Classical and Quantum Initial Conditions for Higgs Inflation
CClassical and Quantum Initial Conditions for Higgs Inflation
Alberto Salvio and Anupam Mazumdar , Departamento de F´ısica Te´orica, Universidad Aut´onoma de Madridand Instituto de F´ısica Te´orica IFT-UAM / CSIC, Madrid, Spain. Consortium for fundamental Physics, Lancaster University, Lancaster, LA1 4YB, UK IPPP, Durham University, Durham, DH1 3LE, UK
Report number: IFT-UAM / CSIC-15-063
Abstract
We investigate whether Higgs inflation can occur in the Standard Model starting from natural initial conditions or not. TheHiggs has a non-minimal coupling to the Ricci scalar. We confine our attention to the regime where quantum Einstein gravitye ff ects are small in order to have results that are independent of the ultraviolet completion of gravity. At the classical levelwe find no tuning is required to have a successful Higgs inflation, provided the initial homogeneity condition is satisfied. Onthe other hand, at the quantum level we obtain that the renormalization for large non-minimal coupling requires an additionaldegree of freedom that transforms Higgs inflation into Starobinsky R inflation, unless a tuning of the initial values of therunning parameters is made. Keywords:
Higgs boson, Inflation, Standard Model.
1. Introduction
Inflation [1, 2, 3] is perhaps one of the most natural wayto stretch the initial quantum vacuum fluctuations to the sizeof the current Hubble patch, seeding the initial perturbationsfor the cosmic microwave background (CMB) radiation andlarge scale structure in the universe [4] (for a theoreticaltreatment, see [5]). Since inflation dilutes all matter it is per-tinent that after the end of inflation the universe is filled withthe right thermal degrees of freedom, i.e. the Standard Model(SM) degrees of freedom (for a review on pre- and post-inflationary dynamics, see [6]). The most economical way toachieve this would be via the vacuum energy density storedwithin the SM Higgs, whose properties are now being mea-sured at the Large Hadron Collider (LHC) [7, 8]. Naturally,the decay of the Higgs would create all the SM quarks andleptons observed within the visible sector of the universe. Al-beit, with just alone SM Higgs and minimal coupling to grav-ity, it is hard to explain the temperature anisotropy observedin the CMB radiation without invoking physics beyond theSM .However, a very interesting possibility may arise withinthe SM if the Higgs were to couple to gravity non-minimally- such as in the context of extended inflation [10], which has Within supersymmetry it is indeed possible to invoke the flat directioncomposed of the Higgses to realize inflation with minimal gravitational in-teraction, see [9], which can explain the current CMB observations. recently received particular attention after the Higgs discov-ery at the LHC in the context of Higgs inflation [11]. Bytuning this non-minimal coupling constant, ξ , between theRicci scalar of the Einstein-Hilbert term and the SM Higgs,it is possible to explain su ffi cient amount of e-folds of in-flation and also fit other observables such as the amplitudeof temperature anisotropy and the spectral tilt in the CMBdata. Indeed, this is very nice and satisfactory, except thatthe non-minimal coupling, ξ , turns out to be very large (atthe classical level ξ ∼ ) in order to explain the CMB ob-servables. This e ff ectively redefines the Planck’s constantduring inflation, and invites new challenges for this model,whose consequences have been debated vigorously in manypapers, such as [12].One particular consequence of such large non-minimalcoupling is that there is a new scale in the theory, ¯ M Pl / √ ξ ,lower than the standard reduced Planck mass, ¯ M Pl ≈ . × GeV. Typically inflation occurs above this scale, theHiggs field takes a vacuum expectation value (VEV) above¯ M Pl / √ ξ in order to sustain inflation su ffi ciently. In fact, theinflaton potential, in the Einstein frame, approaches a con-stant plateau for su ffi ciently large field values. E ff ectively,the