ≈ g ⊙c 2 H ≈ 3 × 10−12Even though this power appears rather large, it is only a fraction of about 3 × 10 −5of the solar emission in the optical range.b) Assume ω = 10 3 s −1 , m = 10 kg, r = 1 m which leads toP ≈ 7.4 × 10 −8 W .17.7 Chapter 71. p = F A = mgA = 1.013 × 105 N m 2 ,mA = p g=1.013 × 1059.81kgm 2 ≈ 10 326 kg/m2 ≈ 1.03 kg/cm 2 .2. p = p 0 e −20/7.99 m 8.29 × 103 kg kg≈ 82.9hPa, ≈ ≈ 845A 9.81 m2 m 2 = 84.5g/cm2 .3. The differential sea-level muon spectrum can be parameterized by∫N(E)dE ∼ E −γ dE with γ = 3 ⇒ I(E) = N(E)dE ∼ E −2 ,where I(E) → 0forE →∞. The thickness of the atmosphere varies with zenith angleliked(θ) = d(0)cos θ .For ‘low’ energies (E

17.7 Chapter 7 3574. N(> E,R) = A(aR) −γ , see (7.15) ,NR =−γAa−γ R −(γ +1) ,NN=−γR−1 R =−γ RR ,traditionally R = 100 g/cm 2 .With γ = 2 (exponent of the integral sea-level spectrum):∣ N∣∣∣ ∣ N ∣ = S200 g/cm2N ∣ = = 2 × 10 −2 , e.g., for 100 m w.e.R5. ϑ r.m.s. = 13.6MeV √ x(1 + 0.038 ln x )βcp X 0 X 0≈ 4.87 × 10 −3 (1 + 0.27) ≈ 6.19 × 10 −3 ≈ 0.35 ◦ .6. The lateral separation is caused bya) transverse momenta in the primary interaction,b, c) multiple scattering in the air and rock.a) p T ≈ 350 MeV/c ,ϑ = p Tp≈350 MeV/c100 GeV/c = 3.5 × 10−3 ,average displacement:x 1 ≈ 70 m .b) Multiple-scattering angle in air,(see [2])x 1 = ϑh,whereh is the production height (20 km),ϑ air ≈ 6.90 × 10 −4 (1 + 0.123) ≈ 7.75 × 10 −4 ≈ 0.044 ◦ ,x 2 = 15.5m.c) Multiple scattering in rock,x 3 ≈ 0.8m.√⇒ x = x1 2 + x2 2 + x2 3 ≈ 72 m .7. Essentially only those geomagnetic latitudes are affected where the geomagnetic cutoffexceeds the atmospheric cutoff. This concerns latitudes between 0 ◦ and approximately50 ◦ . The average latitude can be worked out from〈λ〉 =∫ 50 ◦0 ◦ λ cos 4 λ dλ∫ 50 ◦0 ◦ cos 4 λ dλ ≈ 18.4◦corresponding to an average geomagnetic cutoff of 〈E〉 ≈ 12 GeV. For zero field allparticles with E ≥ 2 GeV have a chance to reach sea level. Their intensity is (ε =E/GeV)

358 17 SolutionsN 1 (> 2GeV) = a∫ ∞2ε −2.7 dε =−(a/1.7)ε −1.7∣ ∣∞= 0.181 a.With full field on, only particles with E > 〈E〉 = 12 GeV will reach sea level forlatitudes between 0 ◦ and 50 ◦ :N 2 (> 12 GeV) ≈ 0.0086 a.These results have to be combined with the surface of the Earth that is affected. Assumingisotropic incidence and zero field, the rate of cosmic rays at sea level is proportionalto the surface of the Earth,Φ 1 = AN 1 = 4πR 2 N 1 .With full field on, only the part of the surface of Earth is affected for which 0 ◦ ≤ λ ≤50 ◦ . The relevant surfaces can be calculated from elementary geometry yielding a fluxΦ 2 = A(50 ◦ –90 ◦ )N 1 + A(0 ◦ –50 ◦ )N 2 ,whereA(50 ◦ –90 ◦ ) = 2π[(R cos 50 ◦ ) 2 + (R(1 − sin 50 ◦ )) 2]andA(0 ◦ –50 ◦ ) = 4πR 2 − A(50 ◦ –90 ◦ ).This crude estimate leads to Φ 1 /Φ 2 = 3.7, i.e., in periods of transition when the magneticfield went through zero, the radiation load due to cosmic rays was higher by abouta factor of 4.8. In principle, neutrons are excellent candidates. Because they are neutral, they wouldpoint back to the sources of cosmic rays. Their deflection by inhomogeneous magneticfields is negligible. The only problem is their lifetime, τ 0 = 885.7 seconds. At 10 20 eVthe Lorentz factor extends this lifetime considerably toτ = γτ 0 = 1020 eVm n c 2 τ 0 = 9.4 × 10 13 s ̂= 2.99 × 10 6 light-years = 0.916 Mpc .Still the sources would have to be very near (compare λ γp ≈ 10 Mpc), and there is noevidence for point sources of this energy in the close vicinity of the Milky Way.217.8 Chapter 81. Due to the relative motion of the galaxy away from the observer, the energy of the photonappears decreased. The energies of emitted photons and observed photons are related bythe Lorentz transformation,E em = γE obs + γβcp ‖obs = γ(1 + β)E obs .

17.8 Chapter 8 359Since E = hν = hc/λ, one gets1= 1 γ(1 + β) ⇒ λ obs = λ em γ(1 + β) ,λ em λ obsz = λ obs − λ emλ em= γ(1 + β) − 1 ==1 + β√ (1 + β)(1 − β)− 1 =which reduces for β ≪ 1toz = √ (1 + β)(1 + β) − 1 ≈ β.1√ (1 + β) − 11 − β 2√ 1 + β√ 1 − β− 1 ,2. For an orbital motion in a gravitational potential one hasmv 2R= GmM R 2 , ⇒ 1 2 mv2 = 1 2 GmM R ,i.e., E kin = 1 2 E pot. The kinetic energy of the gas cloud is simplyE kin = 3 2 kT M µ .The potential energy can be obtained by integration: the mass in a spherical subvolumeof radius r ism = 4 3 πr3 ϱ.A spherical shell surrounding this volume contains the massdm = 4πr 2 ϱ drleading to a potential energy of gravitation ofGm dmdE = = G 4r r 3 πr3 ϱ 4πr 2 ϱ dr = (4π)2 ϱ 2 r 4 G dr ,3so that the potential energy isE pot =∫ R0dE = (4π)2 ϱ 2 G R53 5 .Using M = 4 3 πR3 ϱ, one obtainsE pot = 3 5 GM2 R .From the relation between E kin and E pot above one gets

360 17 Solutions2E kin = 3kT M µ = 3 5 GM2 RThenand R = 1 GMµ.5 kT) 3G 3 M 3 µ 3M = 4π 3 R3 ϱ = 4π 3 ϱ ( 15and finally(kT ) 3M ≈( ) kT 3/21 √ϱ × 5.46 .µG3.If M cloud >M, the cloud gets unstable and collapses.mv 2R≥ GMmR 2 ⇒ v 2 ≥ GM R .Since the density is constant, M = 4 3 πϱ R3 , the revolution time will beT r = 2πR/v = 2π R √ RR 3/2 √3π√ = 2πGM√ √ = √ ,√ G ϱR 3/2 4π Gϱ34.so thatv =√4π3 R√ Gϱ ∼ Rif ϱ = constant.This behaviour is characteristic of the orbital velocities of stars in galaxies not too faraway from the galactic center (see Fig. 1.17 or Fig. 13.3).Ṙ 2R 2 + k R 2 = 8π 3 Gϱ , Ṙ2 + k = 8π 3 GϱR2 .Differentiating this expression with respect to time gives2Ṙ ¨R = 8π 3 G(˙ϱR2 + 2RṘϱ) ,inserting ˙ϱ from the fluid equation leads to2Ṙ ¨R = 8π 3ṘG(2RṘϱ −3 R (ϱ + P)R2 ).This is equivalent to¨R = 4π 3 G(2Rϱ − 3Rϱ − 3RP) or ¨RR =−4π 3which is the acceleration equation.G(ϱ + 3P) ,

17.8 Chapter 8 3615. Photon mass m = hνc 2 = E c 2 ;energy loss of a photon in a gravitational potential U: E = mU ;reduced photon energy: E ′ = E − E = hν − hν(ν ′ = ν 1 − U )c 2 ,νν= Uc 2 ;c 2 U ⇒gravitational potential:U = GM R⇒6.νν= GMRc 2 ⇒ GMRc 2is dimensionless;using the Schwarzschild radius R S = 2GMνc 2 one hasν= R S2R .This result is, however, only valid far away from the event horizon. The exact resultfrom general relativity reads, see, e.g., [9]: ν/ν = z = 1/ √ 1 − R S /R − 1.xSunRylight beamAcceleration due to gravity at the solar surface: g = GM/R 2 .Assumption: the deflection takes place essentially over the Sun’s diameter 2R. The photonstravel on a parabola:y = g 2 t2 ,x= ct ⇒ y(x) = GM x 22R 2 c 2 .The deflection δ corresponds to the increase of y(x) at x = 2R,dydx = GMR 2 c 2 x ⇒ y′ (2R) = 2GMRc 2= R SR = δ,R = 6.961 × 10 5 km, M = 1.9884 × 10 30 kg ⇒ δ ≈ 4.24 × 10 −6 ≈ 0.87 arcsec,1arcsec= 1 ′′ .This is the classical result using Newton’s theory. The general theory of relativity givesδ ∗ = 2δ = 1.75 arcsec.The solution to this problem may alternatively be calculated following Problem 13.7,see also Problem 3.4.

362 17 Solutions7. An observer in empty space measures times with an atomic clock of frequency ν 0 .Thesignals emitted by an identical clock on the surface of the pulsar reach the observer inempty space with a frequency ν = ν 0 − ν where νin this chapter) ⇒f pulsar= ν 0 − νf empty space ν 0ν 0= Uc 2= GM (see Problem 5Rc2 = 1 − GMRc 2 = 1 − ε.For the pulsar this gives ε = 0.074, i.e., the clocks in the gravitational potential on thesurface of the pulsar are slow by 7.4%. For our Sun the relative slowing-down rate, e.g.,with respect to Earth, is 2 × 10 −6 . At the surface of the Earth clocks run slow by 1.06 ×10 −8 with respect to clocks far away from any mass, from which just 7 × 10 −10 resultsfrom the Earth’s gravitation and the main contribution is caused by the gravitationalpotential of the Sun.8. Gravitational forcedF =−G M(r)dmr 2 =− GM(r)r 2 ϱ(r) dr 4πr 2 , inward forcedp =dF4πr 2 =−GM(r) dpr 2 ϱ(r) dr ,dr =−GM(r) r 2 ϱ(r) . (∗)dmrdrRM(r)On the other hand dpdr≈p(R) − p(r = 0)R=− p . Compare with (∗) and with the re-Rplacement r → R and ϱ(r) → average density ϱ one gets p R = GMR 2 ϱ.If a uniform density ϱ(r) = ϱ is assumed, (∗) can be integrated directly usingM(r) = 4π 3 ϱr3 and thus dpdr =−4π 3 Gϱ2 r,p(0) =∫ 0Rdpdr dr = 4π ∫ R3 Gϱ2 r dr = 4π 30R2Gϱ22 = 1 GM2 R ϱ.This leads to p = 1 {GM 1.3 × 10 142 R ϱ ≈ N/m 2 for the Sun1.7 × 10 11 N/m 2 for the Earth .

17.9 Chapter 9 3639. Schwarzschild radiusR S = 2GM }c 2M = 4 3 πR3 S ϱ ⇒( ) 3 1/34πϱ M = 2GMc 2(M⊙3c 6ϱ =32πG 3 M 2 = 3c 632πG 3 M⊙2 Ma) M≈ 10 11 M ⊙ ⇒ ϱ ≈ 1.8 × 10 −3 kg/m 3b) M= M ⊙ ⇒ ϱ ≈ 1.8 × 10 19 kg/m 3c) M= 10 15 kg ⇒ ϱ ≈ 7.3 × 10 49 kg/m 3 .⇒() 2≈ 1.8 × 10 19 (M⊙M) 3c 2 3M2GM 4π = ϱ.) 2kgm 3 ,10. a) mv2R= GmMR 2= GmR 2 43 πϱR3 ⇒ v 2 = 4 3 πϱGR2 ⇒√4v = R πϱG = 245 km/s forR = 20 000 light-years .3b) For R>20 000 light-years the orbital velocities show Keplerian characteristics,mv 2R≈ GmM R 2 ⇒ v 2 ≈ G M R , M ≈ v2 R G = 1.7 × 1041 kg = 8.6 × 10 10 M ⊙ .c) The energy density of photons in the universe is approximately 0.3eV/cm 3 .Thecritical density amounts to ϱ c = 0.945 × 10 −29 g/cm 3 , which corresponds to anenergy density of ϱ c c 2 = 5.3×10 3 eV/cm 3 , which means ϱ photons /ϱ c c 2 ≈ 5×10 −5 ;i.e., photons contribute only a small fraction to the total Ω parameter.17.9 Chapter 91. For a gas of non-relativistic particles the ideal gas law holds: P = nT . The density isgiven by ϱ = nm.SinceT ≪ m for non-relativistic particles, one has P ≈ 0.The fluid equation for P = 0 reads˙ϱ + 3ṘR ϱ = 0 ⇒ 1 R 3 ddt(ϱR 3) = 0 ⇒ ϱR 3 = const ⇒ ϱ ∼ 1 R 3 .2. Assume P = 0 which gives ϱ ∼ 1 (see Problem 9.1). The last relation can be parameterizedby3 ( ) 3 R0Rϱ = ϱ 0 .RThe Friedmann equation can be approximated:Ṙ 2R 2 + k R 2 = 8π Ṙ2Gϱ ⇒3 R 2 = 8π 3 Gϱ ,since the second term on the left-hand side in the Friedmann equation is small comparedto ϱ (∼ 1 ) for the early universe.R 3

364 17 SolutionsWith the ansatzR = At p ,Ṙ = pAt p−1one getsṘ 2R 2 = p2 A 2 t 2p t −2A 2 t 2p = p2t 2 = 8π ( ) 33 Gϱ R00 = 8π R 3 Gϱ 0R0 3 A−3 t −3p .Comparing the t dependence on the right- and left-hand sides givesp = 2 3⇒ R = At 2/3 .In this procedure the constant of proportionality A is automatically fixed asA = 3√ 6πGϱ 0 R 0 .3. For the early universe one can approximate the Friedmann equation byṘ 2R 2 = 8π 3Gϱ ; with ϱ =π2Since ṘR = H and G = 1m 2 one hasPl√8πH =3 g ∗ T 2.90 m Pl30 g ∗T 4 ⇒ Ṙ2R 2 = 8π 3 Gπ 230 g ∗T 4 = 8π 390 Gg ∗T 4 .4. [G] = m 3 s −2 kg −1 , [c] = ms −1 , [¯h] = Js= kg m 2 s −1 .[ ]¯hGTryc 3 = kg m2 s −1 m 3 s −2 kg −1m 3 s −3 = m 2 .√¯hGTherefore, has the dimension of a length, and this is the Planck length.c 35. The escape velocity from a massive object can be worked out from12 mv2 = G mMR ,where m is the mass of the escaping particle, and M and R are the mass and radius of themassive object. The mass of the escaping object does not enter into the escape velocity.If the escape velocity is equal to the velocity of light, even light cannot escape from theobject. If v is replaced by c this leads toc 2 = 2 GMRorR = 2 GMc 2 ,

17.9 Chapter 9 365which is the Schwarzschild radius. The result of this classical treatment of the problem(even though classical physics does not apply in this situation) accidentially agrees withthe outcome of the correct derivation based on general relativity.By definition the Schwarzschild radius is R S = 2GM and the calculation givesR S ={ 8.9mmforEarth2.95 km for the Sun.c 26.7.GM 2R > 3 2 kT M µM> 3 kT2 µG R = 3 kT2 µG, µ – mass of a hydrogen atom ,( ) 3M 1/3;4πϱ( 32since M = 4 3 πϱR3 : M 3 >ϱ> 34πkTµG) 33M4πϱ ,( )1 3 kT 3M 2 ≈ 3.9 × 10 −10 kg/m 32 µG(k = 1.38 × 10 −23 J/K, M = 2 × 10 30 kg, µ = 1.67 × 10 −27 kg).starγ v(t+ d 1✻c) = dDd 1d 2td✻❅❅❅❅❅❅❄D − d 1❄❄observer attime t 0❄observer attime t 0 + tThe first observation is made at time t 0 . The star is at a distance D from the observer.It moves at an angle γ to the line of sight. During t it has moved d 1 = d cos γ closerto the observer. The first light beam has further to travel. The light was emitted t + d 1cearlier compared to the second measurement. The apparent velocity is( )t + d 1csin γv ∗ = d 2t = d sin γ v=( t= v 1 + d )2ct cot γ sin γ = vt(= v 1 + d )1sin γct(1 + v∗c cot γ )sin γ = v sin γ + v c v∗ cos γ.

366 17 Solutions(Solve for v ∗ : v ∗ 1 − v )c cos γ = v sin γ, v ∗ = v sin γ1 − v ; if v approaches c:ccos γv ∗ sin γ= c1 − cos γ = c 2sin γ 2 cos γ 22sin 2 γ = c cot γ 2 >c for 0

17.10 Chapter 10 367whereσ√hadr 0 – peak cross section,s – center-of-mass energy,Γ Z – total width of the Z,M Z – massofthe Z,δ rad – radiative correction.The peak cross section σhadr 0 for the process e+ e − → Z → hadrons is given byσhadr 0 = 12π Γ e + e −Γ hadrMZ2 ΓZ2 ,where Γ e + e − is the partial e+ e − width of the Z and Γ hadr the partial width for Z →hadrons.The total Z width can be written asΓ Z = Γ hadr + Γ e + e − + Γ µ + µ − + Γ τ + τ − + Γ inv ,where Γ inv describes the invisible decay of the Z into neutrino pairs (ν e ¯ν e , ν µ ¯ν µ , ν τ ¯ν τ ,and possibly other neutrino pairs from a suspected fourth neutrino family).Lepton universality is well established. Therefore,Γ Z = Γ hadr + 3Γ l ¯l + Γ inv .Γ inv= Γ Z− 3 − Γ hadr= Γ Z− 3 − R,(∗)Γ l ¯l Γ l ¯l Γ l ¯l Γ l ¯lwhere R is the usual ratioR = σ(e+ e − → hadrons)σ(e + e − → l + l − .)The experiment provides σhadr 0 , Γ Z,andM Z .The measurement of the peak cross section σhadr 0 and Γ Z together with M Z fixes theproduct Γ l ¯l Γ hadr.The measurement of the peak cross section for the process e + e − → Z → µ + µ − ,σµµ 0 = 12π Γ e + e − Γ µ + µ −MZ2 ΓZ2 ,determines Γ l ¯l , if lepton universality is assumed. Therefore, also Γ hadr is now known,so that Γ inv can be obtained from (∗).In the framework of the Standard Model the width ofZ → ν x ¯ν xis calculated to be ≈ 170 MeV. The LEP experiments resulted inΓ inv ≈ 500 MeVindicating very clearly that there are only 500/170 ≈ 3 neutrino generations.

368 17 Solutions17.11 Chapter 111. This probability can be worked out fromφ = σ Th Nd,where σ Th is the Thomson cross section (665 mb), N the number of target atoms percm 3 ,andd the distance traveled.Since the universe is flat, one has ϱ = ϱ crit . Because of ϱ crit = 9.47 × 10 −30 g/cm 3 andm H = 1.67 × 10 −24 g, and the fact that only 4% of the total matter density is in the formof baryonic matter, one has N = 2.27 × 10 −7 cm −3 . The distance travelled from thesurface of last scattering corresponds to the age of the universe (14 billion years),d = 14 × 10 9 × 3.156 × 10 7 s × 2.998 × 10 10 cm/s = 1.32 × 10 28 cmresulting inφ = 1.99 × 10 −3 ≈ 0.2% .2. The critical density ϱ c = 3H 2 /8πG as obtained from the Friedmann equation has to bemodified if the effect of the cosmological constant is taken into account:ϱ c,Λ = 3H 2 − Λc 2.8πGSince ϱ c,Λ cannot be negative, one can derive a limit from this equation:3H 2 − Λc 2 > 0 , Λ < 3H 2c 2 = 1.766 × 10 −56 cm −2 .This leads to an energy densityϱ ≤3. p 1 =− dE classdV∼ddVc48πG Λ = 8.51 × 10−10 J/m 3 = 5.3GeV/m 3 = 5.3keV/cm 3 .=− ddV(− GmM )= GmM dRdV( 1V 1/3 )=− 1 3 V −4/3 ∼− 1 R 4 ,( 1V 1/3 )( 4π3) 1/3p 2 =− dE ΛdV∼− ddV (−ΛR2 ) ∼ddV V 2/3 = 2 3 V −1/3 ∼+ 1 R .p 1 is an inward pressure due to the normal gravitational pull, whereas p 2 is an outwardpressure representing a repulsive gravity.

17.11 Chapter 11 3694. Planck distributionϱ(ν, T ) = 8πh∫ ∞1c 3 ν3 e hν/kT − 1 , 〈ν〉 = 0νϱ(ν,T)dν∫ ∞0ϱ(ν, T ) dν ;substitution hνkT = x:∫ ∞0∫ ∞0∫ ∞0∫ ∞0ϱ(ν, T ) dν = 8πhc 3= 8πhc 3νϱ(ν,T)dν = 8πhc 3( kTh( kTh( kTh) 3 ∫ ∞0x 3e x − 1kTh dx) 4 ∫ ∞x 30 e x − 1 dx,) 5 ∫ ∞x 4e x − 1 dx ;0x 3e x − 1 dx = 3! ζ(4) , ζ(4) = π 4 /90 , ζ – Riemann’s zeta function,x 4e x dx = 4! ζ(5) , ζ(5) = 1.036 927 7551 ... ;− 1〈hν〉 = h 8πh ( kTh) 5c × 4! ζ(5)3(8πh kTh) 4= h kTc × 3! ζ(4) h × 4 × ζ(5)ζ(4)3= kT × 4 × ζ(5) 360× 90 = kT ζ(5) ≈ 900 µeV .π 4 π 45. a) At present: number density of bb photons: 410/cm 3 , average energy 〈E〉 =900 µeV(from the previous problem). This leads to a first estimate of the present energydensity ofϱ 0 ≈ 0.37 eV/cm 3 .Since, however, the temperature enters with the fourth power into the energy densityone has to useϱ 0 = π 215 T 4for photons (number of degrees of freedom g = 2, see (9.10)). One has to includethe adequate factors of k and ¯hc to get the correct numerical result:ϱ 0 = π 215 T 4 k 4≈ 0.26 eV/cm3(¯hc) 3(k = 8.617 × 10 −5 eV K −1 , ¯hc = 0.197 GeV fm).

