Proc. Neutrino Astrophysics - MPP Theory Group

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150 where I have ignored the cosmological constant for simplicity. Neglecting the curvature term (as one can do at early times, see below), and approximating ˙a ∼ a/t, eq. (2) implies the familiar result that the expansion time scale t ∝ ρ −1/2 . The scale factor a grows monotonically with the cosmic time t and can therefore be used as a time variable. Boundary conditions for (2) are chosen such that a(t0) = 1 at the present cosmic time t0, and t = 0 for a = 0. The present value of the Hubble parameter, H = ˙a/a, is called the Hubble constant H0. K is the curvature of spatial hypersurfaces of space-time, and ρ is the total matter density. The quantity ρcr ≡ 3H2 0 8πG ≈ 2 × 10−29 h 2 g cm −3 is called the critical density, for reasons that will become clear below. The density parameter Ω0 is the current total matter density ρ(t0) = ρ0 in units of ρcr, The Hubble constant is commonly written as Ω0 ≡ ρ0 ρcr (3) . (4) H0 = 100hkm s −1 Mpc −1 ≈ 3.2 × 10 −18 hs −1 , (5) where 0.5 ≤ h ≤ 1 expresses our ignorance of H0. Note that the Hubble parameter has the dimension (time) −1 , so that H −1 provides the natural time scale for the expansion of the Universe. Evaluating Friedmann’s equation (2) at time t0, it follows � c K = H0 � 2 (Ω0 − 1) . (6) In other words, a universe which contains the critical matter density is spatially flat. The first law of thermodynamics, dU + pdV = 0, can be written as d(ρc 2 a 3 ) dt + p d(a3 ) dt = 0 . (7) For ordinary matter, ρc 2 ≫ p ∼ 0, hence ρ ∝ a −3 . For relativistic matter, p = ρc 2 /3, hence ρ ∝ a −4 . Therefore, the matter density changes with a as ρ(a) = ρ0 a −n = ρcr Ω0 a −n , (8) where n = 3 for ordinary matter (“dust”), and n = 4 for relativistic matter (“radiation”). Summarising, we can cast Friedmann’s equation into the form, H 2 (a) = H 2 0 � Ω0 a −n − (Ω0 − 1)a −2� . (9) Since the universe expands, a < 1 for t < t0, and so the expansion rate H(a) was larger in the past. At very early times, a ≪ 1, the first term in eq. (9) dominates because n ≥ 3, and we can write H(a) = H0 Ω 1/2 a −n/2 . (10) This is called the Einstein-de Sitter limit of Friedmann’s equation.
Photon and <strong>Neutrino</strong> Backgrounds The Universe is filled with a sea of radiation. It has the most perfect black body spectrum ever measured, with a temperature of Tγ,0 = 2.73K. Apart from local perturbations and kinematic effects, this Cosmic Microwave Background (CMB) is isotropic to about one part in 10 5 . It is a relic of the photons that were produced in thermal equilibrium in the early Universe. The energy density in the CMB is uγ,0 = 2 π2 30 (kTγ,0) 4 151 (¯hc) 3 , (11) and the equivalent matter density is ργ,0 = uγ,0/c 2 . We have seen before that the matter density in radiation changes with the scale factor like a −4 , hence Tγ(a) = a −1 Tγ,0 . (12) Converting the matter density in the CMB to the density parameter as in eq. (4), we find Ωγ,0 = uγ,0 c 2 ρcr = 2.4 × 10 −5 h −2 . (13) The energy density in radiation today is therefore about five orders of magnitude less than that in ordinary matter. Like photons, neutrinos were produced in thermal equilibrium in the early universe. Because of their weak interaction, they decoupled from the plasma at very early times when the temperature of the universe was kT ∼ 1MeV. Electrons and positrons remained in equilibrium until the temperature dropped below their rest-mass energy, kT ∼ 0.5MeV. The consequent annihilation of electron-positron pairs heated the photons, but not the neutrinos since they had already decoupled. Therefore, the entropy of the electron-positron pairs Se was completely dumped into the entropy of the photon sea Sγ. Hence, (Se + Sγ)before = (Sγ)after , (14) where ‘before’ and ‘after’ refer to the electron-positron annihilation time. Ignoring constant factors, the entropy per particle species is S ∝ gT 3 , where g is the statistical weight of the species. For bosons, g = 1, and for fermions, g = 7/8 per spin state. Before the annihilation, we thus have the total statistical weight gbefore = 4·7/8+2 = 11/2, while after the annihilation, gafter = 2 because only photons remain. From eq. (14), � Tafter Tbefore � 3 = gbefore gafter = 11 4 . (15) After the electron-positron annihilation, the neutrino temperature is therefore lower than the photon temperature by the factor (4/11) 1/3 . In particular, the neutrino temperature today is Tν,0 = � �1/3 4 Tγ,0 = 1.95K . (16) 11 Although the neutrinos are long out of thermal equilibrium, their distribution function remained unchanged since they decoupled, except that their temperature gradually dropped.