inflaton becomes a flat direction, where it does not costany energy for the field to take any VEV beyond this cut-o ff .Given this constraint on the initial VEV of the inflaton andthe new scale, we wish to address two particularly relevantissues concerning the Higgs inflation model [11], one on the Preprint submitted to the arXiv July 3, 2015 a r X i v : . [ h e p - ph ] J u l lassical front and the other on the quantum front. I . Classically, a large VEV of the inflaton does not pose abig problem as long as the initial energy density stored in theinflaton system, in the Einstein frame, is below the cut-o ff of the theory. Since, the potential energy remains boundedbelow this cut-o ff , the question remains - what should be theclassical initial condition for the kinetic energy of the infla-ton?A-priori there is no reason for the inflaton to move slowlyon the plateau, therefore the question we wish to settle in thispaper is what should be the range of phase space allowed fora sustainable inflation to occur with almost a flat potential?The aim of this paper is to address this classical initial con-dition problem . Here we strictly assume homogeneity ofthe universe from the very beginning; we do not raise the is-sue of initial homogeneity condition required for a success-ful inflation; this issue has been discussed earlier in a genericinflationary context in many classic papers (see [15, 16]). Inour paper, instead we look into the possibility of initial phasespace for a spatially flat universe, and study under what pre-inflationary conditions Higgs inflation could prevail. II . At quantum level, the original Higgs model poses a com-pletely di ff erent challenge. A large ξ will inevitably modifythe initial action. One may argue that there will be quan-tum corrections to the Ricci scalar, R , such as a Higgs-loopcorrection - leading to a quadratic in curvature action, i.e. R + α R type correction, where α is a constant, whose mag-nitude we shall discuss in this paper. The analysis is basedon the renormalization group equations (RGEs) of the SMparameters and the gravitational interactions. By restrict-ing for simplicity the study to operators up to dimension4, the RGE analysis will yield a gravitational action thatwill become very similar to the Starobinsky type inflation-ary model [17] .One of the features of theories with curvature squaredterms is that there are extra degrees of freedom involved inthe problem, besides the SM ones and the graviton. Thereis another scalar mode arising from R , which will also par-ticipate during inflation. The question then arises when thisnew scalar degree of freedom becomes dominant dynami-cally, and play the role of an inflaton creating the initial den-sity perturbations? Some single monomial potentials and exponential potentials exhibit aclassic example of late time attractor where the inflaton field approaches aslow roll phase from large initial kinetic energy, see [13, 14]. In principle, large ξ may also yield higher derivative corrections up toquadratic in order, see [18], and also higher curvature corrections, but in thispaper, we will consider for simplicity the lowest order corrections. We willargue that the α R is necessarily generated unless one is at the critical pointof Ref. [19] or invokes a fine-tuning on the initial values of the runningparameters. The aim of this paper will be to address both the classicaland quantum issues.We briefly begin our discussion with essential ingredientsof Higgs inflation in section 2, then we discuss the classicalpre-inflationary initial conditions for Higgs inflation in sec-tion 3. In this section, we discuss both analytical 3.1, andnumerical results 3.2. In section 4, we discuss the quantumcorrection to the original Higgs inflation model, i.e. we dis-cuss the RGEs of the Planck mass in subsection 4.1, SM pa-rameters in 4.2, and the gravitational correction arising dueto large ξ in subsection 4.3, respectively. We briefly discussour results and consequences for inflation in subsection 4.4,before concluding our paper.
