Classical and Quantum Initial Conditions for Higgs Inflation

We investigate whether Higgs inflation can occur in the Standard Model starting from natural initial conditions or not. The Higgs has a non-minimal coupling to the Ricci scalar. We confine our attention to the regime where quantum Einstein gravity effects are small in order to have results that are independent of the ultraviolet completion of gravity. At the classical level we find no tuning is required to have a successful Higgs inflation, provided the initial homogeneity condition is satisfied. On the other hand, at the quantum level we obtain that the renormalization for large non-minimal coupling requires an additional degree of freedom that transforms Higgs inflation into Starobinsky $R^2$ inflation, unless a tuning of the initial values of the running parameters is made.


Introduction
Inflation [1,2,3] is perhaps one of the most natural way to stretch the initial quantum vacuum fluctuations to the size of the current Hubble patch, seeding the initial perturbations for the cosmic microwave background (CMB) radiation and large scale structure in the universe [4] (for a theoretical treatment, see [5]). Since inflation dilutes all matter it is pertinent that after the end of inflation the universe is filled with the right thermal degrees of freedom, i.e. the Standard Model (SM) degrees of freedom (for a review on pre-and postinflationary dynamics, see [6]). The most economical way to achieve this would be via the vacuum energy density stored within the SM Higgs, whose properties are now being measured at the Large Hadron Collider (LHC) [7,8]. Naturally, the decay of the Higgs would create all the SM quarks and leptons observed within the visible sector of the universe. Albeit, with just alone SM Higgs and minimal coupling to gravity, it is hard to explain the temperature anisotropy observed in the CMB radiation without invoking physics beyond the SM 1 .
However, a very interesting possibility may arise within the SM if the Higgs were to couple to gravity non-minimally -such as in the context of extended inflation [10], which has classical front and the other on the quantum front.
I. Classically, a large VEV of the inflaton does not pose a big problem as long as the initial energy density stored in the inflaton system, in the Einstein frame, is below the cut-off of the theory. Since, the potential energy remains bounded below this cut-off, the question remains -what should be the classical initial condition for the kinetic energy of the inflaton?
A-priori there is no reason for the inflaton to move slowly on the plateau, therefore the question we wish to settle in this paper is what should be the range of phase space allowed for a sustainable inflation to occur with almost a flat potential? The aim of this paper is to address this classical initial condition problem 2 . Here we strictly assume homogeneity of the universe from the very beginning; we do not raise the issue of initial homogeneity condition required for a successful inflation; this issue has been discussed earlier in a generic inflationary context in many classic papers (see [15,16]). In our paper, instead we look into the possibility of initial phase space for a spatially flat universe, and study under what preinflationary conditions Higgs inflation could prevail.
II. At quantum level, the original Higgs model poses a completely different challenge. A large ξ will inevitably modify the initial action. One may argue that there will be quantum corrections to the Ricci scalar, R, such as a Higgs-loop correction -leading to a quadratic in curvature action, i.e. R + αR 2 type correction, where α is a constant, whose magnitude we shall discuss in this paper. The analysis is based on the renormalization group equations (RGEs) of the SM parameters and the gravitational interactions. By restricting for simplicity the study to operators up to dimension 4, the RGE analysis will yield a gravitational action that will become very similar to the Starobinsky type inflationary model [17] 3 .
One of the features of theories with curvature squared terms is that there are extra degrees of freedom involved in the problem, besides the SM ones and the graviton. There is another scalar mode arising from R 2 , which will also participate during inflation. The question then arises when this new scalar degree of freedom becomes dominant dynamically, and play the role of an inflaton creating the initial density perturbations? 2 Some single monomial potentials and exponential potentials exhibit a classic example of late time attractor where the inflaton field approaches a slow roll phase from large initial kinetic energy, see [13,14]. 3 In principle, large ξ may also yield higher derivative corrections up to quadratic in order, see [18], and also higher curvature corrections, but in this paper, we will consider for simplicity the lowest order corrections. We will argue that the αR 2 is necessarily generated unless one is at the critical point of Ref. [19] or invokes a fine-tuning on the initial values of the running parameters.
The aim of this paper will be to address both the classical and quantum issues.
We briefly begin our discussion with essential ingredients of Higgs inflation in section 2, then we discuss the classical pre-inflationary initial conditions for Higgs inflation in section 3. In this section, we discuss both analytical 3.1, and numerical results 3.2. In section 4, we discuss the quantum correction to the original Higgs inflation model, i.e. we discuss the RGEs of the Planck mass in subsection 4.1, SM parameters in 4.2, and the gravitational correction arising due to large ξ in subsection 4.3, respectively. We briefly discuss our results and consequences for inflation in subsection 4.4, before concluding our paper.