370 17 Solutionsb) At last scattering (t = 380 000 a, temperature at the time of last scattering: T dec =0.3 eV, see Chap. 11):T dec = 0.3eV/k ≈ 3500 K ,ϱ dec = ϱ 0(TdecT 0) 4≈ 0.26 eV/cm 3 ×( ) 3500 4≈ 0.7TeV/cm 3 .2.7256. Naïvely one would expect the fraction of neutral hydrogen to become significant whenthe temperature drops below 13.6 eV. But this happens only at much lower temperaturesbecause there are so many more photons than baryons, and the photon energy distribution,i.e., the Planck distribution, has a long tail towards high energies. The baryon-tophotonratio, η ≈ 5 × 10 −10 , is extremely small. Therefore, the temperature must besignificantly lower than this before the number of photons with E>13.6 eV is comparableto the number of baryons. Furthermore, interaction or ionization can take place inseveral steps via excited states of the hydrogen atom, the H 2 molecule, or the H + 2 ion.One finds that the numbers of neutral and ionized atoms become equal at a recombinationtemperature of T rec ≈ 0.3 eV (3500 K). At this point the universe transforms froman ionized plasma to an essentially neutral gas of hydrogen and helium.17.12 Chapter 121. The Friedmann equation for k = 0, corresponding to the dominance of Λ, reads, seealso Problem 11.2,H 2 = 8πG3 (ϱ + ϱ v).WithΛ = 8πGc 2 ϱ v ⇒ H 2 − 1 3 Λc2 = 8πG ϱ.3Since ϱ>0 one has the inequalityH 2 − 1 3 Λc2 ≥ 0 or Λ ≤ 3H 2c 2 ≈ 2 × 10 −56 cm −2 .This is just a reflection of the fact that in the visible universe there is no obvious effectof the curvature of space. The size of the visible flat universe being 10 28 cm can beconverted intoΛ 1 ≤ 10 −56 cm −2 .If one assumes on the other hand that Einstein’s theory of relativity is valid down to thePlanck scale, then one would expectΛ 2 ≈ (l 2 Pl )−1 ≈ 10 66 cm −2 .The difference between the two estimates is 122 orders of magnitude.

17.12 Chapter 12 371The related mass densities for the two scenarios are estimated to beϱv 1 = Λ 1c 28πG ≈ 10−52 m −2 × 9 × 10 16 m 2 /s 2 kg s 28π × 6.67 × 10 −11 m 3≈ 5.37 × 10 −27 kg/m 3 = 5.37 × 10 −30 g/cm 3 ,ϱv 2 = Λ 2c 28πG ≈ 5.37 × 1095 kg/m 3 = 5.37 × 10 92 g/cm 3 .2. Starting fromH 2 = 8πG3 (ϱ + ϱ v) ⇒ H 2 − 1 3 Λc2 = 8πG3 ϱand since H = Ṙ/R one obtains (for ϱ = const)ṘR = √13 Λc2 + 8πG3 ϱ ⇒ R = R i exp(√13 Λc2 + 8πG3 ϱt ),which represents an expanding universe.3. The Friedmann equation extended by the Λ term for a flat universe isṘ 2R 2 − 1 3 Λc2 = 8πG3 ϱ.For 1 3 Λc2 ≫ 8πG3ϱ this equation simplifies to√ √Ṙ 1 1R = 3 Λc2 =3 Λ 0c 2 (1 + αt) = a √ √11 + αt with a =3 Λ 0c 2 ,∫ln R = a √ 1 + αt dt + const = a 23 α (1 + αt)3/2 + const ,( )2aR = exp3α (1 + αt)3/2 + const ;boundary condition R(t = 0) = R i( )2aR(t = 0) = exp3α + const = R i ⇒2a3α + const = ln R i ⇒ const = ln R i − 2a3α ,( 2aR = R i exp3α (1 + αt)3/2 − 2a ) ( )2a= R i exp3α3α [(1 + αt)3/2 − 1]with a =√13 Λ 0c 2 .For large t this result shows a dependence likeR ∼ exp(βt 3/2) with β = 2a√ α.3

372 17 Solutions4.( ) 2R 0 t0 3= ,R mr t mrwhere R mr is the size of the universe at the time of matter–radiation equality t mr =50 000 a. With t 0 = 14 billion years, one hasR 0≈ 4280 .R mrThe size of the universe at that time wasR mr ≈ R 04280 ≈ 3.27 × 106 light-years ≈ 3.09 × 10 22 m .Extrapolating to earlier times requires to assume R ∼ √ t, since for times t

17.13 Chapter 13 373(p ′ W −m Wm W + m ep W) 2= p 2 WCentral collisionm Wm 2 e(m W + m e ) 2 .p W ′ − p W =±p Wm W + m esince only the negative sign is physically meaningful.E = 1 ( )2 m W(v1 2 − v′2 1 ) = Ekin W 1 − v′2 1v12 = EWkinm e≈ 1GeV× 2 × 10 −5 = 20 keV .3. Classical Fermi gas of neutrinos:m W + m e⇒ p ′ W = m W − m em W + m ep W ,(1 −( ) )mW − m 2 em W + m eE F = p2= ¯h 2 k 2, k – wave vector .2m ν 2m νIn a quantized Fermi gas there is one k vector per 2π/L if one assumes that the neutrinogas is contained in a cube of side L. Number of states (at T = 0) for 2 spin states:N = 243 πk3 F(2π/L) 3 = V 13π 2 k3 F ⇒ k F = (3π 2 n) 1/3with n = N/V particle density,⇒ E F = ¯h 22m (3π 2 n) 2/3 .Relativistic Fermi gas:E F = p F c = ¯hk F c = ¯hc(3π 2 n) 1/3assuming relativistic neutrinos one would get, e.g., for 300 neutrinos per cm 3 .¯hc = 197.3MeVfm,E F = ¯hc (3π 2 × 300) 1/3 cm −1 = 197.3 × 10 6 eV × 10 −13 cm × 20.71 cm −1≈ 409 µeV .4. Since v ≪ c is expected, one can use classical kinematics:12 mv2 = G mM √2GM⇒ v = ⇒RR√2 × 6.67 × 10v ≈−11 m 3 kg −1 s −2 × 10 10 × 2 × 10 30 kg3 × 10 3 × 3.086 × 10 16 ≈ 1.7 × 10 5 m/s ,mβ ≈ 5.66 × 10 −4 .

374 17 SolutionsThe axion mass does not enter the calculation, i.e., the escape velocity does not dependon the mass.A relativistic treatment, (γ − 1)m 0 c 2 = Gγ m 0 M/R, leads to a similar result:√ √2GMv = βc = 1 − GMR 2Rc 2 .Since this expression is not based on general relativity, it is of limited use.5. Spherically symmetric mass distribution.∫ ra) Mass inside a sphere: M(r) = ϱ(r ′ ) dV ′ = 4πpotential energy of mass m: E (m)pot (r) = Gm ∫ rverification: − ∂E(m) pot (r)= F =−G mM(r)∂rr 2 .0∞∫ r0M(r ′ )r ′2 dr ′ ,ϱ(r ′ )r ′2 dr ′ ;b) Potential energy of mass shell dM = M ′ (r) dr:dE pot = GM ′ (r) drtotal potential energy: E pot = G∫ ∞0∫ rM ′ (r) dr∫ rM(r ′ ∣) ∣∣∣integration by parts: E pot = GM(r)∞ r ′2 dr ′ ∞∞M(r ′ )r ′2 dr ′ ;r=0− G∫ ∞∫ ∞M 2 (r)margin term vanishes for finite masses: E pot =−G0 r 2 dr.c) Mass M on a shell, i.e., M(r < R) = 0andM(r > R) = M:E (shell)pot=−GM 2 ∫ ∞R1dr = GM2r2 r∣∞r=R=−G M2R .0∫ r∞M 2 (r)r 2M(r ′ )r ′2 dr ′ ;d) Homogeneous mass distribution: M(r < R) = Mr 3 /R 3 and M(r > R) = M:∫ E pot(hom) =−G M2 RR 6 r 4 dr + E pot(shell) =−G M2 r 5RR 6 − G M25 ∣ R=−6 5 GM2 R .06. n max m ν = m4 ν v3 f3π 2 ¯h 3 . (*)If n max m ν is assumed to be on the order of a typical dark-matter density (Ω = 0.3) onecan solve (*) for m ν ,r=0dr;m ν =( ) 1/43π 2 ¯h 3 n max m ν.v 3 f

17.14 Chapter 14 375The mass of a typical galaxy can be estimated as M = 10 11 M ⊙ = 2 × 10 41 kg. Withan assumed radius of r = 20 kpc one getsv f = 2.11 × 10 5 m/s .With n max m ν = 0.3 ϱ c ≈ 3 × 10 −30 g/cm 3 one obtainsm ν ≥ 1eV.7.✘ ✘✘✘ ✘ ✘✘✘✘ ✘ ✘✘✘ ✘ ✘δEarth ✗✔ ϑ R✖✕deflector✛ L 1✲✛L 2✲starδ = 2R SRas mentioned earlier (see Problem 8.6); R S = 2GM dc 2 ;for L 2 ≫ L 1 : δ ≈ ϑ = R L 1⇒ R = L 1 δ,δ = 2R SR= 2R SL 1 δ√ √⇒ δ = 2R S 4GM d=L 1 L 1 c 2 ∼ √ M d .Ring radius R E = L 1 δ = √ 2L 1 R S √, L 1 = 10 kpc ≈ 10×10 3 ×3.26×3.15×10 7 s×3×10 8 ms −1 ≈ 3.08 × 10 20 m: R E ≈ 2 × 3.08 × 10 20 m × 3000 m ≈ 1.36 × 10 12 m ;opening angle for the Einstein ring: γ = 2δ = 2R E= 8.8 × 10 −9 ̂= 0.0018 arcsecL 1⇒ too small to be observable ⇒ only brightness excursion visible.17.14 Chapter 141. The weight of a human is proportional to its volume, which in turn is proportional to thecube of its size;W = W 0 R 2 20 R,if one assumes that the ‘height’ of a human is 20 times its ‘radius’. The strength of ahuman is proportional to its cross sectionS = S 0 R 2 .For a mass of 100 kg, an assumed radius of 10 cm, a human has to carry its own weightplus, maybe, an additional 100 kg,

376 17 Solutions100 kg kgW 0 = = 500020 R3 m 3 , S 200 kg0 =R 2 = 20 000 kgm 2 .From this the stability limit (weight ̂= strength) can be derived,5000 kgm 3 × 20 R3 = 20 000 kgm 2 × R2 ,which leads toR max = 1 5 m ⇒ H max = 4m.If the gravity were to double, the strength would be composed of 100 kg and a bodymass of 50 kg (corresponding to 100 kg ‘weight’, since g ∗ = 2g).SinceV = 1 2 V 0 ⇒ R max (g ∗ = 2g) = 3√ 1 Rmax 0 ,2R max (g ∗ = 2g) = 0.1587 m ⇒ H max (g ∗ = 2g) = 3.17 m .2. Carbon is produced in the so-called triple-alpha process. In step one 8 Be is produced inαα collisions4 He + 4 He → 8 Be + γ − 91.78 keV .The 8 Be produced in this step is unstable and decays back into helium nuclei in 2.6 µs.It therefore requires a high helium density to induce a reaction with 8 Be before it hasdecayed,8 Be + 4 He → 12 C ∗ + 7.367 MeV .The excited 12 C ∗ state is unstable, but a few of these excited carbon nuclei emit a γ rayquickly enough to become stable before they disintegrate. The net energy release of thetriple-alpha process is 7.275 MeV.Only at extremely high temperatures (≈ 10 8 K) the bottleneck of carbon productionfrom helium in a highly improbable reaction can be accomplished because of a luckymood of nature: high temperatures are required to overcome the Coulomb barrier forhelium fusion. Also high densities of helium nuclei are needed to make the triple-alphaprocess possible.The reaction rate for the triple-alpha process depends on the number density N α of αparticles and the temperature T of the α plasma:σ(3α → 12 C) ∼ N 3 α1(T 3 exp − ε )Γ γ ,kTwhere ε is a parameter (≈ 0.4 MeV). For high temperatures up to T ≈ 1.5 × 10 9 Ktheexponential wins over the power T 3 . The maximum cross section is derived to be in thefollowing way:

17.14 Chapter 14 3770 = ddT[ 1 (T 3 exp − ε ) ] =(− 3kT T 4 +ε )kT 5 exp(− ε )kTT max = ε3k ≈ 1.5 × 109 K ⇒ σ max (3α → 12 C) ∼ N 3 α⇒( ) 3k 3e −3 Γ γ .εΓ γ is the electromagnetic decay width to the first excited state of 12 C.The conditions for the triple-alpha process can be summarized as follows:• high plasma temperatures (≈ 10 8 K),• high α-particle density,• large, resonance-like cross section σ(3α).3. Gravity sets the escape velocity (v esc ) and the temperature determines the mean speedwith which the molecules are moving (v mol ). The condition for a planet to retain anatmosphere isv esc ≫ v mol .v esc can be obtained from12 mv2 = G mM √planet⇒ v esc = 2GM planet /R planetR planet(= 11.2km/sforEarth).v mol can be calculated from12 mv2 = 3 2 kT ⇒ v mol = √ 3kT/m (= 517 m/sforN 2 for Earth) .For v esc = v mol one obtains2G M planet= 3kTR planet m .Let us assume earthlike conditions (T = 300 K, m = m( 14 N 2 ) = 4.65 × 10 −23 g =4.65 × 10 −26 kg, k = 1.38 × 10 −23 JK −1 ):2G M planetR planet= 267 097 J/kg ⇒ M planetR planet= 2 × 10 15 kg/m .With M = 4 3 πR3 ϱ, and assuming an earthlike density ϱ = 5.5g/cm 3 = 5500 kg/m 3one gets√3MR = 3 4πϱ ≈ 0.035 m × 3√ M/kg ,M planet3√Mplanet≈ 2 × 1015 kgm × 0.035mkg 1/3 = 7.0 × 1013 kg 2/3 .

378 17 SolutionsThis results inM planet ≈ 5.9 × 10 20 kg .The result of this estimate depends very much on earthlike properties and the assumptionthat other effects, like rotational velocity (→ centrifugal force) are negligible.4. a) The long-range forces in three dimensions are the gravitational force and the electromagneticforce, the latter of which reduces in the non-relativistic (static) limit tothe Coulomb force. The force is proportional to the masses (Newton’s gravitationlaw) or to the charges (Coulomb’s law), respectively. Both forces scale with 1/r 2 ,where r is the distance between the two bodies. The surface of a three-dimensionalsphere is 4πr 2 , and the forces are, roughly speaking, proportional to the solid angleunder which the bodies see each other. The surface of an n-dimensional sphere is(already from dimensional considerations) given bys n (r) = s n r n−1 , i.e., g n = n − 1 ,where the constant s n is the surface of the corresponding unit sphere. Isotropic longrangetwo-body forces in n dimensions should therefore scale asF(r) ∼ 1s n (r) ∼ 1r n−1 .b) The potential of a radial force is given by (F · r < 0 for attractive forces, n>2)∫ r∫ (rV(r)= F(r ′ ) dr ′ dr ′ 1 1∼ =−r 0 r 0r ′n−1 n − 2 r n−2 − 1)r n−20∼− 1 + const .rn−2 A particle of mass m, at distance r from a center, and having velocity v on a circularorbit has an angular momentum of L = rp = mvr. (In general, the velocitycomponent perpendicular to r is considered.) The centrifugal force is then given byF c (r) = mv2r= L2mr 3 ,where the last expression is also valid for general motion and L is a constant forcentral forces. The corresponding centrifugal potential therefore reads∫ rV c (r) =− F c (r ′ ) dr ′ =L2r 02mr 2 + const .The effective potential is the sum of V and V c ,V eff (r) = V(r)+ V c (r) =− C nr n−2 +L22mr 2 + const .Circular orbits take place for a vanishing effective radial force,

17.15 Chapter 15 3790 =−F eff (r) = dV eff(r)dr= (n − 2)C nr n−1( ) 1− L2mn−4˜C nmr 3 ⇒ r orb =L 2with ˜C n = (n − 2)C n , n ̸= 4, also applicable for n = 2. Stable orbits are given forpotential minima:∣ ()∣0 < d2 V eff (r) ∣∣∣∣rorbdr 2 = − (n − 1) ( ) 4˜C nr n + 3L2 ∣∣∣∣rorbmr 4 = (4 − n) L2 L2 n−4m m ˜C nThus, stable motion is only possible for n2 ,which is similar to the expression of the potential energy. For n > 2 the limitΛ →∞leads to a finite result, whereas W diverges for λ → 0. In the case n = 2both limits are divergent. Quantum corrections, here for the so-called self-energycorrections, are supposed to become significant for small distances. Therefore, thelimit λ → 0 may diverge, whereas the limit Λ →∞should exist. Thus, n>2dimensions are considered valid from this aspect. The degree of divergence is thensmallest for n = 3.λ.17.15 Chapter 151. There is not yet a solution. If you have solved the problem successfully, you shouldbook a flight to Stockholm because you will be the next laureate for the Nobel Prize inphysics.

381A Mathematical AppendixA.1 Selected Formulae“Don’t worry about your difficulties inmathematics; I can assure you that mineare still greater.”Albert EinsteinThe solution of physics problems often involves mathematics.In most cases nature is not so kind as to allow a precisemathematical treatment. Many times approximations are notonly rather convenient but also necessary, because the generalsolution of specific problems can be very demandingand sometimes even impossible.In addition to these approximations, which often involvepower series, where only the leading terms are relevant, basicknowledge of calculus and statistics is required. In thefollowing the most frequently used mathematical aids shallbe briefly presented.1. Power SeriesBinomial expansion:(1 ± x) m =m(m − 1)1 ± mx + x 2 m(m − 1)(m − 2)± x 3 +···2!3!n m(m − 1) ···(m − n + 1)+ (±1) x n +··· .n!For integer positive m this series is finite. The coefficientsare(m(m − 1) ···(m − n + 1) m= .n!n)If m is not a positive integer, the series is infinite and convergentfor |x| < 1. This expansion often provides a simplificationof otherwise complicated expressions.binomial expansionbinomial coefficients

382 A Mathematical Appendixexamplesfor binomial expansionstrigonometric functionsexponential functionnatural logarithmA few examples for most commonly used binomial expansions:(1 ± x) 1/2 = 1 ± 1 2 x − 1 8 x2 ± 116 x3 − 5128 x4 ±··· ,(1 ± x) −1/2 = 1 ∓ 1 2 x + 3 8 x2 ∓ 516 x3 + 35128 x4 ∓··· ,(1 ± x) −1 = 1 ∓ x + x 2 ∓ x 3 + x 4 ∓··· ,(1 ± x) 4 = 1 ± 4x + 6x 2 ± 4x 3 + x 4 finite .Trigonometric functions:sin x = x − x33! + x55! −···+(−1)n x 2n+1(2n + 1)! ±··· ,cos x = 1 − x22! + x4x2n−···+(−1)n4! (2n)! ±··· ,tan x = x + 1 3 x3 + 215 x5 + 17315 x7 +··· , |x| < π 2 ,cot x = 1 x − x 3 − x345 − 2x5 −··· , 0 < |x|

A.1 Selected Formulae 383∫e x dx = e x ,exponential function∫x e ax dx = eax(ax − 1) ,a2 ∫dx1 + e ax = 1 a ln e ax1 + e ax ,∫sin x dx =−cos x,trigonometric functions∫cos x dx = sin x,∫tan x dx =−ln cos x,∫ln x dx = x ln x − x.natural logarithm3. Specific Integrals∫ π/20cos x sin x dx = 1 2 ,∫ ∞0∫ ∞0∫ ∞0∫ ∞0∫ 10e −ax2 dx = 1 2√ πa ,x dxe x − 1 = π 26 ,x dxe x + 1 = π 2sin axxtriginometricGaussianexponentials12 ,{ π2for a>0dx =sin ax/x− π ,2for a

384 A Mathematical AppendixPoisson distributionGaussian distributionPoisson:f(r,µ)= µr e −µ, r = 0, 1, 2,... , µ>0 ;r!mean: 〈r〉 =µ, variance: σ 2 = µ.Gaussian:f(x,µ,σ 2 ) = 1{ }σ √ 2π exp (x − µ)2−2σ 2 , σ > 0 ;mean: 〈x〉 =µ, variance: σ 2 .Landau distributionApproximation for the Landau distribution:L(λ) = √ 1 {exp − 1 }2π 2 (λ + e−λ ) ,where λ is the deviation from the most probable value.5. Errors and Error Propagationmean valuevariancestandard deviationindependent, uncorrelatedvariablesMean value of n independent measurements:〈x〉 = 1 nn∑x i ;i=1variance of n independent measurements:s 2 = 1 nn∑(x i −〈x i 〉) 2 = 1 ni=1n∑xi 2 −〈x〉2 ,i=1where s is called the standard deviation. A best estimate forthe standard estimation of the mean issσ = √ .n − 1If f(x,y,z) and σ x , σ y , σ z are the function and standarddeviations of the independent, uncorrelated variables, then( ) ∂f 2 ( ) ∂f 2 ( ) ∂f 2σf 2 = σx 2 ∂x+ σy 2 ∂y+ σz 2 ∂z.If D(z) is the distribution function of the variable z aroundthetruevaluez 0 with expectation value 〈z〉 and standarddeviation σ z , the quantity

