Higgs Inflation

Home , Alessandro Strumia, Andrei Linde, Inflaton

Higgs inﬂation

Javier Rubio

Institut fur¨ Theoretische Physik, Ruprecht-Karls-Universitat¨ Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany

—————————————————————————————————————————— Abstract

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 nonminimally 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.

—————————————————————————————————————————— arXiv:1807.02376v3 [hep-ph] 13 Mar 2019

Email: [email protected]

1 Contents

1 Introduction and summary3

2 General framework7 2.1 Induced gravity ...... 7 2.2 Higgs inﬂation from approximate scale invariance ...... 8 2.3 Tree-level inﬂationary predictions ...... 11

3 Effective ﬁeld theory interpretation 14 3.1 The cutoff scale ...... 14 3.2 Relation between high- and low-energy parameters ...... 16 3.3 Potential scenarios and inﬂationary predictions ...... 17 3.4 Vacuum metastability and high-temperature effects ...... 21

4 Variations and extensions 22 4.1 Palatini Higgs inﬂation ...... 23 4.2 Higgs-Dilaton model ...... 24

5 Concluding remarks 26

6 Acknowledgments 26

2 1 Introduction and summary

Inflation is nowadays a well-established paradigm [1–6] able to explain the flatness, homogeneity and isotropy of the Universe and the generation of the primordial density fluctuations seeding structure formation [7–10]. In spite of the phenomenological success, the inflaton’s nature re- mains unknown and its role could be played by any particle physics candidate able to imitate a slowly-moving scalar field in the very early Universe. In spite of dedicated searches, the only outcome of the Large Hadron Collider (LHC) experiments till date is a scalar particle with mass [11–13]

mH = 125.18 ± 0.16 GeV (1.1) and properties similar to those of the Standard Model (SM) Higgs. The mass value (1.1) is certainly particular since it allows to extend the SM up to the Planck scale without leaving the perturbative regime [14]. The main limitation to this appealing scenario is the potential instability of the electroweak vacuum at high energies. Roughly speaking, the value of the Higgs self-coupling following from the SM renormalization group equations decreases with energy till a given scale and starts increasing thereafter. Whether it stays positive all the way up to the Planck scale, or turns negative at some intermediate scale µ0 depends, mainly, on the interplay between the Higgs mass mH and the top quark Yukawa coupling yt extracted from the reconstructed Monte-Carlo top mass in collider experiments [15], cf. Fig.1. Neglecting the effect of gravitational corrections, crit 1 the critical value yt separating the region of absolute stability from the metastability/instability regions is given by [16] m /GeV − 125.7 α (m ) − 0.01184 ycrit = 0.9244 ± 0.0012 H + 0.0012 s Z , (1.2) t 0.4 0.0007 with αs(mZ ) the strong coupling constant at the Z boson mass. Within the present experimental and theoretical uncertainties [15–17], the SM is compatible with vacuum instability, metastability and absolute stability [18–22], with the separation among the different cases strongly depending on the ultraviolet completion of gravity [23–25], 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 Higgs self-interaction significantly exceeds the value λ ∼ 10−13 leading to 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 [26–28]. The simplest way out is to modify the Higgs field kinetic term in the large-field regime. In Higgs inflation2 this is done by including a direct coupling to the Ricci scalar R [32], namely3 Z √ δS = d4x −g ξH†HR , (1.3)

1The metastability region is deﬁned as the parameter space leading to vacuum instability at energies below the Planck scale but with an electroweak vacuum lifetime longer than the age of the Universe. 2An alternative possibility involving a derivative coupling among the Einstein tensor and the Higgs kinetic term was considered in Refs. [29–31]. 3 Prior to Ref. [32], the effect of non-minimal couplings had been extensively studied in the literature (see for instance Refs. [33–46]), but never with the SM Higgs as the inﬂaton.

3 mH=125.5 GeV 0.14 y =0.9176, m =170.0 0.12 t t yt=0.9235, mt=171.0 0.1 yt=0.9294, mt=172.0 0.08 yt=0.9359, mt=173.1 yt=0.9413, mt=174.0 0.06 yt=0.9472, mt=175.0 λ 0.04 0.02 0 -0.02 -0.04 100000 1e+10 1e+15 1e+20 µ, GeV

Figure 1: The running of the Higgs self-coupling following from the Standard Model renormalization group equations for several values of the top quark Yukawa coupling at the electroweak scale and a fixed Higgs boson mass mH = 125.5 GeV [49]. with H the Higgs doublet and ξ a dimensionless constant to be fixed by observations. The inclusion of the non-minimal coupling (1.3) can be understood as an inevitable consequence of the quantization of the SM in a gravitational background, where this term is indeed required for the renormalization of the energy-momentum tensor [47, 48]. 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 with no radial Higgs component [52, 53]. 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 type and strength of the interactions among the Higgs field and other SM particles is experimentally known, the duration of this entropy production era can be computed in detail [54–58], leading to precise inflationary predictions in perfect agreement with observations [59]. The situation becomes more complicated when quantum corrections are included. The shape of the inflationary potential depends then on the values of the Higgs mass and top Yukawa coupling at the inflationary scale. In addition to the plateau already existing at tree-level [60–63], the renormalization group enhanced potential can develop a quasi-inflection point along the inflationary trajectory [49,60,61,63–68] or a hilltop [61,68,69], with different cases giving rise to different predictions. Although a precise measurement of the inflationary observables could be understood as an interesting consistency check between cosmological observations and particle physics experiments [18, 52, 70–76], the low- to high-energy connection is subject to unavoidable ambiguities related to the non-renormalizability of the model [49, 61, 63, 66, 67, 77–82]. 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 infla-

4 Mt=172.38±0.66 GeV 127

126.5

126

125.5 , GeV h M 125

124.5

124 0.91 0.92 0.93 0.94 0.95 0.96

yt(µ=173.2 GeV)

Figure 2: The SM stability and metastability regions for a renormalization point µ = 173.2 GeV in the MS scheme [16]. The solid red line corresponds to the critical top quark Yukawa coupling (1.2) leading to vacuum instability at a sub-Planckian energy scale µ0, with the dashed lines accounting for uncertainties associated with the strong coupling constant. To the left (right) of these diagonal lines, the SM vacuum is unstable (metastable). The ﬁlled elliptical contours account for the 1σ and 2σ experimental errors on the Higgs mass in Ref. [50] and the CMS (Monte-Carlo) top quark mass in Ref. [51], namely mH = 125.7 ± 0.4 GeV and mt = 172.38 ± 0.10 (stat) ± 0.66 (syst) GeV (note that the current value of the Higgs mass is slightly lower [13]). The additional empty contours illustrate the shifts associated with the theoretically ambiguous relation between the top quark Yukawa coupling and the (Monte-Carlo) top quark mass (cf. Ref. [16] for details).

5 tion. The strength of these jumps encodes the remnants of the ultraviolet completion and cannot be determined within effective field theory approach [49, 80, 83]. If the finite parts are significantly smaller than the associated coupling constants, Higgs inflation leads to a direct connection among the SM parameters measured at collider experiments and the large scale properties of the Universe, provided that the former 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 occur even if the electroweak vacuum is not completely stable [49]. In this article we review the minimalistic Higgs inflation scenario, its predictions, open issues and extensions, and discuss its interplay with the potential metastability of the SM vacuum. The paper is organized as follows:

• The general framework is introduced in Section2. 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 Higgs vacuum expectation value. Having this toy model in mind, we reformulate Higgs inflation in the so-called Einstein frame in which the coupling to gravity is minimal and all non-linearities appear in 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 Section3. 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 controllable relation between low- and high-energy observables. Based on the resulting framework, we analyze the contribution of quantum corrections to the renormalization group enhanced potential and their impact on the inflationary observables. We finish this section by discussing the potential issues of Higgs inflation with the metastability of the SM vacuum.

• Some extensions and alternatives to the simplest Higgs inflation scenario are considered in Section4. In particular, we address the difference between the metric and Palatini formulations of the theory and its extension to a fully scale invariant framework [84–98]. 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 [99–101] 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.

6 2 General framework

The inflationary paradigm is usually formulated in terms of conditions on the local flatness on an arbitrary potential, which can in principle contain a large number of extrema and slopes [102]. 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.

2.1 Induced gravity Let us start by considering an induced gravity scenario Z √ 2 2 4 ξh 1 2 λ 4 1 µν g 2 µ ¯ y ¯ S = d x −g R − (∂h) − h − FµνF − h BµB − iψ∂ψ/ − √ hψψ , (2.1) 2 2 4 4 4 2 involving a scalar field h, a vector field Bµ 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 . µν The quantity FµνF stands for the standard Bµ kinetic term, which for simplicity we take to be Abelian. In this toy model, the effective Newton constant is induced by the scalar field expectation value, 1 G ≡ . (2.2) N,eff 8πξh2

In order for GN,eff to be well-behaved, the non-minimal coupling ξ is restricted to take positive values. This condition is equivalent to require the semi-positive definiteness of the scalar field kinetic term, as can be easily seen by performing a field redefinition h2 → h2/ξ. An important property of the induced gravity action (2.1) is its invariance under scale transformations xµ → x¯µ = α xµ , ϕ(x) → ϕ¯(¯x) = α∆ϕ ϕ(α x) . (2.3)

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 a minimally-coupled frame displaying the standard Einstein-Hilbert term. This Einstein frame is achieved by a Weyl redeﬁnition of the metric4

F 2 M g → Θ g , Θ ≡ ∞ ,F ≡ √P , (2.4) µν µν h2 ∞ ξ together with a Weyl rescaling of the vector and fermion ﬁelds,

−1/2 −3/4 Aµ → Θ Aµ , ψ → Θ ψ . (2.5)

4In spite of its extensive use in the literature, we avoid referring to point-wise rescalings of the metric as “conformal transformations.” For a comprehensive discussion on the differences between Weyl and conformal symmetries, see for instance Refs. [103, 104].

7 After some trivial algebra we obtain an Einstein-frame action5 Z √ 2 2 4 MP 1 2 2 λ 4 1 µν g 2 µ ¯ y ¯ S = d x −g R − M K(Θ)(∂Θ) − F − FµνF − F BµB − iψ∂ψ/ − √ F∞ψψ , 2 2 P 4 ∞ 4 4 ∞ 2 (2.6) containing a non-canonical term for the Θ field. The coefficient of this kinetic term, 1 K(Θ) ≡ , (2.7) 4|a| Θ2 involves a quadratic pole at Θ = 0 and a constant ξ a ≡ − < 0 , (2.8) 1 + 6ξ varying between zero at ξ = 0 and −1/6 when ξ → ∞. The Θ-field kinetic term can be made canonical by performing an additional field redefinition, ! 2p|a|φ Θ−1 = exp , (2.9) MP mapping the vicinity of the pole at Θ = 0 to φ → ∞. The resulting action Z √ 2 2 4 MP 1 2 λ 4 1 µν g 2 µ ¯ y ¯ S = d x −g R − (∂φ) − F − FµνF − F BµB − iψ∂ψ/ + √ F∞ψψ , 2 2 4 ∞ 4 4 ∞ 2 (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 the non-linear realization of the original scale invariance we started with in Eq. (2.1)[105]. The Einstein-frame transition in Eq. (2.4) is indeed equivalent to the spontaneous breaking of dilatations, since we implicitly required the field h to acquire a non-zero expectation value. The canonically normalized scalar field φ is the associated Goldstone boson and as such it is completely decoupled from the matter fields Bµ 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 [85, 87, 91, 106–108] or other non-conformal interactions such as R2 terms [1, 109–111].