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arXiv:astro-ph/9801320v1 30 Jan 199
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Preface This was the fourth worksho
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vi S.J. Hardy Quasilinear Diffusion
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viii
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History of Neutrino Physics: Pauli
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Brief an Oskar Klein, Stockholm, vo
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Recent observations with the Hubble
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MACHO sample of ∼100,000 light cu
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Solar Neutrinos
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14 In the depth range where an abun
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16 as follows: (1) All the detector
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18 [3] J.N. Bahcall and H.A. Bethe,
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20 the opacity has a very slowly ch
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22 Status of the Radiochemical Gall
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24 known true source activity. The
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26 Solar Neutrino Observation with
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28 Events/day/kton/bin 0.3 0.25 0.2
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30 log(Δm 2 (eV 2 log(Δm )) 2 (eV
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32 Table 1: Neutrino event rates in
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34 tank and sphere high purity wate
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36 Measurements of Low Energy Nucle
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38 Figure 1: The S(E) factor of 3 H
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40 Solar Neutrinos: Where We Are an
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42 [9] B. Pontecorvo, Chalk River R
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44 Figure 2: The difference between
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Phenomenology of Supernova Explosio
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Figure 2: Theoretical classificatio
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Supernova Rates Bruno Leibundgut Eu
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galaxies, however, and the statisti
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Figure 1: The motions of pulsars re
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Convection in Newly Born Neutron St
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a b Figure 2: Panel a shows the rec
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ference). The angular variations of
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Figure 1: Neutrino luminosity and e
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[9] G.S. Bisnovatyi-Kogan, Astron.
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and the mass-to-luminosity ratio of
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Figure 1: Time variation of superno
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Figure 3: Energy spectra of the sup
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Supernova Neutrino Opacities G.G. R
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Quasilinear Diffusion of Neutrinos
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Assuming a saturated thermal distri
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Anisotropic Neutrino Propagation in
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So far the damping rate corresponds
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Cherenkov Radiation by Massless Neu
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on ω so that they are well approxi
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2 2 / meff,e m eff 1.0 0.5 0.0 elec
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Gamma-Ray Burst Observations D.H. H
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Figure 1: The cumulative distributi
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[7] Lilly, S., in Critical Dialogue
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) Phase Transitions in Neutron Star
Page 107 and 108: after the collapse began (nb., no e
Page 109 and 110: Models of Coalescing Neutron Stars
Page 111 and 112: Figure 2: Contour plots of the mass
Page 113 and 114: From our torus models we find that
Page 115: High-Energy Neutrinos
Page 118 and 119: 110 in gamma-rays. However, the acc
Page 120 and 121: 112 Atmospheric Muons and Neutrinos
Page 122 and 123: 114 Today, however, charm productio
Page 124 and 125: 116 Atmospheric Neutrinos in Super-
Page 126 and 127: 118 number of events 300 200 100 (a
Page 128 and 129: 120 Acknowledgements D.K. is suppor
Page 130 and 131: 122 Key signatures for νµ nucleon
Page 132 and 133: 124 (a.u) 0.5 0.4 0.3 0.2 0.1 0 0 1
Page 134 and 135: 126 However, in order to operate th
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Page 138 and 139: 130 Ground-Based Observation of Gam
Page 140 and 141: 132 Figure 2: Sensitivities in unit
Page 142 and 143: 134 Crab flux Figure 4: Gamma-ray f
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Page 147 and 148: Helium Absorption and Cosmic Reioni
Page 149 and 150: Non-Standard Big Bang Nucleosynthes
Page 151 and 152: ) Non-standard Neutrino Properties
Page 153 and 154: some heating of the plasma. In cont
Page 155 and 156: Big Bang Nucleosynthesis With Small
Page 157: Neutrinos and Structure Formation i
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Page 163 and 164: spectrum has therefore changed to 1
Page 165 and 166: Figure 4: Growth of structure in CD
Page 167: Future Prospects
Page 170 and 171: 162 Figure 1: Calorimetric β − s
Page 172 and 173: 164 References [1] F. Gatti: Procee
Page 174 and 175: 166 The frequency of Galactic super
Page 176 and 177: 168 Table 1: Comparison of proposed
Page 178 and 179: 170 Neutrinos in Astrophysics José
Page 180 and 181: 172 N eq 7 6.5 6 5.5 5 4.5 4 3.5 3
Page 182 and 183: 174 Figure 4: SN 1987A bounds on FC
Page 184 and 185: 176 [5] A. Joshipura and J. W. F. V
Page 186 and 187: 178 Some Neutrino Events of the 21s
Page 188 and 189: 180 2099: Relic Neutrinos And last
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Page 192 and 193: 184 Tuesday, Oct. 21: Supernova Neu
Page 194 and 195: 186 Michael Altmann Technische Univ
Page 196 and 197: 188 Paolo Gondolo Max-Planck-Instit
Page 198 and 199: 190 Patrizia Meunier Max-Planck-Ins
Page 200 and 201: 192 Torsten Soldner Technische Univ
Page 202 and 203: 194 A Experimental Particle Physics
Page 204: 196 C Astrophysics and Cosmology C1
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