A.2 Mathematics for Angular Variations of the CMB 385∫ 〈z〉+δ1 − α = D(z) dz〈z〉−δrepresents the probability that the true value z 0 lies in the interval±δ around the measured value 〈z〉; i.e., 100(1 − α)%measured values are within ±δ.IfD(z) is a Gaussian distributionone hasconfidence intervalδ 1 − α1σ 68.27%2σ 95.45%3σ 99.73%.In experimental distributions frequently the full width at halfmaximum, z, is easily measured. For Gaussian distributionsz is related to the standard deviation byfull width at half maximumz(fwhm) = 2 √ 2ln2σ z ≈ 2.355 σ z .A.2 Mathematics for Angular Variationsof the CMB“As physics advances farther and fartherevery day and develops new axioms,it will require fresh assistancefrom mathematics.”Francis BaconIn this appendix the mathematics needed to describe thevariations in the CMB temperature as a function of directionis reviewed. In particular, some of the important propertiesof the spherical harmonic functions Y lm (θ, φ) will be collected.More information can be found in standard texts onmathematical methods of physics such as [46].First it will be recollected what these functions areneeded for in astroparticles physics. Suppose a quantity(here the temperature) as a function of direction has beenmeasured, which one can take as being specified by thestandard polar coordinate angles θ and φ. This applies, e.g.,for the directional measurements of the blackbody radiation.But one is not able – or at least it is highly impractical – totry to understand individually every measurement for everydirection. Rather, it is preferable to parameterize the datawith some function and see if one can understand the mostimportant characteristics of this function.spherical harmonicsdependence on the direction

386 A Mathematical Appendix‘periodic functions’important differentialequationsnabla operatorseparation of variablesWhen, however, a function to describe the measuredtemperature as a function of direction is chosen, one cannottake a simple polynomial in θ and φ, because this wouldnot satisfy the obvious continuity requirements, e.g., that thefunction at φ = 0 matches that at φ = 2π. By using sphericalharmonics as the basis functions for the expansion, theserequirements are automatically taken into account.Now one has to remember how the spherical harmonicsare defined. Several important differential equations ofmathematical physics (Schrödinger, Helmholtz, Laplace) canbe written in the form( )∇ 2 + v(r) ψ = 0 ,(A.1)where ∇ is the usual nabla operator, as defined by∂∇ = e x∂x + e ∂y∂y + e ∂z∂z .(A.2)Here v(r) is an arbitrary function depending only on the radialcoordinate r. In separation of variables in spherical coordinates,a solution of the formψ(r,θ,φ) = R(r) Θ(θ) Φ(φ)(A.3)angular partsis tried. Substituting this back into (A.1) gives for the angularpartsd 2 Φdφ 2 =−m2 Φ,d 2 Θdθ 2 + cos θ[]dΘsin θ dθ + l(l + 1) −m2sin 2 Θ = 0 ,θ(A.4)azimuthal solutionThe solution for Θ is proportional to the associated Leg-endre function Plm (cos θ). The product of the two angularparts is called the spherical harmonic function Y lm (θ, φ),polar solutionspherical harmonic functionwhere l = 0, 1,...and m =−l,...,l are separation constants.The solution for Φ isΦ(φ) = 1 √2πe imφ .(A.5)Y lm (θ, φ) = Θ(θ)Φ(φ)√2l + 1 (l − m)!=4π (l + m)! P l m (cos θ)e imφ . (A.6)

A.2 Mathematics for Angular Variations of the CMB 387Some of the spherical harmonics are given below:Y 00 (θ, φ) = √ 1 ,4π(A.7)√3Y 11 (θ, φ) =−8π sin θ eiφ ,(A.8)√3Y 10 (θ, φ) = cos θ,4π (A.9)√15Y 22 (θ, φ) =32π sin2 θ e 2iφ ,(A.10)√15Y 21 (θ, φ) =−8π sin θ cos θ eiφ ,(A.11)√5Y 20 (θ, φ) =16π (3cos2 θ − 1) .(A.12)The importance of spherical harmonics for this investigationis that they form a complete orthogonal set of functions.That is, any arbitrary function f(θ,φ) can be expandedin a Laplace series asf(θ,φ)=∞∑l∑l=0 m=−la lm Y lm (θ, φ) .(A.13)To determine the coefficients a lm , one uses the orthogonalityrelation∫∫sin θ dθ dφY lm (θ, φ)Yl ∗ ′ m ′(θ, φ) = δ l ′ lδ m ′ m .(A.14)If both sides of (A.13) are multiplied by Y l ′ m ′, integrationover θ and φ leads to∫∫a lm =sin θ dθ dφYlm ∗ (θ, φ)f (θ, φ) . (A.15)So, in principle, once a function f(θ,φ) is specified, thecoefficients of its Laplace series can be found.The same spherical harmonics are found in the multipoleexpansion of the potential from an electric charge distribution.The terminology is usually borrowed from this exampleand the terms in the series are referred to as multipolemoments. The l = 0 term is the monopole, l = 1 the dipole,etc.complete orthogonal setof functionsLaplace seriesorthogonality relationcalculation of coefficientsmultipole expansion

388 A Mathematical AppendixLaplace series for thetemperature variationsfinite series: practical limitTo quantify the temperature variations of the CMB, theLaplace series can be used to describeof the CMB T (θ, φ) = T(θ,φ)−〈T 〉= ∑ l∑a lm Y lm (θ, φ) ,l≥1 m=−l(A.16)where 〈T 〉 is the temperature averaged over all directions.Here the sum starts at l = 1, not l = 0, since by constructionthe l = 0 term gives the average temperature, whichhas been subtracted off. In some references one expandsT /〈T 〉 rather than T . This gives the equivalent informationbut with the coefficients simply differing from those in(A.16) by a factor of 〈T 〉.In practice one determines the coefficients a lm up tosome l max by means of a statistical parameter estimationtechnique such as the method of maximum likelihood. Thisprocedure will use as input the measured temperatures andinformation about their accuracy to determine estimates forthe coefficients a lm and their uncertainties.Once one has estimates for the coefficients a lm , one cansummarize the amplitude of regular variation with angle bydefiningC l = 12l + 1l∑m=−l|a lm | 2 . (A.17)angular power spectrumThe set of numbers C l is called the angular power spectrum.The value of C l represents the level of structure found at anangular separationθ = 180◦l. (A.18)The measuring device will in general only be able to resolveangles down to some minimum value; this determines themaximum measurable l.

389B Results from Statistical Physics:Thermodynamics of the EarlyUniverse“Scientists speak of the Law of Inertia or theSecond Law of Thermodynamics as if somegreat legislative in the sky once met and setdown the rules to govern the universe.”Victor J. StengerIn this appendix some results from statistical and thermalphysics will be recalled that will be needed to describe theearly universe. To start with, the Fermi–Dirac and Bose–Einstein distributions for the number of particles per unitvolume of momentum space will be derived:Fermi–Dirac, Bose–Einsteindistributionf(p) = gV 1(2π) 3 e (E−µ)/T ± 1 ,(B.1)where E = √ p 2 + m 2 is the energy and g is the number ofinternal degrees of freedom for the particle, V is the volumeof the system, T is the temperature, and µ is the chemicalpotential. (The Boltzmann constant k is set to unity asusual.) The minus sign in (B.1) is used for bosons and theplus for fermions. These distributions will be required to derivethe number density n of particles, the energy density ϱ,and the pressure P .Some of the relations may differ from those covered in atypical course in statistical mechanics. This is for two mainreasons. First, the particles in the very hot early universetypically have velocities comparable to the speed of light,therefore the relativistic equation E 2 = p 2 + m 2 will beneeded to relate energy and momentum. Second, the temperatureswill be so high that particles are continually beingcreated and destroyed, e.g., through reactions such asγγ ↔ e + e − . This is in contrast to the physics of low-temperaturesystems, where the number of particles in a systemis usually constrained to be constant. The familiar exceptionis blackbody radiation, since massless photons can be createdand destroyed at any non-zero temperature. For a gas ofrelativistic particles it will be found that the expressions forn, ϱ, andP are similar to those for blackbody radiation.chemical potentialrelativistic treatmentvariable particle numbersblackbody radiation

390 B Results from Statistical Physics: Thermodynamics of the Early UniverseB.1 Statistical Mechanics Review“The general connection between energyand temperature may only be establishedby probability considerations.Two systems are in statistical equilibriumwhen a transfer of energy does notincrease the probability.”Max Planckfundamental postulate:equipartition of energyequilibrium distributionN-particle wave functionConsider a system with volume V = L 3 and energy U,which could be a cube of the very early universe. The numberof particles will not be fixed since the temperatures consideredhere will be so high that particles can be continuallycreated and destroyed. For the moment only a single particletype will be considered but eventually the situation will begeneralized to include all possible types.The system can be in any one of a very large numberof possible microstates. The fundamental postulate of statisticalmechanics is that all microstates consistent with theimposed constraints (volume, energy) are equally likely. Agiven microstate, e.g., an N-particle wave function ψ(x 1 ,...,x N ) specifies everything about the system, but this isfar more than one wants to know. To reduce the informationto a more digestible level, one can determine from the microstatethe momentum distribution of the particles, i.e., theexpected number of particles in each cell d 3 p of momentumspace.There will be many microstates that lead to the same distribution,but one distribution in particular will have overwhelminglymore possible microstates than the others. Togood approximation all the others can be ignored and thisequilibrium distribution can be regarded as the most likely.Once it has been found, one can determine from it the otherquantities needed, such as the energy density and pressure.So, to find the equilibrium distribution one needs to determinethe number of possible microstates consistent witha distribution and then find the one for which this is a maximum.This is treated in standard books on statistical mechanics,e.g., [47]. Here only the main steps will be reviewed.It is assumed that the system’s N-particle wave functioncan be expressed as a sum of N terms, each of which is theproduct of N one-particle wave functions of the formψ A (x) ∼ e ip A·x .(B.2)

B.1 Statistical Mechanics Review 391The total wave function is thusψ(x 1 ,...,x N ) == √ 1 ∑P(i,j,...)ψA (x i )ψ B (x j ) ··· ,N!(B.3)where the sum includes all possible permutations of the coordinatesx i . For a system of identical bosons, the factor P isequal to one, whereas for identical fermions it is plus or minusone depending on whether the permutation is obtainedfrom an even or odd number of exchanges of particle coordinates.This results in a wave function that is symmetric forbosons and antisymmetric for fermions upon interchange ofany pair of coordinates. As a consequence, the total wavefunction for a system of fermions is zero if the same oneparticlewave function appears more than once in the productof terms; this is the Pauli exclusion principle.Although the most general solution to the N-particleSchrödinger equation does not factorize in the way ψ hasbeen written in (B.3), this form will be valid to good approximationfor systems of weakly interacting particles. Forhigh-temperature systems such as the early universe, (B.3)is assumed to hold.Further, one assumes that the one-particle wave functionsshould obey periodic boundary conditions in the volumeV = L 3 . The plane-wave form for the one-particlewave functions in (B.2) then implies that the momentumvectors p cannot take on arbitrary values but that they mustsatisfysymmetrization for bosonsantisymmetrizationfor fermionsPauli exclusion principleperiodic boundaryconditions: discretizingmomentump = 2π L (n x,n y ,n z ),(B.4)where n x , n y ,andn z are integers. Thus, the possible momentafor the one-particle states are given by a cubic latticeof points in momentum space with separation 2π/L.For a given N-particle wave function, where N will ingeneral be very large, the possible momentum vectors forthe one-particle states will follow some distribution in momentumspace. That is, one will find a certain number dNof one-particle states for each element d 3 p in momentumspace, andmomentum distributionf(p) = d3 Nd 3 p(B.5)will be called the momentum distribution.

392 B Results from Statistical Physics: Thermodynamics of the Early Universenumber of microstatesA given distribution f(p) could result from a numberof distinct N-particle wave functions, i.e., from a number ofdifferent microstates. All available microstates are equallylikely, but the overwhelming majority of them will correspondto a single specific f(p), the equilibrium distribution.This is what one needs to find.To find this equilibrium momentum distribution, onemust determine the number of microstates t for a givenf(p). To do this, one considers the momentum space to bedivided into cells of size δ 3 p. The number of particles in theith cell isν i = f(p i )δ 3 p.(B.6)The number of possible one-particle momentum states in thecell is δ 3 p divided by the number of states per unit volumetotal number in momentum space, (2π/L) 3 . The total number of one-of one-particle states particle states in δ 3 p is therefore 1δ 3 pγ i = g(2π/L) 3 ,(B.7)where g represents the number of internal (e.g., spin) de-grees of freedom for the particle. For an electron with spin1/2, for example, one has g = 2.It is assumed that the element δ 3 p is large compared tothe volume of momentum space per available state, which is(2π/L) 3 , but small compared to the typical momenta of theparticles. Within this approximation, the set of numbers ν ifor all i contains the same information as f(p).For a system of bosons, there is no restriction on thenumber of particles that can have the same momentum.Therefore, each of the γ i states can have from zero up to ν iparticles. The number of ways of distributing the ν i particlesamong the γ i states is a standard problem of combinatorics(see, e.g., [47]). One obtainsnumber of degreesof freedomsystem of bosons(ν i + γ i − 1)!ν i !(γ i − 1)!(B.8)total number of microstatespossible arrangements. The total number of microstates forthe distribution is therefore1 In many references the number of particles is called n i and thenumber of states g i . Unfortunately, these letters need to be usedwith different meanings later in this appendix, so here ν i and γ iwill be used instead.

B.1 Statistical Mechanics Review 393t BE [f(p)] = ∏ i(ν i + γ i − 1)!ν i !(γ i − 1)!≈ ∏ i(ν i + γ i )!ν i !γ i !,(B.9)where the product extends over all cells in momentum space.The subscript BE in (B.9) stands for Bose–Einstein sincethis relation holds for a collection of identical bosons.For fermions, the antisymmetric nature of the total wavefunction implies that it can contain a given one-particle stateat most only once. Therefore, each of the γ i states in theith cell in momentum space can be occupied either once ornot at all. This implies γ i ≥ ν i . The number of possiblearrangements of ν i particles in the γ i states where each stateis occupied zero or one time is another standard problem ofcombinatorics, for which one findsγ i !ν i !(γ i − ν i )! .(B.10)The total number of combinations for all cells is thust FD [f(p)] = ∏ iγ i !ν i !(γ i − ν i )! ,(B.11)where FD stands for Fermi–Dirac.As the number of microstates t[f(p)] is astronomicallylarge, it is more convenient to work with its logarithm, andfurthermore one can use Stirling’s approximation,system of fermionstotal number of microstateslogarithm of the numberof microstatesln N! ≈N ln N − N,(B.12)valid for large N. Thisgivesln t BE [f(p)] ≈(B.13)≈ ∑ [(ν i + γ i ) ln(ν i + γ i ) − ν i ln ν i − γ i ln γ i ]iandln t FD [f(p)] ≈(B.14)≈ ∑ [γ i ln γ i − ν i ln ν i − (γ i − ν i ) ln(γ i − ν i )]ifor bosons and fermions, respectively.The next step is to find the distribution f(p) which maximizesln t[f(p)]. Before doing this, however, the problemshould be generalized to allow for more than one type ofparticle. As long as a particle’s mass is small compared tomore than one typeof particle

394 B Results from Statistical Physics: Thermodynamics of the Early Universemaximizationenergy conservationthe temperature, it will be continually created and destroyed,and for sufficiently early times this will be true for all particletypes. So one will have a set of distribution functionsf a (p), where the index a = e,µ,τ,u,d,s,... is a labelfor the particle type. One can write this set of functions as avector, f ≡ (f e ,f µ ,f τ ,f u ,f d ,...).In order to find the set of distributions f (p), one needsto maximize the total number of microstates t[f (p)] subjectto two types of constraints. First, it is required that the sumof the energies of all of the particles be equal to the totalenergy U, i.e.,U = ∑ ∫E a f a (p) d 3 pa= ∑ ∫ √p 2 + m 2 a f a(p) d 3 p.(B.15)aSecond, although particles may be created and destroyed,certain quantities are conserved. Suppose that the systemhas a total conserved charge Q (e.g., zero), baryon num-ber B, and lepton number L. Suppose further that in thermalequilibrium the system has N a particles of type a.Theserequirementscan be written asconserved quantitiesQ = ∑ aQ a N a ,(B.16)B = ∑ aL = ∑ aB a N a ,L a N a .(B.17)(B.18)Note that the values N a are not explicitly constrained, butrather only the total Q, B, andL. Thus, the quantity thatone wants to maximize can be expressed asφ(f (p), α Q ,α B ,α L ,β)== ∑ (ln t a [f a (p)]+β U − ∑ ∫)E a f a (p) d 3 paa(+ α Q Q − ∑ ) (Q a N a + α B B − ∑ )B a N aaa(+ α L L − ∑ )L a N a , (B.19)aLagrange multiplierswhere β, α Q , α B ,andα L are Lagrange multipliers. Settingthe derivatives of φ with respect to the Lagrange multipliers

B.1 Statistical Mechanics Review 395to zero ensures that the corresponding constraints are fulfilled.To find the set of distributions f which maximize(B.19), one substitutes f a (p i ) = ν ai /δ 3 p. Then the integralsare converted to sums, and further one has ∑ i ν ai =N a . In addition, the number of microstates can be obtainedfrom (B.13) for bosons and (B.14) for fermions. The derivativeof φ with respect to ν ai is∂φ= ln(γ ai ± ν ai ) − ln ν ai − βE ai∂ν ai− α Q Q a − α B B a − α L L a , (B.20)where γ ai = g a δ 3 p/(2π/L) 3 is the number of states availableto a particle of type a in the cell i, and for the derivationin this section the upper sign refers to bosons and the lowersign to fermions. Setting (B.20) equal to zero and solvingfor ν ai givesdiscretizing momentumintegralssolutionfor number of particlesν ai =γ aiexp [ α Q Q a + α B B a + α L L a + βE ai]∓ 1.(B.21)Re-expressing this in terms of the functions f a (p i ) =ν ai /δ 3 p givesf a (p) =g a (L/2π) 3exp [ α Q Q a + α B B a + α L L a + βE a] ∓ 1.(B.22)The temperature can be defined asdefinition of temperatureT = 1/β ,(B.23)and it can be shown that this has all of the desired propertiesof the usual thermodynamic temperature. Furthermore, thechemical potential for particle type a can be defined aschemical potentialµ a =−T(α Q Q a + α B B a + α L L a ), (B.24)which can be modified in the obvious way to include a differentset of conserved quantities. Note that, although theLagrange multipliers are specific to the system, i.e., are thesame for all particle types, the chemical potential depends onthe charge, baryon number, and lepton number of the particle.In a reaction where, say, a + b ↔ c + d, (B.24) impliesµ a + µ b = µ c + µ d .

396 B Results from Statistical Physics: Thermodynamics of the Early Universeresulting momentumdistributioninternal degrees of freedomUsing these modified names for the Lagrange multipliersgives the desired result for the momentum distribution,f a (p) = g a(L/2π) 3e (E a−µ a )/T ∓ 1 ,(B.25)where one uses the minus sign if particle type a is a bosonand plus if it is a fermion. The number of internal degrees offreedom, g a , is usually 2J + 1 for a particle of spin J ,butitcould include other degrees of freedom besides spin such ascolour.B.2 Number and Energy Densities“There are 10 11 stars in the galaxy.That used to be a huge number. But it’sonly a hundred billion. It’s less than thenational deficit! We used to call themastronomical numbers. Now we shouldcall them economical numbers.”Richard P. Feynmannumber density nFrom the Planck distribution given by (6.81), (B.1) one canproceed to determine the number and energy per unit volumefor all of the particle types. The function (B.25) gives thenumber of particles of type a in a momentum-space volumed 3 p. The number density n is obtained by integrating thisover all of momentum space and dividing by the volumeV = L 3 , i.e.,n = 1 V∫f(p) d 3 p =∫g(2π) 3d 3 pe (E−µ)/T ± 1 , (B.26)n: energy integraln: relativistic limitwhere for clarity the index indicating the particle type hasbeen dropped. Since the integrand only depends on the magnitudeof the momentum through E = √ p 2 + m 2 , one cantake the element d 3 p to be a spherical shell with radius p andthickness dp, sothatd 3 p ̂= 4πp 2 dp.FromE 2 = p 2 + m 2one gets 2E dE = 2p dp and therefore√E 2 − m 2 E dEn =g ∫ ∞2π 2 m e (E−µ)/T . (B.27)± 1The integral (B.27) can be carried out in closed form onlyfor certain limiting cases. In the limit where the particles arerelativistic, i.e., T ≫ m,andalsoifT ≫ µ, one finds

B.2 Number and Energy Densities 397⎧ζ(3)⎪⎨πn =2 gT 3 for bosons,3 ζ(3) ⎪⎩4 π 2 gT 3 for fermions.(B.28)Here ζ is the Riemann zeta function and ζ(3) ≈ 1.20 206 ....Notice that in particle physics units the number density hasdimension of energy cubed. To convert this to a normalnumber per unit volume, one has to divide by (¯hc) 3 ≈(0.2GeVfm) 3 .In the non-relativistic limit (T ≪ m), the integral (B.27)becomes( ) mT 3/2n = g e −(m−µ)/T , (B.29)2πn: non-relativistic limitwhere the same result is obtained for both the Fermi–Diracand Bose–Einstein distributions. One sees that for a nonrelativisticparticle species, the number density is exponentiallysuppressed by the factor e −m/T , the so-called Boltzmannfactor. This may seem counter intuitive, since the densityof air molecules in a room is certainly not suppressed bythis factor, although they are non-relativistic. One must takeinto account the fact that the chemical potentials depend ingeneral on the temperature, and this dependence is exactlysuch that all relevant quantities are conserved. In the case ofthe air molecules, µ varies with temperature so as to exactlycompensate the factor T 3/2 e −m/T .For very high temperatures, to good approximation allof the chemical potentials can be set to zero. The total numberof particles will be large compared to the net values ofany of the conserved quantum numbers, and the constraintseffectively play no rôle.To find the energy density ϱ one multiplies the numberof particles in d 3 p by the energy and integrate over all momenta,E d 3 pϱ =e (E−µ)/T ± 1= g ∫ ∞√E 2 − m 2 E 2 dE2π 2 m e (E−µ)/T . (B.30)± 1As with n, the integral can only be carried out in closed formfor certain limiting cases. In the relativistic limit, T ≫ m,one findsg ∫(2π) 3very high temperaturesenergy density ϱϱ: relativistic limit

