Master's Thesis in Theoretical Physics - Universiteit Utrecht

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Note that we have succeeded in removing the coupling of the Higgs field to gravity, but anadditional term has appeared. Let us simplify this term by making use of ¯∇ λ Ω = ∂ λ Ω, sinceΩ only contains a scalar field. Then we get[ ]−3MP√ 2 −ḡ ¯□lnΩ + ḡ ρλ (∂ ρ lnΩ)(∂ λ lnΩ)[ ( )= −3M 2 √P −ḡḡρλ 1¯∇ ρΩ ∂ λΩ + ( 1 Ω ∂ ρΩ)( 1 ]Ω ∂ λΩ)[= −3M 2 √P −ḡḡρλ− 1 Ω 2 ∂ ρ∂ λ Ω + 1 Ω ¯∇ ρ ∂ λ Ω + 1 ]Ω 2 ∂ ρΩ∂ λ Ω( √−ḡḡ= 3M 2 ¯∇ ρλ 1 )P ρ ∂ λ ΩΩ= −3 1 Ω 2 M2 P√−ḡḡ ρλ ∂ ρ Ω∂ λ Ω, (4.56)where in the last step I have used metric compatibility ¯∇ ρ ḡ µν = 0. Now I use the specificform of Ω 2 from Eq. (4.54) and I find−3 1 Ω 2 M2 P√−ḡḡ ρλ ∂ ρ Ω∂ λ Ω = −3 1 ξ 2 φ 2 √−ḡḡ ρλΩ 4 ∂ ρ φ∂ λ φ. (4.57)We see that this combines neatly with the kinetic term in the action (4.55) and we getS =∫d 4 x √ { 1−ḡ2 M2 ¯R P− 1 M 2 P Ω2 + 6ξ 2 φ 22 M 2 ḡ µν ∂ µ φ∂ ν φ − Ω −4 λP Ω4 4 (φ2 − v 2 ) 2} .We see that we have removed the coupling to gravity by doing the conformal transformation,but as a consequence we created a non-trivial kinetic term. We can actually write this againas a canonical kinetic term by defining a new field χ(φ), such that we haveM 2 P( ) dχ 2ḡ µν ∂ µ φ∂ ν φ. (4.58)− 1 2 ḡµν ∂ µ χ(φ)∂ ν χ(φ) = − 1 2dφWhen√ √√√dχdφ = M 2 P Ω2 + 6ξ 2 φ 2M 2 , (4.59)P Ω4we retrieve our non-trivial kinetic term. In terms of the new field χ the action becomes∫S = d 4 x √ 1−ḡ{2 M2 ¯R P− 1 2 ḡµν ∂ µ χ∂ ν χ − Ω −4 λ 4 (φ(χ)2 − v 2 ) 2} .We could solve Eq. (4.59) for the new field χ in the case where ξ < 0. However, it is moreinsightful to solve the differential equation in two different regimes. First consider theregime where the field φ is small, i.e.φ ≪ M P√−ξ(small field approximation). (4.60)Take a typical value ξ = −10 4 , then we can safely say that the small field approximation isvalid when the Higgs field φ < 10 −2 M P . So this approximation works well up to an energyscale of ∼ 10 16 GeV. In the small field approximation, Ω 2 ≃ 1 anddχ≃ 1,dφχ ≃ φ. (4.61)
ΛM P44Ξ 2VΧΛv 44VΧ00 v(a) Effective potential, φ ≪ M P / √ −ξΧΛM P416Ξ 2Χ end(b) Effective potential, φ ≫ M P / √ −ξΧFigure 4.5: Effective Higgs potential in the Einstein frame. For small field values the effective potentialreduces to the original Mexican hat Higgs potential. However, for large field values the effective potentialbecomes very flat and asymptotically reaches the value λM4 P4ξ 2 . The flatness of the potential ensures that inflationis possible. Inflation ends when the fields become very small, such that χ ≪ 6M P .Therefore, the potential in terms of the field χ becomesV (χ) ≃ λ 4 (χ2 − v 2 ) 2 , (4.62)which is the well-known Mexican hat potential that leads to the Higgs mechanism throughwhich the W and Z bosons become massive. In Fig. 4.5a we show the effective Higgspotential in the Einstein frame for small field values.However, if we now consider another regime whereφ ≫ M P√−ξ(large field approximation), (4.63)we find that Ω 2 ≃ − ξφ2M 2 Pand thatdχdφ ≃ 6 M Pφ ,χ ≃ 6M P ln(√ )−ξφ. (4.64)M PAgain for a typical value ξ = 10 4 , we are in the large field approximation when φ 10 −1 M P .We write for the field φφ(χ) ≃ M (P χ√ exp ). (4.65)−ξ 6MPNote that in the regime where φ ≫ M P the new field χ ≫ 6M P .−ξpotential in the Einstein frame then becomesThe effective HiggsV (χ) ≃ λM4 P 14ξ 2 ( ( ))1 + exp −2 χ 2. (4.66)6MPNote that we have neglected v since v ≪ φ. In Fig. 4.5b we show the form of the potential forlarge values of φ. The flatness of the potential makes sure that the slow-roll approximationis valid and inflation is successful.The large field regime where φ 10 −1 M P is precisely the chaotic inflationary regime, and
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 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 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
Page 67 and 68: the lowest energy state at a later
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
Page 116 and 117:
Note that we could also have calcul
Page 118:
[24] A. O. Barvinsky, A. Y. Kamensh
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Master's Thesis in Theoretical Physics - Universiteit Utrecht

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