2.2 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 shift- symmetric and does not allow for inflation to end. On top of this limitation, 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

5In order to keep the notation as simple as possible, we will not introduce different notations for the quantities deﬁned in different Weyl-related frames. In particular, the implicit Lorentz contractions in this article should be understood to be performed with the metric of the frame under consideration.

8 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 [32] Z √ M 2 S = d4x −g P R + ξH†HR + L , (2.11) 2 SM √ 18 contains two dimensionfull parameters: the reduced Planck MP ≡ 1/ 8πGN = 2.435 × 10 GeV and the Higgs vacuum expectation value vEW ' 250 GeV responsible for the masses within the SM Lagrangian density LSM. Among these two scales, the Planck mass is the most important one at the large field values relevant for inflation. To illustrate how the inclusion of MP modifies the results of the previous√ section, let us consider the graviscalar part of Eq. (2.11) in the unitary gauge H = (0, h/ 2)T , namely Z √ M 2 + ξh2 1 S = d4x −g P R − (∂h)2 − U(h) , (2.12) 2 2 with λ U(h) = (h2 − v2 )2 , (2.13) 4 EW the usual SM symmetry-breaking potential. As in the induced gravity scenario, the inclusion of the non-minimal coupling to gravity changes the strength of the gravitational interaction and makes it dependent on the Higgs field, GN GN,eff = 2 . (2.14) 1 + 8πξGN h In order for the graviton propagator to be well-defined at all h values, the non-minimal coupling ξ must be positive.6 If ξ 6= 0, this requirement translates into a weakening of the effective Newton 2 2 constant at increasing Higgs values. For non-minimal couplings√ in the range 1 ξ MP /vEW, this effect is important in the large-field regime h MP / 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 gµν → Θ gµν with h2 M −1 √P Θ = 1 + 2 ,F∞ ≡ . (2.15) F∞ ξ In the new frame, all the non-linearities associated with the non-minimal Higgs-gravity interaction are moved to the scalar sector of the theory, Z √ M 2 1 S = d4x −g P R − M 2 K(Θ) (∂Θ)2 − V (Θ) , (2.16) 2 2 P which contains now a non-exactly flat potential

4 2 2 2 λF∞ vEW V (Θ) ≡ U(Θ) Θ = 1 − 1 + 2 Θ , (2.17) 4 F∞ 6Models with negative ξ have been considered in the literature [112–114]. In this type of scenarios the gravitational instability at large ﬁeld values can be avoided by replacing the quadratic coupling ξh2 by a designed function 2 ξf(h) remaining smaller than MP during the whole ﬁeld regime.

9 MP

ϕC

ϕC F∞ h

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 φC , 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 with no radial Higgs component. and a non-canonical kinetic sector resulting from the rescaling of the metric determinant and the non-homogeneous part of the Ricci scalar transformation. The kinetic function

1 1 − 6|a|Θ K(Θ) ≡ , (2.18) 4|a|Θ2 1 − Θ 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 induced gravity scenario. This pole leads to an enhanced friction for the Θ field around Θ = 0 and allows for inflation to happen even if the potential V (Θ) is not sufficiently flat. The second pole is a Minkowski pole around which the Weyl transformation equals one and the usual SM action is approximately recovered. To see this explicitly, we carry out an additional field redefinition ,7

1 dφ 2 2 = K(Θ) , (2.19) MP dΘ to recast Eq. (2.16) in terms of a canonically normalized scalar ﬁeld φ. This differential equation admits an exact solution [54] p s s |a|φ 1 − Θ p 6|a|(1 − Θ) = arcsinh − 6|a| arcsinh . (2.20) MP (1 − 6|a|)Θ 1 − 6|a|

In terms of the original ﬁeld h, we can distinguish two asymptotic regimes

7Note that all equations till this point hold even if the non-minimal coupling ξ is ﬁeld-dependent [115, 116].

10  h for φ < φC , 2 φ ' MP h (2.21) √ log 1 + 2 for φ > φC ,  2 |a| F∞ separated by a critical value 2MP (1 − 6|a|) φC ≡ . (2.22) p|a| The comparison between these approximate expressions and the exact ﬁeld redeﬁnition in Eq. (2.20) is shown in Fig.3. The large hierarchy between the transition scale φC and the electroweak scale allows us to identify in practice the vacuum expectation value vEW with φ = 0. In this limit, the Einstein- frame potential (2.17) can be rewriten as λ V (φ) ' F 4(φ) , (2.23) 4 with  φ for φ < φC ,  √ 1 F (φ) ≡ 2 |a|φ 2 (2.24) − M F∞ 1 − e P for φ > φC .

At φ < φC we recover the usual Higgs potential (up to highly suppressed corrections, cf. Section 3.1). At φ > φC the Einstein-frame potential becomes exponentially stretched and approaches p the asymptotic value F∞ at φ > MP /(2 |a|). The presence of MP in Eq. (2.11) modiﬁes also the decoupling properties of the Higgs ﬁeld as compared to those in the induced gravity scenario. In particular, the masses of the intermediate gauge bosons and fermions in the Einstein-frame, ,8

2 2 g 2 y m (φ) ≡ F (φ) , mF (φ) ≡ √ F (φ) , (2.25) B 4 2 coincide with the SM masses in the small ﬁeld regime (φ < φC ) and evolve towards constant p values proportional to F∞ in the large-ﬁeld regime (φ > MP /(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) [52, 53].

2.3 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 > φC , where √ 4 2 |a|φ 2 λF∞ − V (φ) ' 1 − e MP . (2.26) 4

8 ± Here we use a compact notation for the gauge boson couplings, namely g = g2 and g2 cos θw for the B = W and −1 Z bosons respectively, with g1 and g2 the gauge couplings of the U(1)Y and SU(2)L SM groups and θw = tan (g1/g2) the weak mixing angle. The coupling y denotes a generic Yukawa coupling.

11 The statistical information of the primordial curvature fluctuations generated by a single-field model like the one under consideration is mainly 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 [117], we parametrize these spectra in an almost scale-invariant form, k ns−1 k nt Ps = As ,Pt = At , (2.27) k∗ k∗ and compute the inflationary observables

1 V At As = 2 4 , ns = 1 + 2η − 6 , r ≡ = −8nt = 16 , (2.28) 24π MP As with M 2 V 0 2 V 00 ≡ P , η ≡ M 2 , (2.29) 2 V P V the ﬁrst and second slow-roll parameters and the primes denoting derivatives with respect to φ. The quantities in (2.28) should be understood as evaluated at a ﬁeld value φ∗ ≡ φ(N∗), with

φ φ p ! ∗ 1 Z ∗ dφ 1 √ 2 |a|φ N = √ = e2 |a|φ/MP − (2.30) ∗ M 8|a| M P φE 2 P φE the e-fold number at which the reference scale k∗ in Eq. (2.27) exits the horizon, i.e. k∗ = a∗H∗. Here, MP p φE = ln 1 + 2 2|a| , (2.31) 2p|a| stands for the field value at the end of inflation, which is defined, as usual, by the condition (φE) ≡ 1. Equation (2.30) admits an exact inversion, √ h i 2 |a|φ∗/MP −8|a|N¯∗ e = −W−1 −e , (2.32) with W−1 the lower branch of the Lambert function and √ p ! 1 2 |a|φE ¯ 2 |a|φE /MP N∗ ≡ N∗ + e − , (2.33) 8|a| MP a rescaled number of e-folds. Inserting Eq. (2.32) into (2.28) we get the following analytical expressions for the primordial scalar amplitude,

2 4 λ(1 − 6|a|) (1 + W−1) As = 2 2 , (2.34) 12 π |a| (8|a|W−1) its spectral tilt, 1 − W−1 ns = 1 − 16|a| 2 , (2.35) (1 + W−1)

12 and the tensor-to-scalar ratio 128 |a| r = 2 . (2.36) (1 + W−1)

At large |a|N∗, these predictions display an interesting attractor behavior, very similar to that appearing in α-attractor scenarios [118–120] (see also Ref. [102]). Indeed, by taking into account the lower bound on the Lambert function [121],

¯ q −8|a|N∗ ¯ ¯ W−1[−e ] > −8|a|N∗ − 2(8|a|N∗ − 1) , (2.37) we can obtain the approximate expressions9 2 2 n ' 1 − , r ' . (2.38) s ¯ ¯ 2 N∗ |a|N∗ ¯ 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 [59],

10 log 10 As ' 3.094 ± 0.034 . (2.39)

Doing this, we get a relation √ ¯ ξ ' 800N∗ λ , (2.40) ¯ among the non-minimal coupling ξ, the number of e-folds N∗ and the Higgs self-coupling λ. The precise value of the number of e-folds in Eqs. (2.38) and (2.40) depends on the whole post-inflationary expansion and, in particular, on the duration of the heating stage. As the strength of the interactions among the Higgs field and the SM particles is experimentally known, the entropy production following the end of inflation can be computed in detail [54, 55, 57].10 The depletion of the Higgs-condensate is dominated by the non-perturbative production of massive intermediate gauge bosons, which, contrary to the SM fermions, can experience bosonic ampli- fication. Once created, the W ± and Z bosons can decay into lighter 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 time at which the Higgs amplitude approaches the critical value φC where the effective potential becomes quartic or by ii) the moment at which the energy density into relativistic fermions approaches that of the Higgs condensate; whatever happens first. The estimates in Refs. [54, 55, 57] provide a range

13 14 10 GeV . TH . 2 × 10 GeV , (2.41) with the lower and upper bounds associated respectively with the cases i) and ii) above. For the ¯ upper limit of this narrow window, we have N∗ ' N∗ ' 59 and we can rewrite Eq. (2.40) as a relation between ξ and λ, √ ξ ' 47200 λ . (2.42) Note that a variation of the Higgs self-coupling in this equation can be compensated by a change of the a priori unknown non-minimal coupling to gravity. For the tree-level value λ ∼ O(1),

9 Note that the expressions contain N¯∗ rather than N∗. 10This allows, for instance, to distinguish Higgs inﬂation from R2 Starobinsky inﬂation [1, 122].