2. The model
Let us define the Higgs inflation model [11]. The action is S = (cid:90) d x √− g (cid:20) L SM − (cid:18) ¯ M + ξ |H| (cid:19) R (cid:21) , (1)where L SM is the SM Lagrangian minimally coupled to grav-ity, ξ is the parameter that determines the non-minimal cou-pling between the Higgs and the Ricci scalar R , and H is theHiggs doublet. The part of the action that depends on themetric and the Higgs field only (the scalar-tensor part) is S st = (cid:90) d x √− g (cid:20) | ∂ H| − V − (cid:18) ¯ M + ξ |H| (cid:19) R (cid:21) , (2)where V = λ ( |H| − v / is the Higgs potential and v isthe electroweak Higgs VEV. We take a sizable non-minimalcoupling, ξ >
1, because this is required by inflation as wewill see.The non-minimal coupling − ξ |H| R can be eliminatedthrough the conformal transformation g µν → Ω − g µν , Ω = + ξ |H| ¯ M . (3)The original frame, where the Lagrangian has the form in(1), is called the Jordan frame, while the one where gravityis canonically normalized (obtained with the transformationabove) is called the Einstein frame. In the unitary gauge,where the only scalar field is the radial mode φ ≡ (cid:112) |H| ,we have (after the conformal transformation) S st = (cid:90) d x √− g (cid:20) K ( ∂φ ) − V Ω − ¯ M R (cid:21) , (4)where K ≡ (cid:0) Ω + ξ φ / ¯ M (cid:1) / Ω . The non-canonicalHiggs kinetic term can be made canonical through the (in-vertible) field redefinition φ = φ ( χ ) defined by d χ d φ = (cid:114) Ω + ξ φ / ¯ M Ω , (5)2ith the conventional condition φ ( χ = =
0. One can finda closed expression of χ as a function of φ : χ ( φ ) = ¯ M Pl (cid:115) + ξξ sinh − (cid:20) √ ξ (1 + ξ ) φ ¯ M Pl (cid:21) − √ M Pl tanh − (cid:34) √ ξφ (cid:112) ¯ M + ξ (1 + ξ ) φ (cid:35) . (6)Thus, χ feels a potential U ≡ V Ω = λ ( φ ( χ ) − v ) + ξφ ( χ ) / ¯ M ) . (7)Let us now recall how slow-roll inflation emerges. From(5) and (7) it follows [11] that U is exponentially flat when χ (cid:29) ¯ M Pl , which is the key property to have inflation. Indeed,for such high field values the slow-roll parameters (cid:15) ≡ ¯ M (cid:18) U dUd χ (cid:19) , η ≡ ¯ M U d Ud χ (8)are guaranteed to be small. Therefore, the region in fieldconfigurations where χ > ¯ M Pl (or equivalently [11] φ > ¯ M Pl / √ ξ ) corresponds to inflation. We will investigatewhether successful sow-roll inflation emerges also for largeinitial field kinetic energy in the next section. Here we sim-ply assume that the time derivatives are small.All the parameters of the model can be fixed through ex-periments and observations, including ξ [11, 20]. ξ can beobtained by requiring that the measured power spectrum [4], P R = U /(cid:15) π ¯ M = (2 . ± . × − , (9)is reproduced for a field value φ = φ b corresponding to anappropriate number of e-folds of inflation [20]: N = (cid:90) φ b φ end U ¯ M (cid:18) dUd φ (cid:19) − (cid:18) d χ d φ (cid:19) d φ ≈ , (10)where φ end is the field value at the end of inflation, that is (cid:15) ( φ end ) ≈ . (11)For N =
59, by using the classical potential we obtain ξ = (5 . ∓ . × √ λ, ( N =
59) (12)where the uncertainty corresponds to the experimental un-certainty in Eq. (9). Note that ξ depends on N : ξ = (4 . ∓ . × √ λ, ( N =
54) (13) ξ = (5 . ∓ . × √ λ. ( N =
64) (14)This result indicates that ξ has to be much larger than onebecause λ ∼ .