The model
Let us define the Higgs inflation model [11]. The action is where L SM is the SM Lagrangian minimally coupled to gravity, ξ is the parameter that determines the non-minimal coupling between the Higgs and the Ricci scalar R, and H is the Higgs doublet. The part of the action that depends on the metric and the Higgs field only (the scalar-tensor part) is where V = λ(|H| 2 − v 2 /2) 2 is the Higgs potential and v is the electroweak Higgs VEV. We take a sizable non-minimal coupling, ξ > 1, because this is required by inflation as we will see. The non-minimal coupling −ξ|H| 2 R can be eliminated through the conformal transformation The original frame, where the Lagrangian has the form in (1), is called the Jordan frame, while the one where gravity is canonically normalized (obtained with the transformation above) is called the Einstein frame. In the unitary gauge, where the only scalar field is the radial mode φ ≡ 2|H| 2 , we have (after the conformal transformation) where K ≡ Ω 2 + 6ξ 2 φ 2 /M 2 Pl /Ω 4 . The non-canonical Higgs kinetic term can be made canonical through the (invertible) field redefinition φ = φ(χ) defined by with the conventional condition φ(χ = 0) = 0. One can find a closed expression of χ as a function of φ: Thus, χ feels a potential Let us now recall how slow-roll inflation emerges. From (5) and (7) it follows [11] that U is exponentially flat when χ M Pl , which is the key property to have inflation. Indeed, for such high field values the slow-roll parameters are guaranteed to be small. Therefore, the region in field configurations where χ >M Pl (or equivalently [11] φ > M Pl / √ ξ) corresponds to inflation. We will investigate whether successful sow-roll inflation emerges also for large initial field kinetic energy in the next section. Here we simply assume that the time derivatives are small. All the parameters of the model can be fixed through experiments and observations, including ξ [11,20]. ξ can be obtained by requiring that the measured power spectrum [4], is reproduced for a field value φ = φ b corresponding to an appropriate number of e-folds of inflation [20]: where φ end is the field value at the end of inflation, that is For N = 59, by using the classical potential we obtain where the uncertainty corresponds to the experimental uncertainty in Eq. (9). Note that ξ depends on N: This result indicates that ξ has to be much larger than one because λ ∼ 0.1 (for precise determinations of this coupling in the SM see Refs. [21,22]).

Pre-inflationary dynamics: classical analysis
Let us now analyze the dynamics of this classical system in the homogeneous case without making any assumption on the initial value of the time derivativeχ. We will assume that the universe is sufficiently homogeneous to begin inflation.
In the Einstein frame S st is given by: where U is the Einstein frame potential given in Eq. (7). Let us assume a universe with three dimensional translational and rotational symmetry, that is a Friedmann-Robertson-Walker (FRW) metric Then the Einstein equations and the scalar equations imply the following equations for a(t) and the spatially homogeneous field χ(t) where H ≡ȧ/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), which is therefore dependent. Thus, we have to solve the following system with initial conditions wheret is some initial time before inflation and χ,Π andā are the initial conditions for the three dynamical variables. In the case k = 0 the previous system can be reduced to a single second order equation. Indeed, by setting k = 0 in Eq. (18) and inserting it in Eq. (17), one obtains This equation has to be solved with two initial conditions (for χ andχ). The initial condition for a is not needed in this case as its overall normalization does not have a physical meaning for k = 0. We confine our attention to the regime where quantum Einstein gravity corrections are small: such that we can ignore the details of the ultraviolet (UV) completion of Einstein gravity 4 . However, we do not always require to be initially in a slow-roll regime. The first and second conditions in (22) come from the requirement that the energy-momentum tensor is small (in units of the Planck scale) so that it does not source a large curvature; the third condition ensures that the three-dimensional curvature is also small. The first condition is automatically fulfilled by the Higgs inflation potential, Eq. (7): the quartic coupling 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 (approximately) de Sitter space, which is maximally symmetric; therefore we do not consider them as a fine-tuning in the initial conditions. In de Sitter we have to set k = 0 andḢ = 0, which then impliesχ = 0 from Eq. (19). Notice also that we cannot start from an exact de Sitter, given Eq. (17): the potential U is almost, but not exactly flat in the large field case (see Eq. (7)).
In order for the Higgs to trigger inflation sooner or later one should have a slow-roll regime, where the kinetic energy is small compared to the potential energy,χ 2 /2 U, and the field equations are approximatelẏ The conditions for this to be true arė We will use these conditions rather than the standard 1 and η 1 as we do not assume a priori a small kinetic energy.

