Higgs inflation

The properties of the recently discovered Higgs boson together with the absence of new physics at collider experiments allows us to speculate about consistently extending the Standard Model of particle physics all the way up to the Planck scale. In this context, the Standard Model Higgs non-minimally coupled to gravity could be responsible for the symmetry properties of the Universe at large scales and for the generation of the primordial spectrum of curvature perturbations seeding structure formation. We overview the minimalistic Higgs inflation scenario, its predictions, open issues and extensions and discuss its interplay with the possible metastability of the Standard Model vacuum.


Introduction and summary
Inflation is nowadays a well-established paradigm [1][2][3][4][5][6], able to explain the properties of the Universe at large scales and the generation of the primordial density fluctuations seeding structure formation [7][8][9][10]. The nature of the inflaton field remains unknown and its role could be played by any particle physics candidate able to imitate a slowly-moving scalar condensate.
In spite of dedicated searches, the only outcome of LHC experiments is a scalar particle with mass m H = 125.09 ± 0.21 ± 0.11 GeV [11,12], and properties that closely resemble those of the Standad Model (SM) Higgs. Among the different values that the Higgs mass could have taken, Nature has chosen one that may allow to extend the SM all the way till the Planck scale while staying in the perturbative regime [13]. The main limitation to this appealing possibility is the stability of the electroweak vacuum. Roughly speaking, the value of the Higgs self-coupling following from the SM renormalization group equations decreases with energy to eventually arrive to a minimum and start increasing thereafter. Whether it stays positive all the way up to the Planck scale, or becomes negative at some intermediate scale µ 0 depends, mainly, on the interplay between the Higgs mass m H and the top Yukawa coupling y t extracted from the reconstructed Monte-Carlo top mass in collider experiments [14]. Neglecting the effect of potential higher dimensional operators, the critical value of the top Yukawa coupling separating the region of absolute stability from the metastability/instability 1 regions is given by [15] y crit with α s (m Z ) the strong coupling constant at the Z boson mass. Within the present experimental and theoretical uncertainties [14][15][16], the SM is still compatible with stability, metastability or instability at the Planck scale [17][18][19][20][21], with the separation among the different cases strongly depending on the ultraviolet completion of gravity [22][23][24], cf. Fig. 2.
In the absence of physics beyond the SM, it is certainly tempting to identify the recently discovered Higgs boson with the inflaton condensate. Unfortunately, the self-coupling of the Higgs field highly exceeds the value λ ∼ 10 −13 needed to support a sufficiently long inflationary period without generating an excessively large spectrum of primordial density perturbations [6]. The situation is unchanged if one considers the renormalization group enhanced potential following from the extrapolation of the SM renormalization group equations to the inflationary scale [25][26][27]. The simplest way out is to modify the Higgs kinetic term in the large-field regime. In Higgs inflation 2 this is done by including a non-minimal coupling of the Higgs field to gravity [31], with R the Ricci scalar, H the Higgs doublet and ξ a dimensionless constant to be fixed by observations. The inclusion of (1.2) can be understood as an inevitable consequence of the quantization 1 The metastability region is defined as the region in parameter space where the lifetime of the SM vacuum exceeds the age of the Universe. 2 An alternative possibility involving a derivative coupling among the Einstein tensor and the Higgs kinetic term was considered in Refs. [28][29][30]. 3 Prior to Ref. [31], the modifications of Einstein's theory of Relativity by non-minimal couplings had been widely studied in the literature [32][33][34][35][36][37][38][39][40][41][42][43], but never with the SM Higgs playing the role of the inflaton. of the SM in a gravitational background, where it is indeed required for the renormalization of the energy-momentum tensor [44,45]. When written in the Einstein frame, the Higgs inflation scenario displays two distinct regimes. At low energies, it approximately coincides with the SM minimally coupled to gravity. At high energies, it becomes a chiral SM in which the radial component of the Higgs field has been integrated out [48,49]. In this latter regime, the effective Higgs potential becomes exponentially flat, allowing for inflation with the usual chaotic initial conditions. The associated predictions depend only on the number of e-folds of inflation, which is itself related to the duration of the heating stage. As the coupling of the Higgs field to the SM particles is experimentally known, the duration of this entropy production era can be computed in detail [50][51][52][53][54], leading to precise predictions for the inflationary observables that are in perfect agreement with observations [55].
The situation becomes more complicated when quantum corrections are taken into account. The shape of the inflationary potential depends then on the precise value of the Higgs mass and top quark Yukawa coupling at the inflationary scale. In addition to the plateau already existing at tree-level [56][57][58][59], the renormalization group enhanced potential can develop a quasi-inflection point along the inflationary trajectory [46,56,57,[59][60][61][62][63][64] or a hilltop [57,64,65], with different cases giving rise to different inflationary predictions.
Although a precise measurement of the inflationary observables could be understood as an interesting consistency check between cosmological observations and particle physics experiments [17,48,[66][67][68][69][70][71][72], the relation between low-and high-energy parameters in Higgs inflation is subject to unavoidable ambiguities related to the non-renormalizability of the SM non-minimally coupled to gravity [46,57,59,62,63,[73][74][75][76][77][78]. In particular, the finite parts of the counterterms needed to renormalize the tree level action lead to localized jumps in the SM renormalization group equations when connected to the chiral phase of Higgs inflation. The strength of these jumps should be understood as a remnant of a given ultraviolet completion and cannot be determined from the effective field theory itself [46,76,79]. If the jumps are significantly smaller   [15]. The red diagonal line stands for the critical value (1.1) of the top Yukawa coupling y crit t (m H ) leading to a negative Higgs self-coupling at an energy scale µ 0 below the Planck scale, with the dashed lines accounting for uncertainties associated with the strong coupling constant. To the left (right) of these lines, the SM is unstable (metastable). The filled elliptical contours account for the top Yukawa coupling following from to the CMS determination of the Monte-Carlo top quark mass m t = 172.38±0.10 (stat)±0.65 (syst) GeV in Ref. [47]. The additional empty elliptical contours illustrate the shifts associated with the ambiguous relation between these two quantities. than the associated coupling constants at that scale, Higgs inflation provides a clear connection among the properties of the Universe at large scales and the value of the SM parameters, as long as these do not give rise to vacuum instability. On the contrary, if the jumps in the coupling constants are large, the relation between high-and low-energy physics is lost, but Higgs inflation can surprisingly take place even if the SM vacuum is not completely stable [46].
In this article we review the minimalistic Higgs inflation scenario, its predictions, open issues and extensions, and the interplay with the potential metastability of the electroweak vacuum. The paper is organized as follows: • The general framework is introduced in Section 2. To illustrate the effect of non-minimal couplings, we consider an induced gravity scenario in which the effective Newton constant is completely determined by the vacuum expectation value of the Higgs field. Having this toy model in mind, we reformulate Higgs inflation in the so-called Einstein frame in which the gravitational part of the action takes the usual Einstein-Hilbert form and all non-linearities are moved to the scalar sector of the theory. After emphasizing the pole structure of the Einstein-frame kinetic term and its role in the asymptotic flatness of the Higgs inflation potential, we compute the tree-level inflationary observables and discuss the decoupling properties of the SM degrees of freedom.
• The limitations of Higgs inflation as a fundamental theory are reviewed in Section 3. In particular, we present a detailed derivation of the cutoff scales signaling the violation of perturbative unitarity in different scattering processes and advocate the interpretation of Higgs inflation as an effective field theory to be supplemented by an infinite set of higher dimensional operators. Afterwards, we adopt a self-consistent approach to Higgs inflation and formulate the set of assumptions leading to a partially controllable link between low-and high-energy observables. Based on the resulting framework, we analyze the contribution of radiative corrections to the renormalization group enhanced potential and their impact on the inflationary observables. We finish this section by considering the potential issues of Higgs inflation with the metastability of the SM vacuum.
• Some extensions and alternatives to the simplest Higgs inflation scenario are presented in Section 4. In particular, we address the difference between the metric and Palatini formulations of the theory and its extension to a fully scale invariant framework [80][81][82][83][84][85]. The inflationary predictions in these models are put in one to one correspondence with the pole structure of the Einstein-frame kinetic term, allowing for an easy comparison with the results of the standard Higgs inflation scenario.
Overall, we intend to complement the existing monographs in the literature [86][87][88] by i) providing a further insight on the classical formulation of Higgs inflation and by ii) focusing on the uncertainties associated with the non-renormalizability of the theory and their impact on model building.

