Flattening the inflaton potential beyond minimal gravity
aa r X i v : . [ h e p - ph ] A ug Flattening the inflaton potential beyond minimal gravity
Hyun Min Lee ,⋆ Department ofphysics, Chung-Ang University, Seoul 06974, Korea.
Abstract.
We review the status of the Starobinsky-like models for inflation beyond mini-mal gravity and discuss the unitarity problem due to the presence of a large non-minimalgravity coupling. We show that the induced gravity models allow for a self-consistentdescription of inflation and discuss the implications of the inflaton couplings to the Higgsfield in the Standard Model.
Prepared for the proceedings of the 13th International Conference on Gravitation,Ewha Womans University, Korea, 3-7 July 2017.
It was precisely five years ago since the discovery of Higgs boson on July 4, 2012 [1] that I gave thistalk in the 2017 International Conference on Gravitation. Together with the detection of gravitationalwaves coming from the merger of binary black holes on September 14, 2015 [2], we have the fortuneto encounter the memorable moments of triumphs of the Standard Model (SM) and General Relativityin our century. Even with such a huge success of our understanding fundamental interactions fromsubatomic scales to cosmological distances, there remain open questions on the validity of our theoriesat high energies and in the early Universe.Cosmological inflation solves problems of horizon, homogeneity, flatness, cosmic relics, andstructure formation, requiring the period of an exponential expansion with a scalar field “inflaton”just after Big Bang. From the measurement of Cosmic Microwave Background (CMB) anisotropiesby Planck satellite [3], almost scale-invariant and Gaussian scalar perturbations are needed, meaningthat canonical single field inflation models are favored. Inflation could also lead to primordial gravitywaves in the CMB polarizations at the detectable level in the future CMB experiments.Almost scale-invariant CMB anisotropies require the inflation potential to be flat for a long periodof time, i.e. N =
60 e-foldings. In most of large field inflation models, the inflaton field makes atrans-Planckian excursion during inflation. The question is then whether inflation potential remainsflat for entire range of inflaton fields such that UV physics can be ignored.In this article, we review on a class of inflation models beyond minimal gravity and discuss theunitarity problems due to a large non-minimal coupling. We show that in induced gravity models, theinflaton potential is justified against the quantum corrections due to physics below the Planck scale. ⋆ e-mail: [email protected] Inflation beyond minimal gravity
In Einstein gravity, the Lagrangian for a real scalar inflaton φ is described by L E = √− g R −
12 ( ∂φ ) − V ( φ ) ! . (1)In order to realize a su ffi ciently long period of inflation, we need to introduce small masses orcouplings in the inflaton potential. For instance, we meed m ∼ GeV for quadratic potential, V ( φ ) = m φ ; λ ∼ − for quartic potential, V ( φ ) = λφ .In the case of scalar-tensor gravity, the general Lagrangian for a real scalar inflaton contains anon-minimal coupling F ( φ ) as below, L J = √− g R + F ( φ ) R −
12 ( ∂φ ) − U ( φ ) ! . (2)Then, if the e ff ective Planck mass is field-dependent as M P ( φ ) ∼ F ( φ ) with F ( φ ) ∼ U ( φ ) duringinflation, gravity becomes weaker and makes the e ff ective inflation potential V ∼ U / F asympoticallyconstant, without a need of small couplings.It is known that higher curvature terms can be introduced to drive inflation without a scalar field,because the degree of freedom in gravity becomes then larger than two. The Starobinsky model [4]introduces an R term in the Lagrangian as below, L Star = √− g R + ξ R ! . (3)Then, it can be shown that the model is dual to a scalar-tensor gravity at the classical level, after theintroduction of a Lagrange multiplier φ , as L dual = √− g R + ξ f ( φ ) R − U ( φ ) ! (4)for f ( φ ) , U ( φ ) satisfying U ( φ ) = f ( φ ), which is called the Starobinsky condition. Thus, there are aclass of scalar-tensor theories, that are equivalent to the Starobinsky model. As discussed previously,the dual theory automatically satisfies the condition for the asymptotic flat e ff ective potential of theinflaton, due to the functional relation, U ( φ ) = f ( φ ). For instance, for U ( φ ) = λφ , we would need f ( φ ) ∼ φ .As a result, after rescaling the metric and the scalar field for canonical kinetic terms by g µν → g µν / Ω with Ω = + ξ f ( φ ) and d χ d φ = q Ω ′ Ω , respectively, the inflaton Lagrangian in the scalar dualtheory becomes the following form, L dual = √− g E R −
12 ( ∂χ ) − ξ (cid:16) − e − χ/ √ (cid:17) ! . (5)The Starobinsky model leads to inflationary predictions for the spectral index and tensor-to-scalar ra-tio, n s = − N , r = N , with ξ ∼ for the normalization of the CMB anisotropies. Therefore, theresults are well consistent with the Planck data [3]. On the other hand, in the SM Higgs inflation withnon-minimal coupling [5], the Higgs kinetic term can be ignored during inflation, so it is classicallyequivalent to the Starobinsky model, leading to the similar predictions for inflation. However, the con-sistency of the Starobinsky model and the Higgs inflation below the Planck scale might be questionedat the quantum level [6, 7], due to the presence of a large non-minimal coupling ξ . Induced gravity and unitarity problem
