Master's Thesis in Theoretical Physics - Universiteit Utrecht

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6 The fermion sector 716.1 Fermion action in Minkowski space time . . . . . . . . . . . . . . . . . . . . . . 726.2 Fermion action in curved space times . . . . . . . . . . . . . . . . . . . . . . . . . 756.2.1 Fermion propagator in curved space time . . . . . . . . . . . . . . . . . . 776.2.2 Solving the chiral Dirac equation . . . . . . . . . . . . . . . . . . . . . . . 786.3 One-loop fermion self-energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.3.1 Renormalization of the one-loop self-energy . . . . . . . . . . . . . . . . 846.3.2 Influence on fermion dynamics . . . . . . . . . . . . . . . . . . . . . . . . 866.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 927 Summary and outlook 95Acknowledgements 101A Conventions 103B General Relativity in an expanding universe 105B.1 General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105B.2 General Relativity in a flat FLRW universe . . . . . . . . . . . . . . . . . . . . . 106B.3 General Relativity in a conformal FLRW universe . . . . . . . . . . . . . . . . . 107Bibliography 109
Chapter 1IntroductionCosmology is in many respects the physics of extremes. On the one hand it handles thelargest length scales as it tries to describe the dynamics of the whole universe. On thesescales gravity is the dominant force and general relativistic effects become very important.Cosmology also deals with the longest time scales. It describes the evolution of the universeduring the last 13.7 billion years and tries to make predictions for the future of ouruniverse. In fact most cosmologists are working on events that happened in the very earlyuniverse with extreme temperatures and densities. These conditions are so extreme thatwe will most likely never be able to recreate them. Extreme conditions are also present andcan be observed today in our universe in for example supernovae, black holes, quasars andhighly energetic cosmic rays.Many cosmologists actually try to explain the large scale effects from microscopic theories.A main field in this sense is the field of baryogenesis, where the matter-antimatter of theuniverse is tried to be explained. This matter-antimatter asymmetry is tiny, but is the reasonwhy our universe now only consists of matter and no antimatter. Sakharov[1] realizedthat this asymmetry can be dynamically generated in particle physics models where particlesdecay in a tiny CP violating way. Another field of research that tries to explain alarge scale phenomenon from a microscopic effect is the dark energy sector. The cosmologicalconstant functions as a uniform energy density in our universe and is incredibly small.However, our universe is so big that the dark energy now dominates our universe and is infact the source for the recent acceleration of our universe.Combining the extremely large with the incredibly small is perhaps most evident when wetry to do quantum mechanics in an expanding universe. In a heuristic picture a particleantiparticlepair can be created out of the vacuum for a very short time through the Heisenberguncertainty principle. The expansion of the universe however can elongate the existenceof such a pair because it increases the spatial distance between the particles. In fact,when the expansion of the universe is rapid enough, the particles can have a lifetime longerthan the lifetime of the universe. Although heuristic, this picture shows that particles canbe created out of the vacuum though the expansion of the universe. These particles formtiny inhomogeneities in the early universe, and in fact these leave their imprint on thespectrum of the cosmic microwave background radiation, which has been measured withincredible precision by the WMAP mission[2]. The inhomogeneities are actually the originof the formation of stars and planets.As mentioned above, a rapid expansion of the universe is necessary for particle creation.This is most evident in the inflationary era, where the universe exponentially grows to asize much larger than the visible universe today. Cosmological inflation, proposed by AlanGuth in 1981 [3], is a beautiful combination of general relativity in an expanding universe1
Page 1: INFLATION AND BARYOGENESIS THROUGH
Page 5: AbstractIn this thesis we consider
Page 10 and 11: and particle physics. The Einstein
Page 12 and 13: 2.2 An expanding universeWe are now
Page 14 and 15: Here, H 0 = H(t = 0) and we have ch
Page 16 and 17: Since the spatial background is hom
Page 18 and 19: into neutral hydrogen at a temperat
Page 20 and 21: horizon during inflation isl phys (
Page 22 and 23: VΦSlowrollregime0 MpΦFigure 3.1:
Page 24 and 25: wavelengths that are larger than th
Page 26 and 27: In the Standard Model, the only sca
Page 28 and 29: esearch of density perturbations in
Page 30 and 31: Finally, we can also vary the actio
Page 32 and 33: such that we get for Eqs. (4.18)Fro
Page 34 and 35: Φt252015105N8060402000 200 400 600
Page 36 and 37: 4.2 The conformal transformationThe
Page 38 and 39: Eq. (4.44) is invariant under a con
Page 40 and 41: Note that we have succeeded in remo
Page 42 and 43: Figure 4.6: WMAP measurements of th
Page 44 and 45: Note that the matrices Ω and ˜Ξ
Page 46 and 47: and we can writeξ ++ =ξ −− =
Page 48 and 49: value of ξ. With the explicit expr
Page 50 and 51: 0.600.800.550.75Φ1t0.50Φ2t0.700.4
Page 53 and 54: Chapter 5Quantum field theory in an
Page 55 and 56: with equation of motion0 = ¨φk +
Page 57 and 58: Eq. (5.18). If we now act with the
Page 59 and 60:
(5.28) with α = 1 and β = 0, and
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5.2 Quantization of the nonminimall
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If we now substitute the expansion
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the adiabatic regime, whereas the t
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the lowest energy state at a later
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wherey ≡ y ++ (x, x ′ ) = −(|
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Now we plug all these expansions in
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of the field and expand the action
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whereΦ k =( ) ( ) ( ) ( )φk,aa φ
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This again exactly the same express
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Chapter 6The fermion sectorIn the p
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In this representation the projecti
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These relations allow us to rewrite
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6.2.1 Fermion propagator in curved
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We now again rescale the field χL/
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φψψFigure 6.1: Feynman diagram f
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Note that the fact that the scalar
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The next step is to get rid of the
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such that we can findχ(x ′ ) = e
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where in all the equations z = k∆
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We can now compare this equation to
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Chapter 7Summary and outlookCosmolo
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esulting action as the action of a
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In future work we hope to study som
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Appendix AConventions• Throughout
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Finally, we use the Bianchi identit
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Note that we could also have calcul
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[24] A. O. Barvinsky, A. Y. Kamensh
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Master's Thesis in Theoretical Physics - Universiteit Utrecht

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