Master's Thesis in Theoretical Physics - Universiteit Utrecht

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ContentsAbstractv1 Introduction 12 Cosmology 32.1 The Cosmological Principle and Einstein’s equations . . . . . . . . . . . . . . . 32.2 An expanding universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Einstein equations from an action principle . . . . . . . . . . . . . . . . . . . . . 63 Cosmological Inflation 93.1 Evolution of the universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Cosmological puzzles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3 Inflation as solution of the puzzles . . . . . . . . . . . . . . . . . . . . . . . . . . 113.4 Inflationary models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.5 Inflationary dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.6 Constraints on inflationary models . . . . . . . . . . . . . . . . . . . . . . . . . . 154 Nonminimal inflation 194.1 Real scalar field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.1.1 Dynamical equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.1.2 Analytical solutions of the dynamical equations . . . . . . . . . . . . . . 224.1.3 Numerical solutions of the field equation . . . . . . . . . . . . . . . . . . 254.2 The conformal transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.3 The Higgs boson as the inflaton . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.4 The two-Higgs doublet model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.4.1 Dynamical equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.4.2 Numerical solutions of the field equations . . . . . . . . . . . . . . . . . . 404.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 Quantum field theory in an expanding universe 455.1 Quantum field theory in Minkowski space . . . . . . . . . . . . . . . . . . . . . . 465.1.1 Bogoliubov transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.1.2 In and out regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.2 Quantization of the nonminimally coupled inflaton field . . . . . . . . . . . . . 535.2.1 Adiabatic vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.2.2 Solutions of the mode functions . . . . . . . . . . . . . . . . . . . . . . . . 575.2.3 Scalar propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.3 Quantization of the two-Higgs doublet model . . . . . . . . . . . . . . . . . . . . 645.3.1 Propagator for the two-Higgs doublet model . . . . . . . . . . . . . . . . 675.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69vii
Page 1: INFLATION AND BARYOGENESIS THROUGH
Page 5: AbstractIn this thesis we consider
Page 9 and 10: Chapter 1IntroductionCosmology is i
Page 11 and 12: Chapter 2Cosmology2.1 The Cosmologi
Page 13 and 14: Table 2.1: Scaling of the energy de
Page 15 and 16: andδR = δ(g µν R µν )= R µν
Page 17 and 18: Chapter 3Cosmological Inflation3.1
Page 19 and 20: If we consider the evolution of the
Page 21 and 22: solution to the homogeneity puzzle.
Page 23 and 24: In general, we can define the so-ca
Page 25 and 26: Figure 3.2: WMAP measurements of th
Page 27 and 28: Chapter 4Nonminimal inflationIn thi
Page 29 and 30: where G µν = R µν − 1 2 R g
Page 31 and 32: and the equation of motion for φ f
Page 33 and 34: 0.41200.310080Φt0.2N60400.1200.000
Page 35 and 36: 0.00010.00010.000080.00008Ht0.00006
Page 37 and 38: such that the kinetic term transfor
Page 39 and 40: We now consider a chaotic inflation
Page 41 and 42: ΛM P44Ξ 2VΧΛv 44VΧ00 v(a) Effe
Page 43 and 44: 4.4 The two-Higgs doublet modelIn t
Page 45 and 46: Implicitly we have defined two new
Page 47 and 48: such that the potential term is giv
Page 49 and 50: 0.50.80.40.6Φ1t0.30.20.1Φ2t0.40.2
Page 51: N12001000800600400200Ht0.000350.000
Page 54 and 55: great triumphs of inflation is that
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We now define the vacuum state that
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of motion (5.4). If we now substitu
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This allows us to write the field e
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We could also use the conformal FLR
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5.2.1 Adiabatic vacuumSuppose now w
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expanding universe with ɛ ≪ 1 th
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(5.61), we will find a nonzero prop
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Note that for D = 4, the factor Γ(
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When we now look at the scalar prop
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Note that this is a matrix equation
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The functions Φ (1)kand in 4 dimen
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small quantum fluctuation. We quant
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In section 6.2.1 we calculate the p
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Thus, the mass term of the fermion
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and satisfy therefore {γ µ ,γ ν
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to obtain the solution for the Feyn
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This term is precisely the derivati
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Table 6.1: Feynman rules for the sc
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Note that this term is proportional
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Σ ren (x + ; x − ′ ), which co
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( )We can now rewrite ln|µ 2 (∆
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is the retarded Green function of t
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of the two Higgs bosons can be coup
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drives inflation is called the infl
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asymptotic past.In the Bunch-Davies
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AcknowledgementsFirst of all I woul
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Appendix BGeneral Relativity in an
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B.3 General Relativity in a conform
Page 117 and 118:
Bibliography[1] A. D. Sakharov, Pis
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Master's Thesis in Theoretical Physics - Universiteit Utrecht

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