Master's Thesis in Theoretical Physics - Universiteit Utrecht

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Since the spatial background is homogeneous according to the cosmological principle, ourscalar field also has to be spatially homogeneous, i.e φ(x) = φ(t). This allows us to identifyby using Eq. (2.3)ρ = 1 2 ˙φ 2 + V (φ)p = 1 2 ˙φ 2 − V (φ). (2.35)Now we use conservation of the energy momentum tensor, ∇ µ T µν = 0, which lead us to theenergy conservation law Eq. (2.13). Using the expressions for ρ and p for a real homogeneousscalar field, we finddV (φ)¨φ + 3H ˙φ + = 0. (2.36)dφNote that we could just as well have derived the equation of motion for the field φ from theaction (2.33). This would give usdV (φ)−□φ + = 0, (2.37)dφwhere□φ = 1 −g∂ µ −gg µν ∂ ν φ. (2.38)Using the flat FLRW metric with g µν = (−1, a 2 , a 2 , a 2 ), we recover Eq. (2.36). Since we canderive the two equations (2.11) and (2.12) from the action (2.29) and we can express thesetwo equations in terms of ρ and φ, we find that the equation of motion relates these twoequations. Thus we have three equations: two Einstein equations derived by varying theaction with respect to the metric g µν and one equation of motion for φ. In chapter 4 we willactually solve the field equations for a specific matter action, and we will use the fact thatonly two of these equations are independent. To be precise, we will solve the equation foronly H and the field φ because we will be able to eliminate Ḣ from these equations.
Chapter 3Cosmological Inflation3.1 Evolution of the universeIn the early days of cosmology the universe was thought to have undergone a certain evolution.The universe starts off with a Big Bang. Since the Big Bang itself is a singularityin space-time, we can not say too much about the Big Bang. The only thing we can sayis that general relativity became important only at energies below the Planck scale withM P ≃ 2.4 × 10 18 GeV, where M P is the reduced Planck mass, defined asM 2 P = 18πG N. (3.1)The reason is that above these energies the entire universe was so small (of the order of thePlanck length l p = 1.6 × 10 −35 m that quantum effects completely dominate the universe.Therefore above these energies we need a quantum theory of gravity, which so far has notyet been constructed. We call this era with extremely high energies and small length scalesthe Planck era.At some point in this very early universe particles are created, which is a process that isnot yet understood. For example, in Grand Unified Models heavy particles decay to thelighter quarks, whereas in inflation it is the inflaton that decays. Although we then stilldo not know when or how these other particles were created, it is safe to say that at hightemperatures T ∼ 10 16 GeV the universe was filled with a dense fluid of particles. At thesetemperatures the particles move at velocities close to the speed of light and are thereforecalled relativistic. The relativistic fluid of particles obeys and equation of state p = 1 3 ρ.Since these particles essentially behave like radiation, this era is called radiation era.As the universe expands, it cools down and it undergoes a number of phase transitions.For example at the electroweak phase transition (T ∼ 100 GeV) the gauge group of theStandard Model is spontaneously broken through the Higgs mechanism and the W and Zbosons acquire a mass whereas the photon remains massless. At the QCD transition (T ∼160 MeV) the quarks are confined into baryons and mesons and for example the protonsand neutrons form. Finally, at the scale T ∼ 160 MeV nucleosynthesis takes place, wherethe lighter elements are created from protons and neutrons.At a temperature of about T ∼ 1 eV the energy density in matter and radiation are roughlyequal. Since for radiation p = 1 3 ρ and for matter p = 0, we see from Eq. (2.15) that ρ r ∝ a −4and ρ m ∝ a −3 . Since the scale factor always increases (the expansion rate of the universeH is positive), we see that after the matter-radiation equality the energy density of matterdominates the total energy density. Therefore, this era is called matter era. Perhapsthe most important event during this era was the recombination of protons in electrons9
Page 1: INFLATION AND BARYOGENESIS THROUGH
Page 5: AbstractIn this thesis we consider
Page 8 and 9: 6 The fermion sector 716.1 Fermion
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 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
Page 61 and 62: 5.2 Quantization of the nonminimall
Page 63 and 64: If we now substitute the expansion
Page 65 and 66: 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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