398 B Results from Statistical Physics: Thermodynamics of the Early Universeϱ: non-relativistic limit⎧π⎪⎨230 gT 4 for bosons,ϱ =⎪⎩ 7 π 28 30 gT 4 for fermions.In the non-relativistic limit one has(B.31)ϱ = mn ,(B.32)average energy per particlewith the number density n given by (B.29).From the number and energy densities one can obtainthe average energy per particle, 〈E〉 =ϱ/n.ForT ≫ m onefinds⎧π 4⎪⎨ T ≈ 2.701 T for bosons,30 ζ(3)〈E〉 =7π⎪⎩4(B.33)180 ζ(3) T ≈ 3.151 T for fermions.In the non-relativistic limit, the average energy, written asthe sum of mass and kinetic terms, reads〈E〉 =m + 3 2 T,(B.34)which is dominated by the mass m for low T .B.3 Equations of State“If your theory is found to be against thesecond law of thermodynamics, I giveyou no hope; there is nothing for it butto collapse in deepest humiliation.”Arthur Eddingtonfirst law of thermodynamicsFinally in this appendix an equation of state will be derived,that is, a relation between energy density ϱ and pressure P .This will be needed in conjunction with the acceleration andfluid equations in order to solve the Friedmann equation forR(t).There are several routes to the desired relation. The approachthat starts from the first law of thermodynamics isthe most obvious one,dU = T dS − P dV ,(B.35)

B.3 Equations of State 399which relates the total energy U, temperature T ,entropyS,pressure P ,andvolumeV of the system. The differentialdU can also be written asdU =( ∂U∂S)VdS +( ) ∂UdV ,∂V S(B.36)where the subscripts indicate what is kept constant whencomputing the partial derivatives. Equating the coefficientsof dV in (B.35) and (B.36) gives the pressure,( ) ∂UP =− . (B.37)∂V SRecall that the entropy is simply the logarithm of the totalnumber of microstates Ω, and that to good approximationthis is given by the number of microstates of the equilibriumdistribution t[f(p)]. Thatis,pressureentropy andnumber of microstatesS = ln Ω ≈ ln t[f(p)] .(B.38)The important thing to notice here is that the entropy isentirely determined by the distribution f(p). Therefore, tokeep the entropy constant when computing (∂U/∂V ) S , onesimply needs to regard the distribution f(p) as remainingconstant when V is changed.The total energy U is∫U = Ef(p) d 3 p,(B.39)total energyand the pressure is therefore( ) ∫∂U ∂EP =− =−∂V∂V f(p) d3 p.S(B.40)The derivative of E with respect to the volume V = L 3 is∂E∂V = ∂E ∂p∂p ∂V = ∂E ∂p∂p ∂L/ ∂V∂L .(B.41)One has ∂V /∂L = 3L 2 and furthermore E = √ p 2 + m 2 ,so∂E∂p = 1 (p 2 + m 2) −1/2 p 2p =2E .(B.42)From (B.4) one gets that p ∼ L −1 , and therefore ∂p/∂L =−p/L. Substituting these into (B.41) gives

400 B Results from Statistical Physics: Thermodynamics of the Early Universegeneral expressionfor the pressure∂E∂V = p ( ) −p 1E L 3L 2 = −p23EV .(B.43)Putting this into (B.40) provides the general expression forthe pressure,P = 1 ∫ p23V E f(p) d3 p.(B.44)In the relativistic limit the particle’s rest mass can beneglected, so, E = √ p 2 + m 2 ≈ p. Equation (B.44) thenbecomesP = 1 ∫Ef(p) d 3 p.(B.45)3VBut the total energy U is (B.39)∫U = Ef(p) d 3 p(B.46)pressurein the relativistic limitand ϱ = U/V, so the final result for the pressure for a gasof relativistic particles is simplyP = ϱ 3 .(B.47)non-relativistic limit:ideal gas lawvacuum energy densityfrom a cosmological constantnegative pressureThis is the well-known result from blackbody radiation, butone realizes here that it applies for any particle type in therelativistic limit T ≫ m.In the non-relativistic limit, the pressure is given by theideal gas law,P = nT .(B.48)In this case, however, the energy density is simply ϱ = mn,so for T ≪ m one has P ≪ ϱ and in the acceleration andfluid equations one can approximate P ≈ 0.Finally, the case of vacuum energy density from a cosmologicalconstant can be treated,ϱ v =Λ8πG .(B.49)If one takes U/V = ϱ v as constant, then the pressure is( ) ∂UP =− =− U ∂V S V =−ϱ v . (B.50)Thus, a vacuum energy density leads to a negative pressure.

401C Definition of Equatorialand Galactic Coordinates“In the technical language of astronomy, therichness of the star field depends mainly onthe galactic latitude, just as the Earth’s climatedepends mainly on the geographic latitude,and not to any great extent on the longitude.”Sir James JeansOptical astronomers mostly prefer equatorial coordinates,like right ascension and declination, while astrophysicistsoften use galactic coordinates.For equatorial coordinates, which are centered on theEarth, the plane of the Earth’s equator is chosen as the planeof reference. This plane is called the celestial equator. Thereis another plane which is defined by the motion of the Eartharound the Sun. This plane has an inclination with respect tothe celestial equator of 23.5 degrees. The plane of the Earth’sorbit is called the ecliptic. At the periphery these two planesintersect in two points. The one where the Sun crosses thecelestial equator from the south is the vernal equinox.The coordinate measured along the celestial equator eastwardfrom the vernal equinox is called right ascension,usuallynamed α. The distance perpendicular to the celestialequator is named declination, named δ. Right ascensionvaries from 0 to 360 degrees, or sometimes – for convenience– from 0 to 24 hours. Declination varies from −90to +90 degrees (see Fig. C.1).equatorial coordinateseclipticright ascensiondeclinationnorth celestial poledeclinationδequatoreclipticvernalequinoxright ascentionαFig. C.1Definition of the equatorialcoordinates right ascension anddeclination

402 C Definition of Equatorial and Galactic Coordinatesgalactic coordinateslongitudelatitudedistance from the SunIn contrast, galactic coordinates are fixed on our galaxy.The spherical coordinates (r,l,b) are centered on the Sun,with l being the galactic longitude and b the latitude. r is thedistance of the celestial object from the Sun. The galacticlongitude l is the angle between the direction to the galacticcenter and the projection of the direction to the star onto thegalactic plane. It counts from the direction to the galacticcenter (l = 0 degrees) via the galactic anticenter (180 degrees,away from the Sun) back to the galactic center. Thelatitude b varies from the +90 degrees (perpendicular abovethe galactic plane) to −90 degrees (Fig. C.2).starFig. C.2Definition of the galacticcoordinates latitude and longitudegalactic planerSunblgalactic center

403D Important Constants for Astroparticle Physics 1“Men who wish to know about the world must learn about it in itsparticular details.”Heraclitusrelativeuncertaintyvelocity of light c 299 792 458 m/s exactgravitational constant G 6.6742 × 10 −11 m 3 kg −1 s −2 1.5 × 10 −4Planck’s constant h 6.626 0693 × 10 −34 Js 1.7 × 10 −7= 4.135 6675 × 10 −15 eV s¯h = h/2π 1.054 571 68 × 10 −34 Js 1.7 × 10 −7Planck mass √¯hc/G = 6.582 119 15 × 10 −16 eV s 8.5 × 10 −82.176 45 × 10 −8 kg 7.4 × 10 −5Planck length √ G¯h/c 3Planck time √ G¯h/c 5 = 1.220 90 × 10 19 GeV/c 21.616 24 × 10 −35 m 7.4 × 10 −55.391 19 × 10 −44 s 7.4 × 10 −5elementary charge e 1.602 176 53 × 10 −19 C 8.5 × 10 −8fine-structure constant α =4πε e20 ¯hc1/137.035 999 11 3.3 × 10 −9Rydberg energy Ry 13.605 6923 eV 8.5 × 10 −8electron mass m e 9.109 3826 × 10 −31 kg 1.7 × 10 −70.510 998 918 MeV/c 2 8.6 × 10 −8proton mass m p 1.672 621 71 × 10 −27 kg 1.7 × 10 −7= 938.272 029 MeV/c 2 8.6 × 10 −8neutron mass m n 1.674 9287 × 10 −27 kg 6 × 10 −7= 939.565 36 MeV/c 2 3 × 10 −7m n − m p = 1.293 3317 MeV/c 2 6 × 10 −6Avogadro constant N A 6.022 1415 × 10 23 mol −1 1.7 × 10 −7Boltzmann constant k 1.380 6505 × 10 −23 JK −1 1.8 × 10 −6= 8.617 343 × 10 −5 eV K −1Stefan–Boltzmann constant σ 5.670 400 × 10 −8 Wm −2 K −4 7 × 10 −6electron volt, eV 1.602 176 53 × 10 −19 J 8.5 × 10 −81 see also Eidelman et al., Phys. Letters B592, 1 (2004)

404 D Important Constants for Astroparticle Physicsrelativeuncertaintystandard atmosphere, atm 101 325 Pa exactacceleration due to gravity g 9.806 65 m s −2 exactHubble constant H 0 71 km s −1 Mpc −1 5%age of the universe t 0 13.7 × 10 9 a 1.5%Hubble distance c/H 0 4 225 Mpc 5%critical density ϱ c = 3H0 2/8πG 9.469 × 10−30 g/cm 3 ≈ 10%density of galaxies≈ 0.02 Mpc −3temperature of the blackbody radiation 2.725 K 4 × 10 −4number density of blackbody photons 410.4cm −3 1.2 × 10 −3astronomical unit, AU 1.495 978 706 60 × 10 11 m 1.3 × 10 −10parsec, pc (1 AE/1arcsec) 3.085 677 5807 × 10 16 m 1.3 × 10 −10light-year, LY0.3066 pc = 0.9461 × 10 16 mSchwarzschild radiusof the Sun: 2GM⊙/c 2 2.953 250 08 km 3.6 × 10 −4mass of the Sun M⊙ 1.988 44 × 10 30 kg 1.5 × 10 −4solar constant 1 360 W/m 2 0.2%solar luminosity L⊙ 3.846 × 10 26 W 0.2%mass of the Earth M ♁ 5.9723 × 10 24 kg 1.5 × 10 −4radius of the Earth R ♁6.378 140 × 10 6 mSchwarzschild radiusof the Earth: 2GM ♁ /c 2 0.887 056 22 cm 1.5 × 10 −4velocity of the solar systemabout the galactic center 220 km/s 9%distance of the Sunfrom the galactic center 8.0 kpc 6%matter density of the universe Ω m 0.27 18%baryon density of the universe Ω b 0.044 9%dark-matter density of the universe Ω dm 0.22 18%radiation density of the universe Ω γ 4.9 × 10 −5 10%neutrino density of the universe Ω ν ≤ 0.015 95% C.L.dark-energy density of the universe Λ 0.73 5%total energy density of the universe Ω tot 1.02 2%(Ω tot ≡ Ω)number density of baryons n b 2.5 × 10 −7 /cm 3 4%baryon-to-photon ratio η 6.1 × 10 −10 3%0 ◦ C 273.15 K

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407Further Reading“Education is the best provision of old age.”AristotleClaus Grupen, Astroteilchenphysik: das Universum im Licht der kosmischen Strahlung,Vieweg, Braunschweig, 2000.Lars Bergström and Ariel Goobar, Cosmology and Particle Astrophysics, Wiley, Chichester,1999.H. V. Klapdor-Kleingrothaus and K. Zuber, Particle Astrophysics, IOP, 1997.G. Börner, The Early Universe: Facts and Fiction, Springer, 1988.S. Weinberg, The First Three Minutes, BasicBooks, New York, 1993.Alan Guth, The Inflationary Universe, Vintage, Vancouver, USA, 1998.G. Cowan, Lecture Notes on Particle Physics, RHUL Physics Dept. course notes forPH3520, 2002.Web site of the European Organisation for Nuclear Research (CERN): www.cern.ch.H. Georgi and S. L. Glashow, Phys. Rev. Lett. 32 (1974) 438.Web site of the Super-Kamiokande experiment:www-sk.icrr.u-tokyo.ac.jp/doc/sk.M. Shiozawa et al., (Super-Kamiokande collaboration), Phys. Rev. Lett. 81 (1998) 3319.G. Cowan, Statistical Data Analysis, Oxford University Press, 1998.A. V. Filippenko and A. G. Riess, astro-ph/9807008 (1998).T. D. Lee and C. N. Yang, Phys. Rev. 104 (1956) 254.C. S. Wu et al., Phys. Rev. 105 (1957) 1413.J. H. Christenson, J. W. Cronin, V. L. Fitch and R. Turlay, Phys. Rev. Lett. 13 (1964) 138.Harold P. Furth, Perils of Modern Living, originally published in the New Yorker Magazine,1956.Ken Nollett, www.phys.washington.edu/~nollett.David N. Schramm and Michael S. Turner, Rev. Mod. Phys. 70 (1998) 303.Yuri I. Izotov and Trinh X. Thuan, Heavy-element abundances in blue compact galaxies,Astrophys. J. 511 (1999) 639–659.Scott Burles, Kenneth M. Nollet, James N. Truran and Michael S. Turner, Phys. Rev. Lett.82 (1999) 4176; astro-ph/9901157.

408 Further ReadingWeb site of the ALEPH collaboration: alephwww.cern.ch, and the ALEPH publicpages, alephwww.cern.ch/Public.html.C. L. Bennett et al., Astrophys. J. Suppl. 148 (2003) 1; map.gsfc.nasa.gov.Keith A. Olive, Primordial Big Bang Nucleosynthesis, astro-ph/9901231 (1999).R. Hagedorn, Relativistic Kinematics, W. A. Benjamin Inc. Reading, Mass., 1963.Otto C. Allkofer, Fortschr. d. Physik 15, 113–196, 1967.S. Hayakawa, Cosmic Ray Physics, Wiley-Interscience, New York, 1969.Albrecht Unsöld, Der Neue Kosmos, Springer, Heidelberg, 1974.Otto C. Allkofer, Introduction to Cosmic Radiation, Thiemig, München, 1975.D. J. Adams, Cosmic X-ray Astronomy, Adam Hilger Ltd., Bristol, 1980.Joseph Silk, The Big Bang – The Creation and Evolution of the Universe, Freeman, NewYork, 1980.Frank H. Shu, The Physical Universe: An Introduction to Astronomy, Univ. Science Books,Mill Valley, California, 1982.Rodney Hillier, Gamma-Ray Astronomy, Oxford Studies in Physics, Clarendon Press, Oxford1984.O. C. Allkofer, P. K. F. Grieder, Cosmic Rays on Earth, Fachinformationszentrum Karlsruhe,1984.Hans Schäfer, Elektromagnetische Strahlung – Informationen aus dem Weltall, Vieweg,Wiesbaden, 1985.W. D. Arnett et al., Supernova 1987A, Ann. Rev. Astron. Astrophysics 27 (1989) 629–700.Thomas K. Gaisser, Cosmic Rays and Particle Physics, Cambridge University Press, Cambridge,1990.John Gribbin, Auf der Suche nach dem Omega-Punkt, Piper, München, 1990.Martin Pohl, Reinhard Schlickeiser, Exoten im Gamma-Licht, Astronomie heute, Bild derWissenschaft 3, S. 92/93, 1993.Neil Gehrels et al., The Compton Gamma-Ray Observatory, Scientific American, 38–45,Dec. 1993.Charles D. Dermer, Reinhard Schlickeiser, Astrophys. J. 416 (1993) 458–484.J. W. Rohlf, Modern Physics from α to Z 0 , J. Wiley & Sons, New York, 1994.T. Kifune, The Prospects for Very High Energy Gamma-Ray Astronomy, Inst. f. CosmicRay Research, Univ. Tokyo, ICRR-326-94-21, 1994.H. V. Klapdor-Kleingrothaus, A. Staudt, Teilchenphysik ohne Beschleuniger, Teubner,Stuttgart, 1995.Kitty Ferguson, Das Universum des Stephen Hawking, Econ, Düsseldorf, 1995.

Further Reading 409Ulf Borgeest, Karl-Jochen Schramm, Bilder von Gravitationslinsen, Sterne und Weltraum,Nr. 1, 1995, S. 24–31.Peter L. Biermann, The Origin of Cosmic Rays, MPI Radioastronomie Bonn, MPIfR 605,astro-ph/9501003, Phys. Rev. D51 (1995) 3450.Daniel Vignaud, Solar and Supernovae Neutrinos, 4th School on Non-Accelerator ParticleAstrophysics, Trieste 1995, DAPNIA/SPP 95-26, Saclay.Thomas K. Gaisser, Francis Halzen, Todor Stanev, Phys. Rep. 258, No. 3, 1995.Lexikon der Astronomie, Spektrum-Verlag, Heidelberg, 1995.S. P. Plunkett et al., Astrophys. Space Sci. 231 (1995) 271; astro-ph/9508083, 1995.Mordehai Milgrom, Vladimir Usov, Astrophys. J. 449 (1995) 37; astro-ph/9505009, 1995.V. Berezinsky, High Energy Neutrino Astronomy, Gran Sasso Lab. INFN, LNGS-95/04,1995.John V. Jelley, Trevor C. Weekes, Ground-Based Gamma-Ray Astronomy, Sky & Telescope,Sept. 95, 20–24, 1995.Volker Schönfelder, Exotische Astronomie mit dem Compton-Observatorium, Physik in unsererZeit 26, 262–271, 1995.A. C. Melissinos, Lecture Notes on Particle Astrophysics, Univ. of Rochester 1995, UR-1841, Sept. 1996.Kitty Ferguson, Gottes Freiheit und die Gesetze der Schöpfung, Econ, Düsseldorf, 1996.Kitty Ferguson, Prisons of Light – Black Holes, Cambridge University Press, Cambridge,1996.Claus Grupen, Particle Detectors, Cambridge University Press, Cambridge, 1996.Michel Spiro, Eric Aubourg, Experimental Particle Astrophysics, Int. Conf. on High EnergyPhysics, Varsovie (Pologne), 1996.Karl Mannheim, Dieter Hartmann, Burkhardt Funk, Astrophys. J. 467 (1996) 532–536;astro-ph/9605108, 1996.G. Battistoni, O. Palamara, Physics and Astrophysics with Multiple Muons, Gran SassoLab. INFN/AE 96/19, 1996.John Ellis, Nucl. Phys. Proc. Suppl. 48 (1996) 522–544, CERN-TH/96-10, astro-ph/9602077, 1996.H. V. Klapdor-Kleingrothaus, K. Zuber, Teilchenastrophysik, Teubner, Stuttgart, 1997, andParticle Astrophysics, Inst. of Physics Publ., 2000.Georg G. Raffelt, Astro-Particle Physics, Europhysics Conference on High-Energy Physics,August 1997, Jerusalem, hep-ph/9712548, 1997.Francis Halzen, The Search for the Source of the Highest Energy Cosmic Rays, Univ. ofMadison Preprint MAD-PH 97-990, astro-ph/9704020, 1997.

410 Further ReadingG. Burdman, F. Halzen, R. Gandhi, Phys. Lett. B417 (1998) 107-113, Univ. of MadisonPreprint MAD-PH 97-1014, hep-ph/9709399, 1997.Paolo Lipari, Cosmology, Particle Physics and High Energy Astrophysics, Conf.Proc.Frontier Objects in Astrophysics and Particle Physics, F. Giovannelli and G. Mannocchi(Eds.), Vol 57, p. 595, 1997.Georg G. Raffelt, Dark Matter: Motivation, Candidates and Searches, Proc. 1997 EuropeanSchool of High Energy Physics, Menstrup, Denmark, hep-ph/9712538, 1997.C. N. de Marzo, Nucl. Phys. Proc. Suppl. 70 (1999) 515–517, physics/9712039, 1997.Hinrich Meyer, Photons from the Universe: New Frontiers in Astronomy, Univ. Wuppertal,Germany, hep-ph/9710362, 1997.Particle Data Group (S. Eidelman et al.), Review of Particle Physics, Phys. Lett. B592(2004) 1–1109.R. K. Bock, A. Vasilescu, The Particle Detector Briefbook, Springer, Berlin, Heidelberg,1998.Jonathan Allday, Quarks, Leptons and the Big Bang, Inst. of Physics Publ., Bristol, 1998.Craig J. Hogan, The Little Book of the Big Bang, Copernicus, Springer, New York, 1998.Lawrence M. Krauss, A New Cosmological Paradigm: The Cosmological Constant andDark Matter, hep-ph/9807376, Case Western Reserve University, Cleveland, 1998.Francis Halzen, Ice Fishing for Neutrinos, AMANDA Homepage,http://amanda.berkeley.edu/www/ice-fishing.html.Jochen Greiner, Gamma-Ray Bursts: Old and New, Astroph. Inst. Potsdam, astro-ph/9802222, 1998.Volker Schönfelder, Gammastrahlung aus dem Kosmos,Phys.Bl.54, 325–330, 1998.K. S. Capelle et al., Astropart. Phys. 8 (1998) 321–328, astro-ph/9801313, 1998.Laura Whitlock, Gamma-Ray Bursts,http://imagine.gsfc.nasa.gov/docs/science/know_l1/burst.html.Glennys R. Farrar, Can Ultra High Energy Cosmic Rays be Evidence for New ParticlePhysics?, Rutgers Univ., USA, astro-ph/9801020, 1998.Super-Kamiokande Collaboration, Phys. Lett. B433 (1998) 9–18, hep-ex/9803006, 1998.M. S. Turner, Publ. Astron. Soc. Pac. 111 (1999) 264–273, astro-ph/9811364, 1998.J. P. Henry, U. G. Briel, H. Böhringer, Die Entwicklung von Galaxienhaufen, Spektrum derWissenschaft, 2/1999, S. 64–69.Guido Drexlin, Neutrino-Oszillationen, Phys. Bl., Heft 2, S. 25–31, 1999.G. Veneziano, Challenging the Big Bang: A Longer History of Time, CERN-Courier, March1999, p. 18.