3. Pre-inflationary dynamics: classical analysis
Let us now analyze the dynamics of this classical systemin the homogeneous case without making any assumption onthe initial value of the time derivative ˙ χ . We will assume thatthe universe is su ffi ciently homogeneous to begin inflation.In the Einstein frame S st is given by: S st = (cid:90) d x √− g (cid:20) ( ∂χ ) − U − ¯ M R (cid:21) , (15)where U is the Einstein frame potential given in Eq. (7).Let us assume a universe with three dimensional trans-lational and rotational symmetry, that is a Friedmann-Robertson-Walker (FRW) metric ds = dt − a ( t ) (cid:20) dr − kr + r ( d θ + sin θ d ϕ ) (cid:21) , (16)with k = , ± a ( t ) and the spatially homo-geneous field χ ( t )¨ χ + H ˙ χ + U (cid:48) = , (17)˙ a + ka − ˙ χ + U M = , (18) ka − ˙ H − ˙ χ M = . (19)where H ≡ ˙ a / a , a dot denotes a derivative with respect to t and a prime is a derivative with respect to χ . Notice that Eq.(17) tells us that χ cannot be constant before inflation unless U is flat. From Eqs. (17) and (18) one can derive (19), whichis therefore dependent.Thus, we have to solve the following system with initialconditions ˙ Π + H Π + U (cid:48) = , Π (¯ t ) = Π , ˙ χ = Π , χ (¯ t ) = χ, ˙ a + k = a M ( Π + U ) , a (¯ t ) = ¯ a , (20)where ¯ t is some initial time before inflation and χ , ¯ Π and ¯ a are the initial conditions for the three dynamical variables. Inthe case k = k = χ + (cid:115) χ + U M ˙ χ + U (cid:48) = , ( k = . (21)3his equation has to be solved with two initial conditions (for χ and ˙ χ ). The initial condition for a is not needed in this caseas its overall normalization does not have a physical meaningfor k = U (cid:28) ¯ M , ˙ χ (cid:28) ¯ M , | k | a (cid:28) ¯ M (22)such that we can ignore the details of the ultraviolet (UV)completion of Einstein gravity . However, we do not al-ways require to be initially in a slow-roll regime. The firstand second conditions in (22) come from the requirementthat the energy-momentum tensor is small (in units of thePlanck scale) so that it does not source a large curvature; thethird condition ensures that the three-dimensional curvatureis also small. The first condition is automatically fulfilledby the Higgs inflation potential, Eq. (7): the quartic cou-pling λ is small [24, 21, 22] and the non-minimal coupling ξ is large (see Eq. (12)). The second and third conditions in(22) are implied by the requirement of starting from an (ap-proximately) de Sitter space, which is maximally symmetric;therefore we do not consider them as a fine-tuning in the ini-tial conditions. In de Sitter we have to set k = H = χ = U is almost, but not exactly flat in the large fieldcase (see Eq. (7)).In order for the Higgs to trigger inflation sooner or laterone should have a slow-roll regime, where the kinetic energyis small compared to the potential energy, ˙ χ / (cid:28) U , andthe field equations are approximately˙ a + ka ≈ U M , ˙ χ ≈ − H U (cid:48) , (slow-roll equations).(23)The conditions for this to be true are˙ χ (cid:28) U , | ¨ χ | (cid:28) | H ˙ χ | (slow-roll regime). (24)We will use these conditions rather than the standard (cid:15) (cid:28) η (cid:28) Let us assume, for simplicity, that the parameter k in theFRW metric vanishes, i.e. a spatially flat metric, and con-sider the case ˙ χ (cid:29) U , such that the potential energy can The conditions in (22) may not be necessary in scenarios where grav-itational interactions are softened at energies much below ¯ M Pl and remainsmall at and above ¯ M Pl [23]. However, here we do not want to rely on anyspecific quantum gravity theory. (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Χ M Pl (cid:80) M Pl Initial field and momentum conditions for Higgs inflation N (cid:61) (cid:72) upper (cid:76) N (cid:61) (cid:72) lower (cid:76) N (cid:61) Figure 1:
Initial conditions χ and Π for the Higgs field and its mo-mentum Π ≡ ˙ χ respectively. The thickness of the lines correspondsto 2 σ uncertainty in the value of the power spectrum, Eq. (9). be neglected compared to the kinetic energy. In this case,combining Eqs. (18) and (19) gives˙ H + H + ka = , ( ˙ χ (cid:29) U ) , (25)which for spatially flat curvature, k =
0, leads to H ( t ) = ¯ H + H ( t − ¯ t ) , ( ˙ χ (cid:29) U , k = , (26)where ¯ H ≡ H (¯ t ). By inserting this result into Eq. (19), wefind ˙ χ = M ¯ H (cid:2) + H ( t − ¯ t ) (cid:3) , ( ˙ χ (cid:29) U , k = . (27)that is the kinetic energy density scales as 1 / t by taking intoaccount the time dependence of H . This result [15] tells usthat an initial condition with large kinetic energy is attractedtowards one with smaller kinetic energy, but it also showsthat dropping the potential energy cannot be a good approxi-mation for arbitrarily large times. Moreover, notice that Eqs.(26) and (27) imply ¨ χ = − H ˙ χ (28)so the dynamics is not approaching the second condition in(24). Therefore, the argument above is not conclusive andwe need to solve the equations with U included in order tosee if the slow-roll regime is an attractor. We studied numerically the system in (20) assuming k =