General framework
The conditions for inflation are usually formulated as conditions of the local flatness on an arbitrary potential, which can in principle contain a large number of extrema and slopes [89]. This flatness is usually related to the existence of some approximate shift-symmetry, which for the purposes of Higgs inflation is convenient to reformulate as a non-linear realization of approximate scale-invariance.

Induced gravity
Let us start by considering an induced gravity scenario involving a scalar field h, a vector field B µ and a fermion field ψ, with interactions similar to those appearing in the SM of particle physics when written in the unitary gauge H = (0, h/ √ 2) T . The quantity F µν F µν stands for the standard B µ kinetic term, which for simplicity we take to be Abelian. In this toy model, the effective Newton constant is determined by the expectation value of the scalar field h, In order for G N,eff to be well-behaved, the non-minimal coupling ξ is restricted to take positive values. This condition is equivalent to the semi-positive definiteness of the scalar field kinetic term, as can be easily seen by performing a field redefinition h 2 → h 2 /ξ. An important property of the the induced gravity action (2.1) is its invariance under scale transformations Here α is an arbitrary constant, ϕ(x) compactly denotes the various fields in the model and ∆ ϕ 's are their corresponding scaling dimensions. The consequences of this dilatation symmetry are more easily understood in the so-called Einstein frame, in which the gravitational part of the action takes the usual Einstein-Hilbert form. This is achieved by performing a Weyl transformation of the metric 4 together with the corresponding Weyl rescaling of the vector and fermion fields, After some trivial algebra we obtain an Einstein-frame action 5 containing a non-canonical term for the Θ field. The coefficient of this kinetic term, involves a quadratic pole at Θ = 0 and a constant 8) varying between zero at ξ = 0 and −1/6 when ξ → ∞. The Θ-field kinetic term can be brought to a canonical form by performing an additional field redefinition, mapping the vicinity of the pole at Θ = 0 to φ → ∞. The resulting action (2.10) is invariant under shift transformations φ → φ+C, with C a constant. The exponential mapping in Eq. (2.9) indicates that such translational symmetry is nothing else than a non-linear realization of the original scale invariance we started with in Eq. (2.1) [92]. The transition to the Einstein frame is indeed analogous to the spontaneous breaking of scale symmetry, since we implicitly required the field h to acquire a non-zero expectation value in Eq. (2.4). The canonically normalized scalar field φ can be interpreted as the associated Goldstone boson and as such it is completely decoupled from the matter fields B µ and ψ. The non-minimal coupling to gravity effectively replaces h by F ∞ in all dimension-4 interactions involving conformal degrees of freedom. Note, however, that this decoupling statement does not apply to scale-invariant extensions including additional scalar fields [80,81,84,[93][94][95] or other non-conformal interactions such as R 2 terms [1, 96-98].