13 the non-minimal coupling must be significantly larger than one, but still much smaller than the 2 2 32 value ξ ∼ MP /vEW ∼ 10 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 φC separating the low- and high-energy regimes, r2 M φ ' P , (2.43) C 3 ξ and collapses the inflationary predictions to the attractor values [32] 2 12 n ' 1 − ' 0.966 , r ' ' 0.0034 , (2.44) s ¯ ¯ 2 N∗ N∗ in very good agreement with the latest results of the Planck collaboration [59]. 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 in order to account for the Weyl factor relating the two frames [41, 42, 123–139].

3 Effective ﬁeld theory interpretation

The presence of gravity makes Higgs inﬂation perturbatively non-renormalizable [77–80] and forbids its interpretation as an ultraviolet complete theory. The model should be therefore understood as an effective description valid up to a given cut-off scale Λ [80, 82]. This cutoff could either indicate the onset of a strongly coupled regime to be studied within the model by non- perturbative techniques (such as resummations, lattice simulations or functional renormalization studies) [140–143] or the appearance of new degrees of freedom beyond the initially-assumed SM content [144, 145].

3.1 The cutoff scale A priori, the cutoff scale of Higgs inflation could coincide with the Planck scale, where gravitational effects should definitely taken into account. Although quite natural, the identification of these two energy scales may not be theoretically consistent, since other interactions could lead to violations of tree-level unitarity at a lower energy scale. An estimate11 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 [80, 147]. 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 fields around their background values g¯µν and h, ¯ gµν =g ¯µν + γµν , h = h + δh , (3.1)

11This procedure does not take into account possible cancellations among scattering diagrams, as those taking place, for instance, in models involving a singlet scalar ﬁeld not minimally coupled to gravity [146].

14 we obtain the following quadratic Lagrangian density for the perturbations γµν and δh

M 2 + ξh¯2 L(2) = P γµν γ + 2∂ γµν∂ργ − 2∂ γµν∂ γ − γ γ 8 µν ν µρ ν µ 1 − (∂ δh)2 + ξh¯∂ ∂ γλρ − γ δh , (3.2) 2 µ λ ρ

µν with γ =g ¯ γµν 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 [71, 72, 74, 148]. To identify the different cutoff scales one must ﬁrst diagonalize the kinetic terms. This ˆ can be done by performing a redeﬁnition of the perturbations (γµν, δh) → (ˆγµν, δh) with ¯ 1 2ξhg¯µν ˆ γµν = γˆµν − δh , (3.3) p 2 ¯2 p 2 ¯2 2 ¯2 MP + ξh (MP + ξh )(MP + (1 + 6ξ)ξh ) s M 2 + ξh¯2 δh = P δhˆ . (3.4) 2 ¯2 MP + (1 + 6ξ)ξh

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, q ¯ 2 ¯2 ΛP(h) ≡ MP + ξh , (3.5) which coincides with the effective Planck scale in Eq. (2.12). For scalar-graviton interactions, the ˆ 2 ¯ leading-order higher-dimensional operator is (δh) γ/ˆ ΛS(h), where 2 ¯2 ¯ MP + (1 + 6ξ)ξh ΛS(h) ≡ . (3.6) p 2 ¯2 ξ MP + ξh 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. One could study, for instance, the scattering of intermediate W ± and Z bosons. Since we are working in the unitary gauge, it is sufficient to consider the 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 a tree-level unitarity violation at a scale

p 2 ¯2 ¯ MP + ξ(1 + 6ξ)h ΛG(h) ≡ √ . (3.7) 6ξ ¯ ¯ Note that the above scales depend on the background field h. For small field values (h . MP /ξ), the cutoffs (3.5), (3.6) and (3.7) coincide with those obtained√ by naively expanding the theory around the electroweak scale, namely ΛP ' MP , ΛS ' 6ΛG ' MP /ξ [77–79, 146, 149]. At ¯ large field values, (h & MP /ξ), the suppression scale depends on the particular process under ¯ √ ¯ consideration. For MP /ξ h MP / ξ the graviscalar cutoff ΛS grows quadratically till h ' √ ¯ MP / ξ, where it becomes linear in h and traces the dynamical Planck mass in that regime, ΛP ' √ ¯ ξh. On the other hand, the gauge cutoff ΛG smoothly interpolates between ΛG ∼ MP /ξ at ¯ ¯ ¯ ¯ ¯ h . MP /ξ and ΛG ∼ gh at h & MP /ξ. Note that all cutoffs scales become linear in h at h & MP /ξ.

15 This means that any operator ∆L constructed out of them, the Higgs field and some Wilson coefficients cn approaches a scale-invariant form at large field values, namely ¯ ¯ X cn On[h] X cn On[h] X cn ∆L ≡ ' √ ∼ √ h4 , n > 4 . (3.8) ¯ n−4 ( ξh¯)n−4 ( ξ)n−4 n Λ(h) n n

3.2 Relation between high- and low-energy parameters 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 to be included on top of the tree-level action is the one generated by the theory itself via radiative corrections [49, 80]. 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 involving a field- dependent subtraction point [52] 2 2 2 µ ∝ MP + ξh , (3.9) fits pretty well with the approximate scale-symmetry of Eq. (2.11) at large-field values.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 µ2Θ coincides with the standard field-independent 2 2 prescription of renormalizable field theories, µ Θ ∝ MP . A general counter-term in dimensional regularization contains a finite part δL and a divergent part in the form of a pole in = (4 − d)/2, with d the dimension of spacetime. 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 encodes the remnants of a particular ultraviolet completion and cannot be determined within the effective field theory approach [49, 80, 83]. From a quantitative point of view, the most relevant δL terms are related to the Higgs and top-quark interactions. In the Einstein-frame at one loop, they take the form [49] " # 1 2 δLF = δλ F 02 + F 00F − δλ F 4 , δLψ = δy F 02F + δy F 00(F 4)00 ψψ¯ , (3.10) 1 a 3 b 1 a b

12The use of other schemes such as Pauli-Villars regularization or standard dimensional regularization with ﬁeld- independent subtraction point leads to dilatation-symmetry breaking and the consequent bending of the Higgs inﬂa- tion plateau due to radiative corrections, see for instance Refs. [71, 72, 74, 148].

16 where the primes denote again derivatives with respect to φ. Note that these operators differ, as expected, from those appearing in the tree-level action. This means that, while the contribution δλb can be removed by a self-coupling redefinition, the finite parts δλa, δya and δyb should be promoted to new couplings constants. 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 must be again promoted to novel couplings with their own renormalization group equations. The itera- tion of this scheme leads to a renormalized action including an infinite set of higher-dimensional operators constructed out of the function F and its derivatives. For small field values, the function F becomes approximately linear (F ≈ φ, F 0 = 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 eliminated by a redefinition of the low energy couplings, as happens in a renormalizable theory. When evolving towards the inflationary region, the function 0 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 p of the SM couplings at the transition region φC < φ < 3/2 MP , such that the SM masses at the electroweak scale cannot be unambiguously related to their inflationary counterparts without a precise knowledge of the ultraviolet completion [49, 83, 150]. If the finite contributions are of the same order as the loops generating them, the tower of higher dimensional operators generated by radiative corrections can be truncated [49]. In this case, the effect of the 1-loop threshold corrections can be imitated by an effective change13 [49] " # 1 2 λ(µ) → λ(µ) + δλ F 02 + F 00F − 1 , y (µ) → y (µ) + δy F 02 − 1 , (3.11) a 3 t t a with λ(µ) and yt(µ) given by 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 [49, 67, 80].

3.3 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 is given by the one in Eq. (2.26) but with the Higgs self-coupling λ replaced by its corresponding running value λ(φ), λ(φ) V (φ) = F 4(φ) . (3.12) 4 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. [115]—but rather assuming it to be constant during inflation. This is indeed a reasonable approximation since the one-loop beta function determining the running of ξ [52, 151], ∂ 1 3 β (µ) = µ ξ = − ξ g02 + 3g2 − 6y2 , (3.13) ξ ∂µ 16π2 2 t

13 This replacement implicitly neglects the running of the ﬁnite parts δλa and δya in the transition region φC < φ < p 3/2 MP .

17 −2 is rather small for realistic values of the couplings constant at the inﬂationary scale, βξ ∝ O(10 ) [67, 116] (see also Ref. [152]). 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 [17,21,153]. In the vicinity of the minimum of λ(φ), we can use the approximation [60]

m (φ) λ(φ) = λ + b log2 t , (3.14) 0 q with the parameters λ0, q and b depending on the inﬂationary values of the Einstein-frame Higgs and top quark masses, according to the ﬁtting formulas [60]

∗ ∗ λ0 = 0.003297 [(mH − 126.13) − 2(mt − 171.5)] , ∗ ∗ q = 0.3MP exp []0.5(mH − 126.13) − 0.03(mt − 171.5)] , (3.15) −6 ∗ ∗ b = 0.00002292 − 1.12524 × 10 [(mH − 126.13) − 1.75912(mt − 171.5)] ,

∗ ∗ with mH and mt 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 Higgs and top quark mass values at that scale and can be well-approximated by b ' 2.3 × 10−5. The choice √ m (φ) y F (φ) ξF (φ) t = α · √t ≡ , (3.16) q 2 q κMP with α = 0.6 optimizes the convergence of perturbation theory [52, 73], while respecting the asymptotic symmetry of the tree-level action (2.12) and its non-linear shift-symmetric Einstein- frame 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) allows us to distinguish three regimes:

i) Non-critical regime/Universal: If λ0 b/(16κ), the effective potential (3.12) is almost independent of the radiative logarithmic correction and can be well approximated by its tree- level form (2.26). Consequently, the inﬂationary observables retain their tree-level values [60, 61, 63, 67], cf. Fig.4.

ii) Critical regime: If λ0 & b/(16κ), the first two derivatives of the potential are approximately zero, V 0 ' V 00 ' 0, leading to the appearance of a quasi-inflection point at √ r3 e φ = log √ M . (3.17) I 2 e − 1 P Qualitatively, the vast majority of inflationary e-folds in this scenario takes place in the vicinity of the inflection point φI, while the inflationary observables depend on the form of the potential as some value φ∗ > φI . −6 Given the small value of the Higgs self-coupling in this scenario, λ0 ∼ O(10 ), the nonminimal coupling ξ can be significantly smaller than in the universal regime, ξ ∼ O(10), while still satisfying the normalization condition (2.39)[60, 64, 65, 154]. This drastic decrease of the non-minimal coupling alleviates the tree-level unitary problems discussed in Section 3.1 by raising the cutoff scale.