0; this case is realistic and is the simplest one: it does not4equire an initial condition for a . We found that even for aninitial kinetic energy density Π of order 10 − ¯ M (which weregard as the maximal order of magnitude to have negligiblysmall quantum gravity), one should start from an initial fieldvalue χ of order 10 ¯ M Pl to inflate the universe for an appro-priate number of e-folds, i.e. N =
59. This value of χ isonly one order of magnitude bigger than the one needed inthe ordinary case, Π (cid:28) U ( χ ) ∼ − ¯ M Pl , where the initialkinetic energy is much smaller than the potential energy.Fig. 1 presents these results more quantitatively. Therethe initial conditions for Π have been chosen to be negativebecause positive values favor slow-roll even with respect tothe case where the initial kinetic energy is much smaller thanthe potential energy: this is because the potential in Eq. (7)is an increasing function of χ for χ (cid:29) v .We conclude that at the classical level Higgs inflation doesnot su ff er from a worrisome fine-tuning problem for the ini-tial conditions.
4. Quantum corrections
The theory in Eq. (1) is not renormalizable. This meansthat quantum corrections ∆Γ at a given order in perturbationtheory can generate terms that are not combinations of thosein the classical action S . In formulae the (quantum) e ff ectiveaction is given by: Γ = S + ∆Γ (29)where S + ∆Γ cannot generically be reproduced by substitut-ing the parameters in S with some renormalized quantities.A UV completion requires the existence of additional de-grees of freedom that render the theory renormalizable oreven finite. Much below the scale of this new physics, thee ff ective action can be approximated by an expansion of theform ∆Γ = (cid:90) d x √− g ( δ L + δ L + . . . ) (30)where δ L n represents a combination of dimension n opera-tors.We consider the one-loop corrections generated by allfields of the theory, both the matter fields and gravity. Ourpurpose is to apply it to inflationary and pre-inflationary dy-namics. We approximate ∆Γ by including all operators up todimension 4: ∆Γ ≈ (cid:90) d x √− g ( δ L + δ L ) . (31)This is the simplest approximation that allows us to includethe dynamics of the Higgs field and possess scale invarianceat high energies and high Higgs field values (up to running e ff ects). We have δ L = − δ ¯ M R (32) δ L = α R + β (cid:18) R − R µν R µν (cid:19) + δ Z H | ∂ H| − δλ |H| − δξ |H| R + . . . (33)where for each parameter p c in the classical action we haveintroduced a corresponding quantum correction δ p and thedots represent the additional terms due to the fermions andgauge fields of the SM. Notice that we have added general quantum corrections that are quadratic in the curvature ten-sors as they are also possible dimension 4 operators. Theseare parameterized by two dimensionless couplings α and β .We have neglected v as it is very small compared to inflation-ary energies.Our purpose is now to determine the RGEs for the renor-malized couplings p = p c + δ p as well as for the new couplings α and β generated by quan-tum corrections. Indeed the RGEs encode the leading quan-tum corrections. We will use the dimensional regularization(DR) scheme to regularize the loop integrals and the modi-fied minimal subtraction (MS) scheme to renormalize awaythe divergences. This as usual leads to a renormalizationscale that we denote with ¯ µ . In the absence of the dimensionful parameter v , the onlypossible contributions to the RGE of ¯ M Pl are the rainbow andthe seagull diagram contributions to the graviton propagatordue to gravity itself: the rainbow topology is the one of Fig.2, while the seagull one is obtained by making the two ver-tices of Fig. 2 coincide without deforming the loop.The seagull diagram vanishes as it is given by combina-tions of loop integrals of the form (cid:90) d d k k µ k ν k + i (cid:15) , (cid:90) d d k k + i (cid:15) , (34)where d is the space-time dimension in DR. These types ofloop integrals vanish in DR. The rainbow diagram does notcontribute to the RGE of ¯ M Pl either. The reason is that eachgraviton propagator carries a factor of 1 / ¯ M and each gravi-ton vertex carries a factor of ¯ M (because the graviton ki-netic term − ¯ M R / M ): the rainbow di-agram has two graviton propagators and two vertices, there-fore this contribution is dimensionless and cannot contribute R µνρσ R µνρσ is a linear combination of R , R µν R µν and a total derivative.