Further Reading 411M. A. Bucher, D. N. Spergel, Was vor dem Urknall geschah, Spektrum der Wissenschaft,3/1999, S. 55.C. J. Hogan, R. P. Kirshner, N. B. Suntzeff, Die Vermessung der Raumzeit mit Supernovae,Spektrum der Wissenschaft, 3/1999, S. 40.L. M. Krauss, Neuer Auftrieb für ein beschleunigtes Universum, Spektrum der Wissenschaft,3/1999, S. 47.J. Trümper, ROSAT und seine Nachfolger, Phys. Bl. 55/9, S. 45, 1999.Craig J. Hogan, Rev. Mod. Phys. 72 (2000) 1149–1161.H. Blümer, Die höchsten Energien im Universum, Physik in unserer Zeit, S. 234–239, Nov.1999.Brian Greene, “The Elegant Universe”, Vintage 2000, London, 1999.H. Völk, H. Blümer & K.-H. Kampert, H. Krawczynski et. al., Ch. Spiering, M. Simon, J.Jochum & F. von Feilitzsch: Phys. Bl., Schwerpunkt Astroteilchenphysik, März 2000,S. 35–68.Paul J. Steinhardt, “Quintessential Cosmology and Cosmic Acceleration”, Selection ofreview papers under the heading “Dark Energy and the Cosmological Constant”http://pancake.uchicago.edu/~carroll/reviewarticles.html.Peter K. F. Grieder, Cosmic Rays at Earth, Elsevier Science, 2001.S. Hayakawa, Cosmic Ray Physics, Wiley Interscience, N.Y., 1969.K. C. Cole, TheHoleintheUniverse, The Harvest Book/Harcourt Inc., San Diego, 2001.Brian Greene, The Fabric of the Cosmos: Space, Time and the Texture of Reality, Alfred A.Knopf, N.Y., 2004.John A. Bahcall, Neutrino Astronomy, Cambridge University Press, 1990.Rodney Hillier, Gamma Ray Astronomy, Oxford Studies in Physics, 1984.John A. Peacock, Cosmological Physics, Cambridge University Press, 2003.Martin Rees, Just Six Numbers: The Deep Forces That Shape the Universe, Basic Books,New York, 2001.Martin Rees, New Perspectives in Astrophysical Cosmology, Cambridge University Press,2000.S. M. Bilenky, The History of Neutrino Oscillations, hep-ph/0410090, 2004.A. V. Filippenko and A. G. Riess, Phys. Rep. 307 (1999) 31.

413Photo Credits“Equipped with his five senses, man explores the universe aroundhim and calls the adventure science.”Edwin Powell Hubble{1} T. Credner and S. Kohle, University of Bonn, Calar Alto Observatory{2} Prof. Dr. D. Kuhn, University of Innsbruck, Austria{3} Courtesy of The Archives, California Institute of Technology, Photo ID 1.22-5{4} C. Butler and G. Rochester, Manchester University{5} Prof. Dr. A. A. Penzias, Bell Labs, USA{6} D. Malin, Anglo-Australian Observatory{7} Particle Data Group, European Physical Journal C3, (1998), 144;http://pdg.lbl.gov{8} Prof. Dr. Y. Totsuka, Institute for Cosmic Ray Research, Tokyo, Japan{9} T. Kajita, Y. Totsuka, Rev. Mod. Phys. 73 (2001) p. 85{10} Y. Fukuda et al., Phys. Rev. Lett. 81 (1998) 1562–1567{11} Prof. Dr. R. Davis, Brookhaven National Laboratory, USA{12} R. Svoboda and K. Gordan (Louisiana State University),http://antwrp.gsfc.nasa.gov/apod/ap980605.html.Equivalent figure in: A. B. McDonald et al., Rev. Sci. Instrum. 75 (2004) 293–316,astro-ph/0311343{13} AMANDA Collaboration, Christian Spiering, private communication 2004{14} Dr. D. Heck, Dr. J. Knapp, Forschungszentrum Karlsruhe, Germany{15} Cangaroo Collaboration, Woomera, Australia{16} CGRO Project Scientist Dr. Neil Gehrels, Goddard Space Flight Center, USA (1999){17} BATSE-Team, NASA, M. S. Briggs,(http://www.batse.msfc.nasa.gov/batse/grb/skymap/){18} Prof. Dr. J. Trümper, Max-Planck Institute for Extraterrestrial Physics, München,Garching{19} ESA/XMM-Newton,http://xmm.vilspa.esa.es/external/xmm_science/gallery/publiccourtesy by L. Strüder

414 Photo Credits{20} ESA/XMM-Newton/Patrick Henry et al., courtesy by L. Strüder{21} Prof. Dr. M. Simon, University of Siegen, Germany{22} R. Tcaciuc, University of Siegen, 2004{23} ALEPH Collaboration, CERN, courtesy by P. Dornan, Imperial College{24} Frejus-Experiment, France/Italy, courtesy by Chr. Berger, RWTH Aachen{25} Prof. Dr. P. Sokolsky, University of Utah, Salt Lake City, USA{26} LBNL-53543, R. A. Knop et al., Astrophys. J. 598 (2003) 102–137;http://supernova.lbl.gov/, courtesy by Saul Perlmutter{27} Courtesy of the COBE Science Working Group{28} Courtesy of the WMAP Science Team{29} Dr. L. J. King, University of Manchester, UK; NASA; King et al. 1998, MNRAS 295,L41{30} Prof. Dr. G. G. Raffelt, Max-Planck Institute for Physics (Werner-Heisenberg Institute),München, Germany

415Index ∗“A document without an index is like a country without a map.”Mike UnwallaAbell 754, 132absolute brightness, 299absolute zero, 299absorption– coefficient, 46– line, neutrino, 275abundance, 299– cosmic, 307– deuterium, 221, 224– helium-4 ( 4 He), 22, 213,221, 222, 224, 227– light nuclei, 214, 222, 224– lithium-7 ( 7 Li), 223–225– measurement, 225– predicted, 224– primordial, 15– – deuterium, 223– – helium-3 ( 3 He), 224– – helium-4 ( 4 He), 22– – light nuclei, 222accelerated expansion, 252,254, 285acceleration– by sunspot pairs, 65– due to gravity, 404– equation, 183, 252, 299, 400– in gravitational potentials,72– mechanism, 63, 68, 312– of particles in binaries, 71– of particles in pulsars, 69,294– of particles in supernovae,67, 167, 294– of protons, 105– parameter, 186, 281, 295,299– shock, 66, 67, 68, 74, 167accelerator, 299– cosmic, 294– data, number of neutrinofamilies, 227– experiments, 293– physics, 293accretion disk, 72, 130, 155,299acoustic peak, 241, 242, 299acoustic-horizon distance, 241active galactic nuclei, 17, 73,110, 118, 131, 132, 168,294, 300active galaxy, 299activity, potassium-40 ( 40 K),106Adams, D., 265additional Higgs field, 210additional neutrino family, 228adiabatic expansion, 182Advanced X-ray AstrophysicsFacility, see AXAFage– of life, 288– of the universe, 299, 404aggregation, gravitational, 266AGN, see active galactic nucleiair scintillation, 53, 116,159–161air shower– Cherenkov technique, 55,79, 114, 116, 119, 300– development, 145– extensive, 9, 10, 79, 154,156, 311– – correlations, 162, 163– – longitudinal development,158– – measurement technique,158, 160, 161– measurement, 159Air Watch, 160, 161ALEPH, 155α– decay, 300– particle, 10, 78Alpha Magnetic Spectrometer,206Alvarez, L. W., 7AMANDA, 88, 107, 300Anderson, C. D., 8, 9Andromeda Nebula, 271, 300angle– azimuthal, 235– Cabibbo, 31– Cherenkov, 55– horizon distance, 241– mixing, 31, 91– polar, 235Ångström, 300angular– (diameter) distance, 240– power spectrum, cosmicmicrowave background, 236,238, 239, 242, 243, 385, 388– resolution, COBE andWMAP, 238– separation, 236∗ Underlined page numbers refer to main entries. Page numbers in italics apply to the glossary.

416 Indexanisotropy, 85, 86– cosmic microwavebackground, 235, 238, 241– dipole, 235, 236, 310– – vanishing, 237– small-angle, 237ankle, see cosmic raysannihilation, 112, 300– of positrons, 120, 207, 218,219, 326– of WIMPs, 277– phase, 171– p ¯p, 112, 206, 295– radiation, 296anomalies, quantum, 210, 327ANTARES, 277anthropic principle, 290, 300antibaryon, 295, see alsobaryonanticarbon, 84anticoincidence counter, 114anticolour, 26, see also colourantigalaxies, 296antigravitation, 300, see alsogravitationantihelium, 84antilepton, 295, see also leptonantimatter, 300, see alsoantiparticle– primary (cosmic), 84, 295– primordial, 296antineutron, 215antinuclei, 214– primary (cosmic), 84, 206antiparticle, 295, 300– in cosmic rays, 84antiproton, 10, 215, 295– primary, 84, 143– production, 84antiquark, 301, see also quarkantisymmetrization, 391apastron, 301aphelion, 301apogee, 301apparent– light curve, 270, 271– magnitude, 301Armenteros, R., 9ASTRO-E, 132astrobiology, 222, 287, 301astronomical unit (AU), 267,301, 404astronomy– bio-, 288, 303– γ , 55, 108, 314– – horizon, 116– infrared, 109, 317– neutrino, 8, 86, 151, 294– proton, 86– radar (radio), 109, 328– ultraviolet, 109– with gravitational waves,133– X-ray, 56, 109, 123, 336astroparticle– interactions, 50,58– physics, outlook, 293– propagation, 58astrophysics, 1– imprints on data, 197asymmetry– baryon–antibaryon,205, 206, 208, 302– matter–antimatter, 211, 296,321asymptotic freedom, 16, 21,301atmosphere– column density, 142– particle composition, 145– transformation in, 143, 144atmospheric– Cherenkov technique, seeair shower Cherenkovtechnique– cutoff, 6– depth, relation to zenithangle, 143– neutrinos, 88, 294– – deficit, 272– particle flux, 144atom, 233atomic– mass, 301– number, 301attenuation coefficient, mass,113attractor, large, 318AU, see astronomical unitAuger experiment, 83, 158,169Auger, P.-V., 9aurora, 2, 301average– density, 260– energy per particle, 398– temperature, 236AXAF, 131, 301axion, 278, 279, 294, 301– coupling to photons, 279– decay, 279– mass, theoretical, 279azimuthal angle, 235B 0 meson– CP violation, 210– decay, 29, 278background– microwave, cosmic,see cosmic microwavebackground– radiation, see cosmicmicrowave background– radioactivity, 277– X-ray radiation, 132Baksan scintillator experiment,2balloon experiment, 3, 79, 124,143, 284baryogenesis, 206, 211, 215,301baryon, 25, 268, 295, 302– anti-, 295– asymmetry of the universe,205, 206, 208, 302– density, 214, 215, 224, 225– energy density, 225, 243– fraction of the universe, 225– Λ, 9– – decay, 30– number, 30, 295, 302, 394– – conservation, 215, 216– – non-zero, 207

Index 417– – processes violating, 204,208, 210, 215– Ω − ,25– -to-photon ratio, 209, 214,215, 220, 224–227, 233, 243– – measured, 243baryonic matter, 265, 268, 269,275, 281, 283, 295BATSE, 121, 302beam-dump experiment, 302Becquerel, H., 3Bell, J., 8bending of light, see gravitationallensing, microlensingberyllium, 78– -7( 7 Be), 94, 213– -8( 8 Be), 95β decay, 27, 29, 302Bethe– –Bloch formula, 51, 302– –Weizsäcker cycle, 302– –Weizsäcker formula, 302Bethe, H. A., 9Biermann, L., 10Big Bang, 302– echo of the, 12– hot, 171– hypothesis, 12– model, 213, 225, 295– neutrino echo, 104– neutron, 104– nucleosynthesis, 22, 172,209, 213, 214, 217, 222, 226– – era, 215– – measurements, 227– – start, 215– – timeline, 218– proton, 104– quark production, 295– theory, 302– weak interactions, 104Big Crunch, 172, 285, 302Big Rip, 285, 303binary, 71, 72, 105, 303– γ rays, 105– X rays, 130, 155binary star, see binarybinding energy, 303– deuterium, 220– neutral hydrogen, 233– nuclear, 323binding, gravitational, seegravitational bindingbinomial distribution, 383binomial expansion andcoefficients, 381bioastronomy, 288, 303biological evolution, 288, 289bioluminescence, 106Birkhoff’s theorem, 178black hole, 5, 110, 118, 131,168, 269, 303– as WIMP candidate, 278– evaporation, 14– γ burst, 122– mini, 122, 278, 294, 321– particle jet, 119– primordial, see black hole,mini– supermassive, 332– X-ray source, 130blackbody, 303– neutrino, 273– photon, 59, 82, 273– radiation, 82, 126, 229,264, 303, 307, 389, seealso cosmic microwavebackground– – 2.7 Kelvin, 12, 104, 116,266, 283– – inhomogeneities, 283– – measurement, 284– – spectrum, 59– – temperature variations,284– – X rays, 125blazar, 118, 168, 294, 303BL-Lacertae object, 168, 303Bloch, Bethe–, formula, 51,302blue supergiant, 100blueshift, 303–cosmicmicrowavebackground, 237B meson, see B 0 mesonbolometer, 277, 304Boltzmann, see also Stefan–Boltzmann– constant, 304– factor, 213Boomerang experiment, 243,284boron, 78– -8( 8 B), 95Bose–Einstein distribution,195, 304, 389, 393boson, 24, 304, 389, 392, 393,395, 396– fermionic partner, 276– GUT, 203– Higgs, 276, 316– supersymmetry, 276– symmetrization, 391– W ± , 335– X, 249, 336– Y , 249, 336bosonic partner, 276Bothe, W., 6bottom quark, 15, 304bottom–up scenario, 283b quark, see bottom quarkBradt, H. L., 10Brahe, T., 130, 267, 304brane, 33, 304bremsstrahlung, 53, 111, 114,125, 304– muon, 54– photons, 111– probability for, 158– spectrum, 125– thermal, 125brightness– absolute, 299– excursion, 270– – duration, 271– increase, 270– relation to redshift, 186broken supersymmetry, 276brown star (dwarf), 270, 271,304– mass spectrum, 270, 271burning– of carbon, 100– of helium, 66

418 Index– of hydrogen, 94, 222, 223,317– – star, 223– of oxygen, neon, silicon,sulphur, 100Burst and Transient SourceExperiment, see BATSEburst(er), see γ , X-ray, Zburst(er)Butler, C. C., 9Cabibbo– angle, 31– –Kobayashi–Maskawamatrix, 31Cabibbo, N., 31Calabi–Yau space, 33, 304calorimeter– crystal scintillation, 113,114– electromagnetic, 57, 113– sandwich, 58calorimetric measurement, 277CANGAROO, 116carbon, 78, 130– anti-, 84– burning, 100– cycle, 304– Greisen–Zatsepin–Kuzmincutoff, 165– primary energy spectrum, 79– production, 287Carlson, J. F., 9cascade, 330– electromagnetic, 10, 51, 55,114, 115, 143, 145, 150, 157– – longitudinal development,157– – width, 144– electron in, 157– extensive air shower, see airshower, extensive– hadronic, 55, 143, 145, 147,154, 166– – width, 144– high-energy, 165– lateral distribution, 158– lateral spreading, 144– longitudinal development,158– muon shower in the ALEPHexperiment, 155– positron in, 157Casimir effect, 181, 189, 304cataclysmic variables, 130, 305causal– contact, 239, 246, 259– isolation, 250CBI data, 243CBR (cosmic backgroundradiation), see cosmicmicrowave background andblackbody radiationCCD camera, 127, 128, 271– X-ray, 56CDF collaboration, 15CDMS experiment, 278celestial equator, 401Centaurus A, 305center, galactic, 108, 117, 118center-of-mass– energy, 37, 38– system, 44centrifugal force, 267Cepheid variables, 305CERN, 305CGRO, 16, 117, 121, 168, 305Chadwick, J., 8Chamberlain, O., 11Chandra, 305, see also AXAFChandrasekhar– limit, 174– mass, 305Chandrasekhar, S., 16, 131channel-plate multiplier, 129channeltron, 127, 128characteristic X rays, 95charge, 305, 394– colour, 306– conjugation, 29, 278, 295,305– conservation, 305– magnetic, 249– ratio, of muons, 148charge-coupled device, seeCCD cameracharged higgsino, 276chargino, 276charm quark, 14, 305chemical composition– of cosmic rays, see cosmicrays– ofthesolarsystem,see solarsystemchemical potential, 193, 305,389, 395, 397– high temperatures, 397Cherenkov– angle, 55– cone, 115– counter, see Cherenkovdetector– detector, 106, 116– – ice, 106– – water, 56, 96, 106, 158– effect, 89, 306– light, see Cherenkovradiation– radiation, 49, 55, 114, 158,159– ring, 56– technique, atmospheric, 55,79, 114, 116, 300– telescope, 16, 119CKM matrix, see Cabibbo–Kobayashi–MaskawamatrixClay, J., 6closed universe, 184, 306cloud– chamber,4,8– gas– – cold, 269– – hot, 269– – ultracold, 272– magnetic, 63, 68, 69cluster, see also galaxy cluster– size, 284– stellar, 131, 331– super-, 131, 265, 282, 283– – local, 162, 319– temperature, 284– Virgo, 169, 335

Index 419CMB, see cosmic microwavebackgroundCOBE satellite, 12, 232,235–238, 266, 283, 306cold dark matter, 278, 279,282, 283, 306cold gas clouds, 269cold particle, 282collapse, gravitational, 66, 69,70, 314collider, 306collision, see scatteringcollisions, Hoffmann, 6colour, 25, 26, 306– charge, 306– -neutral hadrons, 205column density, atmosphere,142comet tail, 10compactification, 306COMPTEL, 306Compton– effect, 56, 306– Gamma-Ray Observatory,see CGRO– scattering, 113– – inverse, 111, 125, 317– telescope, 57Compton, A. H., 7, 111confidence interval, 385confinement, 30, 306conservation– law, particle physics, 30– of baryon number, 215, 216– of charge, 305– of energy, 178– of entropy, 219– of helicity, 28, 42– of mass, 280conserved quantity, 307, 394constant, dynamical, 281constellation, 307constraint, 394contraction, Lorentz, 319contrast, density, 261conversion electron, 307coordinates– equatorial, 401– galactic, 401, 402– spherical, 235Copernicus, N., 266corona, 307COS-B, 307cosmic abundance, 307cosmic accelerator, 294cosmic antimatter, 295Cosmic Background Explorer,see COBEcosmic background radiation,307, see cosmic microwavebackground and blackbodyradiationcosmic mass density, 269cosmic microwave background,172, 205, 229, 245,295– angular power spectrum,236, 238, 239, 242, 243,385, 388– anisotropy, 235, 241– blueshift, 237– discovery, 231– formation, 233– measurement, 232– – angular resolution, 238– prediction and observation,232– properties, 231– redshift, 237– spectrum, 232– temperature, 225, 233–235– – anisotropy, 238– – isotropy, 259– – measurement, 237, 238– – variations, 225, 261, 284,388cosmic radiation, see cosmicrayscosmic rays, 277, 287, 293,307– acceleration model, 63– ankle of primary spectrum,80, 82– antinuclei, 206– antiparticle, 84– at sea level, 147– chemical composition, 77,78, 84, 163– elemental abundance, 78– energy density, 71– energy spectrum, 73, 74, 79,80– extragalactic, 82– high-energy, 169– highest-energy, 163, 294– intensity, 146– knee of primary spectrum,80, 81, 167– neutrino flux, 104– origin, 85, 104, 110, 167,294– – extragalactic, 118, 132–– γ , point-like, 117– primary, 10, 77, 141, 142,326– – antimatter, 84– – antinuclei, 84– – antiproton, 84, 143– – charged component, 78– – momentum spectrum, 143–– powerlaw,75– – transformation in theatmosphere, 144– primordial, 77– propagation in theatmosphere, 142– secondary, 141, 329– soft component, 144– source, see cosmic rays,origin– toe of primary spectrum, 82– underground, 151cosmic strings, 169, 307cosmic textures, 307cosmoarcheology, 197, 307cosmogony, 307cosmographic map, 238, 307cosmological– constant, 180, 187, 198, 251,279, 280, 283–285, 289,295, 308, 400– limits on neutrino masses,273, 274– neutrinos, 104

420 Index– parameters, 243– – determination, 238– principle, 175, 308– redshift, 308cosmology, 171, 308– equations, 213, 214cosmoparticle physics, 204,290, 308cosmos, see universecounter– anticoincidence, 114– Cherenkov, see Cherenkovdetector– Geiger, 124– proportional, 127, 128– semiconductor, 56, 113, 127– silicon semiconductor, 128Cowan, C. L., 8CP– invariance, 278, 308– problem, strong, 279– symmetry, see CP invariance– transformation, 29– violation, 278, 295CPT– invariance, 29, 308– symmetry, see CPTinvariancec quark, see charm quarkCrab Nebula, 2, 11, 16, 108,117, 119, 308Crab Pulsar, 130critical– density, 172, 185, 225, 246,247, 256, 263, 266, 269,273, 279, 308– temperature, 249, 250cross section, 35, 46, 49, 308– differential, 47– nuclear interaction, 49, 213– reaction of protons andneutrons, 216cryogenic detection techniques,58‘cryptons’, 169crystal– calorimeter, 113, 114– detectors, 57– spectrometer, 127– ultrapure, 277current, charged and neutral,see interaction, weakcurvature, 308– early universe, 284– negative, 241– parameter, 178, 199curve, rotational, see rotationalcurvecyclotron mechanism, 64, 308Cygnus– X1, 14, 309– X3, 11, 108, 117, 155, 309DAMA collaboration, 278dark energy, 17, 32, 186, 265,280, 283, 295, 309– present-day, 262dark matter, 17, 172, 265, 266,272, 280–282, 285, 294, 309– circumstantial evidence, 267– cold, 278, 279, 282, 283,306– dark stars, 269– hot, 273, 282, 283, 316– LSP, 276– neutrinos, 272, 273– non-baryonic, 266– resumé, 283– WIMPs, 275Davis experiment, 13, 96, 309Davis, R., 13, 96de Broglie wavelength, 309decay, 309– α, 300– axion, 279– B 0 meson, 29, 278– β, 27, 29, 302– D meson, 149– kaon, 29, 144, 150, 278– Λ baryon, 30– muon, 28, 43– – electron spectrum, 44– neutrino, 103– neutron, 27, 28, 217, 218,219, 323– nucleus, γ rays, 112– pion, 28, 42, 88, 111, 112,144, 147, 150– three-body, 43– tritium, 102– two-body, 41– width, Z, 22, 275deceleration parameter, 186declination, 309, 401decoupling– distance, 235– of neutrinos, 218– photons from matter, 233,234– temperature, 222, 233, 234– time, 234deflection of light, seegravitational lensing,microlensingdeflector, 270degeneracy pressure, 309degenerate matter, 309degrees of freedom, 309– effective number, 196, 214,218, 219, 226– internal, 194, 389, 396deleptonization, 100, 309∆ resonance, 59, 82, 164density– average, 260– baryon, 214, 215, 224, 225– column, atmosphere, 142– contrast, 261– critical, see critical density– effect, 53– electron flux, 141– energy, see energy density– fluctuations, 237, 240, 256,260, 309– galaxy, 261– mass, see mass density– matter, see matter density– monopole, 259– nucleon, 214– – BBN phase, 231– number, see number density– parameter, see Ω parameter– photon, 225– position dependent, 260