Higgs inflation from approximate scale invariance
Although the toy model presented above contains many of the key ingredients of Higgs inflation, it is not phenomenologically viable. In particular, the Einstein-frame potential is completely shiftsymmetric and does not allow for a graceful inflationary exit. On top of this, the scalar field φ is completely decoupled from all conformal fields, excluding the possibility of entropy production and the eventual onset of a radiation-dominated era. All these phenomenological limitations are intrinsically related to the exact realization of scale invariance and as such they should be expected to disappear once a (sizable) dimensionfull parameter is included into the action. This is precisely what happens in Higgs inflation. The total Higgs inflation action [31] the usual SM symmetry-breaking potential. As in the induced gravity scenario, the presence of the non-minimal coupling to gravity modifies the strength of the gravitational interaction and makes it dependent on the Higgs field, (2.14) In order to ensure a well-defined graviton propagator for all field values, the non-minimal ξ must be positive. 6 If ξ = 0, this requirement translates into a weakening of the effective Newton constant for increasing Higgs values. For non-minimal couplings in the range 1 ξ M 2 P /v 2 EW , this effect is important in the large-field regime h M P / √ ξ, but completely negligible otherwise. As we did in Section 2.1, it is convenient to reformulate Eq. (2.12) in the Einstein frame by performing a Weyl transformation g µν → Θg µν with In the new frame, the non-linearities associated with the non-minimal coupling to gravity are moved to the the scalar sector of the theory, which contains now a non-flat potential and a non-canonical kinetic term resulting from the rescaling of the metric determinant and the non-homogeneous part of the Ricci scalar transformation. The kinetic function shares some similarities with that in Eq. (2.7). In particular, it contains two poles located respectively at Θ = 0 and Θ = 1. The first pole is an inflationary pole, like the one appearing in the Figure 3: Comparison between the approximate expressions in Eq. (2.21) (dashed black and blue lines) and the exact solution (2.20) (solid red). Below the critical scale φ C , Higgs inflation coincides, up to highly suppressed corrections, with the SM minimally coupled to gravity. Above that scale, the Higgs field starts to decouple from the SM particles. The decoupling becomes efficient at a scale F ∞ , beyond which the model can be well approximated by a chiral SM in which the radial component of the Higgs field has been integrated out.
induced gravity scenario. This pole leads to an enhanced friction for the Θ field around Θ = 0 and allows for successful inflation to happen even if the non-constant potential V (Θ) is not sufficiently flat. The second pole is a Minkowski pole around which the Weyl transformation equals one and the SM non-minimally coupled to gravity is approximately recovered. To see this explicitly, we can perform an additional field redefinition , 7 to recast Eq. (2.16) in terms of a canonically normalized field φ. This differential equation admits an exact solution [50] |a|φ In terms of the original field h, we can distinguish two asymptotic regimes separated by a critical value The comparison between these approximate expressions and Eq. (2.20) is shown in Fig. 3. The large difference between the electroweak scale and the transition scale φ C at which the Higgs behavior is significantly modified allows us to identify, for all practical purposes, the vacuum expectation value v EW with φ = 0. In this limit, the Einstein-frame potential (2.17) can be rewriten as At φ < φ C we recover the usual Higgs potential (up to highly suppressed corrections, cf. Section 3.1). At φ > φ C the Einstein-frame potential becomes exponentially stretched and approaches the asymptotic value F ∞ at φ > M P /(2 |a|). The presence of M P in Eq. (2.11) modifies also the decoupling properties of the Higgs field as compared to those in the induced gravity scenario. In the Einstein-frame, the masses of the intermediate gauge bosons and fermions, 8 coincide with the SM masses in the small field regime (φ < φ C ) and evolve towards constant values proportional to F ∞ in the large-field regime (φ > M P /(2 |a|)). The transition to the Einstein-frame effectively replaces h by F (φ) in all (non-derivative) SM interactions. This behavior allows us to describe the Einstein-frame matter sector in terms of a chiral SM with "vacuum expectation value" F (φ) [48,49].