5o the RGE of a dimensionful quantity. We conclude that¯ M Pl does not run in this case. This argument assumes thatthe graviton wave function renormalization is trivial, whichwe have checked to be the case at the one-loop level at hand. Having neglected v all SM parameters are dimensionlessand thus cannot receive contributions from loops involvinggraviton propagators (that carry a factor of 1 / ¯ M ). There-fore, the SM RGEs apply and can be found (up to the three-loop level) in a convenient form in the appendix of Ref. [22]. Finally, we consider the RGEs for ξ , α and β . The one of ξ does not receive contribution from loops involving gravitonpropagators as they carry a factor of 1 / ¯ M and ξ is dimen-sionless. So the RGE of ξ receives contribution from the SMcouplings and ξ itself only [25, 26]:(4 π ) d ξ d ln ¯ µ = (1 + ξ ) (cid:18) y t − g − g + λ (cid:19) , (35)where y t is the top Yukawa coupling and g , g and g Y = √ / g are the gauge couplings of SU(3) c , SU(2) L and U(1) Y respectively.The RGEs of α and β receive two contributions: one frompure gravity loops (a rainbow and a seagull diagram), whichwe denote with β g , and one from matter loops, β m :(4 π ) d α d ln ¯ µ = β g α + β m α , (36)(4 π ) d β d ln ¯ µ = β g β + β m β . (37)One finds [27] β g α = − , β g β = , (38)and in the SM [26] β m α = − (1 + ξ ) , β m β = . (39) Let us start this section by commenting on fine-tuningsin the couplings, a relevant issue as inflation is motivatedby cosmological fine-tuning problems. The first equationin (39) has an important implication; the Feynman diagramthat leads to this contribution is given in Fig. 2. GenericallyHiggs inflation requires a rather large value of ξ , which im-plies a strong naturalness bound | α | (cid:38) ξ π . (40) Figure 2:
The leading loop diagram that generates the R term inthe e ff ective action. The dashed lines correspond to the Higgs field,while the external double lines represent gravitons. A large value of ξ is necessary at the classical level (see Eq.(12) and the corresponding discussion). At quantum levelone can obtain smaller values, but still ξ (cid:29) ξ (cid:38)
10 to fulfill the most recent observationalbounds, r (cid:46) . ξ is large [28].Since ξ (cid:29) R term with such a large coe ffi cient may participate in infla-tion. Therefore, we write the following e ff ective action: Γ = (cid:90) d x √− g (cid:20) L e ff SM − (cid:18) ¯ M + ξ |H| (cid:19) R + α R (cid:21) , (41)where the L e ff SM part corresponds to the e ff ective SM action.The scalar-tensor e ff ective action is Γ st = (cid:90) d x √− g (cid:20)
12 ( ∂φ ) − V e ff − (cid:0) ¯ M + ξφ (cid:1) R + α R (cid:21) . Here we have neglected the wave function renormalizationof the Higgs because ξ is large and we have fixed the unitarygauge. Moreover, V e ff is the SM e ff ective potential.As well-known, the R term corresponds to an additionalscalar. In order to see this one can add to the action the term − (cid:90) d x √− g α (cid:16) R + ω α (cid:17) , where ω is an auxiliary field: indeed by using the ω fieldequation one obtains immediately that this term vanishes. Onthe other hand, after adding that term Γ st = (cid:90) d x √− g (cid:20)
12 ( ∂φ ) − V − f R − ω α (cid:21) , (42)where f ≡ ¯ M + ω + ξφ .6ote that we have the non-canonical gravitational term − f R /
2. Like we did in section 2, we can go to the Einsteinframe (where we have instead the canonical Einstein term − ¯ M R E /