18 N 60 50 40 30 20 10 0

ns =0.968, r =0.065

-1 10 κ =1.01 -7 κ =1.02 10 n =0.968, r =0.027 κ =1.03 s κ =1.04

κ =1.05 ns =0.968, r =0.015 10-8 =1 06 R

r κ . P κ =1.07 κ =1.08 =0 968 =0 007 -2 κ =1.09 ns . , r . 10 -9 κ =1.1 10

ns =0.968, r =0.004 -10 10 -3 1 5 9 13 17 21 0.94 0.95 0.96 0.97 0.98 10 10 10 10 10 10 10 1 ns k[Mpc− ]

Figure 4: (Left) The tensor-to-scalar ratio r and the spectral tilt nS following from the effective potential (3.12)[67]. 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). The blue contours indicate the latest 68% and 95% C.L. Planck constraints on the r-ns plane [59]. (Right) The power spectrum PR as a function of the number of e-folds before the end of inflation and the associated comoving scale k in inverse megaparsecs [67]. 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 [59].

19 Figure 5: (Left) Spectral tilt running αs ≡ d ln ns/d ln k in critical Higgs inflation as a function of the tensor-to-scalar ratio r and the spectral-tilt ns [68]. (Right) Scale dependence of the spectral- 2 2 tilt βs ≡ d ln ns/d ln k in the same case [68]. The purple dots indicate the universal/non-critical Higgs inflation regime. The boundaries on the right-hand side of the figures correspond to the constraint on the number of e-folds following the heating estimates in Refs. [54, 55, 57]. For lower heating efficiencies, the boundaries move to the left, decreasing the spectral tilt but not significantly affecting the tensor-to-scalar ratio [68].

For small ξ values, the tensor-to-scalar ratio can be rather large, r ∼ O(10−1) [60,64,65,154] (see also Ref. [116]). Note, however, that although CMB data seems to be consistent with the primordial power spectrum at large scales, the simple expansion in (2.27) cannot accurately describe its global behavior since the running of the spectral tilt αs ≡ d ln ns/d ln k and its 2 2 scale dependence βs ≡ d ln ns/d ln k also become considerably large, cf. Fig.5. 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 contribution in the equation of motion for the inflation field is not negligible as compared with the Hubble friction [155–157]. In this regime, even the classical treatment is compromised since stochastic effects can no longer be ignored [158–161]. If we restrict ourselves to situation in which the slow-roll approximation is satisfied during the whole inflationary trajectory, [67]14 the height and width of the generated bump at fixed spectral tilt are correlated with the tensor-to-scalar ratio r, cf. Fig.4. Contrary to some claims in the literature [115], the maximum amplitude of the power-spectrum compatible max with the 95% C.L Planck ns − r contours [67] is well below the critical threshold PR ' 10−2 −10−3 needed for primordial black hole formation [162–164] (see, however, Refs. [161] and [165]). This conclusion is unchanged if one considers the effect of non-instantaneous threshold corrections [67], which could potentially affect the results given the numerical

14The onset of the slow-roll regime prior to the arrival of the field to the inflection point and its dependence on pre-inflationary conditions was studied in Ref. [152], where a robust inflationary attractor was shown to exist.

20 m 8 T= 8 x 10 13 GeV t =172 GeV m = 125.5 GeV 13 10 h T= 7 x 10 GeV ξ= 1500 6 T= 6 x 10 13 GeV m T= 5 x 10 13 GeV 5 t =173 GeV 4 T=0 0

λ m

3 2 0 t U /

=174 GeV 7 10 10 0 -5 δλ=- 0.0071 -2 -10 δλ=- 0.0142 -4 δλ=- 0.0216 -15 10-10 10-8 10-6 10-4 10-2 100 0.00 0.05 0.10 0.15 0.20 0.25 3 μ/MP 10 ϕ/MP

Figure 6: (Left) Illustrative values of the 1-loop threshold corrections needed to restore the asymptotic behavior of the universal Higgs inflation potential at large field values for electroweak SM 9 10 12 pole masses leading to SM vacuum instability at µ0 ∼ 10 , 10 and 10 GeV. (Right) Comparison between the potential following from the set of parameters leading to the red line in the previous plot and the thermally-corrected effective potential accounting for the backreaction effects of −3 4 the decay products created during the heating stage. The normalization factor U0 = (10 MP ) account for the typical energy density at the end of inflation.

proximity of the inflection point (3.17) to the upper boundary of the transition region, φ ' p 3/2MP . iii) Hilltop regime: If λ . b/(16κ) the potential develops a new minimum at large field values [62, 68]. This minimum is separated from the electroweak minimum by a local maximum where hilltop inflation can take place [166, 167]. 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 [61,68].

3.4 Vacuum metastability and high-temperature effects The qualitative classification of scenarios and predictions presented in the previous section depends on the inflationary values of the Higgs and top quark masses 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 δλa 9 10 needed to restore the universal/non-critical Higgs inflation scenario beyond µ0 ∼ 10 , 10 and 1012 GeV are shown in Fig.6. For a detailed scan of the parameter space see Refs. [61, 63]. 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 evolve in the trans-Planckian field regime, inflating the Universe while moving towards

21 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 is strongly dependent on the ratio of the Higgs energy density to the second minimum depth. 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 allowing the Higgs field to evolve towards the desired electroweak vacuum [55]. The one-loop finite temperature corrections to be added on top of the Einstein-frame renormalization group enhanced potential take the form [168]

X Z d3k k2/a2 + m2 ∆V = T ln 1 ± exp − i , (3.18) (2π)3a3 T i=B,F with the plus and minus signs corresponding respectively to fermions and bosons and mB,F standing for the Einstein-frame masses in Eq. (2.25). The most important contributions in Eq. (3.18) are associated with the top quark and the electroweak bosons, with the corresponding coupling constants yt and g evaluated at µyt = 1.8 T and µg = 7 T , in order to minimize the radiative corrections [169]. 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 unwanted secondary vacuum at large field values disappears [54, 55], see 15 16 Fig6. The stabilization becomes favored for increasing µ0 values and holds even if this scale is as low as 1010 GeV [55]. The thermally-corrected potential enables the Higgs field to relax to the SM vacuum. After the heating stage, the temperature decreases as the Universe expands and the secondary minimum 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 SM vacuum significantly exceeds the life of the Universe [70, 170–172]. Universal/non-critical Higgs inflation with a graceful exit can therefore take place for electroweak SM pole masses leading to vacuum metastability at energies below the inflationary scale. [55]. The situation changes completely if one considers the critical Higgs inflation scenario. In this case, the energy of the Higgs condensate is comparable to 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 [173]. The success of critical Higgs inflation requires therefore the absolute stability of the electroweak vacuum [55].

4 Variations and extensions

Many variations and extensions of Higgs inﬂation have been considered in the literature, see for instance Refs. [144, 174–204]. In what follows we will restrict ourselves to those proposals that

15A detailed scan of the parameter space assuming instantaneous conversion of the inﬂaton energy density into a thermal bath was performed in Ref. [63]. 16 The larger µ0 is, the shallower and narrower the “wrong” minimum becomes, cf. Fig.6.

22 are more closely related to the minimalistic spirit of the original scenario. In particular, we will address a Palatini formulation of Higgs inﬂation and the embedding of the model to a fully scale invariant framework.

4.1 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 depending on the metric tensor and the inclusion of a York-Hawking-Gibbons term ensuring the cancellation of a total derivative term with no-vanishing variation at the boundary [205, 206]. One could alternatively consider a Palatini formulation of gravity in which the metric tensor and the connection are treated as independent variables and no additional boundary term is required to obtain the equations of motion [207]. 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 [207], this is not true for scalar-tensor theories as Higgs inflation. To see this µν explicitly let us consider the Higgs inflation action in Eq. (2.12) with R = g Rµν(Γ, ∂Γ) and Γ a 17 non-Levi-Civita connection. Performing a Weyl rescaling of the metric gµν → Θ gµν with Θ given by Eq. (2.15) we obtain an Einstein-frame action Z √ M 2 1 S = d4x −g P gµνR (Γ) − M 2 K(Θ)(∂Θ)2 − V (Θ) , (4.1) 2 µν 2 P containing a potential (2.17) and a non-canonical kinetic term with 1 1 K(Θ) ≡ , (4.2) 4|a|Θ2 1 − Θ and |a| ≡ ξ . (4.3) Note that the kinetic function (4.2) differs from that obtained in the metric formulation, see Eq. (2.18). In particular, it does not contain the part associated with the non-homogeneous transformation of the Ricci scalar, since R = R(Γ) is now invariant under Weyl rescalings. For the purposes of inflation, this translates into a modification of the residue of the inflationary pole at Θ = 0 with respect to the metric case. While the metric value of |a| in Eq. (2.18) is bounded from above (cf. Eq. (2.8)), it can take positive arbitrary values in the Palatini formulation (cf. Eq. (4.3)). Performing a field redefinition √ 1 dφ 2 aφ 2 = K(Θ) −→ h(φ) = F∞ sinh , (4.4) MP dΘ MP to canonically normalize the h-field kinetic term, we can rewrite the graviscalar action (4.1) at φ vEW as Z √ M 2 1 S = d4x −g P R − (∂φ)2 − V (φ) , (4.5) 2 2

17For a recent generalization built from the Higgs, the metric and the connection and involving only up to two derivatives see Ref. [208].

23 with √ λ 4 aφ V (φ) = F (φ) ,F (φ) ≡ F∞ tanh . (4.6) 4 MP 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 low-energy quartic potential and an asymptotically flat potential at large field values. 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 a strong impact on the inflationary observables. In the large-field regime they read 2 2 n ' 1 − , r ' , (4.7) s ¯ ¯ 2 N∗ ξN∗ with √ ¯ 1 2 aφE N∗ ≡ N∗ + cosh , (4.8) 16|a| MP a rescaled number of e-folds and M √ φ = √P arcsinh( 32a) , (4.9) E 2 a p √ the inflaton value at the end of inflation√ ((φend) ≡ 1), with φE = 3/2 arcsinh(4/ 3) MP corre- 4 sponding to the ξ → ∞ limit and φE = 2 2 MP to the end of inflation in a minimally coupled λφ theory. A relation between the non-minimal coupling ξ, the self-coupling λ and the number of ¯ e-folds N∗ can be obtained by taking into account the amplitude of the observed power spectrum in Eq. (2.39), 6 ¯ 2 ξ ' 3.8 × 10 N∗ 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 [209] only if ξ . 10, which, assuming ¯ −9 N ' 59, requires a very small coupling λ . 10 . For a discussion of unitarity violations in the Palatini formulation see Ref. [210].