Tree-level inflationary predictions
The flattening of the Einstein-frame potential (2.23) due to the Θ = 0 pole, allows for inflation with the usual slow-roll conditions even if the potential V (Θ) is not sufficiently flat. Let us compute the inflationary observables in the corresponding region φ > φ C , where (2.26) Since we are dealing with a single-field model of inflation, the statistical information of the primordial fluctuations generated during inflation is dominantly encoded in the two-point correlation functions of scalar and tensor perturbations, or equivalently in their Fourier transform, the power spectra. Following the standard approach [104], we parametrize these spectra in an almost scale-invariant form, and compute the inflationary observables the standard slow-roll parameters and the primes denoting derivatives with respect to the scalar field φ. The quantities A s , n s , α s and r in these expressions should be understood as evaluated at the field value φ * ≡ φ(N * ) at which the reference pivot scale k * in Eq. (2.27) exits the horizon, i.e. at k * = a * H * . Here, denotes the corresponding number of e-folds and stands for the value of φ at the end of inflation, which is defined, as usual, by the condition (φ E ) ≡ 1. Equation (2.30) admits an exact inversion, with W −1 the lower branch of the Lambert function, and a rescaled number of e-folds accounting for the details of the end of inflation. Inserting Eq. (2.32) into (2.28) we obtain the following analytical expressions for the amplitude of the primordial spectrum of scalar perturbations, its spectral tilt, 35) and the tensor-to-scalar ratio (2.36) At large |a|N * , these predictions display an interesting attractor behavior, very similar to that appearing in α-attractor scenarios [105][106][107] (for a connection between this approach and the existence of stationary points along the inflationary potential see for instance Ref. [89]). Indeed, by taking into account that [108], we can obtain the approximate expressions 9 (2.38) at 8|a|N * 1. The free parameter |a| (or equivalently the non-minimal coupling ξ) can be fixed by combining Eq. (2.34) with the normalization of the primordial spectrum at large scales [55], among the non-minimal coupling ξ, the Higgs self-coupling λ and the number of e-foldsN * . The precise number of e-folds to be inserted in this expression depends on the duration of the heating stage.
As the couplings of the Higgs/inflaton field to the SM particles are experimentally known, the entropy production following the end of inflation can be computed in detail, removing the usual uncertainty in the number of e-folds [50,51,53]. 10 The depletion of the Higgs condensate is dominated by the non-perturbative production of W ± and Z bosons, which, contrary to the SM fermions, can experience bosonic amplification. Once created, the gauge bosons tend to decay into the SM fermions with a decay probability proportional to the instantaneous expectation value of the Higgs field φ(t). The onset of the radiation-domination era is determined either by: i) the moment in which the amplitude of the Higgs field approaches the critical value φ C at which the effective potential becomes quartic or ii) the time at which the energy density into relativistic fermions equals that of the Higgs condensate; whatever happens first. The estimates in Refs. [50,51,53] provide a range with the lower and upper bound associated respectively with the cases i) and ii) above. For the upper limit of this narrow window, we haveN * N * 59 and we can rewrite Eq. (2.40) as a relation between ξ and λ, ξ 47200 Note that a variation of the Higgs self-coupling in this equation can always be compensated by a change in the a priori unknown non-minimal coupling, with smaller values of λ associated with smaller values of ξ. For the tree-level value λ ∼ O(1), the non-minimal coupling must be rather large ξ ∼ O(10 4 ), but still significantly smaller than the value ξ ∼ M 2 P /v 2 EW ∼ 10 32 leading to sizable modifications of the effective Newton constant at low energies. In this regime, the parameter |a| is very close to its maximum value 1/6. This effective limit simplifies considerably the expression for the critical scale φ C separating the low-and high-energy regimes, which are in excellent agreement with the Planck collaboration results [55]. Note that, although computed in the Einstein frame, these predictions could have been alternatively obtained in the non-minimally coupled frame (2.12), provided a suitable redefinition of the slow-roll parameters and the number of e-folds in order to account for the Weyl factor relating the two frames [40,41,[110][111][112][113][113][114][115][116][117][118][119][120][121][122][123][124][125].

Effective field theory interpretation
The presence of the non-minimal coupling to gravity makes Higgs inflation perturbatively nonrenormalizable [73][74][75][76], which in turn forbids the interpretation of the model as an ultraviolet complete theory. Higgs inflation should be therefore understood as an effective description valid up to a given cut-off scale Λ [76,78]. This scale could either signal the onset of a strong coupling regime to be treated inside the SM by non-perturbative methods (such as resummations or lattice simulations) [126][127][128] or the appearance of additional degrees of freedom beyond the initiallyassumed SM content [129,130].

The cutoff scale
A priori, the cutoff scale of Higgs inflation could be as large as the Planck scale, where gravitational interactions should definitely become important. Although quite natural, the identification of these two energy scales may turn out to be theoretically inconsistent since other processes can break tree-level unitarity at lower energies. An estimate 11 of the cutoff scale can be obtained by expanding the fields around their background values, such that all kind of higher dimensional operators appear in the resulting action [76,132]. The computation is technically simpler in the original frame (2.11). In order to illustrate the procedure let us consider the graviscalar sector in Eq. (2.12). Expanding the metric and the Higgs field around their background valuesḡ µν andh, we obtain the following quadratic Lagrangian density for the perturbations γ µν and δh with γ =ḡ µν γ µν denoting the trace of the metric excitations. For non-vanishing ξ, the last term in this equation mixes the trace of the metric perturbation with the scalar perturbation δh [67,68,70,133]. In order to properly identify the different cutoff scales we must first diagonalize the kinetic terms. This can be done by performing a redefinition of the perturbations (γ µν , δh) → (γ µν , δĥ) with Once Eq. (3.2) has been reduced to a diagonal form, we can proceed to read the cutoff scales.
The easiest one to identify is that associated with purely gravitational interactions, which coincides with the effective Planck scale in Eq. (2.12). For scalar-graviton interactions, the leading-order higher-dimensional operator is Although we have focused on the graviscalar sector of the theory, the lack of renormalizability associated with the non-minimal coupling to gravity permeates all SM sectors involving the Higgs field. Let us consider, for instance, the symmetry breaking sector responsible for the masses of the intermediate W ± and Z bosons. Since we are working in the unitary gauge, it is sufficient to consider the scattering of non-Abelian vector fields with longitudinal polarization. The modification of the Higgs kinetic term at large field values changes the delicate pattern of cancellations in the SM and leads to tree-level unitarity violation at a scale Note that the cutoff scales