2) by performing a conformal transformation, g µν → ¯ M f g µν . (43)One obtains [31] Γ st = (cid:90) d x √− g (cid:20) L φ z − U e ff − ¯ M R (cid:21) , (44)where L φ z ≡ M z ( ∂φ ) + ( ∂ z ) , U e ff ( φ, z ) ≡
36 ¯ M z (cid:20) V e ff ( φ ) + α (cid:18) z − ¯ M − ξφ (cid:19) (cid:21) and we have introduced the new scalar z = √ f .Notice that when α →
0, the potential U e ff forces z =
6( ¯ M + ξφ ) and we recover the Higgs inflation action. Forlarge α (as dictated by a large ξ ), this conclusion cannot bereached. The absence of runaway directions in U e ff requires α > λ >
0, which is possible within the pure SM(without gravity [33]), although in tension with the mea-sured values of some electroweak observables [22, 29]. Ref.[31] studied a system that includes (44) as a particular case .It was found that inflation is never dominated by the Higgs,because its quartic self-coupling λ (which we assume to bepositive for the argument above) is unavoidably larger thanthe other scalar couplings, taking into account its RG flow.Even assuming that the Higgs has a dominant initial value, inour two-field context inflation starts only after the field evo-lution has reached an attractor where φ is subdominant. Wehave checked that this happens also when ξ is large.Therefore, the predictions are closer to those of Starobin-sky inflation, which are distinct from the Higgs inflation ones[35].
5. Conclusions
In conclusion, we have studied two di ff erent aspects ofstandard Higgs inflation - to seek how fine-tuned the initialconditions should be to fall into a slow-roll attractor solutionin an approximate exponentially flat Higgs potential in theEinstein frame. We started with a large kinetic energy, and Such tension, however, can be be eliminated by adding to the SM well-motivated new physics, which solve its observational problems [32]. Ref. [31] has an additional scalar which, however, can be consistentlydecoupled by taking its mass large enough. For another treatment of thedynamical system in (44) see Ref. [34]. we found that for an initial kinetic energy density of order10 − ¯ M (this is the maximum allowed order of magnitude toavoid quantum gravity corrections) the inflaton VEV shouldbe ∼
10 ¯ M Pl to sustain inflation long enough to give rise toenough e-folds.In the second half of the paper, we focused on the questionof viability of Higgs inflation in presence of large ξ , typi-cally required for explaining the observed CMB power spec-trum and the right tilt. We found that one would incur quan-tum corrections (at the lowest order) to the Ricci scalar, i.e.quadratic in Ricci scalar, α R , with a universality bound on α given by Eq. (40), unless the initial value of α is fine-tuned.If one includes this R term in the e ff ective action, both theHiggs and a new scalar degree of freedom are present. Bytaking ξ ∼ − and using the bound in Eq. (40), thepotential would be e ff ectively determined by the Starobin-sky scalar component z , and the CMB predictions would bedi ff erent from that of Higgs inflation. Acknowledgments
We thank Mikhail Shaposhnikov for valuable correspon-dence and Alexei A. Starobinsky for discussing. AM is sup-ported by the STFC grant ST / J000418 /
1. AM also acknowl-edges the kind hospitality from IPPP, Durham, during thecourse of this work. AS is supported by the Spanish Ministryof Economy and Competitiveness under grant FPA2012-32828, Consolider-CPAN (CSD2007-00042), the grant SEV-2012-0249 of the “Centro de Excelencia Severo Ochoa” Pro-gramme and the grant HEPHACOS-S2009 / ESP1473 fromthe C.A. de Madrid.
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