4.2 Higgs-Dilaton model The existence of robust predictions in (non-critical) Higgs inflation is intimately related to the emerging dilatation symmetry of its tree-level action at large field values. The uplifting of Higgs inflation to a completely√ scale-invariant setting was considered in Refs. [84–98]. In the unitary gauge H = (0, h/ 2)T , the graviscalar sector of the Higgs-Dilaton model considered in these papers takes the form Z √ ξ h2 + ξ χ2 1 1 S = d4x −g h χ R − (∂h)2 − (∂χ)2 − V (h, χ) , (4.11) 2 2 2 with λ 2 U(h, χ) = h2 − αχ2 + βχ4 (4.12) 4

24 a scale-invariant version of the SM symmetry-breaking potential and α, β positive dimensionless parameters. The existence of a well-defined gravitational interactions at all field values requires the non-minimal gravitational couplings to be positive-definite, i.e. ξh, ξχ > 0. In the absence of gravity, the ground state of Eq. (4.11) is determined by the scale-invariant potential (4.12). For α 6= 0 and β = 0, the vacuum manifold extends along the flat directions h0 = ±αχ0. Any solution with χ0 6= 0 breaks scale symmetry spontaneously and induces non-zero values for the effective Planck mass and the electroweak scale.18 The relation between these highly hierarchical scales is 2 2 −32 set by fine-tuning α ∼ v /MP ∼ 10 . 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 2 2 2 scenario, let us perform a Weyl rescaling gµν → MP /(ξhh + ξχχ )gµν followed by a field redefinition [95]

2 2 2 2 −2 (1 + 6ξh)h + (1 + 6ξχ)χ 2γΦ a (1 + 6ξh)h + (1 + 6ξχ)χ γ Θ ≡ 2 2 , exp ≡ 2 , (4.13) ξhh + ξχχ MP a¯ MP with s ξ ξ ξ γ ≡ χ , a ≡ − h , a¯ ≡ a 1 − χ . (4.14) 1 + 6ξχ 1 + 6ξh ξh After some algebra, we obtain a rather simple Einstein-frame action [91, 95]

Z √ M 2 1 1 S = d4x −g P R − M 2 K(Θ)(∂Θ)2 − Θ(∂Φ)2 − U(Θ) , (4.15) 2 2 P 2 containing a potential

λ M 4 1 + 6¯a2 U(Θ) = U (1 − Θ)2 ,U ≡ P , (4.16) 0 0 4 a¯ and a non-canonical, albeit diagonal, kinetic sector. The kinetic function for the Θ ﬁeld,

1 c 1 − 6|a¯|Θ K(Θ) = + , (4.17) 4 |a¯|Θ2 |a¯|Θ − c 1 − Θ contains two “inflationary” poles at Θ = 0 and Θ = c/|a¯| and a “Minkowski” pole at Θ = 1, where the usual SM action is approximately recovered. As in the single field case, the “Minkowski” pole does not play a significant role during inflation and can be neglected for all practical purposes. Interestingly, the field-derivative space becomes in this limit a maximally symmetric hyperbolic manifold with Gaussian curvature a < 0 [91]. Inflation takes place in the vicinity of the inflationary poles. During this regime, the kinetic term of the Φ-field is effectively suppressed and the dilaton rapidly approaches a constant value Φ = Φ0. This effective 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 dilaton or Goldstone of dilatations. As first shown in Ref. [85] (see also Refs. [93, 97]), the equation of

18 Among the possible values of β in the presence of gravity, the case β = 0 seems also preferred [85, 211, 212], see also Refs. [213–218].

25 motion for this field coincides with the scale-current conservation equation, effectively restricting the evolution to constant Φ ellipsoidal trajectories in the {h, χ} plane. Given this emergent single- field dynamics, no non-gaussianities nor isocurvature perturbations are significantly generated during inflation [85, 96]. If the Θ variable is dominated by the Higgs component (ξh ξχ), the spectral tilt and the tensor-to-scalar ratio take the compact form

32 c2 n ' 1 − 8 c coth (4cN ) , r ' csch2 (4cN ) , (4.18) s ∗ |a| ∗ 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.

5 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 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 finding of a relatively light Higgs boson in the Large Hadron Collider 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 till the Planck scale and provide a solid theoretical basis for describing the early Universe. The Higgs field itself could lead to inflation if a minimalistic coupling to the Ricci scalar 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 at tree level. On top of that, the experimental knowledge of the Standard Model couplings re- duces the usual uncertainties associated with the heating stage and allows us to obtain precise predictions in excellent agreement with observations. Note, however, the mere existence of gravity makes the theory non-renormalizable and forces its interpretation as an effective field theory. Even in a self-consistent approach to Higgs inflation, the finite parts of the counterterms needed to make the theory finite obscure the connection between low- and high-energy observables. If these unknown coefficients 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 when the Standard Model vacuum is not completely stable.

6 Acknowledgments

The author acknowledges support from the Deutsche Forschungsgemeinschaft through the Open Access Publishing funding programme of the Baden-Wurttemberg¨ Ministry of Science, Research and Arts and the Ruprecht-Karls-Universitat¨ Heidelberg as well as through the project TRR33

26 “The Dark Universe”. He thanks Guillem Domenech, Georgios Karananas and Martin Pauly for useful comments and suggestions on the manuscript.

References [11] Georges Aad et al. Observation of a new particle in the search for the Standard Model [1] Alexei A. Starobinsky. A New Type of Higgs boson with the ATLAS detector at the Isotropic Cosmological Models Without Sin- LHC. Phys. Lett., B716:1–29, 2012. gularity. Phys. Lett., B91:99–102, 1980. [,771(1980)]. [12] Serguei Chatrchyan et al. Observation of a new boson at a mass of 125 GeV with the [2] Alan H. Guth. The Inflationary Universe: A CMS experiment at the LHC. Phys. Lett., Possible Solution to the Horizon and Flatness B716:30–61, 2012. Problems. Phys. Rev., D23:347–356, 1981. [13] M. Tanabashi et al. Review of Particle Physics. [3] Viatcheslav F. Mukhanov and G. V. Chibisov. Phys. Rev., D98(3):030001, 2018. Quantum Fluctuations and a Nonsingular Universe. JETP Lett., 33:532–535, 1981. [14] Mikhail Shaposhnikov and Christof Wet- [Pisma Zh. Eksp. Teor. Fiz.33,549(1981)]. terich. Asymptotic safety of gravity and the Higgs boson mass. Phys. Lett., B683:196– [4] Andrei D. Linde. A New Inflationary Universe 200, 2010. Scenario: A Possible Solution of the Hori- zon, Flatness, Homogeneity, Isotropy and Pri- [15] Mathias Butenschoen, Bahman Dehnadi, An- mordial Monopole Problems. Phys. Lett., dre H. Hoang, Vicent Mateu, Moritz Preisser, 108B:389–393, 1982. and Iain W. Stewart. Top Quark Mass Calibra- tion for Monte Carlo Event Generators. Phys. [5] Andreas Albrecht and Paul J. Steinhardt. Cos- Rev. Lett., 117(23):232001, 2016. mology for Grand Unified Theories with Ra- diatively Induced Symmetry Breaking. Phys. [16] Fedor Bezrukov and Mikhail Shaposhnikov. Rev. Lett., 48:1220–1223, 1982. Why should we care about the top quark Yukawa coupling? J. Exp. Theor. Phys., [6] Andrei D. Linde. Chaotic Inflation. Phys. Lett., 120:335–343, 2015. [Zh. Eksp. Teor. 129B:177–181, 1983. Fiz.147,389(2015)]. [7] S. W. Hawking. The Development of Irreg- [17] Jose R. Espinosa, Mathias Garny, Thomas ularities in a Single Bubble Inflationary Uni- Konstandin, and Antonio Riotto. Gauge- verse. Phys. Lett., 115B:295, 1982. Independent Scales Related to the Stan- [8] Alexei A. Starobinsky. Dynamics of Phase dard Model Vacuum Instability. Phys. Rev., Transition in the New Inflationary Universe D95(5):056004, 2017. Scenario and Generation of Perturbations. [18] Fedor Bezrukov, Mikhail Yu. Kalmykov, Phys. Lett., 117B:175–178, 1982. Bernd A. Kniehl, and Mikhail Shaposhnikov. [9] Misao Sasaki. Large Scale Quantum Fluctua- Higgs Boson Mass and New Physics. JHEP, tions in the Inflationary Universe. Prog. Theor. 10:140, 2012. [,275(2012)]. Phys., 76:1036, 1986. [19] Giuseppe Degrassi, Stefano Di Vita, Joan [10] Viatcheslav F. Mukhanov. Quantum Theory of Elias-Miro, Jose R. Espinosa, Gian F. Giudice, Gauge Invariant Cosmological Perturbations. Gino Isidori, and Alessandro Strumia. Higgs Sov. Phys. JETP, 67:1297–1302, 1988. [Zh. mass and vacuum stability in the Standard Eksp. Teor. Fiz.94N7,1(1988)]. Model at NNLO. JHEP, 08:098, 2012.

27 [20] Dario Buttazzo, Giuseppe Degrassi, [31] Jacopo Fumagalli, Sander Mooij, and Pier Paolo Giardino, Gian F. Giudice, Marieke Postma. Unitarity and predictive- Filippo Sala, Alberto Salvio, and Alessandro ness in new Higgs inﬂation. JHEP, 03:038, Strumia. Investigating the near-criticality of 2018. the Higgs boson. JHEP, 12:089, 2013. [32] Fedor L. Bezrukov and Mikhail Shaposhnikov. [21] Jose R. Espinosa, Gian F. Giudice, Enrico The Standard Model Higgs boson as the inﬂa- Morgante, Antonio Riotto, Leonardo Sen- ton. Phys. Lett., B659:703–706, 2008. atore, Alessandro Strumia, and Nikolaos Tetradis. The cosmological Higgstory of the [33] Peter Minkowski. On the Spontaneous Origin vacuum instability. JHEP, 09:174, 2015. of Newton’s Constant. Phys. Lett., 71B:419– 421, 1977. [22] Jose R. Espinosa. Implications of the top (and Higgs) mass for vacuum stability. PoS, [34] A. Zee. A Broken Symmetric Theory of Grav- TOP2015:043, 2016. ity. Phys. Rev. Lett., 42:417, 1979. [23] Vincenzo Branchina and Emanuele Messina. [35] Lee Smolin. Towards a Theory of Space-Time Stability, Higgs Boson Mass and New Physics. Structure at Very Short Distances. Nucl. Phys., Phys. Rev. Lett., 111:241801, 2013. B160:253–268, 1979.

[24] Vincenzo Branchina, Emanuele Messina, and [36] B. L. Spokoiny. INFLATION AND GEN- Alessia Platania. Top mass determination, ERATION OF PERTURBATIONS IN BROKEN Higgs inﬂation, and vacuum stability. JHEP, SYMMETRIC THEORY OF GRAVITY. Phys. 09:182, 2014. Lett., 147B:39–43, 1984.