Index 421– proton flux, 141departure from thermalequilibrium, 202, 210depth–intensity relation, 152,153detection– of electrons, 114– of neutrinos, 49, 51, 56, 106,107– of particles, 49,51– of photons, 50, 56, 128– of radiation, 49– of WIMPS, 277– of X rays, 56, 126, 127, 128detector– characteristic features, 60– Cherenkov, see Cherenkovdetector– cryogenic, 58– crystal,see crystal detector– neutrino, 277– particle, see particle detector– sampling, 159– semiconductor, seesemiconductor detector– semiconductor pixel, seesemiconductor pixel detector– silicon semiconductor,see silicon semiconductordetector– (water) Cherenkov, see(water) Cherenkov detectordeuterium, 94, 213, 220, 222,309– abundance, 221, 224– – primordial, 223– binding energy, 220– fraction, 224– fusion, 231– Lyman-α line, 224– production, 219–221, 223,288– stability, 288– synthesis, 218– -to-hydrogen ratio, 224development of life, 287deviation, standard, 384differential cross section, 47differential equations, 386differential gravitation, 310dilution– of matter density, 280– of monopoles, 260dimension– extended, 311–extra,32– number, 289dipole– anisotropy, 235, 236, 310– – vanishing, 237– moment, neutron, 279– term, 236Dirac, Fermi–, distribution, seeFermi–Dirac distributionDirac, P. A. M., 8directionality, in neutrinodetection, 96disk, 310– accretion, 72, 130, 155, 299– galactic, 118, 267, 271distance– angular (diameter), 240– from the Sun, 402– horizon, 234, 241, 242– – acoustic, 241– Hubble, 259, 404– ladder, 310– luminosity, 173, 187, 320– of decoupling, 235– proper, 240distortions of images, 272distribution– binomial, 383– Bose–Einstein, 195, 304,389, 393– equilibrium, 390, 392, 399– Fermi–Dirac, 195, 312, 389,393– Gaussian, 384– Landau, 53, 384– momentum, 390, 391, 396– of microstates, 390– Planck, 59, 325– Poisson, 384– probability, 383– thermal, of velocities, 202– zenith-angle, muon, 148,153, 154D meson, decay and lifetime,149domain wall, 169, 310DONUT experiment, 13Doppler effect, 310, see alsoblueshift and redshiftdouble– image, 5, 270– pulsar, 310– ratio, 89– star,see binarydoublet (isospin), 29down quark, 27, 310d quark, see down quarkdust, galactic, 269dwarf– brown, 270, 304– star, 269– white, 6, 130, 335dynamical constant, 281dynamics– of galactic clusters, 269, 283– of galaxies, 17, 269, 272,281, 283– of the universe, 131, 266Earl, J. A., 11early universe, see universe,earlyEAS, see air shower, extensiveeast–west effect, 7, 310ecliptic, 401Eddington, A. S., 5effect– Casimir, 181, 189, 304– Cherenkov, 55, 89, 306– Compton, 56, 306– density, 53– Doppler, 310, see alsoblueshift and redshift– east–west, 7, 310– latitude, 6, 318– micro-gravitational-lens, seemicrolensing– MSW, 98, 322– photoelectric, 49, 56, 113,128, 325

422 Indexeffective number of degreesof freedom, 196, 214, 218,219, 226efficiency of energy generationin stars, 289EGRET, 310eigenstate– mass, 31, 90– weak interaction, 31, 90Einstein– Bose–, distribution, seeBose–Einstein distribution– Observatory, 129– ring, 270, 271Einstein, A., 5, 21, 280, 295electromagnetic– calorimeter, 57, 113– cascade, see cascade,electromagnetic– interaction, see interaction– radiation, 160, see also γand X rays, photonelectron, 21, 111, 310– annihilation with positrons,see annihilation of positrons– atmospheric flux, 144– β decay, 29– bremsstrahlung, 53, 125– capture, 94, 95, 310– collision with positron, 226– conversion, 307– detection, 114– evidence, 89– flux density, 141– in rock, 154– in shower, 157– interaction, 218, 219– – with neutrino, 27– knock-on, 150– mass, temperature, 218– neutrino, 150– – energy spectrum, 150– number, 311– – density, 215– –positron pair production,see pair production– primary, 10, 84– radiation length in air, 143– reaction, scattering, seeelectron interaction– secondary, 144, 150– solar wind, 141– spectrum from muon decay,44– synchrotron radiation, 125– volt, 311, 403electroweak– force, 293– interaction, see interaction– scale, 204, 293– theory, 13, 311– unification, 171element– abundance, in cosmic rays,78– formation, 77, 120– light, abundance, 214– – primordial, 222– production, in supernovae,66, 112, 120– radioactive, primordial, 287– synthesis, see nucleosynthesiselementary particles, seeparticle– Standard Model, seeStandard Modelelliptical galaxy, 311Elster, J., 3endotherm fusion, 66energy– average per particle, 398– binding, 303– – deuterium, 220– – neutral hydrogen, 233– – nuclear, 323– center-of-mass, 37, 38– conservation, 178– consumption, world, 102–dark,see dark energy– density, 194, 214, 262, 311,389, 396, 397, 400– – baryon, 225, 243– – dependence on scalefactor, 230– – gravitational wave, 263– – matter, 243– – matter domination, 230– – monopole, 250– – of cosmic rays, 71– – perturbations, 266– – photon, 250– – photon contribution, 232– – present-day, 230, 256– – scalar field, 254– – spatial variations, 256– – total, of the universe, 195,214, 219, 243, 245, 246– – vacuum, see vacuumenergy density– equipartition of, 390– Fermi, 274, 312– generation in stars, 10– – efficiency, 289– high– – cascade, 165– – events, 163, 166, 168, 169– – muon, 155– – neutrino, 82, 95, 105, 166,168– – photon, 82, 165– highest– – cosmic rays, 163, 294– – event, nucleus, 163–– events,82– – observed, 293– – proton, 163– loss– – of charged particles, 51, 53– – of muons, 54, 151, 152– measurement, 159– –momentum tensor, 180– muon– – definition, 54– – determination, 106– Planck, 192, 294– potential, 280– –range relation, 152– recoil, 277– Rydberg, 403– spectrum– – of cosmic rays, 73, 74, 79,80– – of electron neutrinos, 150

Index 423– – of muon neutrinos, 150– – of muons, 146, 147, 154– – of primary nuclei, 79– – of protons, 146– threshold, 37– total, 252, 394, 399– – relativistic, 36– vacuum, see vacuum energy– zero-point, 253, 336entropy, 399– conservation, 219equality of matter and radiation,see matter–radiationequalityequation– acceleration, 183, 252, 299,400– differential, 386– fluid, 182, 200, 312, 400– Friedmann, 177, 178, 179,183, 198, 247, 251, 313– – matter dominance, 230– – solving, 199– of cosmology, 213, 214– of state, 182, 198, 252, 398–– parameter, see wparameter– rate, 222equator, celestial, 401equatorial coordinates, 401equilibrium– distribution, 390, 392, 399– number density, 217– thermal, 214–217, 222, 394– – departure from, 202, 210equipartition of energy, 390EROS, 17, 271, 311error (propagation), 384escape velocity, 274, 311evaporation of neutrons, 323even–even nuclei, 79even–odd nuclei, 79event– high-energy, 163, 166, 168,169– highest-energy, 82, 294– – nucleus, 163– horizon, 14, 118, 192, 311– rate, 49evolution– biological, 288, 289– stellar, 287exchange particle, 39excitation, 51– of fields, 253expansion, see also scale factor– accelerated, 252, 254, 285– adiabatic, 182– binomial, 381– discovery, 6– exponential, 252, 254, 266,281, see also inflation– Hubble, 12, 173, 240– model, 311– multipole, 236, 387– non-relativistic, 184– quasi-exponential, 263– rate, 176, 201, 214,216–219, 226, 257, 258, 281– – constant, 252Expérience pour la Recherched’Objets Sombres, seeEROSexperiment– accelerator, 293– ALEPH, 155– AMANDA, 88, 107, 300– ANTARES, 277– Auger, 83, 158, 169– Baksan scintillator, 2– balloon, 3, 79, 124, 143, 284– beam-dump, 302– Boomerang, 243, 284– CDMS, 278– Davis, 13, 96, 309– DONUT, 13–fixed-target,312– Fly’s Eye, 160, 166, 313– Frejus, 156– GALLEX, 13, 96, 313– HERA, 166– IMB, 103– Kamiokande, 13, 96, 101,103, 318– Maxima, 243, 284– radiochemical, 95– SAGE, 13, 96, 329– satellite, 79, 124– SNO, 18, 283– Super-Kamiokande, 17, 88,96, 283, 332– underground, 161Explorer I satellite, 7explosion, thermonuclear, 130exponential expansion, 252,254, 266, 281, see alsoinflationexponential function, 382extended dimension, 311extensive air shower, see airshower, extensiveextra dimensions, 32extragalactic– matter sample, 77– neutrinos, 104– radiation, 82, 311– sources, 118, 132extrasolar– planets, 18– X rays, 124extraterrestrial intelligence, 18false vacuum, 255, 279, 311families, 311– neutrino, see neutrino,families and generation– of particles, 21Fermi– acceleration mechanism, 10,68, 312– constant, 216– –Dirac distribution, 195,312, 389, 393– energy, 274, 312Fermi, E., 10fermion, 23, 24, 312, 389, 393,395, 396– antisymmetrization, 391– bosonic partner, 276– gas, relativistic, 274– generation, 15– relativistic, 215– supersymmetry, 276fermionic partner, 276

424 IndexFeynman diagram, 26, 98, 216,276, 312Feynman graph, see Feynmandiagramfield– excitation, quantum, 253– Higgs, see Higgs field– inflaton, 255– magnetic, see magnetic field– scalar, see scalar field– theory– – interaction, 255– – potential, 254– – quantum, 253, 327fine-tuned parameters, 222,289fission, nuclear, 312, 323fixed-target experiment, 312flare, 312– solar, 330flash, X-ray, 130flat geometry, 241flat rotational curve, 267, 268flat universe, see universe, flatflatness problem, 246, 248,256, 312flavour, 23, 25, 312– neutrino, 91, 98, 101– quark, 28, 30– – violation, 31fluctuations– density, 237, 240, 256, 260,309– potential, 284– quantum, 181, 256, 266,281, 282fluid equation, 182, 200, 312,400fluorescence technique, 79,159flux– atmospheric, of particles,144– density, electron and proton,141– muon, at sea level, 147– neutrino, cosmic, 104– of gamma rays, 207– WIMP, 278Fly’s Eye, 160, 166, 313– technique, 53, 159, 161Forbush, S. E., 10force– centrifugal, 267– electroweak, 293– gravitational, 267–Lorentz,6formation– neutron star, 66, 100–ofcosmicmicrowavebackground, 233– of elements, 77, 120– of galaxies, 13, 237, 281,282, 284– of structure, universe, 261,266four– -momentum, 40, 313– – vector, 39– -vector, 39, 313Fourier series, coefficients, 261fragmentation, 78, 313frame, rest, local, 237free parameters, StandardModel, 287freedom, asymptotic, 16, 21,301freeze-out temperature,neutron-to-proton ratio,217–219, 226, 228Freier, P., 10Frejus experiment, 156Friedmann– equation, 177, 178, 179,183, 198, 247, 251, 313– – matter dominance, 230– – solving, 199– –Lemaître universes, 313full width at half maximum(FWHM), 385functions– orthogonal, 387– spherical harmonics, 385,386– trigonometric and exponential,382fundamental particle, 313fusion, 10, 66, 79, 94, 130,313, 323– deuterium, 231– endotherm, 66– of hydrogen, 288– processes, solar, 94, 95– successive, 66galactic– center, 108, 117, 118– coordinates, 401, 402– disk, 118, 267, 271– dust, 269– γ rays, 117– gas, 267– halo, 267, see halo, galactic– latitude, 402– longitude, 402– lower limits on neutrinomasses, 274– magnetic field, 81, 313– neutrinos, 104– nucleus, 267– – active, see active galacticnucleigalaxy, 129, 131, 176, 177,265, 269– 3C134, 162– 3C273, 108– active, 299– Andromeda, 271, 300– at high redshift, 132– cluster, 129, 131, 265, 282,306, 313– – Abell 754, 132– – dynamics, 269, 283– – X rays from, 130– density, 261– dynamics, 17, 269, 272, 281,283– elliptical, 311– formation, 13, 237, 281,282, 284– groups, 131– M81, 130– M87, 320– Markarian, 17, 108, 119,165, 321

Index 425– mass density, 267, 268– metal-poor, 222– NGC 6503, 267– proto-, 282– radio, 328– rotational curve, 267– Seyfert, 329– starburst,331– super-, 86, 168Galilei transformation, 313GALLEX, 13, 96, 313gallium– arsenide, 128– experiment, 96γ astronomy, 108, 314– horizon, 116– inTeV,55γ burster, 15, 120, 162, 314– black hole as, 122– discovery, 120– neutron stars as, 122– quasi-periodic, 123γ rays, 109, 314– flux, 207– from pion decay, 111– galactic, 117– line emission, 112, 120– measurement, 113– nucleus decay, 112– of binaries, 105– point source, 117– production mechanisms, 110– supernova, 110, 118γ satellite, 11γ -ray burster, see γ bursterγγ interaction, 82, 86, 106,116Gamow, G., 12gas– cloud– – cold and hot, 269– – ultracold, 272– galactic, 267– ideal, law, 400– proportional chamber, 128– relativistic fermions, 274Gassendi, P., 2gaugino, 276Gaussian distribution, 384Geiger counter, 124Geitel, H., 3Gell-Mann, M., 13, 14Geminga, 117general relativity, 178, 314– principle, 314– theory, 5, 280, 314generation, 15, 22, 23, 28, 314– fourth, lepton, 275– neutrino, see neutrino,families and generationgeomagnetic cutoff, 6geometry– flat, 241– of the universe, 241germanium, 128Giacconi, R., 11giant, red, 6, 174, 328Glashow, S., 13GLAST, 314gluino, 276, 314gluon, 26, 276, 314– residual interaction, 26Golden, R. L., 11graceful exit problem, 255Gran Sasso laboratory, 278grand unification, 314, see alsoGrand Unified TheoryGrand Unified Theory, 32, 197,204, 254, 293, 296, 315– bosons, 203– creation, 293– epoch, 315– particles, 169– scale, 204, 250, 293, 315gravitation, 289, see alsointeraction, gravitative– differential, 310– repulsive, 284, 289– singularity, 14– super-, 32, 33gravitational– aggregation, 266– binding, 282– – of neutrinos, 273– – of WIMP, 277, 278– collapse, 66, 69, 70, 314– constant, 403– force, 267– instabilities, 241, 266, 282,314– lens,315– lensing, 5, 270–272, see alsomicrolensing– – weak, 272– redshift, 315– wave, 14, 263, 315– – astronomy, 133– – background, 264– – energy density, 263gravitino, 276, 315graviton, 24, 276, 315gravity, 315– acceleration due to, 404– Newtonian, 178– particle acceleration, 72– quantum, 171, 191, 293, 327Great Wall, 315Greisen–Zatsepin–Kuzmincutoff, 59, 87, 161, 164, 315– carbon, 165Gross, D. J., 16group, local, 319GUT, see Grand UnifiedTheorygyroradius, 315hadron, 25, 315– colour-neutral, 205– in rock, 154– interaction length in air, 143– interaction, inelasticity, 157hadronic– cascade, see cascade,hadronic– matter, 21Halley, E., 2halo, galactic, 267, 269, 271,313– mass content, 271– metal-poor star, 223– neutrino, 275– object, non-luminous, 270– WIMP, 278Harari, H., 86

426 IndexHarrison–Zel’dovich spectrum,261Hawking– radiation, 14, 316– temperature, 316Hawking, S., 14, 293HEAO, 129, 316heat, specific, 277heavy neutrino, 275Heisenberg’s uncertaintyrelation, see uncertaintyrelationHeisenberg, W., 8, 22Heitler, W., 9helicity, 28, 29, 97– conservation, 28, 42– suppression, 28, 150heliocentric, 316heliosphere, 316helium, 79, 125, 130, 220, 222,270– anti-, 84– burning, 66– from fusion reaction, 94– primary energy spectrum, 79– production, 218helium-3 ( 3 He), 94, 213– primordial abundance, 224helium-4 ( 4 He), 94, 213– abundance, 213, 221, 222,224, 225, 227– – primordial, 22– mass fraction, 221, 226, 227– – predicted, 226– – primordial, 223– reaction chain, 220– synthesis, 220HERA, 166Hertzsprung–Russell diagram,5, 316HESS telescope, 119Hess, V. F., 3Hewish, A., 8hidden quantum number, 25hierarchy of masses, 98, 273,274Higgs– boson, 276, 316– field, 204, 249, 254, 259,316– – additional, 210– mechanism, 31– potential, 254higgsino, charged and neutral,276High Energy AstronomyObservatory, see HEAOHigh Resolution Imager, seeHRIhigh-energy– events, 163, 166, 168, 169– muon, 155– neutrino, 82, 95, 105, 166,168– photon, 82, 165highest observed energy, 293highest-energy– cosmic rays, 163– events, 82, 294– proton, 163Hoffmann’s collisions, 6Hoffmann, G., 6holographic universe, 32homogeneity, universe, 175– early, 266horizon, 316– distance, 234, 241, 242– event, 14, 118, 192, 311– γ astronomy, 116– particle, 239, 240, 250, 258,324– problem, 246, 258, 316Hot Big Bang, 171hot dark matter, 273, 282, 283,316hot gas clouds, 269hot particle, 282h parameter, 175HRI, 129, 130, 317Hubble– constant, 175, 186, 187, 225,243, 263, 317, 404– diagram, 187– distance, 259, 404– expansion, 12, 173, 240– law, 175, 317– parameter, 176, 186, 187– telescope, 16, 281, 282, 317Hubble, E. P., 6Hulse, R. A., 14hydrogen, 125, 270– binding energy, 233– burning, 94, 222, 223, 317– – star, 223– fusion, 288– Lyman-α line, 224– neutral, 233– primary energy spectrum, 79hypernova, 122hyperon, 9ice Cherenkov detector, 106IceCube, 277ideal gas law, 400identification of particles, 147image– distortions, 272– double, 5, 270– sensor, silicon, 128IMB, 2, 101, 103, 317impact parameter, 270implosion, 317imprints on astrophysical data,197index of refraction– in the keV range, 126– of air, 114index, scalar spectral, 261inelasticity, 157inertia, 317inflation, 171, 205, 245, 251,253, 254, 256, 259, 260,262, 263, 266, 281, 317– causal contact, 259– end of, 255–257– graceful exit problem, 255– gravitational waves, 263– models, 261– new, 255, 256– number of e foldings, 258– Ω parameter, 257– period of, 258– start of, 257, 258– time, 263inflationary epoch, 172

Index 427inflaton (field), 255infrared– astronomy, 109, 317– photons, 116– radiation, 106– slavery, 21, 317inhomogeneities– blackbody radiation, 283– in the universe, 265, 266initial conditions of theuniverse, 260instabilities, gravitational, 241,266, 282, 314integrals– indefinite, 382– specific, 383intensity attenuation, photon,113intensity of cosmic-rayparticles, 146, see alsodepth–intensity relationinteraction, 24, 255, 317– astroparticle, 50, 58– electromagnetic, 24, 293,310– – of nuclei, 50– – parity, 29– electroweak, 13, 24, 216,293– γγ, 82, 86, 106, 116– gravitational, 24, 275, 315,see also gravitation– – gravitino, 276– – graviton, 24, 276, 315– – weakness, 32– hadron, inelasticity, 157– kinematics, 44– length, 46, 143, 317– mechanism, 51– neutrino–air, 51– neutrino–neutron, 27– neutrino–nucleon, 51, 89– nuclear, 54– – cross section, 49, 213– probability, 46, 49– processes, 26, 51– properties, 25– proton–air, 50, 88– rate, 46– residual, 26, 328– strong, 24, 332– – asymptotic freedom, 16,21, 301– – carrier, 15–– CP violation, 278– – GUT, 293– – of nuclei, 50– – parity, 29– – strangeness, 29, 332– superweak, 167– unified, 32– united, 31– weak, 24, 27, 275, 293, 295,335– – Big Bang, 104– – carrier, 16– – charge conjugation, 29,278– – charged current, 27, 305–– CP violation, 278– – eigenstate, 31, 90– – in supernovae, 101– – neutral current, 27, 322– – of nuclei, 50– – parity, 29, 278– – strangeness, 29– – strength, 216– WIMP, 277interchange particles, 27internal degrees of freedom,194, 389, 396International Space Station,207interstellar medium, 317interval, confidence, 385invariance– CP, 278, 308– CPT, 29, 308– time reversal, 334inverse Compton scattering,111, 125, 317inverse reaction of protons andneutrons, 217ionization, 51, 114, 277, 318– minimum, 53iron, 78, 79, 100, 132, 151, 163– group, 66– line, 132– primary energy spectrum, 79– production, 287– solid-, momentumspectrometer, 148Irvine–Michigan–Brookhaven,see IMBisomer, 318isospin, 29, 30, 318– doublet, triplet, multiplet, 29isotope, 318isotopic shift of wavelengths,224isotropic radiation, 235isotropy– temperature, cosmicmicrowave background, 259– universe, 175Jeans mass, 318Jesse, W. P., 10jet, 119, 318Johnson, T. H., 7K ± ,K 0 , see kaonKamiokande, 13, 96, 101,103, 318, see alsoSuper-KamiokandeKant, I., 3, 281kaon, 318– as secondary particle, 143– at sea level, 150– CP violation, 210– decay, 29, 144, 150, 278– discovery, 9– lifetime, 144– mass, 42– strangeness, 29Kepler, J., 100, 267Keplerian motion, 267kinematics– interaction, 44– relativistic, 35K meson, see kaonknee, see cosmic raysknock-on electron, 150Kobayashi, M., 31Kohlhörster, W., 4, 6

