Master's Thesis in Theoretical Physics - Universiteit Utrecht

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)