We consider a general gravity Lagrangian including the general inflaton kinetic term in the following[8], L gen = √− g Ω ( φ ) R − K ( φ )( ∂φ ) − U ( φ ) ! . (6)Then, after rescaling the metric by g µν → g µν / Ω , the above Lagrangian becomes in Einstein frame as L gen = √− g E R − K Ω + Ω ′ Ω ! ( ∂φ ) − V ! (7)with V = U / Ω . Identifying the canonical inflation field by d χ d φ = p K / Ω + Ω ′ / Ω , the slow-roll condition is ensured for ǫ = ( dV /χ/ V ) = ( Ω U ′ / ( Ω ′ U ) − ≪
1, constraining U ( φ ) ∼ Ω [1 + O ( √ ǫ )]. Furthermore, for Starobinky-like inflation, we take Ω ′ / Ω ≫ K and 12 Ω ′ R = U ′ . (8)As a result, we get Ω = + R V I and U = R V I , fixing the potential completely [8] to U = V I ( Ω − (9)with V I being the vacuum energy during inflation. Then, during inflation, the Lagrangian (7) witheq. (9) becomes the same form as eq. (5) in the Starobinsky model, but it takes a more general formdue to the general kinetic terms for inflation, thus leading to Starobinsky-like models.Choosing K = Ω and the potential [8] as follows, Ω = ξ f ( φ ) , U = λ (cid:16) f ( φ ) − ξ − (cid:17) , (10)we can generate the Planck mass by the inflaton VEV such that f ( h φ i ) = ξ − , as in the induced gravitymodel [9]. Then, for f ( φ ) = φ n , the inflaton kinetic term in Einstein frame becomes in vacuum [8] L kin √− g E = − K Ω + Ω ′ Ω ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) φ = h φ i ( ∂φ ) = − n ξ / n ( ∂φ ) . (11)In this case, defining the canonical inflaton field χ in vacuum by φ = q
23 1 n ξ − / n χ , we obtain theinflaton potential in Einstein frame, V = U / Ω , with f ( φ ) = φ n , as V = λξ h − ( ξφ n ) − i = λξ − ! n / n n χ − n . (12)Therefore, after expanding the inflaton around h φ i = ξ − / n , we find that the interaction terms forthe canonical inflaton field χ are suppressed by the Planck scale, so there is no violation of unitaritybelow the Planck scale [8]. The induced gravity model with f ( φ ) = φ and a mixing quartic couplingbetween φ and Higgs field has been proposed as a unitarization of the original Higgs inflation withnon-minimal coupling [10]. According to the above discussion, we note that there are a general classof induced gravity models unitarizing the Higgs inflation. The discussion with a large inflaton VEVhas been extended to the case with a general form of the kinetic term K ( φ ) [11].n the other hand, in universal attractor models [12] where K = Ω = + ξ f ( φ ) and U = λ f ( φ )are taken, a small or vanishing inflation VEV is assumed. In this case, although the inflationarypredictions are the same as in induced gravity models, the non-minimal coupling does not rescalethe inflaton field in vacuum such that the interaction terms in the inflaton potential are suppressed by Λ UV = M P /ξ / ( n − , which hints at the premature violation of unitarity below the Planck scale [8].This class of models contains the Higgs inflation with non-minimal coupling for n = n = φ and the SM Higgs scalar h , we canchoose the non-minimal couplings and the scalar potential in eq. (10) [10] as follows, ξ f ( φ, h ) = ξ φ φ + ξ h h , (13) U ( φ, h ) = λ φ ( φ − w ) + λ h ( h − v ) + λ H φ ( h − v )( φ − w ) , (14)with the inflation VEV given by w ∼ / p ξ φ ≫ v . Then, it has been shown that a nonzero mixingquartic coupling λ H φ improves the stability of electroweak vacuum by inducing a tree-level shift inthe e ff ective quartic coupling, λ e ff = λ h − λ H φ λ φ [15, 16], which is inferred from the measured Higgsmass, m h = √ λ e ff v . Furthermore, the inflaton decay into a pair of Higgs bosons reheats the Universeafter inflation, thanks to the mixing quartic coupling λ H φ , leading to the reheating temperature in therange of 10 GeV . T R . GeV [17].
We have shown that the inflaton potential can be made flat without a small coupling, as gravity be-comes weaker due to an inflaton-dependent e ff ective gravity coupling during inflation. We have re-visited the Starobinsky model and its scalar-dual theories as well as the Higgs inflation with a largenon-minimal coupling, all of which are consistent with Planck data. Generalizing to Starobinsky-likemodels in scalar-tensor theories, we have identified the induced gravity models to be consistent atthe quantum level below the Planck scale. We have also discussed the implications of the couplingbetween the inflaton and the Higgs field for vacuum stability and reheating. Acknowledgments
The author would like to thank Cli ff Burgess, Jose Espinosa, Gian Giudice and Mike Trott for collab-oration and discussion. The work is supported in part by Basic Science Research Program throughthe National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science andTechnology (NRF-2016R1A2B4008759).
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