Master's Thesis in Theoretical Physics - Universiteit Utrecht

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expanding universe with ɛ ≪ 1 the conformal time η runs from −∞ to 0, see the discussionbelow Eq. (2.21). We could also write the two independent solutions as a linear combinationof J ν (−kη) and N ν (−kη),H (1)ν (−kη) = J ν(−kη) + iN ν (−kη)H (2)ν (−kη) = J ν(−kη) − iN ν (−kη), (5.78)where H (1,2)ν (−kη) are the Bessel functions of the third kind, called Hankel functions. TheseHankel functions proof to be useful in our analysis, as we will see shortly. Combining theabove, we find that the two fundamental solutions of Eq. (5.62) areδϕ (1)k (η) = 12k√ π2δϕ (2)k (η) = √− πη4 H(2) ν√√−kηH(1)ν (−kη) = − πη4 H(1) ν(−kη)(−kη). (5.79)In the first line I have explicitly written out the normalization factors, which have beenchosen such that the Wronskian of the two solutions satisfies[]W δϕ (1)k (η),δϕ(2) k (η) = δϕ (1)k (η)δϕ(2)′ (η) − δϕ(2)kk (η)δϕ(1)′ k(η)= i. (5.80)( )In general we have the relation H ν(1) ∗(z) = H(2)ν(z ∗ ). Since in this case both the index ν∗and the coordinate z = −kη are real, we find that δϕ (2) (η) = δϕ(1)∗(η). Of course this will bek kdifferent when for example ν is complex, as we will see when we quantize the two-Higgsdoublet model. If we now look at the original mode fluctuations δφ k , we can write the twofundamental solutions asδφ (1)kδφ (2)kδϕ(1)k(η) = (η)= 1 a(η) aδϕ(2)k(η) = (η)a(η)√− πη4 H(1) νThe Wronskian between these two solutions is then simply[]W δφ (1)k (η),δφ(2) k (η)(−kη)= δφ (1)∗ (η). (5.81)k=ia 2 . (5.82)The general mode functions in Eq. (5.57) can now be expressed as linear combinations ofthe fundamental solutions in Eq. (5.81) in the following wayδφ k = α k δφ (1)k + β kδφ (2)kδφ ∗ k= α ∗ k δφ(2) k+ β∗ k δφ(1) k , (5.83)where I have used that δφ (2)∗ = δφ (1) . These mode functions must satisfy the normalizationk kcondition in Eq. (5.60), and by using the relation (5.82) we find the condition|α k | 2 − |β k | 2 = 1. (5.84)The question is now: can we make some natural choice for the parameters α k and β k ? Ingeneral we cannot do this, because the lowest energy state at one instant of time will not be
the lowest energy state at a later time in an expanding universe. However, in the previoussection we showed that in the limit η → −∞ we can define a natural vacuum state, theBunch-Davies vacuum, that minimizes the energy and remains the lowest energy state to avery good approximation. So let us take the limit η → −∞. The modes δϕ k from Eq. (5.79)then becomeδϕ (1)k (η → −∞) = 12ke −ikηThe energy in the mode δϕ k from Eq. (5.66) then isδϕ (2)k (η → −∞) = 12ke ikη . (5.85)E k (η → −∞) = 1 2(|δϕ′k |2 + ω 2 k |δϕ k| 2)∣ ∣ ∣η→−∞= 1 2 k(|α k| 2 + |β k | 2 )= 1 2 k(1 + 2|β k| 2 ). (5.86)So, the energy is minimized when α k = 1, β k = 0. Precisely these conditions lead to theBunch-Davies vacuum, which corresponds to zero particles and energy in the infinite asymptoticpast.We note that the adiabatic analysis we made in this section is only valid for those modeswith k > Ha = 1η(1−ɛ). If k < Ha, the state is nonadiabatic and we can no longer define theBunch-Davies vacuum to be the natural vacuum state.5.2.3 Scalar propagatorIn the previous sections we have looked at particle production due to the expansion of theuniverse. We calculated the particle number with respect to the vacuum, and we have seenthat we cannot define a unique zero particle vacuum. However, in the adiabatic regime,which is the infinite asymptotic past, we showed that particle production is minimal. Wecould define a natural vacuum state, the Bunch-Davies vacuum, with zero particles asη → −∞.Outside of the adiabatic regime this analysis breaks down and we cannot define a naturalvacuum state anymore. There is however another important quantity, which is the amplitudeof field fluctuations. This quantity is well defined even for quantum states where wecannot unambiguously define the notion of a particle. The amplitude of field fluctuationswith respect to some state is described by the expression〈n|δ ˆϕ(x,η)δ ˆϕ(x ′ ,η)|n〉. (5.87)For simplicity we take the state to be the vacuum state |n〉 = |0〉. Note that Eq. (5.87) is theequal time scalar propagator,〈0|δ ˆϕ(x)δ ˆϕ(x ′ )|0〉 (5.88)The propagator describes the probability for a particle to travel from a space time pointx = (x,η) to another space time point x ′ = (x ′ ,η ′ ). Note that even though the vacuum containszero particles, particle-antiparticle pairs can be created from the vacuum by the Heisenberguncertainty principle. This means that the amplitude of field fluctations can be nonzero ifwe calculate (5.87) with respect to the vacuum |0〉. Indeed, if we substitute the expansion
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INFLATION AND BARYOGENESIS THROUGH
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AbstractIn this thesis we consider
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6 The fermion sector 716.1 Fermion
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and particle physics. The Einstein
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2.2 An expanding universeWe are now
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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
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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: the adiabatic regime, whereas the t
Page 69 and 70: wherey ≡ y ++ (x, x ′ ) = −(|
Page 71 and 72: Now we plug all these expansions in
Page 73 and 74: of the field and expand the action
Page 75 and 76: whereΦ k =( ) ( ) ( ) ( )φk,aa φ
Page 77 and 78: This again exactly the same express
Page 79 and 80: Chapter 6The fermion sectorIn the p
Page 81 and 82: In this representation the projecti
Page 83 and 84: These relations allow us to rewrite
Page 85 and 86: 6.2.1 Fermion propagator in curved
Page 88 and 89: We now again rescale the field χL/
Page 90 and 91: φψψFigure 6.1: Feynman diagram f
Page 92 and 93: Note that the fact that the scalar
Page 94 and 95: The next step is to get rid of the
Page 96 and 97: such that we can findχ(x ′ ) = e
Page 98 and 99: where in all the equations z = k∆
Page 100 and 101: We can now compare this equation to
Page 103 and 104: Chapter 7Summary and outlookCosmolo
Page 105 and 106: esulting action as the action of a
Page 107: In future work we hope to study som
Page 111: Appendix AConventions• Throughout
Page 114 and 115: 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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