Master's Thesis in Theoretical Physics - Universiteit Utrecht

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Eq. (4.44) is invariant under a conformal transformation for a massless scalar field and forthe special value ξ = 1 6. We say that such a scalar field is conformally coupled. This meansthat for a massless conformally coupled scalar field we transform back to the Minkowskimetric by a conformal transformation and the action will be the same with this new metric.Note that we only considered the matter part of the action, but we also have to include theusual the Einstein-Hilbert action. This part is not invariant under a conformal transformation.However, since the field φ does not occur in this part, the dynamics of φ (the equationof motion) are not altered by this part. So the equation of motion for a massless conformallycoupled scalar is in an expanding FLRW universe equal to the equation in a Minkowskispace time. As a consequence, a massless conformally coupled scalar field does not feel theexpansion of the universe. Other examples are the photon in 4 dimensions or the masslessfermion in D-dimensions. The latter we will actually show in chapter 6.This concludes our general introduction of the conformal transformation. Let us for a momentreturn to the statements in the introduction to this chapter. We mentioned that Fakirand Unruh calculated the amplitude of density perturbations in Ref. [16]. In the minimalinflationary model this amplitude is found to be δρ/ρ ∝ λ, see Eq. (3.20). However, inthe nonminimal case we also have to include the ξRφ 2 term in the potential, which actsas a mass term. The power spectrum in Eq. (3.20) was calculated for a minimally coupledinflaton field. The question is how to calculate the spectrum of density perturbations in thenonminimal case. Fortunately we can do a trick such that we can still use the power spectrumfor a minimally coupled inflaton field. We perform a specific conformal transformationthat removes the coupling of the inflaton field to gravity. In the next section we will actuallyperform this conformal transformation, and we will see that we can write our action againin the minimally coupled form, but the potential will me modified. From this potential onecan then easily calculate the amplitude of density perturbations, which then gives the relationδρ/ρ ∝ √ λ/ξ 2 . For more on this see for example [17, 16, 26]. Let us now quickly go tothe next section and apply the conformal transformation to rewrite our original action.4.3 The Higgs boson as the inflatonIn this section we will explain the idea by Bezrukov and Shaposhnikov [4]. As mentionedin the previous section, a scalar field is an essential ingredient for inflationary models. Theonly known scalar particle in the Standard Model is the Higgs boson. The potential for theHiggs field isV (H) = λ(H † H) 2 (4.47)where H is the Higgs doublet( φ0)H =φ + , (4.48)where φ 0 and φ + are complex scalar fields. We can fix a gauge (the unitary gauge) in whichH =( φ20), (4.49)where φ is now a real scalar field with vacuum expectation value (VEV) 〈φ〉 = v. The potentialthen becomes the familiar "Mexican hat" potentialV (φ) = 1 4 λ(φ2 − v 2 ) 2 (4.50)
We now consider a chaotic inflationary scenario, where the Higgs field φ has a large initialvalue of O(M P ). Since v ≪ M P , we can safely neglect the Higgs VEV v in the potential (4.50).Thus, the Higgs potential during chaotic inflation is effectively a quartic potential with selfcouplingλ. The value of λ is not yet known, but we can calculate an allowed range. Takingthe quadratic part of the Higgs potential (4.50), we find that the Higgs mass is m H = λv 2 .The Higgs mass is expected to lie in the range [114 − 185] GeV. The lower bound is set byexperiments, whereas the upper bound is needed for stability of the Standard Model all theway up to the Planck scale. We can infer from this that the quartic self-coupling of theHiggs boson must be λ ∼ 10 −1 .For successful inflation in Linde’s chaotic inflation scenario, the self-coupling of the scalarfield must be very small λ ≃ 10 −12 in order to produce the correct amplitude of densityfluctuations. We calculated this in section 3.6 where we found that δρ/ρ ∝ λ. Such asmall self-coupling excludes the Higgs boson with λ ∼ 10 −1 as a candidate for the inflaton.In this section we include a nonminimal coupling term in the Higgs action. This changesthe potential of our theory and therefore the constraints on λ. As we will see, the amplitudeof density perturbations δρ/ρ is proportional to λ/ξ, such that the self-coupling can bemuch larger if ξ is sufficiently large. Specifically the quartic self-coupling can be λ ∼ 10 −1if ξ ∼ −10 4 . This allows the Higgs boson to be the inflaton, which is a very nice feature ofnonminimal inflation, because we can now explain inflation from the Standard Model.The total action we thus consider consist of the Standard Model action, the Einstein-Hilbertaction and the nonminimal coupling term.∫S = d 4 x −g{L SM + 1 2 M2 P R − ξH† HR}, (4.51)where L M is the Standard Model Lagrangian. If we again take the unitary gauge for theHiggs doublet H and neglect all the gauge and fermion interactions for now, we find theaction for the Higgs boson∫S = d 4 x { 1−g2 (M2 P − ξφ2 )R − 1 2 gµν ∂ µ φ∂ ν φ − λ }4 (φ2 − v 2 ) 2 , (4.52)where again the Higgs VEV appears with a value v = 246 GeV. This is the action in theso-called Jordan frame. We can now get rid of the nonminimal coupling of φ to gravity bymaking a conformal transformation to the Einstein frame. The reason that we do this isto calculate the power spectrum from Eq. (3.20), which gives us constraints on the inflatonpotential. The power spectrum was derived for a minimally coupled inflaton field, andtherefore we will perform a conformal transformation that removes the nonminimal couplingterm from the action. To do the conformal transformation we first write our actionas∫S =d 4 x { 1−g2 M2 P Ω2 R − 1 2 gµν ∂ µ φ∂ ν φ − λ }4 (φ2 − v 2 ) 2 , (4.53)whereΩ 2 = M2 P − ξφ2M 2 . (4.54)PWe see that if we now perform the conformal transformation to the new metric ḡ µν = Ω 2 g µν ,the action becomes (see also Eq. (4.39)),∫S = d 4 x √ { 1−ḡ2 M2 ¯R P− 1 2 Ω−2 ḡ µν ∂ µ φ∂ ν φ − Ω −4 λ 4 (φ2 − v 2 ) 2[ ]}−3M 2 P ¯□lnΩ + ḡ ρλ (∂ ρ lnΩ)(∂ λ lnΩ) . (4.55)
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 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
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/
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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
Page 98 and 99:
where in all the equations z = k∆
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We can now compare this equation to
Page 103 and 104:
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
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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