428 IndexKoshiba, M., 13kpc, 318, see also parseckrypton, 56laboratory system, 37, 44Lagrange multipliers, 394Lambda baryon, see Λ baryonΛ baryon, 9– decay, 30Landau distribution, 53, 384Landau, L. D., 8Laplace series, 235, 387, 388large attractor, 318Large Electron–PositronCollider, see LEPLarge Hadron Collider, seeLHCLarge Magellanic Cloud, 86,100, 120, 270, 319large-scale structure of theuniverse, 265, 272– development, 266Las Campanas observatory,100last scattering– surface of, 235, 240– time of, 239latitude– effect, 6, 318– galactic, 402Lattes, C. M. G., 9law– conservation, particlephysics, 30– first, of thermodynamics,398– Hubble, 175, 317– ideal gas, 400– radiation, Planck, 126, 325– Stefan–Boltzmann, 126,194, 331Lederman, L. M., 13, 15Lemaître, Friedmann–,universes, 313length– of interaction, see interactionlength– of radiation, see radiationlength– Planck, 192, 325, 403lens, gravitational, 315, seealso gravitational lensing,microlensingLEP, 15, 22, 227, 275, 319leptogenesis, 206, 319lepton, 21, 30, 214, 295, 319– anti-, 295– fourth generation, 275– mass hierarchy, 98, 273, 274– number, 28, 30, 319, 394–– tau,333– supersymmetry, 276LHC, 276, 293, 319life– age of, 288– development of, 287lifetime– D meson, 149– kaon, 144– muon, 88– neutrino, 16, 103– neutron, 217, 220, 221, 288– pion, 144light– bending and deflection,see gravitational lensing,microlensing– Cherenkov, see radiation,Cherenkov– curve, apparent, 270, 271– northern, 2, 301– scintillation, 160, 161– velocity of, 35, 335, 403– -year, 319, 404light elements, see light nucleilight nuclei, 94– abundances, 214– – predicted, 224– – primordial, 222– mass fractions, 223– nucleosynthesis, 220lightest particle, supersymmetric,see supersymmetry,LSPlighthouse model, 319line– absorption, neutrino, 275– emission, γ rays, 112, 120– hydrogen, Lyman-α, 224– iron, X rays, 132– of sight, 270lithium, 78– -6( 6 Li), 213– -7( 7 Li), 94, 213– – abundance, 223–225– -to-hydrogen ratio, 223LMC, see Large MagellanicCloudlocal– group, 319– supercluster, 162, 319logarithm, natural, 382longitude, galactic, 402Lorentz– contraction, 319– factor, 35– force, 6– transformation, 44, 45, 319low temperatures, 277lower mass limit, galactic, forneutrinos, 274LSP, see supersymmetry, LSPluminosity, 320– distance, 173, 187, 320luminous matter and stars, 269lunar X rays, 124, 132Lyman-α line, hydrogen, 224MACHO, 17, 270–272, 280,320Magellanic Cloud, 123, 320,see also Large MagellanicCloudmagic numbers, 79, 320MAGIC telescope, 119magnetar, 123, 320magnetic– charge, 249– cloud, 63, 68, 69– field– – galactic, 81, 313– – planetary, 287– – solar, 141– moment, neutrino, 97– monopole, see monopole,magnetic

Index 429magnitude, 187, 320– apparent, 301magnitudo, see magnitudemain-sequence star, 320Mairan, J.-J. d’Ortous de, 3map, cosmographic, 238, 307Markarian galaxies, see galaxy,MarkarianMaskawa, T., 31mass, 321– atomic, 301– attenuation coefficient, 113– axion, theoretical, 279– Chandrasekhar, 305– conserved, 280– density– – cosmic, 269– – galaxy, 267, 268– – neutrino, 274– – universe, 269– difference, of neutron andproton, 215, 217– eigenstate, 31, 90– electron, temperature, 218– fraction– – helium-4, see helium-4mass fraction– – light nuclei, 223– galactic halo, content, 271– hierarchy, 98, 273, 274– Jeans, 318– kaon, 42– missing, 321– muon, 42– neutrino, see neutrino mass– nucleon, 214, 221– number, 321– pion, 39, 42– Planck, 192, 325, 403– quark, 24, 288– rest, 35, 39, 328– shell, 39– spectrum, of brown stars,270, 271– WIMP, 278Massive Compact Halo Object,see MACHOmassive quark stars, 271matter– anti-, see antimatter– asymmetry with antimatter,211, 296, 321– baryonic, 265, 268, 269,275, 281, 283, 295–dark,see dark matter– degenerate, 309– density, 243, 280– – dilution, 280– – total, universe, 275– dominance, 229, 230, 239,240, 247, 259, 260, 296–– Ω parameter, 257– energy density, 243– extragalactic, 77– hadronic, 21– luminous, 269– non-baryonic, 281– non-luminous, 269, 283– non-relativistic, 200– oscillation, 321– photons decoupling from,233, 234– –radiation equality, 229,230, 247, 259, 260, 321– – before, 239– visible, 265, 269Maxima experiment, 243, 284Mayor, M., 18McNaught, R., 100mean free path, 233, 234mean value, 384measurement– abundances, 225– air shower, 159– axion decay, 279– baryon-to-photon ratio, 243– Big Bang nucleosynthesis,227– blackbody radiation, 284– calorimetric, 277–cosmicmicrowavebackground, 232, 237, 238– – angular resolution, 238– energy, 159– nucleus, 50–ofγ rays, 113– power spectrum, 262– proton, 50– scintillation light, 160– technique of extensive airshowers, 158, 160, 161medium, interstellar, 317meson, 25, 321, see also kaon,pion, B 0 meson– charmed, 14, 149, 150metal-poor star and galaxy,222, 223metric tensor, 179metric, Robertson–Walker,179, 329Meyer, P., 11micro-gravitational-lens effect,see microlensingmicrolensing, 17, 270, 271,321microstates– distribution, 390– number, 390, 392, 393– – logarithm, 399– – total, 392microwave background, cosmic,see cosmic microwavebackgroundMikheyev, S. P., 98Milky Way, see galaxymini black hole, see black hole,miniminimum-ionizing particles,53missing mass, 321mixing angle and matrix, 31,91Mk 421, Mk 501, see galaxy,Markarianmodel– cosmology, see StandardCosmological Model– for cosmic-ray acceleration,63– lighthouse, 319– nucleus, shell, 79– of expansion, 311– of inflation, 261– of the universe, 321

430 Index– standard, see StandardModelmodulation, solar, 141momentum– distribution, 390, 391, 396– four-, 40, 313– space, 389–391, 393, 396– – division, 392– spectrometer, solid-iron, 148– spectrum– – muon, 146– – muon, at sea level, 148– – primary cosmic rays, 143– – proton, 146– tensor, energy–, 180– transverse, 144– vector, 391– – four-, 39monopole– magnetic, 169, 245, 294,320– – as topological defect, 249– – density, 259– – dilution, 260– – energy density, 250– – number density, 250, 260– – search, 250– – stability, 249– problem, 248, 251, 259, 321– term, 236Moon in X-ray light, 132Mpc, 322, see also parsecMSW effect, 98, 322M theory, 33, 322Muirhead, H., 9multiplate spark chamber, 113multiple scattering, 144multiplet– isospin, 29– super-, 276multiplicity, 322multipole expansion, 236, 387multiverse, 290, 322multiwire proportionalchamber, 56, 129muon, 9, 21, 322– at sea level, 147– atmospheric flux, 144– bremsstrahlung, 54– charge ratio, 148– decay, 28, 43– – electron spectrum, 44– depth–intensity relation,152, 153– directions, 156– discovery, 9–energy– – definition, 54– – determination, 106– – loss, 54, 151, 152– – spectrum, 146, 147, 154– evidence, 89– flux at sea level, 147– high-energy, 155– in an extensive air shower,158– in rock, 152, 154– inclined horizontal direction,147– lifetime, 88– mass, 42– momentum spectrum, 146– – at sea level, 148– neutrino, 88–90, 150, seealso neutrino– – deficit, 91, 272, 294– – energy spectrum, 150– – mass, 274– number, 322– ratio, proton-to-, 149– shower in the ALEPHexperiment, 155– spectrum at sea level, 147– zenith-angle distribution,148, 153, 154nabla operator, 386NACHO, 272natural logarithm, 382natural radioactivity, 277nebula, see galaxy– planetary, 325Neddermeyer, S., 9negative– curvature, 241– pressure, 186, 198, 252, 284,322, 400neon burning, 100neutral– higgsino, 276– hydrogen, 233neutralino, 172, 276, 322neutrino, 8, 21, 282, 322– absorption line, 275– as (hot) dark matter, 272,273, 282– as fermion gas, 274– astronomy, 8, 86, 151, 294– atmospheric, 88, 272, 294– blackbody, 273– blazar as source, 168– burst of SN 1987A, 100–102– cosmic flux, 104– cosmological, 104– decay, 103– decoupling, 218– deficit, 96– – atmospheric, 272– – muon, 91, 272, 294–– solar,13– detection, 49, 51, 56, 106,107– detectors, 277– distinguishing of ν e /ν µ ,89– dominance, 282– echo of the Big Bang, 104– electron, 150– energy spectrum, 150– evidence, 88– extragalactic, 104–– burst,16– families, 214, see alsoneutrino, generation– – additional, 228– – number of, 226, 227– – number of, acceleratordata, 227– – number of, light,equivalent, 227– flavour, see neutrino,generation– galactic, 104– – halo, 275– generation, 13, 15, 22, 91,98, 101

Index 431– gravitational binding, 273– heavy, 275– high-energy, 82, 95, 105,166, 168– in rock, 154– interaction, 218– – with air, 51– – with electron, 27– – with neutron, 27– – with nucleon, 51, 89– lifetime, 16, 103– light, relativistic, 283– magnetic moment, 97– mass, 16, 23, 97, 102, 272,275, 294– – density, 274– – muon, 274– – non-zero, 17– – tau, 274– mass limit– – cosmological, 273, 274– – direct, 272– – fourth generation, 275– – lower, 273– – lower, galactic, 274– – tau, 273– massive, 17– mixing, see neutrino,oscillation– muon, 88–90, 150– number, 272– – density, 215, 216, 218, 274– – density, primordial, 272– oscillation, 13, 17, 24, 90,272, 274, 293, 294, 323– – in matter, 97– – mixing matrix, 91– – vacuum, 323– postulate, 8– range, 58– reaction, scattering, seeneutrino interaction– solar, 1,94, 95, 96, 104– – puzzle, 96– spectrum, 107– sterile, 99– supernova, 100, 104– tau, 92– telescope, 88, 106, 107– temperature, 196, 219neutron, 21, 213, 323– anti-, 215– attachment, 66– Big Bang, 104– composition, see neutron,quark content– decay, 27, 28, 217, 218, 219,323– dipole moment, 279– discovery, 8– evaporation, 323– freeze-out temperature, seefreeze-out temperature– interaction, 213, 216– – with neutrino, 27– isospin, 29– lifetime, 217, 220, 221, 288– number, 215– – density, 216, see alsofreeze-out temperature– primary, 145– quark content, 25, 288– reaction, see neutroninteraction– star, 8, 69, 110, 269, 323– – formation, 66, 100–– γ burst, 122– – in binary, 71– – rotating, see pulsar– – X-ray source, 130– -to-proton ratio, 213, 216,218, 220, 222, 226, see alsofreeze-out temperature– – temperature dependence,220new inflation, 255, 256Newton observatory, see XMMNewtonian gravity, 178nitrogen fluorescence, 159Niu, K., 14‘no hair’ theorem, 323non-baryonic (dark) matter,266, 281non-luminous– matter, 269, 283– object, 270non-relativistic matter, 200non-zero baryon number, 207northern light, 2, 301nova, 323nuclear– binding energy, 323– fission, 312, 323– fusion, 323, see also fusion– interaction, 54– – cross section, 49, 213– reaction rate, 213, 222, 231nucleon, 21, 214, see alsoproton and neutron– at sea level, 149– density, 214– – BBN phase, 231– interaction with neutrino,51, 89– mass, 214, 221– number, 221, 225– primary, 145– -to-photon ratio, 220nucleosynthesis, 120, 222, 288– Big Bang, see Big Bangnucleosynthesis– deuterium, 218– helium-4 ( 4 He), 220– of light nuclei, 220– primordial, see Big Bangnucleosynthesis– stellar, 222nucleus, 50, 165– active galactic, see activegalactic nuclei– anti-, see antinuclei– β decay, 27– electromagnetic interaction,50– even–even and even–odd, 79– events of highest energy, 163– formation, 77, 120– galactic, 267– γ decay, 112– light, see light nuclei– magic, 79– measurement, 50– odd–even and odd–odd, 79

432 Index– primary, energy spectrum,79– recoil, 277– shell model, 79– strong interaction, 50– synthesis, see (Big Bang)nucleosynthesis– weak interaction, 50number– atomic, 301– baryon, see baryon number– density, 194, 214, 216, 217,396– – electron, 215– – equilibrium, 217– – monopole, 250, 260– – neutrino, 215, 216, 218,274– – neutrino, primordial, 272– – neutron, 216, see alsofreeze-out temperature– – of particles, 389– – proton, 216– – WIMP, 277– effective, degrees offreedom, 196, 214, 218, 219,226– electron, 311– equivalent, light neutrinofamilies, 227– fraction, light nuclei, 223– internal degrees of freedom,194, 389, 396– lepton, 28, 30, 319, 394–– tau,333– magic, 79, 320– mass, 321– muon, 322– of e foldings duringinflation, 258– of dimensions, 289– of microstates, 390, 392, 393– – logarithm, 399– – maximization, 394– of neutrino families, 226,227– – accelerator data, 227– of neutrinos, 272– of neutrons, 215– of nucleons, 221, 225– of one-particle states, 392– of photons, 221– of protons, 215– of relativistic particles, 222– of states, 395– of X-ray sources, 129– particle, variable, 389– quantum, see quantumnumberobservatory– CGRO, see CGRO– Einstein, 129–HEAO,see HEAO– Las Campanas, 100– Newton, see XMMOcchialini, G. P. S., 9odd–even nuclei, 79odd–odd nuclei, 79OGLE, 271, 324Olbert’s paradox, 324Ω parameter, 241, 247, 266,281, 283, 285, 289, 295– at Planck time, 248– during inflation, 257– matter dominance, 257Ω − baryon, 25one-particle state, 392open universe, 184, 242, 324Oppenheimer, J. R., 9Optical Gravitational LensExperiment, see OGLEorbit, 267, 324orbital velocity, 267origin of cosmic rays, 85, 104,110, 167, 294– extragalactic, 118, 132– point-like γ sources, 117orthogonal functions, 387orthogonality relation, 387oscillation hypothesis, length,and model, see neutrinooscillationoscillation, matter, 321OSSE, 324outlook, 293oxygen, 78, 79– burning, 100pair annihilation, seeannihilationpair creation, see pairproductionpair production, 38, 40, 50, 57,59, 113, 114, 165, 324– direct, 54parallax, 324parameter– acceleration, 186, 281, 295,299– curvature, 178, 199– deceleration, 186– density, see Ω parameter– h, 175– Hubble, 176, 186, 187– impact, 270– w, 252, 254, 335parameters– cosmological, 243– – determination, 238– (fine-)tuning, 221, 222, 289– free, Standard Model, 287parity, 29, 324– R, 276, 277, 329– violation, 29, 278, 295parsec (pc), 81, 324, 404particle, 295– acceleration– – in binaries, 71– – in gravitational potentials,72– – in pulsars, 294– – in supernovae, 67, 167,294– α, 10, 78– anti-, see antiparticle– at sea level, 150– average energy, 398– charged, energy loss, 51, 53– cold, 282– composition in theatmosphere, 145– content of the universe, 226– dark matter, 282– detection, 49,51

Index 433– exchange, 39– family, 21– flux, atmospheric, 144– fundamental, 313– GUT, 169– horizon, 239, 240, 250, 258,324– hot, 282– identification, 143, 147– in rock, 154– interchange, 27– jet, 119, 168– minimum ionizing, 53– numbers, variable, 389– periodic table, 23, 24– physics, conservation law,30– primary, 50, 145– primordial, 277, 326– pseudoscalar, 279– relativistic, 214– – number of, 222– sampling, 116– secondary, 143, 144, 150– supersymmetric, 17, 203,276– – creation, 276– – lightest, 276– – primordial, 277– type, 393, 396– virtual, 39, 335– Yukawa, 336partner, bosonic and fermionic,276Pauli principle, 69, 325, 391Pauli, W., 8pc, see parsecpeak, acoustic, 241, 242, 299Penzias, A., 12, 232, 264perchlorethylene, 95periastron, 325– rotation, 15perigee, 325perihelion, 325– rotation of the planetMercury, 15periodic table of elementaryparticles, 23, 24Perl, M. L., 9perturbations in the energydensity, 266Peters, B., 10PETRA, 325Pfotzer maximum, 9, 325Pfotzer, G., 9phase transition, 204, 250– early universe, 248phonon, 253photino, 276, 325photoelectric effect, 49, 56,113, 128, 325photomultiplier, 49, 53, 89,106, 127, 128, 159, 160photon, 26, 27, 214, 325, seealso Xandγ rays– blackbody, 59, 82, 273– bremsstrahlung, 111– contribution to energydensity, 232– decoupling from matter,233, 234– density, 225– detection, 50, 56, 128– energy density, 250– from electromagneticcascades, 150– high-energy, 82, 165– in rock, 154– infrared, 116– intensity attenuation, 113– interaction, 219– – with photon, see γγinteraction– mean free path, 233, 234– number, 221– pair annihilation, 112– pair production, 38, 59– radiation length in air, 143– ratio, baryon-to-, seebaryon-to-photon ratio– ratio, nucleon-to-, 220– real, 40– space-like, 40– starlight, 82, 106, 111, 116– temperature, 199, 219, 231– time-like, 41– virtual, 40photoproduction– of electron–positron pair, 40– of pions, 39, 82, 161, 164physics– accelerator, 293– astro-, 1– astroparticle, outlook, 293– beyond the Standard Model,293– cosmoparticle, 204, 290, 308– of particle and radiationdetection, 49– particle, conservation law,30– statistical, 389– thermal, 213, 214pion, 325– as secondary particle, 143– at sea level, 150– decay, 28, 42, 88, 111, 112,144, 147, 150– discovery, 9– isospin, 30– lifetime, 144– mass, 39, 42– photoproduction, 39, 59, 82,161, 164– production, 88, 106, 111,112, 157– quark content, 25– tertiary, 147pixel detector, semiconductor,57pixel-lensing technique, 271Planck– constant, 22, 109, 403– distribution, 59, 325– energy, 192, 294– length, 192, 325, 403– mass, 192, 325, 403– radiation law, 126, 325– scale, 191, 293– tension, 325– time, 191, 192, 248, 325,403–– Ω parameter at, 248PLANCK project, 243

434 Indexplanet, extrasolar, 18planetary nebula, 325plasma, primordial, 240, 241plethora of universes, 290Poisson distribution, 384polar angle, 235Politzer, H. D., 16Pontecorvo, B., 13positron, 326– annihilation, 120, 207, 218,219, 326– collision with electron, 226– discovery, 8– generation, 84– in rock, 154– in shower, 157– primary, 84– secondary, 144, 150potassium-40 ( 40 K) activity,106potential– chemical, 193, 305, 389,395, 397– – high temperatures, 397– energy, 280– fluctuations, 284– Higgs, 254– in field theory, 254Powell, C. F., 9power law, 261– primary cosmic rays, 75power series, 381power source, 63power spectrum, 261, 284, 326– angular, cosmic microwavebackground, 236, 238, 239,242, 243, 385, 388– measurement, 262predicted– 4 He mass fraction, 226– abundances, 224pressure, 214, 252, 389, 399– degeneracy, 309– general expression, 400– negative, 186, 198, 252, 284,322, 400– non-relativistic, 400– radiation, 66, 241, 327– relativistic limit, 400– scalar field, 254– wave, 284primordial– abundance, 15– – deuterium, 223– – helium-3 ( 3 He), 224– – helium-4 ( 4 He), 22– – light elements, 222– antimatter, 296– black hole, see black hole,mini– cosmic rays, 77– elements, mass and numberfraction, 223– helium-4 ( 4 He) massfraction, 223– neutrinos, number density,272– nucleosynthesis, 326,see also Big Bangnucleosynthesis– particle, 326– plasma, 240, 241– radioactive elements, 287– supersymmetric particles,277principle– anthropic, 290, 300– cosmological, 175, 308– of general relativity, 314– of relativity, 328– Pauli, 69, 325, 391– symmetry, 181– uncertainty, 22, 181, 316,334probability– distributions, 383– interaction, 46, 49production– of antiprotons, 84– of carbon, 287– of deuterium, 219–221, 223,288– of electron and positron, 38,40, 57, 59, 113– of elements in supernovae,66, 112– of γ rays, 110– of helium, 218– of iron, 287– of neutron stars, 66, 100– of pions, 88, 106, 111, 112,157– – by photons, 39, 59, 82,161, 164– of positrons, 84– of protons, 38– of quarks in the Big Bang,295– of stars,66– of supersymmetric particles,276– of X rays, 109, 123, 124– pair, 50, 114, 165, 324– – direct, 54propagation of astroparticles,58propagation of errors, 384proper distance, 240proportional chamber– gas, 128– multiwire, 56, 129proportional counter, 127, 128,see also gas proportionalchamber and multiwireproportional chamberprotogalaxy, 282proton, 21, 161, 162, 213, 295,326– acceleration, 105– annihilation with antiproton,112, 206, 295– anti-, see antiproton– astronomy, 86– at sea level, 147– atmospheric flux, 144– Big Bang, 104– composition, see proton,quark content– energy spectrum, 146– flux density, 141– highest energy, 163– in solar wind, 141– interaction, 213, 216– – with air, 50, 88