Relation between high-and low-energy parameters
Any sensible computation in Higgs inflation requires the inclusion of an infinite set of higherdimensional operators suppressed by a given cutoff scale. In what follows we will assume that the ultraviolet completion of the theory respects the original symmetries of the tree level action, and in particular the approximate scale invariance of Eq. (2.12) in the large-field regime and the associated shift-symmetry of its Einstein-frame formulation. This strong assumption forbids the generation of dangerous higher-dimensional operators that would completely spoil the predictivity of the model. In some sense, this requirement is not very different from the one implicitly assumed in other inflationary models involving trans-Planckian field displacements. The minimal set of higher-dimensional operators respecting the symmetries of the tree level action is the one generated by the theory itself via radiative corrections [46,76]. According to the standard rules of quantum field theory, the cancellation of the loop divergences stemming from the original action requires the inclusion of an infinite set of counterterms with a very specific structure. As in any other non-renormalizable theory, the outcome of this subtraction procedure depends on the renormalization scheme, with different choices corresponding to different assumptions about the ultraviolet completion of the theory. Among the different subtractions setups, a dimensional regularization scheme with an asymptotically scale-invariant subtraction point [48] fits pretty well with the approximate scale-symmetry of Eq. (2.11) in the large-field regime. 12 Given this frame and scheme, the minimal set of higher-dimensional operators generated by the theory can be computed in any Weyl-related frame provided that all fields and dimensionfull parameters are appropriately rescaled. The computation becomes particularly simple in the Einstein-frame, where the Weyl-rescaled renormalization point µ 2 Θ coincides with the standard field-independent prescription of renormalizable field theories, µ 2 Θ ∝ M 2 P . A general counter-term in dimensional regularization contains a finite part δL and a pole in = (4 − d)/2, with d the spacetime dimension. The coefficient of the pole is chosen to cancel the loop divergences stemming from the original action. Once this divergent part is removed, we are left with the finite contribution δL. The strength of this term should be understood as a remnant of a given ultraviolet completion and cannot be determined from the effective field theory itself [46,76,79]. From a quantitative point of view, the most relevant δL contributions are those associated with the Higgs and top-quark interactions. In the Einstein-frame at one loop, they take the form [46] with the primes denoting, as before, derivatives with respect to the scalar field φ. Note that these operators differ, as expected, from those in the tree-level action. This means that, while the finite part δλ b can be reabsorbed into the definition of the Higgs-self coupling, the finite parts δλ a , δy a and δy b should be promoted to new couplings constants with their own renormalization group equations. Once the associated operators are added to the tree-level action, the re-evaluation of radiative corrections will generate additional contributions beyond the original one-loop result. These contributions come together with new finite parts that should again be promoted to novel coupling constants with their own renormalization group equations. The iteration of this scheme gives rise to a renormalized action containing an infinite tower of higher-dimensional operators constructed out of F and its derivatives. For small field values, the function F becomes approximately linear (F ≈ φ, F = 1) and one recovers the SM non-minimally coupled to gravity up to highly suppressed interactions. In this limit, the coefficients of the infinite set of counterterms can be reabsorbed into the definition of the low energy couplings, as usually done in renormalizable field theories. When evolving towards the inflationary region, the function F becomes approximately constant (F ∞ = F ∞ , F = 0) and some of the previously absorbed finite parts are dynamically subtracted. The unknown finite parts modify therefore the running of the SM couplings at the transition region φ C < φ < 3/2 M P , such that the masses of the SM particles at the electroweak scale cannot be unequivocally related to their inflationary values without the precise knowledge of the ultraviolet completion [46,79,135]. If the finite parts appearing at every order in perturbation theory are small and of the same order as the loops generating them, the tower of higher dimensional operators generated by radiative corrections can be truncated [46]. In this case, the effect of the 1-loop threshold corrections can be imitated by an effective change 13 [46] with λ(µ) and y t (µ) evaluated using the SM renormalization group equations. We emphasize, however, that the truncation of the renormalization group equations is not essential for most of the results presented below, since, within the self-consistent approach to Higgs inflation, the functional form of the effective action is almost insensitive to it [46,63,76].