Analytic approximations in simple cases
Let us assume, for simplicity, that the parameter k in the FRW metric vanishes, i.e. a spatially flat metric, and consider the caseχ 2 U, such that the potential energy can be neglected compared to the kinetic energy. In this case, combining Eqs. (18) and (19) giveṡ which for spatially flat curvature, k = 0, leads to whereH ≡ H(t). By inserting this result into Eq. (19), we findχ that is the kinetic energy density scales as 1/t 2 by taking into account the time dependence of H. This result [15] tells us that an initial condition with large kinetic energy is attracted towards one with smaller kinetic energy, but it also shows that dropping the potential energy cannot be a good approximation for arbitrarily large times. Moreover, notice that Eqs. (26) and (27) so the dynamics is not approaching the second condition in (24). Therefore, the argument above is not conclusive and we need to solve the equations with U included in order to see if the slow-roll regime is an attractor.

Numerical studies
We studied numerically the system in (20) assuming k = 0; this case is realistic and is the simplest one: it does not require an initial condition for a. We found that even for an initial kinetic energy density Π 2 of order 10 −3M4 Pl (which we regard as the maximal order of magnitude to have negligibly small quantum gravity), one should start from an initial field value χ of order 10M Pl to inflate the universe for an appropriate number of e-folds, i.e. N = 59. This value of χ is only one order of magnitude bigger than the one needed in the ordinary case, Π 2 U(χ) ∼ 10 −10M Pl , where the initial kinetic energy is much smaller than the potential energy. Fig. 1 presents these results more quantitatively. There the initial conditions for Π have been chosen to be negative because positive values favor slow-roll even with respect to the case where the initial kinetic energy is much smaller than the potential energy: this is because the potential in Eq. (7) is an increasing function of χ for χ v.
We conclude that at the classical level Higgs inflation does not suffer from a worrisome fine-tuning problem for the initial conditions.

Quantum corrections
The theory in Eq. (1) is not renormalizable. This means that quantum corrections ∆Γ at a given order in perturbation theory can generate terms that are not combinations of those in the classical action S . In formulae the (quantum) effective action is given by: where S + ∆Γ cannot generically be reproduced by substituting the parameters in S with some renormalized quantities. A UV completion requires the existence of additional degrees of freedom that render the theory renormalizable or even finite. Much below the scale of this new physics, the effective action can be approximated by an expansion of the form where δL n represents a combination of dimension n operators. We consider the one-loop corrections generated by all fields of the theory, both the matter fields and gravity. Our purpose is to apply it to inflationary and pre-inflationary dynamics. We approximate ∆Γ by including all operators up to dimension 4: This is the simplest approximation that allows us to include the dynamics of the Higgs field and possess scale invariance at high energies and high Higgs field values (up to running effects). We have where for each parameter p c in the classical action we have introduced a corresponding quantum correction δp and the dots represent the additional terms due to the fermions and gauge fields of the SM. Notice that we have added general 5 quantum corrections that are quadratic in the curvature tensors as they are also possible dimension 4 operators. These are parameterized by two dimensionless couplings α and β.
We have neglected v as it is very small compared to inflationary energies. Our purpose is now to determine the RGEs for the renormalized couplings p = p c + δp as well as for the new couplings α and β generated by quantum corrections. Indeed the RGEs encode the leading quantum corrections. We will use the dimensional regularization (DR) scheme to regularize the loop integrals and the modified minimal subtraction (MS) scheme to renormalize away the divergences. This as usual leads to a renormalization scale that we denote withμ.

RGE of the Planck mass
In the absence of the dimensionful parameter v, the only possible contributions to the RGE ofM Pl are the rainbow and the seagull diagram contributions to the graviton propagator due to gravity itself: the rainbow topology is the one of Fig.  2, while the seagull one is obtained by making the two vertices of Fig. 2 coincide without deforming the loop.
The seagull diagram vanishes as it is given by combinations of loop integrals of the form where d is the space-time dimension in DR. These types of loop integrals vanish in DR. The rainbow diagram does not contribute to the RGE ofM Pl either. The reason is that each graviton propagator carries a factor of 1/M 2 Pl and each graviton vertex carries a factor ofM 2 Pl (because the graviton kinetic term −M 2 Pl R/2 is proportional toM 2 Pl ): the rainbow diagram has two graviton propagators and two vertices, therefore this contribution is dimensionless and cannot contribute 5 R µνρσ R µνρσ is a linear combination of R 2 , R µν R µν and a total derivative. 5 to the RGE of a dimensionful quantity. We conclude that M Pl does not run in this case. This argument assumes that the graviton wave function renormalization is trivial, which we have checked to be the case at the one-loop level at hand.