Index 435– – with proton, 50– isospin, 29– measurement, 50– momentum spectrum, 146– number, 215– – density, 216– pion production, 59, 88, 106,111, 112– primary, 10, 50, 59, 78, 82,143, 145, 149, 168– production, 38– –proton chain, 326– quark content, 25, 288– ratio, neutron-to-, seeneutron-to-proton ratio– reaction, scattering, seeproton interaction– stability, 288– structure function, 166– -to-muon ratio, 149protostar, 326Proxima Centauri, 326pseudoscalar, 326– particle, 279PSPC, 326pulsar, 69, 110, 118, 326– Crab, 130– creation rate, 71– Cygnus X3, 155, see alsoCygnus X3– discovery, 8– double, 310– in a binary, 71, 105– particle acceleration, 69, 294– Vela, 2, 117, 129, 334QCD, see quantum chromodynamicsquantity, conserved, 307, 394quantum, 326– anomalies, 210, 327– chromodynamics, 13, 21,278, 279, 327– – scale, 204– field excitation, 253– field theory, 253, 327– – potential, 254– fluctuations, 181, 256, 266,281, 282–foam,327– gravitation, 171, 191, 293,327– mechanics, 327– – tunneling, 255, 256– number, 28– – hidden, 25– theory of gravity, seequantum gravitationquark, 13, 14, 21, 25, 30, 327– anti-, 301– asymptotic freedom, 16, 21,301– baryon number, 30– bottom, 15, 304– charm, 14, 305– colour, 25– confinement, 30, 306– content– – neutron, 25, 288– – pion, 25– – proton, 25, 288– down, 27, 310– flavour, 23, 28, 30– – violation, 31– generation, see quark flavour– mass, 24, 288– mixing, 31– nugget, 294– production in Big Bang, 295– sea, 25– spectator, 27, 30– stars, massive, 271– strange, 29, 30, 332– supersymmetry, 276– top, 15, 334– up, 27, 334– valence, 25quasar, 118, 132, 168, 327– 3C134, 162– 3C273, 108– discovery, 11– redshift, 12quasi-exponential expansion,263Queloz, D., 18quintessence, 284, 327Rabi, I. I., 9radiation– annihilation, 296– at sea level, 147– belts, 7, 141, 142, 327, 334– blackbody, see also cosmicmicrowave background, seeblackbody radiation– bremsstrahlung, seebremsstrahlung– Cherenkov, see Cherenkovradiation– cosmic microwave background,see also blackbodyradiation, see cosmicmicrowave background– cosmic rays underground,151– detection, 49– dominance, 196, 214, 248,258, 260–– Ω parameter, 257– electromagnetic, 160, seealso γ and X rays, photon– equality with matter, seematter–radiation equality– era, 327, see also radiationdominance– extragalactic, 82, 311– γ rays, see γ rays– Hawking, 14, 316– infrared, 106– isotropic, 235– length, 54, 143, 327– Planck law, 126, 325– pressure, 66, 241, 327– synchrotron, see synchrotronradiation– underground, 151– X-ray, see Xray(s)radio– (radar) astronomy, 109, 328– galaxy, 328radioactive elements,primordial, 287radioactivity, 3– natural, 277radiochemical experiment, 95

436 Indexrange– relation, energy–, 152– straggling, 152rate– equations, 222– event, 49– interaction, 46– of expansion, 176, 201, 214,216–219, 226, 257, 258, 281– – constant, 252– of pulsar creation, 71– reaction, 202, 213, 216, 217,226– – nuclear, 213, 222, 231ratio– baryon to photon, seebaryon-to-photon ratio– charge, of muons, 148– deuterium to hydrogen, 224– double, 89– lithium to hydrogen, 223– nucleon to photon, 220– proton to muon, 149rays, cosmic, γ ,X,see cosmic,γ ,Xray(s)reaction, see also scattering– chain to helium-4 ( 4 He), 220– neutron, 213, 216– of electrons– – and neutrinos, 218– – and photons, 219– of protons and neutrons,213, 216– – cross section, 216– – inverse, 217– proton, 213, 216– rate, 202, 213, 216, 217, 226– – nuclear, 213, 222, 231– thermonuclear, 333real photon, 40recoil, energy and nucleus, 277recollapse of the universe, 251recombination, 284, 328– atoms and electrons, 233– temperature, 233, 328– time, 233, 234red giant, 6, 174, 328red supergiant, 100redshift, 6, 11, 173, 224, 234,281, 328–cosmicmicrowavebackground, 237– cosmological, 308– gravitational, 315– high, galaxies at, 132– quasar, 12– relation to brightness, 186reflection, total of X rays, 127refraction, index of– keV range, 126– of air, 114Reines, F., 8relation– depth–intensity, 152, 153– energy–range, 152– expansion rate andtemperature, 219– temperature and time, 219– uncertainty, 22, 181, 316,334relativistic– fermion gas, 274– fermions, 215– kinematics, 35– particle, 214– – number, 222Relativistic Heavy Ion Collider(RHIC), 205relativity– general, 178, 314– – principle, 314– – theory, 5, 280, 314– principle, 328– special, theory, 5, 35, 331repulsive gravitation, 284, 289residual interaction, 26, 328rest frame, local, 237rest mass, 35, 39, 328Richter, B., 14Riemann zeta function, 194,397right ascension, 328, 401Robertson–Walker metric, 179,329Rochester, G. D., 9rocket flight, 124Röntgen, W. C., 3, 123ROSAT, 16, 129–131, 329Rossi curve, 7Rossi, B., 7rotation, periastron andperihelion, 15rotational curve, 267–269– flat, 267, 268– galaxy NGC 6503, 267– of planets, 267R parity, 276, 277, 329Rubbia, C., 16Russell, Hertzsprung–,diagram, 5, 316Rutherford scattering, 26Rutherford, E., 3Rydberg energy, 403SAGE, 13, 96, 329Sakharov conditions, 209, 210,329Salam, A., 13sampling– detectors, 159– of particles, 116Sanduleak, 100, 102sandwich calorimeter, 58sapphire, 277SAS-2, SAS-3, 329satellite– COBE,see COBE satellite– experiment, 79, 124– Explorer I, 7– γ ,11– ROSAT, 16, 129–131, 329– WMAP,see WMAP satellite– X-ray, 11, 127, 128, 129scalar field, 253– energy density, 254– pressure, 254scalar spectral index, 261scale– electroweak, 204, 293– factor, 177, 185, 230, 234,257, 329– – exponential increase, 252– – relation to temperature,199– – time dependence, 230, 231

Index 437– GUT, 204, 250, 293, 315– Planck, 191, 293– QCD, 204scattering, see also reaction– Compton, 113– – inverse, 111, 125, 317– cross section, differential, 47– electron–positron, 226– multiple, 144– neutrino–electron, 27– proton–proton, 50– Rutherford, 26– surface of last, 235, 240– Thomson, 234– time of last, 239Schein, M., 10Schwartz, M., 13Schwarzschild radius, 192, 329Schwarzschild, K., 5, 14scintillation, 277, 329– crystal calorimeter, 113, 114– in air, 53, 116, 159–161– mechanism, 53scintillator experiment,Baksan, 2sea quark, 25Segrè, E. G., 11selectron, 329semiconductor– counter, 56, 113, 127– – silicon, 128– pixel detector, 57separation– angular, 236– of variables, 386series– Fourier, 261– Laplace, 235, 387, 388– power, 381Seyfert galaxy, 329SGR, 123, 329shell model, 79Shelton, I., 100shock– acceleration, 66, 67, 68, 74,167– front, 67, 329– wave, 330shower, see cascade–air,see air showersilicon, 128– burning, 100– image sensor, 128– semiconductor counters, 128singularity, 14, 330, see alsoblack holeslepton, 276, 330Sloan Digital Sky Survey, 261Small Magellanic Cloud, 330small-angle anisotropy, 237SMC, see Small MagellanicCloudSmirnov, A. Yu., 98SNAP, 330SNO experiment, 18, 283SNR, see supernova remnantSoft-Gamma-Ray Repeater,see SGRsolar– flare, 330– modulation, 141– neutrinos, 1, 94, 95, 96, 104– – deficit, 13– – puzzle, 96–system– – chemical composition, 78– – rotational curves, 267– wind, 10, 141, 330solid-iron momentumspectrometer, 148sound waves in primordialplasma, 240, 241, 284sources of cosmic rays, seeorigin of cosmic raysSoviet–American GalliumExperiment, see SAGEspace-like photon, 40spaghettification, 330spallation, 330spark chamber, multiplate, 113special relativity, 5, 35, 331specific heat, 277spectator quark, 27, 30spectral index, scalar, 261spectrometer– crystal, 127– momentum, solid iron, 148spectrum– blackbody radiation, 59– bremsstrahlung, 125– cosmic microwavebackground, 232– cosmic rays, primary– – ankle of, 80, 82– – knee of, 80, 81, 167–– toeof,82– electron, from muon decay,44– energy, see energy spectrum– Harrison–Zel’dovich, 261– mass, of brown stars, 270,271– momentum, see momentumspectrum– muon, at sea level, 147– neutrino, 107– power, see power spectrum– X rays, 125spherical coordinates, 235spherical harmonics, 385, 386spin, 331, 396spontaneous symmetrybreaking, 204, 331squark, 276, 331s quark, see strange quarkstandard– candle, 2, 173, 281, 331– rock, 152Standard Cosmological Model,172, 173, 245standard deviation, 384Standard Model, 13, 21, 226,294, 331– free parameters, 287– of electroweak interactions,216– physics beyond, 293– supersymmetric extension,276star– as X-ray source, 132– binary, see binary– brown, 270, 271– cluster, 131, 331

438 Index– dark, 269– double, see binary– dwarf, see dwarf– energy generation, 10– hydrogen burning, 223– luminous, 269– main sequence, 320– mass spectrum, 270, 271– metal-poor, 223– neutron, see also pulsar, seeneutron star– of antimatter, 84– production of newgeneration, 66– proto-, 326– quake, 123, 331– quark, massive, 271– spot, 64starburst galaxies, 331starlight photons, 82, 106, 111,116state– equation of, 182, 198, 252,398–– parameter, see wparameter– number, 395– one-particle, 392static universe, 280statistical physics (mechanics),389, 390steady-state universe, 12, 331Stefan–Boltzmann– constant, 126, 403– law, 126, 194, 331Steinberger, J., 13stellar– cluster, 131, 331– evolution, 287– nucleosynthesis, 222– wind, 331sterile neutrinos, 99Stirling approximation, 393Störmer, C., 6storage ring, 37, 331strange quark, 29, 30, 332strangeness, 29, 332string, 332– cosmic, 169, 307– super-, theory, 289– theory, 31, 332strip, superconducting, 277strong CP problem, 279strong interaction, seeinteractionstructure function, proton, 166structure of the universe, 260,283– formation, 261, 266– growth, 260– large-scale, 265, 272– – development, 266SUGRA, see supergravitysulphur burning, 100Sun– cycle, 141– distance from, 402– fusion process, 94, 95– in X-ray light, 128– magnetic field, 141– neutrinos, see solar neutrinossunspot, 64, 332– cycle, 79, 332–pair,65supercluster, 131, 265, 282,283, 332, see also galaxycluster– local, 162, 319superconducting strip, 277supergalactic plane, 162, 168,332supergalaxy, 86, 168supergiant, blue and red, 100supergravity, 32, 33Super-Kamiokande, 17, 88, 96,283, 332superluminal speed, 332supermultiplet, 276supernova, 281, 332–asγ source, 110, 118– element production, 66, 112,120– explosion, 100, 287– γ burst, 122– neutrino, 100, 104– particle acceleration,67, 167, 294, see alsoshock-wave acceleration– remnant, 69, 108, 123, 129,130, 330, 332– SN 1987A, 1, 16, 100, 103,120, 330– – discovery, 100– SNR 1572, 130– total number, 270– type-Ia, 174, 186– Vela, 2, 108– weak interaction processes,101– X rays, 129Supernova Cosmology Project,281superpartner, 276, 333superstring theories, 289supersymmetric particle, 17,203, 276– creation, 276– lightest, 276– primordial, 277supersymmetry, 276, 294, 333– broken, 276– Feynman diagram, 276– LSP, 276superweak interaction, 167surface of last scattering, 235,240SUSY, see supersymmetrysymmetrization, 391symmetry, see also invariance– breaking, 333– – mechanisms, 249– – spontaneous, 204, 331– C and CP, violation, 210– matter and antimatter,microscopic, 211– principle, 181synchrotron, 333– radiation, 110, 125, 333– – X rays, 125synthesis, see nucleosynthesistachyon, 333tail of comet, 10Tarantula Nebula, 16, 100

Index 439tau, 9, 21, 333– lepton number, 333– neutrino, 92– – mass, 274– – mass limits, 273Tau Boötis, 18Taylor, J. H., 14technique– Cherenkov, atmospheric, 55,79, 114, 116, 119, 300– cryogenic, 58– extensive air showermeasurement, 158, 160, 161– fluorescence, 79, 159– Fly’s Eye, 53, 159, 161– pixel-lensing, 271telescope– Cherenkov, 16, 119– Compton, 57– HESS, 119– Hubble, 16, 281, 282, 317– MAGIC, 119– neutrino, 88, 106, 107– Wolter, 127– X-ray, 56, 127temperature, 219, 226, 389,399– average, 236– clusters, 284– cosmic microwavebackground, see cosmicmicrowave backgroundtemperature– critical, 249, 250– decoupling, 222, 233, 234– definition, 395– dependence, 214, 217, 220,226– deuterium production, 220,221– electron mass, 218– freeze-out, see freeze-outtemperature– freeze-out, neutron-toprotonratio, 217–219, 226,228– Hawking, 316– low, 277– map, 285– neutrino, 196, 219– photon, 199, 219, 231– recombination, 233, 328– relation to scale factor, 199– rise, 277– time dependence, 231– uniform, of the universe, 246tension, Planck, 325tensor– energy–momentum, 180– metric, 179ter Haar, D., 10TeV γ astronomy, 55textures, cosmic, 307theory– electroweak, 13, 311–field,see (quantum) fieldtheory– M, 33, 322– of relativity– – general, 5, 280, 314– – special, 5, 35, 331– string, 31, 332– superstring, 289Theory of Everything, 31, 290,294, 333thermal– bremsstrahlung, 125– distribution of velocities,202– equilibrium, 214–217, 222,394– – departure from, 202, 210– physics, 213, 214– X rays, 125thermodynamics, 193, 389– first law, 398thermonuclear– explosion, 130– reaction, 333Thomson scattering, 234Thomson, J. J., 3t’Hooft, G., 13three-body decay, 43threshold energy, 37time, 219– decoupling, 234– dilatation, 333– -like photon, 41– of inflation, 263– of last scattering, 239– Planck, 191, 192, 248, 325,403– recombination, 233, 234– -reversal invariance, 334Ting, S. C. C., 14TOE, see Theory of Everythingtoeofcosmicrays,82Tomasini, G., 9top quark, 15, 334top–down scenario, 282topological defect, 169, 249,334total energy, 252, 394, 399– density of the universe, 195,214, 219, 243, 245, 246– relativistic, 36total matter density, universe,275total reflection, X rays, 127t quark, see top quarktracking chamber, 113transformation– between neutrons andprotons, 213– CP,29– Galilei, 313– in the atmosphere, 143, 144– Lorentz,44, 45, 319transverse momentum, 144trigonometric functions, 382triple-alpha process, 334triplet (isospin), 30tritium, 213, 334– decay, 102true vacuum, 255, 280tuning (fine-) of parameters,221, 222, 289tunneling, 255, 256two-body decay, 41type-Ia supernova, 174, 186ultracold gas clouds, 272ultrapure crystal, 277ultraviolet astronomy, 109

440 Indexuncertainty– principle, see uncertaintyrelation– relation, 22, 181, 316, 334underground– cosmic rays, 151– experiments, 161unification– electroweak, 171– grand, 314, see also GrandUnified TheoryUnified Theory, 334, see alsoGrand Unified Theoryuniform temperature, 246universe– age, 299, 404– alternatives, 287– baryon asymmetry,205, 206, 208, 302– baryon fraction, 225– closed, 184, 306– clumpiness, 256– dynamics, 131, 266– early, 104, 179, 191, 287,293, 296– – curvature, 284– – homogeneity, 266– – phase transition, 248– – thermal history, 203– – thermodynamics, 193, 389– energy density, total, 195,214, 219, 243, 245, 246– eras, 245– expansion, see also inflation,see expansion– flat, 184, 242, 243, 247, 266,279, 281, 285, 289, 295– Friedmann–Lemaître, 313– geometry, 241– holographic, 32– homogeneity, 175– inflation, see inflation– inhomogeneities, 265, 266– initial conditions, specific,260– isotropy, 175– mass density, 269– matter asymmetry, density,dominance, see matterasymmetry, density,dominance– models, 321– neutrino-dominated, 282– open, 184, 242, 324– particle content, 226– plethora, 290– radiation dominance, 196,214, 248, 258, 260– recollapse, 251– static, 280– steady-state, 12, 331– structure, see structure of theuniverse– uniform temperature, 246up quark, 27, 334u quark, see up quarkvacuum– energy, 205, 243, 251, 253,257, 262, 263, 285– – density, 172, 180, 243,251, 252, 263, 279, 280,295, 334, 400– – experimental evidence,186– expectation value of theHiggs field, 204– false, 255, 279, 311– neutrino oscillation, 323– true, 255, 280– velocity of light, 35, 403valence quark, 25Van Allen, J. A., 7Van Allen belts, 7, 141, 142,327, 334van der Meer, S., 16variables– cataclysmic, 130, 305– Cepheid, 305– independent, uncorrelated,384– separation of, 386variance, 384variation– spatial, energy density, 256– temperature, cosmic microwavebackground, 225,261, 284, 388vector– four-, 39, 313– – momentum, 39– momentum, 391Vela– pulsar, 2, 117, 129, 334– supernova, 2, 108– X1, 2, 11, 117, 129velocity– escape, 274, 311– of light, 35, 335, 403– orbital, 267– superluminal, 332– thermal distribution, 202Veltman,M.J.G.,13vernal equinox, 401violation– baryon number, 204, 208,210, 215– of C and CP, 210– of CP, 278, 295– – strong, 278– of parity P, 29, 278, 295– quark flavour, 31Virgo cluster, 169, 335virtual– particle, 39, 335– photon, 40virtuality, 40, 335visible matter, 265, 269Vogt, R., 11volume, 399W ± , 335, see also interaction,weak– discovery, 16water Cherenkov detector, 56,96, 106, 158wave– gravitational, 14, 263, 315– – background, 264– – energy density, 263– pressure, 284– shock, see shock wave– sound, in primordial plasma,240, 241, 284

Index 441wave function, 335– N-particle, 390wavelength, 181, 261– de Broglie, 309– shift, isotopic, 224weak gravitational lensing, 272weak interaction, seeinteraction, weakWeakly Interacting MassiveParticle, see WIMPWeber, J., 14Weinberg, S., 13Weizsäcker– Bethe–, cycle, 302– Bethe–, formula, 302Weizsäcker, C. F., 10white dwarf, 6, 130, 335Wilczek, F., 16Wilson chamber, see cloudchamberWilson, C. T. R., 3Wilson, R. W., 12, 232, 264WIMP, 275, 276, 280, 282,283, 285, 294, 335– annihilation (signal), 277– flux, 278– gravitational binding, 277,278– halo, 278– interaction and detection,277– mass, 278– number density, 277– primordial, 278wind– solar, 10, 141, 330– stellar, 331wino, 276WMAP satellite, 235, 237–239, 243, 266, 283, 284,335Wolfenstein, L., 98Wollan, E. O., 10Wolter telescope, 127Wolter, H., 127world energy consumption,102wormhole, 335w parameter, 198, 252, 254,335Wright, T., 281Wulf,Th.,3X boson, 249, 336Xrays– blackbody radiation, 125– by synchrotron radiation,125– characteristic, 95– detection, 56, 126, 127, 128– direction of incidence, 126– discovery, 3, 124– extrasolar, 124– from binaries, 130, 155– from black holes, galacticclusters, neutron stars, 130– from stars, 132– from supernovae, 129– lunar, 124, 132– number of sources, 129– penetration power, 123– production, 109, 123, 124– solar, 128– spectrum, 125– thermal, 125– total reflection, 127X-ray– astronomy, 56, 109, 123,336– background radiation, 132– burster,336– CCD, 56, 127, see also CCD– flashes, 130– satellite, 11, 127, 128, 129– telescope, 56, 127xenon, 56, 127XMM-Newton, 131, 336X-ray Multi-Mirror Mission,see XMMY boson, 249, 336ylem, 336Yukawa– particle, 336– postulate, 8Yukawa, H., 8Z, 336, see also interaction,weak– burst, 275– decay width, 22, 275– discovery, 16– exchange, 27– resonance, 22, 226, 227Zatsepin, see Greisen–Zatsepin–Kuzmin cutoffZeeman splitting, 64Zel’dovich, Harrison–,spectrum, 261zenith angle, relation toatmospheric depth, 143zenith-angle distribution ofmuons, 148, 153, 154zero, absolute, 299zero-point energy, 253, 336zeta function, 194, 397zino, 276Zweig, G., 13, 14