Potential scenarios and inflationary predictions
To describe the impact of radiative corrections on the inflationary predictions, we will make use of the renormalization group enhanced potential. This means that we will take as effective potential the one in Eq. (2.26) but with the Higgs self-coupling λ replaced by its corresponding running value λ(φ), Note that we are not promoting the non-minimal coupling ξ within F (φ) to a running coupling ξ(φ), as done, for instance, in Ref. [102], but rather assuming it to be constant during inflation. This is indeed a good approximation since the running of ξ following from the one-loop beta function in the SM non-minimally coupled to gravity [48,136], is rather small for realistic values of the couplings constant at the inflationary scale, β ξ ∝ O(10 −2 ) [63,103] (see also Ref. [137]). Although, strictly speaking, the renormalization group enhanced potential is not gauge invariant, the gauge dependence is small during slow-roll inflation, especially in the presence of extrema [16,20,138]. In the vicinity of the minimum of λ(φ), we can use the approximation [56] with the parameters λ 0 , q and b depending on the Einstein-frame Higgs and top quark masses at the inflationary scale according to the fitting formulas [56] with m * H and m * t in GeV. As seen in the last expression, the parameter b, standing for the derivative of the beta function for λ at the scale of inflation, is rather insensitive to the precise value of Higgs and top quark masses at that scale and can be well-approximated by b 2.3 × 10 −5 . The choice with α = 0.6 optimizes the convergence of perturbation theory [48,69], while respecting the asymptotic symmetry of the tree-level action (2.12) and its non-linear shift-symmetric Einsteinframe realization. In the second equality, we have introduced an effective parameter κ to facilitate the numerical computation of the inflationary observables. A simple inspection of Eqs. (3.12) and (3.14) reveals the existence of three different regimes: i) Universal/Non-critical regime: For λ 0 b/(16κ), the effective potential (3.12) is almost independent of the precise values of the parameters appearing within the logarithmic correction and can be well approximated by its tree-level form (2.26). In this regime, the spectral tilt and the tensor-to-scalar ratio coincide with their tree-level values [56,57,59,63], cf. Fig. 4.
ii) Critical regime: If λ 0 b/(16κ), both the first and the second derivative of the potential are approximately equal to zero, V V 0, leading to the appearance of a quasi-inflection point at Qualitatively, the vast majority of inflationary e-folds in this scenario takes place in the vicinity of the inflection point φ I , while the inflationary observables are related to the form of the potential as some value φ * > φ I .
Given the small value of the Higgs self-coupling in this scenario, λ 0 ∼ O(10 −6 ), the nonminimal coupling ξ can be rather small, ξ ∼ O(10), while still satisfying the normalization condition (2.39) [56,60,61,139]. This drastic decrease of the non-minimal coupling alleviates the tree-level unitary problems discussed in Section 3.1, and avoids the introduction of an ultraviolet completion at energy scales significantly below the Planck scale.  Figure 4: (Left) The tensor-to-scalar ratio r and the spectral tilt n S following from the effective potential (3.12) [63]. The non-minimal coupling ξ varies between 10 and 100 along the lines of constant κ, with larger values corresponding to smaller tensor-to-scalar ratios. The star in the lower part of the plot stands for the universal values in Eq. (2.44). (Right) The power spectrum P R as a function of the number of e-folds before the end of inflation and the associated comoving scale k in inverse megaparsecs [63]. The monotonic curve at the bottom of the plot corresponds to the universal/non-critical Higgs inflation scenario. The upper non-monotonic curves are associated with different realizations of the critical Higgs inflation scenario. The shaded regions stand for the latest 68% and 95% C.L. constraints provided by the Planck collaboration.
For small ξ values, the tensor-to-scalar ratio can become rather large, r ∼ O(10 −1 ) [56,60,61,139] (see also Ref. [103]). Note, however, that although the direct comparison of CMB data with the primordial spectrum at large scales displays a reasonable consistency, the global behavior of the spectrum cannot be accurately described by the simple expansion in (2.27), since the running of the spectral tilt α s ≡ d ln n s /d ln k and its running β s ≡ d 2 ln n s /d ln k 2 also become considerably large, cf. Fig. 5.
On top of the large-scale modifications, the non-monotonic evolution of the slow-roll parameter in the vicinity of the inflection point leads to the enhancement of the spectrum of primordial density fluctuations at small and intermediate scales. It is important to notice at this point that the standard slow-roll condition may break down if the potential becomes extremely flat and the inertial term in the inflaton equation of motion is not negligible with respect to the Hubble friction term [140][141][142]. In this regime, even the classical treatment is compromised since stochastic effects cannot longer be ignored [143][144][145][146]. If we restrict ourselves to situation in which the slow-roll approximation is satisfied during the whole inflationary trajectory, [63] 14 the height and width of the generated bump at fixed spectral tilt are correlated with the value of the tensor-to-scalar ratio r, cf. Fig. 4. Contrary to some claims in the literature [102], the maximum amplitude of the power-spectrum compatible with the 95% C.L Planck n s − r contours [63] is well below the critical threshold P max R 10 −2 − 10 −3 needed for primordial black hole formation [147][148][149]. This conclusion is unchanged if one Figure 5: (Left) Running of the spectral tilt α s ≡ d ln n s /d ln k in critical Higgs inflation as a function of the tensor-to-scalar ratio r and the spectral-tilt n s [64]. (Right) Running of the running of the spectral-tilt β s ≡ d 2 ln n s /d ln k 2 in the same case. The grey dot corresponds to the universal/non-critical Higgs inflation regime. [64] considers the effect of non-instantaneous threshold corrections [63], which could potentially affect the results given the numerical proximity of the inflection point (3.17) to the upper boundary of the transition region, φ 3/2M P .
iii) Hilltop regime: If λ b/(16κ) the potential develops a new minimum at large field values [58,64]. This minimum is separated from the electroweak minimum by a a local maximum where hilltop inflation can take place [150,151]. This scenario is highly sensitive to the initial conditions since the inflaton field must start on the electroweak vacuum side and close enough to the local maximum in order to support an extended inflationary epoch. On top of that, the fitting formulas in (3.15) may not be accurate enough for this case, since they are based on an optimization procedure around the λ(φ) minimum. The tensor-to-scalar ratio in this scenario differs also from the universal/non-critical Higgs inflation regime, but contrary to the critical case, it is decreased to 2 × 10 −5 < r < 1 × 10 −3 , rather than increased [57,64].