[25] Vincenzo Branchina, Emanuele Messina, and [37] T. Futamase and Kei-ichi Maeda. Chaotic In- Marc Sher. Lifetime of the electroweak vac- flationary Scenario in Models Having Non- uum and sensitivity to Planck scale physics. minimal Coupling With Curvature. Phys. Rev., Phys. Rev., D91:013003, 2015. D39:399–404, 1989. [26] Gino Isidori, Vyacheslav S. Rychkov, Alessan- [38] D. S. Salopek, J. R. Bond, and James M. dro Strumia, and Nikolaos Tetradis. Gravita- Bardeen. Designing Density Fluctuation tional corrections to standard model vacuum Spectra in Inflation. Phys. Rev., D40:1753, decay. Phys. Rev., D77:025034, 2008. 1989. [27] Yuta Hamada, Hikaru Kawai, and Kin-ya Oda. [39] R. Fakir and W. G. Unruh. Induced gravity Minimal Higgs inflation. PTEP, 2014:023B02, inflation. Phys. Rev., D41:1792–1795, 1990. 2014. [28] Malcolm Fairbairn, Philipp Grothaus, and [40] R. Fakir and W. G. Unruh. Improvement on Robert Hogan. The Problem with False Vac- cosmological chaotic inflation through non- uum Higgs Inflation. JCAP, 1406:039, 2014. minimal coupling. Phys. Rev., D41:1783– 1791, 1990. [29] Cristiano Germani and Alex Kehagias. New Model of Inflation with Non-minimal Deriva- [41] Nobuyosi Makino and Misao Sasaki. The tive Coupling of Standard Model Higgs Bo- Density perturbation in the chaotic infla- son to Gravity. Phys. Rev. Lett., 105:011302, tion with nonminimal coupling. Prog. Theor. 2010. Phys., 86:103–118, 1991. [30] Cristiano Germani and Alex Kehagias. Cos- [42] Redouane Fakir, Salman Habib, and William mological Perturbations in the New Higgs In- Unruh. Cosmological density perturbations flation. JCAP, 1005:019, 2010. [Erratum: with modified gravity. Astrophys. J., 394:396, JCAP1006,E01(2010)]. 1992.

28 [43] David I. Kaiser. Primordial spectral indices [54] Juan Garcia-Bellido, Daniel G. Figueroa, and from generalized Einstein theories. Phys. Javier Rubio. Preheating in the Standard Rev., D52:4295–4306, 1995. Model with the Higgs-Inflaton coupled to gravity. Phys. Rev., D79:063531, 2009. [44] J. L. Cervantes-Cota and H. Dehnen. Induced gravity inflation in the SU(5) GUT. Phys. Rev., [55] F. Bezrukov, D. Gorbunov, and M. Shaposh- D51:395–404, 1995. nikov. On initial conditions for the Hot Big Bang. JCAP, 0906:029, 2009. [45] Jorge L. Cervantes-Cota and H. Dehnen. In- duced gravity inflation in the standard model [56] F. Bezrukov, D. Gorbunov, and M. Shaposh- of particle physics. Nucl. Phys., B442:391– nikov. Late and early time phenomenology 412, 1995. of Higgs-dependent cutoff. JCAP, 1110:001, 2011. [46] Eiichiro Komatsu and Toshifumi Futamase. Constraints on the chaotic inflationary sce- [57] Joel Repond and Javier Rubio. Combined nario with a nonminimally coupled ’in- Preheating on the lattice with applications to flaton’ field from the cosmic microwave Higgs inflation. JCAP, 1607(07):043, 2016. background radiation anisotropy. Phys. [58] Yohei Ema, Ryusuke Jinno, Kyohei Mukaida, Rev., D58:023004, 1998. [Erratum: Phys. and Kazunori Nakayama. Violent Preheat- Rev.D58,089902(1998)]. ing in Inflation with Nonminimal Coupling. [47] Curtis G. Callan, Jr., Sidney R. Coleman, and JCAP, 1702(02):045, 2017. Roman Jackiw. A New improved energy - [59] Y. Akrami et al. Planck 2018 results. X. Con- momentum tensor. Annals Phys., 59:42–73, straints on inflation. arXiv:1807.06211. 1970. [60] Fedor Bezrukov and Mikhail Shaposhnikov. [48] N. D. Birrell and P. C. W. Davies. Quantum Higgs inflation at the critical point. Phys. Fields in Curved Space. Cambridge Mono- Lett., B734:249–254, 2014. graphs on Mathematical Physics. Cambridge Univ. Press, Cambridge, UK, 1984. [61] Jacopo Fumagalli and Marieke Postma. UV (in)sensitivity of Higgs inflation. JHEP, [49] Fedor Bezrukov, Javier Rubio, and Mikhail 05:049, 2016. Shaposhnikov. Living beyond the edge: Higgs inflation and vacuum metastability. Phys. [62] Jacopo Fumagalli. Renormalization Group in- Rev., D92(8):083512, 2015. dependence of Cosmological Attractors. Phys. Lett., B769:451–459, 2017. [50] K. A. Olive et al. Review of Particle Physics. Chin. Phys., C38:090001, 2014. [63] Vera-Maria Enckell, Kari Enqvist, and Sami Nurmi. Observational signatures of Higgs in- [51] CMS Collaboration. Combination of the CMS flation. JCAP, 1607(07):047, 2016. top-quark mass measurements from Run 1 of the LHC, Report number CMS-PAS-TOP-14- [64] Kyle Allison. Higgs xi-inflation for the 125- 015. 126 GeV Higgs: a two-loop analysis. JHEP, 02:040, 2014. [52] F. Bezrukov and M. Shaposhnikov. Standard Model Higgs boson mass from inflation: Two [65] Yuta Hamada, Hikaru Kawai, Kin-ya Oda, loop analysis. JHEP, 07:089, 2009. and Seong Chan Park. Higgs Inflation is Still Alive after the Results from BICEP2. Phys. [53] Sukanta Dutta, Kaoru Hagiwara, Qi-Shu Yan, Rev. Lett., 112(24):241301, 2014. and Kentaroh Yoshida. Constraints on the electroweak chiral Lagrangian from the preci- [66] Javier Rubio. Higgs inflation and vacuum sta- sion data. Nucl. Phys., B790:111–137, 2008. bility. J. Phys. Conf. Ser., 631:012032, 2015.

29 [67] Fedor Bezrukov, Martin Pauly, and Javier Ru- [79] C. P. Burgess, Hyun Min Lee, and Michael bio. On the robustness of the primordial Trott. Comment on Higgs Inflation and Natu- power spectrum in renormalized Higgs infla- ralness. JHEP, 07:007, 2010. tion. JCAP, 1802(02):040, 2018. [80] F. Bezrukov, A. Magnin, M. Shaposhnikov, [68] Syksy Rasanen and Pyry Wahlman. Higgs in- and S. Sibiryakov. Higgs inflation: consis- flation with loop corrections in the Palatini tency and generalisations. JHEP, 01:016, formulation. JCAP, 1711(11):047, 2017. 2011. [69] Vera-Maria Enckell, Kari Enqvist, Syksy Rasa- [81] Damien P. George, Sander Mooij, and nen, and Eemeli Tomberg. Higgs inflation at Marieke Postma. Quantum corrections in the hilltop. JCAP, 1806(06):005, 2018. Higgs inflation: the real scalar case. JCAP, 1402:024, 2014. [70] J. R. Espinosa, G. F. Giudice, and A. Riotto. Cosmological implications of the Higgs mass [82] Damien P. George, Sander Mooij, and measurement. JCAP, 0805:002, 2008. Marieke Postma. Quantum corrections in Higgs inflation: the Standard Model case. [71] A. O. Barvinsky, A. Yu. Kamenshchik, and JCAP, 1604(04):006, 2016. A. A. Starobinsky. Inflation scenario via the Standard Model Higgs boson and LHC. JCAP, [83] C. P. Burgess, Subodh P. Patil, and Michael 0811:021, 2008. Trott. On the Predictiveness of Single-Field Inflationary Models. JHEP, 06:010, 2014. [72] Andrea De Simone, Mark P. Hertzberg, and Frank Wilczek. Running Inflation in the Stan- [84] Mikhail Shaposhnikov and Daniel Zen- dard Model. Phys. Lett., B678:1–8, 2009. hausern. Scale invariance, unimodular gravity and dark energy. Phys. Lett., B671:187– [73] Fedor L. Bezrukov, Amaury Magnin, and 192, 2009. Mikhail Shaposhnikov. Standard Model Higgs boson mass from inflation. Phys. Lett., [85] Juan Garcia-Bellido, Javier Rubio, Mikhail B675:88–92, 2009. Shaposhnikov, and Daniel Zenhausern. Higgs-Dilaton Cosmology: From the Early to [74] A. O. Barvinsky, A. Yu. Kamenshchik, the Late Universe. Phys. Rev., D84:123504, C. Kiefer, A. A. Starobinsky, and C. F. 2011. Steinwachs. Higgs boson, renormalization group, and naturalness in cosmology. Eur. [86] Diego Blas, Mikhail Shaposhnikov, and Phys. J., C72:2219, 2012. Daniel Zenhausern. Scale-invariant alternatives to general relativity. Phys. Rev., [75] L. A. Popa and A. Caramete. Cosmological D84:044001, 2011. Constraints on Higgs Boson Mass. Astrophys. J., 723:803–811, 2010. [87] Fedor Bezrukov, Georgios K. Karananas, Javier Rubio, and Mikhail Shaposhnikov. [76] Alberto Salvio. Higgs Inflation at NNLO after Higgs-Dilaton Cosmology: an effective field the Boson Discovery. Phys. Lett., B727:234– theory approach. Phys. Rev., D87(9):096001, 239, 2013. 2013. [77] J. L. F. Barbon and J. R. Espinosa. On the [88] Juan Garcia-Bellido, Javier Rubio, and Naturalness of Higgs Inflation. Phys. Rev., Mikhail Shaposhnikov. Higgs-Dilaton cos- D79:081302, 2009. mology: Are there extra relativistic species? Phys. Lett., B718:507–511, 2012. [78] C. P. Burgess, Hyun Min Lee, and Michael Trott. Power-counting and the Validity of [89] Javier Rubio and Mikhail Shaposhnikov. the Classical Approximation During Inflation. Higgs-Dilaton cosmology: Universality versus JHEP, 09:103, 2009. criticality. Phys. Rev., D90:027307, 2014.