RGEs of SM parameters
Having neglected v all SM parameters are dimensionless and thus cannot receive contributions from loops involving graviton propagators (that carry a factor of 1/M 2 Pl ). Therefore, the SM RGEs apply and can be found (up to the threeloop level) in a convenient form in the appendix of Ref. [22].

RGEs of gravitational couplings
Finally, we consider the RGEs for ξ, α and β. The one of ξ does not receive contribution from loops involving graviton propagators as they carry a factor of 1/M 2 Pl and ξ is dimensionless. So the RGE of ξ receives contribution from the SM couplings and ξ itself only [25,26]: where y t is the top Yukawa coupling and g 3 , g 2 and g Y = √ 3/5g 1 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 from pure gravity loops (a rainbow and a seagull diagram), which we denote with β g , and one from matter loops, β m : One finds [27] β g α = − and in the SM [26] β m

Higgs-to-Starobinsky inflation
Let us start this section by commenting on fine-tunings in the couplings, a relevant issue as inflation is motivated by cosmological fine-tuning problems. The first equation in (39) has an important implication; the Feynman diagram that leads to this contribution is given in Fig. 2. Generically Higgs inflation requires a rather large value of ξ, which implies a strong naturalness bound A large value of ξ is necessary at the classical level (see Eq. (12) and the corresponding discussion). At quantum level one can obtain smaller values, but still ξ 1 [28,29]. A possible exception is Higgs inflation at the critical point [19]; however, ξ 10 to fulfill the most recent observational bounds, r 0.1 [30]. Moreover, in previous analysis of Higgs inflation at the critical point the wave function renormalization of the Higgs field has been neglected, an approximation that is under control when ξ is large [28].
Since ξ 1 generically, (40) indicates that an additional R 2 term with such a large coefficient may participate in inflation. Therefore, we write the following effective action: where the L eff SM part corresponds to the effective SM action. The scalar-tensor effective action is Here we have neglected the wave function renormalization of the Higgs because ξ is large and we have fixed the unitary gauge. Moreover, V eff is the SM effective potential. As well-known, the R 2 term corresponds to an additional scalar. In order to see this one can add to the action the term where ω is an auxiliary field: indeed by using the ω field equation one obtains immediately that this term vanishes. On the other hand, after adding that term where and we have introduced the new scalar z = √ 6 f . Notice that when α → 0, the potential U eff forces z 2 = 6(M 2 Pl + ξφ 2 ) and we recover the Higgs inflation action. For large α (as dictated by a large ξ), this conclusion cannot be reached. The absence of runaway directions in U eff requires α > 0 and λ > 0, which is possible within the pure SM (without gravity [33]), although in tension 6 with the measured values of some electroweak observables [22,29]. Ref.
[31] studied a system that includes (44) as a particular case 7 . It was found that inflation is never dominated by the Higgs, because its quartic self-coupling λ (which we assume to be positive for the argument above) is unavoidably larger than the other scalar couplings, taking into account its RG flow. Even assuming that the Higgs has a dominant initial value, in our two-field context inflation starts only after the field evolution has reached an attractor where φ is subdominant. We have checked that this happens also when ξ is large.
Therefore, the predictions are closer to those of Starobinsky inflation, which are distinct from the Higgs inflation ones [35].

Conclusions
In conclusion, we have studied two different aspects of standard Higgs inflation -to seek how fine-tuned the initial conditions should be to fall into a slow-roll attractor solution in an approximate exponentially flat Higgs potential in the Einstein frame. We started with a large kinetic energy, and we found that for an initial kinetic energy density of order 10 −3M4 Pl (this is the maximum allowed order of magnitude to avoid quantum gravity corrections) the inflaton VEV should be ∼ 10M Pl to sustain inflation long enough to give rise to enough e-folds.
In the second half of the paper, we focused on the question of viability of Higgs inflation in presence of large ξ, typically required for explaining the observed CMB power spectrum and the right tilt. We found that one would incur quantum corrections (at the lowest order) to the Ricci scalar, i.e. quadratic in Ricci scalar, αR 2 , with a universality bound on α given by Eq. (40), unless the initial value of α is fine-tuned. If one includes this R 2 term in the effective action, both the Higgs and a new scalar degree of freedom are present. By taking ξ ∼ 10 2 − 10 4 and using the bound in Eq. (40), the potential would be effectively determined by the Starobinsky scalar component z, and the CMB predictions would be different from that of Higgs inflation.