Vacuum metastability and high-temperature effects
The qualitative classification of scenarios and predictions presented in the previous section depends on the Higgs and top quark masses at the inflationary scale and holds independently of the value of their electroweak counterparts. In particular, any pair of couplings following from the SM renormalization group equations can be connected to a well-behaved pair of couplings in the chiral phase by a proper choice of the unknown threshold corrections. This applies also if the SM vacuum is not completely stable. Some examples of the 1-loop threshold correction δλ a needed to restore the universal/non-critical Higgs inflation scenario beyond µ 0 ∼ 10 9 , 10 10 and 10 12 GeV are shown in Fig. 6. For a detailed scan of the parameter space see Refs. [57,59].
The non-trivial interplay between vacuum stability and threshold corrections generates an additional minimum at large field values. Provided the usual chaotic initial conditions, the Higgs field will start its evolution in the trans-Planckian field regime, inflating the Universe while moving towards smaller field values. Since the new minimum is significantly wider and deeper than the electroweak one, it seems likely that the Higgs field will finish its post-inflationary evolution there. Note, however, that this conclusion strongly depends on the ratio between the energy stored in the Higgs condensate and the depth of the second minimum. If this ratio is large, the entropy production at the end of inflation may significantly modify the shape of the potential, triggering its stabilization and the evolution of the Higgs field towards the desired electroweak vacuum [51]. At one-loop, the finite temperature corrections to be added on top of the Einstein-frame renormalization group enhanced potential take the form [152] with the plus sign corresponding to fermions and the minus sign to bosons and m B,F standing for the Einstein-frame masses in Eq. (2.25). The most important contributions in Eq. (3.18) come from the the top quark and the electroweak bosons, with the corresponding coupling constants y t and g evaluated at µ yt = 1.8 T and µ g = 7 T , in order to minimize the radiative corrections [153]. A detailed analysis of the universal/non critical Higgs inflation scenario reveals that the temperature of the decay products generated during the heating stage exceeds generically the temperature at which the secondary minimum at large field values disappears [50,51], see Fig 6. 15 The stabilization becomes favored for increasing µ 0 values 16 and holds even if this scale is as low 15 A detailed scan of the parameter space assuming instantaneous conversion of the inflaton energy density into a thermal bath was performed in Ref. [59]. 16 The larger µ 0 is, the shallower and narrower the "wrong" minimum becomes, cf. Fig. 6.
as 10 10 GeV [51]. The thermally-corrected potential enables the Higgs field to relax to the SM vacuum. After the heating stage, the temperature decreases due to the expansion of the Universe and the secondary minimum at large field values reappears, first as a local minimum and eventually as the global one. When that happens, the Higgs field is already trapped in the electroweak vacuum. Although the barrier separating the two minima prevents a direct decay, the Higgs field could still tunnel to the global minimum. The probability for this to happen is, however, very small and the lifetime of the electroweak vacuum highly exceeds the life of the Universe [66,[154][155][156]. Universal/Non-critical Higgs inflation with a graceful exit can therefore take place even if the SM vacuum is not completely stable [51]. The situation changes completely if one considers the critical Higgs inflation scenario. In this case, the energy stored in the Higgs condensate is comparable the depth of the secondary minimum and symmetry restoration does not take place. Unless the initial conditions are extremely fine-tuned, the Higgs field will relax to the minimum of the potential at Planckian values, leading with it to the inevitably collapse of the Universe [157]. Critical Higgs inflation does, therefore, require the absolute stability of the SM vacuum [51].

Variations and extensions
Many variations and extensions of Higgs inflation have been considered in the literature, see for instance Refs. [129,. In what follows we will restrict ourselves to those proposals that are more closely related to the spirit of the original scenario. In particular, we will address a Palatini formulation of Higgs inflation and the uplifting of the model to a fully scale invariant framework.

Palatini Higgs inflation
In the usual formulation of Higgs inflation presented in Section 2.2, the action is minimized with respect to the metric. This procedure implicitly assumes the existence of a Levi-Civita connection expressed in terms of the metric tensor and its derivatives, and the inclusion of a York-Hawking-Gibbons term ensuring the cancellation of a total derivative term with no-vanishing variation at the boundary [185,186]. One could alternatively consider a Palatini formulation of gravity in which the metric and the connection are taken to be independent variables and no additional boundary term is required to obtain the equations of motion. Roughly speaking, this formulation corresponds to assuming an ultraviolet completion involving different gravitational degrees of freedom.
Although the metric and Palatini formulations of General Relativity give rise to the same equations of motion, this is not true for scalar-tensor theories as Higgs inflation. To see this explicitly let us consider the Higgs inflation action in Eq. (2.12) with R = g µν R µν (Γ, ∂Γ) and Γ a non-Levi-Civita connection. Performing a Weyl rescaling of the metric g µν → Θg µν with Θ given by Eq. (2.15) we obtain an Einstein-frame action the kinetic term for the h field can be brought to a canonical form. The graviscalar action (4.1) at φ v EW becomes The comparison of the latest expression with Eq. (2.24) reveals some important differences between the metric and Palatini formulations. In both cases, the effective Einstein-frame potential smoothly interpolates between a quartic potential at small field values and an asymptotically flat potential in the large-field regime. Note, however, that the transition in the Palatini case is rather direct and does not involve the quadratic piece appearing in the metric formulation. On top of that, the flatness of the asymptotic plateau is different in the two cases, due to the effective change in |a|. The Palatini dependence |a| = ξ has an strong impact on the inflationary observables. In the large-field regime they read a rescaled number of e-folds and the value of inflaton field at the end of inflation ( (φ end ) ≡ 1), with φ E = 3/2 arcsinh(4/ √ 3) M P corresponding to the ξ → ∞ limit and φ E = 2 √ 2 M P to the end of inflation in a minimally coupled λφ 4 theory. A relation between the non-minimal coupling ξ, the self-coupling λ and the number of e-foldsN * can be obtained by taking into account the amplitude of the observed power spectrum in Eq. (2.39), ξ 3.8 × 10 6N 2 * λ . (4.10) A simple inspection of Eq. (4.7) reveals that the predicted tensor-to-scalar ratio in Palatini Higgs inflation is within the reach of current or future experiments [187] only if ξ 10, which, assuminḡ N 59, requires a very small coupling λ 10 −9 . For a discussion of unitarity violations in the Palatini formulation see Ref. [188].