30 [90] Manuel Trashorras, Savvas Nesseris, and [100] Fedor Bezrukov and Mikhail Shaposhnikov. Juan Garcia-Bellido. Cosmological Con- Inflation, LHC and the Higgs boson. Comptes straints on Higgs-Dilaton Inflation. Phys. Rev., Rendus Physique, 16:994–1002, 2015. D94(6):063511, 2016. [101] Ian G. Moss. Higgs boson cosmology. Con- [91] Georgios K. Karananas and Javier Rubio. temp. Phys., 56(4):468–476, 2015. On the geometrical interpretation of scale- [102] Michal Artymowski and Javier Rubio. End- invariant models of inflation. Phys. Lett., lessly flat scalar potentials and α-attractors. B761:223–228, 2016. Phys. Lett., B761:111–114, 2016. [92] Pedro G. Ferreira, Christopher T. Hill, and [103] Georgios K. Karananas and Alexander Monin. Graham G. Ross. Scale-Independent Infla- Weyl vs. Conformal. Phys. Lett., B757:257– tion and Hierarchy Generation. Phys. Lett., 260, 2016. B763:174–178, 2016. [104] Georgios K. Karananas and Alexander Monin. [93] Pedro G. Ferreira, Christopher T. Hill, and Weyl and Ricci gauging from the coset con- Graham G. Ross. Weyl Current, Scale- struction. Phys. Rev., D93(6):064013, 2016. Invariant Inflation and Planck Scale Genera- tion. Phys. Rev., D95(4):043507, 2017. [105] Csaba Csaki, Nemanja Kaloper, Javi Serra, and John Terning. Inflation from Bro- [94] Pedro G. Ferreira, Christopher T. Hill, and ken Scale Invariance. Phys. Rev. Lett., Graham G. Ross. No fifth force in a scale in- 113:161302, 2014. variant universe. Phys. Rev., D95(6):064038, 2017. [106] David I. Kaiser. Conformal Transformations with Multiple Scalar Fields. Phys. Rev., [95] Santiago Casas, Martin Pauly, and Javier D81:084044, 2010. Rubio. Higgs-dilaton cosmology: An inflation-dark-energy connection and fore- [107] David I. Kaiser, Edward A. Mazenc, and Evan- casts for future galaxy surveys. Phys. Rev., gelos I. Sfakianakis. Primordial Bispectrum D97(4):043520, 2018. from Multifield Inflation with Nonminimal Couplings. Phys. Rev., D87:064004, 2013. [96] Pedro G. Ferreira, Christopher T. Hill, Jo- [108] David I. Kaiser and Evangelos I. Sfakianakis. hannes Noller, and Graham G. Ross. Infla- Multifield Inflation after Planck: The Case tion in a scale invariant universe. Phys. Rev., for Nonminimal Couplings. Phys. Rev. Lett., D97(12):123516, 2018. 112(1):011302, 2014.

[97] Pedro G. Ferreira, Christopher T. Hill, and [109] D. S. Gorbunov and A. G. Panin. Scalaron Graham G. Ross. Inertial Spontaneous Sym- the mighty: producing dark matter and metry Breaking and Quantum Scale Invari- baryon asymmetry at reheating. Phys. Lett., ance. Phys. Rev., D98(11):116012, 2018. B700:157–162, 2011.

[98] Santiago Casas, Georgios K. Karananas, [110] D. S. Gorbunov and A. G. Panin. Free scalar Martin Pauly, and Javier Rubio. Scale- dark matter candidates in R2-inflation: the invariant alternatives to general relativity. light, the heavy and the superheavy. Phys. III. The inflation–dark-energy connection, Lett., B718:15–20, 2012. arXiv:1811.05984. [111] Dmitry Gorbunov and Anna Tokareva. R2- [99] Fedor Bezrukov. The Higgs field as an infla- inflation with conformal SM Higgs field. ton. Class. Quant. Grav., 30:214001, 2013. JCAP, 1312:021, 2013.

31 [112] Matti Herranen, Tommi Markkanen, Sami [123] Eiichiro Komatsu and Toshifumi Futamase. Nurmi, and Arttu Rajantie. Spacetime curva- Complete constraints on a nonminimally cou- ture and the Higgs stability during inflation. pled chaotic inflationary scenario from the Phys. Rev. Lett., 113(21):211102, 2014. cosmic microwave background. Phys. Rev., D59:064029, 1999. [113] Kohei Kamada. Inflationary cosmology and the standard model Higgs with a small Hub- [124] Shinji Tsujikawa and Burin Gumjudpai. Den- ble induced mass. Phys. Lett., B742:126–135, sity perturbations in generalized Einstein sce- 2015. narios and constraints on nonminimal couplings from the Cosmic Microwave Back- [114] Daniel G. Figueroa, Arttu Rajantie, and Fran- ground. Phys. Rev., D69:123523, 2004. cisco Torrenti. Higgs field-curvature coupling and postinflationary vacuum instability. Phys. [125] Seoktae Koh. Non-gaussianity in nonmini- Rev., D98(2):023532, 2018. mally coupled scalar field theory. J. Korean Phys. Soc., 49:S787–S790, 2006. [115] Jose Maria Ezquiaga, Juan Garcia-Bellido, and Ester Ruiz Morales. Primordial Black [126] Takeshi Chiba and Masahide Yamaguchi. Ex- Hole production in Critical Higgs Inflation. tended Slow-Roll Conditions and Rapid-Roll Phys. Lett., B776:345–349, 2018. Conditions. JCAP, 0810:021, 2008.

[116] Isabella Masina. Ruling out Critical Higgs In- [127] Jan Weenink and Tomislav Prokopec. Gauge flation? Phys. Rev., D98(4):043536, 2018. invariant cosmological perturbations for the nonminimally coupled inflaton field. Phys. [117] Viatcheslav F. Mukhanov, H. A. Feldman, and Rev., D82:123510, 2010. Robert H. Brandenberger. Theory of cosmo- [128] Eanna E. Flanagan. The Conformal frame logical perturbations. Part 1. Classical per- freedom in theories of gravitation. Class. turbations. Part 2. Quantum theory of per- Quant. Grav., 21:3817, 2004. turbations. Part 3. Extensions. Phys. Rept., 215:203–333, 1992. [129] Takeshi Chiba and Masahide Yamaguchi. Conformal-Frame (In)dependence of Cosmo- [118] Sergio Ferrara, Renata Kallosh, Andrei Linde, logical Observations in Scalar-Tensor Theory. and Massimo Porrati. Minimal Super- JCAP, 1310:040, 2013. gravity Models of Inflation. Phys. Rev., D88(8):085038, 2013. [130] Marieke Postma and Marco Volponi. Equiv- alence of the Einstein and Jordan frames. [119] Renata Kallosh, Andrei Linde, and Diederik Phys. Rev., D90(10):103516, 2014. Roest. Superconformal Inflationary α- Attractors. JHEP, 11:198, 2013. [131] Laur Jarv, Piret Kuusk, Margus Saal, and Ott Vilson. Invariant quantities in the scalar- [120] Mario Galante, Renata Kallosh, Andrei Linde, tensor theories of gravitation. Phys. Rev., and Diederik Roest. Unity of Cosmolog- D91(2):024041, 2015. ical Inflation Attractors. Phys. Rev. Lett., 114(14):141302, 2015. [132] Jing Ren, Zhong-Zhi Xianyu, and Hong-Jian He. Higgs Gravitational Interaction, Weak [121] Ioannis Chatzigeorgiou. Bounds on the lam- Boson Scattering, and Higgs Inflation in Jor- bert function and their application to the dan and Einstein Frames. JCAP, 1406:032, outage analysis of user cooperation. CoRR, 2014. abs/1601.04895, 2016. [133] Laur Jarv, Piret Kuusk, Margus Saal, and Ott [122] F. L. Bezrukov and D. S. Gorbunov. Dis- Vilson. Transformation properties and gen- tinguishing between R2-inflation and Higgs- eral relativity regime in scalar-tensor theo- inflation. Phys. Lett., B713:365–368, 2012. ries. Class. Quant. Grav., 32:235013, 2015.

32 [134] Piret Kuusk, Laur Jarv, and Ott Vilson. In- [144] Gian F. Giudice and Hyun Min Lee. Unita- variant quantities in the multiscalar-tensor rizing Higgs Inflation. Phys. Lett., B694:294– theories of gravitation. Int. J. Mod. Phys., 300, 2011. A31(02n03):1641003, 2016. [145] J. L. F. Barbon, J. A. Casas, J. Elias-Miro, and [135] Laur Jarv, Kristjan Kannike, Luca Marzola, J. R. Espinosa. Higgs Inflation as a Mirage. Antonio Racioppi, Martti Raidal, Mihkel JHEP, 09:027, 2015. Runkla, Margus Saal, and Hardi Veermae. [146] Mark P. Hertzberg. On Inflation with Non- Frame-Independent Classification of Single- minimal Coupling. JHEP, 11:023, 2010. Field Inflationary Models. Phys. Rev. Lett., 118(15):151302, 2017. [147] Sergio Ferrara, Renata Kallosh, Andrei Linde, Alessio Marrani, and Antoine Van Proeyen. [136] Daniel Burns, Sotirios Karamitsos, and Apos- Superconformal Symmetry, NMSSM, and In- tolos Pilaftsis. Frame-Covariant Formula- flation. Phys. Rev., D83:025008, 2011. tion of Inflation in Scalar-Curvature Theories. Nucl. Phys., B907:785–819, 2016. [148] A. O. Barvinsky, A. Yu. Kamenshchik, C. Kiefer, A. A. Starobinsky, and [137] Alexandros Karam, Thomas Pappas, and Kyri- C. Steinwachs. Asymptotic freedom in akos Tamvakis. Frame-dependence of higher- inflationary cosmology with a non-minimally order inflationary observables in scalar- coupled Higgs field. JCAP, 0912:003, 2009. tensor theories. Phys. Rev., D96(6):064036, [149] Michael Atkins and Xavier Calmet. Remarks 2017. on Higgs Inflation. Phys. Lett., B697:37–40, [138] Sotirios Karamitsos and Apostolos Pilaftsis. 2011. Frame Covariant Nonminimal Multifield In- [150] Mark P. Hertzberg. Can Inflation be Con- flation. Nucl. Phys., B927:219–254, 2018. nected to Low Energy Particle Physics? JCAP, 1208:008, 2012. [139] Sotirios Karamitsos and Apostolos Pilaftsis. On the Cosmological Frame Problem. In 17th [151] Youngsoo Yoon and Yongsung Yoon. Asymp- Hellenic School and Workshops on Elementary totic conformal invariance of SU(2) and stan- Particle Physics and Gravity (CORFU2017) dard models in curved space-time. Int. J. Corfu, Greece, September 2-28, 2017, 2018. Mod. Phys., A12:2903–2914, 1997.

[140] Ufuk Aydemir, Mohamed M. Anber, and [152] Alberto Salvio. Initial Conditions for Critical John F. Donoghue. Self-healing of unitarity in Higgs Inflation. Phys. Lett., B780:111–117, effective field theories and the onset of new 2018. physics. Phys. Rev., D86:014025, 2012. [153] Jessica L. Cook, Lawrence M. Krauss, An- drew J. Long, and Subir Sabharwal. Is [141] Xavier Calmet and Roberto Casadio. Self- Higgs inflation ruled out? Phys. Rev., healing of unitarity in Higgs inflation. Phys. D89(10):103525, 2014. Lett., B734:17–20, 2014. [154] Yuta Hamada, Hikaru Kawai, Kin-ya Oda, [142] Ippocratis D. Saltas. Higgs inflation and and Seong Chan Park. Higgs inflation quantum gravity: An exact renormalisation from Standard Model criticality. Phys. Rev., group approach. JCAP, 1602:048, 2016. D91:053008, 2015. [143] Albert Escriva` and Cristiano Germani. Be- [155] Kristjan Kannike, Luca Marzola, Martti yond dimensional analysis: Higgs and new Raidal, and Hardi Veermae. Single Field Higgs inflations do not violate unitarity. Phys. Double Inflation and Primordial Black Holes. Rev., D95(12):123526, 2017. JCAP, 1709(09):020, 2017.