Higgs-Dilaton model
The universal predictions of Higgs inflation are intimately related to the approximate scaleinvariance of Eq. (2.12) in the large-field regime. The embedding of Higgs inflation in a fully scale-invariant framework was considered in Refs. [80][81][82][83][84][85]. In the unitary gauge H = (0, h/ √ 2) T , the graviscalar sector of the Higgs-Dilaton model considered in these papers takes the form a scale-invariant version of the SM symmetry-breaking potential and α, β positive dimensionless parameters. The existence of a well-defined graviton propagator for all field values requires the non-minimal gravitational couplings to be positive-definite, i.e. ξ h , ξ χ > 0. In the absence of gravity, the ground state of Eq. (4.11) corresponds to the minima of the scale-invariant potential (4.12). For α = 0 and β = 0, this potential contains two flat directions h 0 = ±αχ 0 , meaning that any solution with χ 0 = 0 leads to the spontaneous symmetry breaking of scale invariance. 17 The non-zero expectation value of the χ field induces the effective Planck mass and the electroweak scale. The relation between these highly hierarchical scales is reproduced by properly fine-tuning the parameter α to α ∼ v 2 /M 2 P ∼ 10 −32 . For this small value, the flat valleys in the potential U (h, χ) are essentially aligned and we can safely approximate α 0 for all inflationary purposes.
To compare the inflationary predictions of this model with those of the standard Higgs-inflation scenario, let us perform a Weyl rescaling g µν → M 2 P /(ξ h h 2 + ξ χ χ 2 )g µν followed by a field redefinition [85] with γ ≡ ξ χ 1 + 6ξ χ , a ≡ − ξ h 1 + 6ξ h ,ā ≡ a 1 − ξ χ ξ h . (4.14) After some algebra, we obtain a rather simple Einstein-frame action [84,85]  and a non-canonical, albeit diagonal, kinetic sector. The kinetic function for the Θ field, contains two "inflationary" poles at Θ = 0 and Θ = c/|ā| and a "Minkowski" pole at Θ = 1, where the usual SM minimally coupled to gravity is approximately recovered. As in the single field case, the "Minkowski" pole can be safely neglected during inflation. Interestingly, the field-derivative manifold in this limit becomes a maximally symmetric hyperbolic space with Gaussian curvature a < 0 [84].
With the standard slow-roll initial conditions, the field Θ will tend to roll-down its potential. While this happens (Θ 0), the Φ-field kinetic term is effectively suppressed and the dilaton "freezes" at its initial value Φ = Φ 0 [80]. This freezing is an immediate consequence of scale invariance. As in the single field case, the shift symmetry Φ → Φ + C in Eq. (4.15) allows us to interpret Φ as the Goldstone boson, or dilaton, associated with the spontaneous breaking of dilatations. As shown in Ref. [80], the equation of motion for this field coincides with the conservation equation for the scale current, leading to an effective restriction of h and χ to ellipsoidal trajectories in the {h, χ} plane with Φ = constant. This means that, in spite of dealing with a two-field model, no isocurvature perturbations nor non-gaussianities are produced [80]. Given the resulting single-field action for Θ, the inflationary observables can be computed using the standard techniques. If the inflationary dynamics is dominated by the Higgs component (ξ h ξ χ ), the spectral tilt and the tensor-to-scalar ratio take the compact form with |a| 1/6 in order to satisfy the normalization condition (2.39). Note that these expressions rapidly converge to the Higgs inflation values (2.38) for 4cN * 1. For increasing c and fixed N * , the spectral tilt decreases linearly and the tensor-to-scalar ratio approaches zero.

Concluding remarks
Before the start of the LHC, it was widely believed that we would find a plethora of new particles and interactions that would reduce the Standard Model to a mere effective description of Nature at energies below the TeV scale. From a bottom-up perspective, new physics was typically advocated to cure the divergences associated with the potential growth of the Higgs self-coupling at high energies. The discovery of a relatively light Higgs boson in the LHC concluded the quest of the Standard Model spectrum while demystifying the concept of naturalness and the role of fundamental scalar fields in particle physics and cosmology. The Standard Model is now a confirmed theory that could stay valid all the way up till the Planck scale and provide a solid theoretical basis for describing the early Universe.
The Higgs field itself could be responsible for inflation if a minimalistic, and at the same time compelling, non-minimal coupling to gravity is added to the Standard Model action. The value of this coupling can be fixed by the normalization of the spectrum of primordial density perturbations, leaving a theory with no free parameters. On top of that, the experimental knowledge of the Standard Model couplings reduces the usual uncertainties associated with the heating stage and allows us to obtain precise predictions for the inflationary observables that are in excellent agreement with observations.
Of course, there is no such thing as a free lunch. The mere existence of gravity makes the theory non-renormalizable and forces its interpretation as an effective field theory. Even in its self-consistent approach to Higgs inflation, the absence of an ultraviolet completion introduces some uncertainties associated with the finite parts of the counterterms needed to make the theory finite. In particular, the unknown coefficients obscure the connection between low-and highenergy observables. If they are small, Higgs inflation provides an appealing relation between the Standard Model parameters and the properties of the Universe at large scales. If they are large, this connection is lost but Higgs inflation can surprisingly take place even if the Standard Model vacuum is not completely stable.

Acknowledgments
The author acknowledges support from the Deutsche Forschungsgemeinschaft through the Open Access Publishing funding programme of the Baden-Württemberg Ministry of Science, Research and Arts and the Ruprecht-Karls-Universität Heidelberg as well as through the project TRR33 "The Dark Universe". He thanks Guillem Domenech, Georgios Karananas and Martin Pauly for useful comments and suggestions on the manuscript.