33 [156] Cristiano Germani and Tomislav Prokopec. [167] Gabriela Barenboim, Wan-Il Park, and On primordial black holes from an inﬂection William H. Kinney. Eternal Hilltop Inﬂation. point. Phys. Dark Univ., 18:6–10, 2017. JCAP, 1605(05):030, 2016.

[157] Juan Garcia-Bellido and Ester Ruiz Morales. [168] Andrei D. Linde. Phase Transitions in Gauge Primordial black holes from single field mod- Theories and Cosmology. Rept. Prog. Phys., els of inflation. Phys. Dark Univ., 18:47–54, 42:389, 1979. 2017. [169] K. Kajantie, M. Laine, K. Rummukainen, and Mikhail E. Shaposhnikov. Generic rules for [158] Alexei A. Starobinsky and Junichi Yokoyama. high temperature dimensional reduction and Equilibrium state of a selfinteracting scalar their application to the standard model. Nucl. field in the De Sitter background. Phys. Rev., Phys., B458:90–136, 1996. D50:6357–6368, 1994. [170] Greg W. Anderson. New Cosmological Con- [159] Vincent Vennin and Alexei A. Starobinsky. straints on the Higgs Boson and Top Quark Correlation Functions in Stochastic Inflation. Masses. Phys. Lett., B243:265–270, 1990. Eur. Phys. J., C75:413, 2015. [171] Peter Brockway Arnold and Stamatis Vokos. [160] Chris Pattison, Vincent Vennin, Hooshyar As- Instability of hot electroweak theory: bounds sadullahi, and David Wands. Quantum dif- on m(H) and M(t). Phys. Rev., D44:3620– fusion during inflation and primordial black 3627, 1991. holes. JCAP, 1710(10):046, 2017. [172] J. R. Espinosa and M. Quiros. Improved [161] Jose Maria Ezquiaga and Juan Garcia-Bellido. metastability bounds on the standard model Quantum diffusion beyond slow-roll: impli- Higgs mass. Phys. Lett., B353:257–266, 1995. cations for primordial black-hole production. [173] Gary N. Felder, Andrei V. Frolov, Lev Kof- JCAP, 1808:018, 2018. man, and Andrei D. Linde. Cosmology with negative potentials. Phys. Rev., D66:023507, [162] Bernard Carr, Florian Kuhnel, and Marit 2002. Sandstad. Primordial Black Holes as Dark Matter. Phys. Rev., D94(8):083504, 2016. [174] Rose N. Lerner and John McDonald. A Unitarity-Conserving Higgs Inflation Model. [163] Simeon Bird, Ilias Cholis, Julian B. Munoz, Phys. Rev., D82:103525, 2010. Yacine Ali-Haimoud, Marc Kamionkowski, Ely D. Kovetz, Alvise Raccanelli, and Adam G. [175] Ido Ben-Dayan and Martin B. Einhorn. Super- Riess. Did LIGO detect dark matter? Phys. gravity Higgs Inflation and Shift Symmetry in Rev. Lett., 116(20):201301, 2016. Electroweak Theory. JCAP, 1012:002, 2010. [176] Kohei Kamada, Tsutomu Kobayashi, [164] Bernard Carr, Martti Raidal, Tommi Tenka- Masahide Yamaguchi, and Jun’ichi nen, Ville Vaskonen, and Hardi Veermae. Pri- Yokoyama. Higgs G-inflation. Phys. Rev., mordial black hole constraints for extended D83:083515, 2011. mass functions. Phys. Rev., D96(2):023514, 2017. [177] Rose N. Lerner and John McDonald. Dis- tinguishing Higgs inflation and its variants. [165] Syksy Rasanen and Eemeli Tomberg. Planck Phys. Rev., D83:123522, 2011. scale black hole dark matter from Higgs inflation, arXiv:1810.12608. [178] Masato Arai, Shinsuke Kawai, and Nobuchika Okada. Higgs inflation in minimal supersym- [166] Lotfi Boubekeur and David. H. Lyth. Hilltop metric SU(5) GUT. Phys. Rev., D84:123515, inflation. JCAP, 0507:010, 2005. 2011.

34 [179] Christian F. Steinwachs and Alexander Yu. [191] Nobuchika Okada and Qaisar Shafi. Higgs Kamenshchik. Non-minimal Higgs Inflation Inflation, Seesaw Physics and Fermion Dark and Frame Dependence in Cosmology. 2013. Matter. Phys. Lett., B747:223–228, 2015. [AIP Conf. Proc.1514,161(2012)]. [192] Rong-Gen Cai, Zong-Kuan Guo, and Shao- [180] Kohei Kamada, Tsutomu Kobayashi, Tomo Jiang Wang. Higgs inflation in Gauss-Bonnet Takahashi, Masahide Yamaguchi, and braneworld. Phys. Rev., D92(6):063514, Jun’ichi Yokoyama. Generalized Higgs 2015. inflation. Phys. Rev., D86:023504, 2012. [193] G. Lazarides and C. Pallis. Shift Symmetry [181] Martin B. Einhorn and D. R. Timothy Jones. and Higgs Inflation in Supergravity with Ob- GUT Scalar Potentials for Higgs Inflation. servable Gravitational Waves. JHEP, 11:114, JCAP, 1211:049, 2012. 2015.

[182] Ross N. Greenwood, David I. Kaiser, and [194] Carsten van de Bruck and Chris Long- Evangelos I. Sfakianakis. Multifield Dynamics den. Higgs Inflation with a Gauss-Bonnet of Higgs Inflation. Phys. Rev., D87:064021, term in the Jordan Frame. Phys. Rev., 2013. D93(6):063519, 2016.

[183] Shinya Kanemura, Toshinori Matsui, and [195] Fuminobu Takahashi and Ryo Takahashi. Takehiro Nabeshima. Higgs inflation in a ra- Renormalization Group Improved Higgs In- diative seesaw model. Phys. Lett., B723:126– flation with a Running Kinetic Term. Phys. 131, 2013. Lett., B760:329–334, 2016. [196] Xavier Calmet and Ibere Kuntz. Higgs [184] Sayantan Choudhury, Trina Chakraborty, and Starobinsky Inflation. Eur. Phys. J., Supratik Pal. Higgs inflation from new Kahler¨ C76(5):289, 2016. potential. Nucl. Phys., B880:155–174, 2014. [197] John Ellis, Hong-Jian He, and Zhong-Zhi Xi- [185] Ichiro Oda and Takahiko Tomoyose. anyu. Higgs Inflation, Reheating and Grav- Quadratic Chaotic Inflation from Higgs itino Production in No-Scale Supersymmetric Inflation. Adv. Stud. Theor. Phys., 8:551, GUTs. JCAP, 1608(08):068, 2016. 2014. [198] Nobuchika Okada and Digesh Raut. [186] Zhong-Zhi Xianyu and Hong-Jian He. Asymp- Inflection-point Higgs Inflation. Phys. totically Safe Higgs Inflation. JCAP, Rev., D95(3):035035, 2017. 1410:083, 2014. [199] Shao-Feng Ge, Hong-Jian He, Jing Ren, and [187] Ichiro Oda and Takahiko Tomoyose. Confor- Zhong-Zhi Xianyu. Realizing Dark Matter and mal Higgs Inflation. JHEP, 09:165, 2014. Higgs Inflation in Light of LHC Diphoton Ex- [188] John Ellis, Hong-Jian He, and Zhong-Zhi Xi- cess. Phys. Lett., B757:480–492, 2016. anyu. New Higgs Inflation in a No-Scale [200] I. G. Marian, N. Defenu, A. Trombettoni, and Supersymmetric SU(5) GUT. Phys. Rev., I. Nandori. Pseudo Periodic Higgs Inflation, D91(2):021302, 2015. arXiv:1705.10276. [189] Hong-Jian He and Zhong-Zhi Xianyu. Ex- [201] Yohei Ema, Mindaugas Karciauskas, Oleg tending Higgs Inflation with TeV Scale New Lebedev, Stanislav Rusak, and Marco Zatta. Physics. JCAP, 1410:019, 2014. Higgs-Inflaton Mixing and Vacuum Stability, arXiv:1711.10554. [190] Kohei Kamada. On the strong coupling scale in Higgs G-inflation. Phys. Lett., B744:347– [202] Yohei Ema. Higgs Scalaron Mixed Inflation. 351, 2015. Phys. Lett., B770:403–411, 2017.

35 [203] Minxi He, Alexei A. Starobinsky, and Jun’ichi [211] Bruce Allen and Antoine Folacci. The Mass- Yokoyama. Inﬂation in the mixed Higgs-R2 less Minimally Coupled Scalar Field in De Sit- model. JCAP, 1805(05):064, 2018. ter Space. Phys. Rev., D35:3771, 1987.

[204] Heng-Yu Chen, Ilia Gogoladze, Shan Hu, [212] Joanna Jalmuzna, Andrzej Rostworowski, Tianjun Li, and Lina Wu. Natural Higgs Infla- and Piotr Bizon. A Comment on AdS collapse tion, Gauge Coupling Unification, and Neu- of a scalar field in higher dimensions. Phys. trino Masses, arXiv:1805.00161. Rev., D84:085021, 2011. [205] James W. York, Jr. Role of conformal three ge- [213] Ignatios Antoniadis, J. Iliopoulos, and T. N. ometry in the dynamics of gravitation. Phys. Tomaras. Quantum Instability of De Sitter Rev. Lett., 28:1082–1085, 1972. Space. Phys. Rev. Lett., 56:1319, 1986. [206] G. W. Gibbons and S. W. Hawking. Action Integrals and Partition Functions in Quantum [214] N. C. Tsamis and R. P. Woodard. Relaxing the Gravity. Phys. Rev., D15:2752–2756, 1977. cosmological constant. Phys. Lett., B301:351– 357, 1993. [207] M. Ferraris, M. Francaviglia, and C. Reina. Variational formulation of general relativity [215] N. C. Tsamis and R. P. Woodard. Strong from 1915 to 1925 Palatini’s method discov- infrared effects in quantum gravity. Annals ered by Einstein in 1925. General Relativity Phys., 238:1–82, 1995. and Gravitation, 14(3):243–254, Mar 1982. [216] Ignatios Antoniadis, Pawel O. Mazur, and [208] Syksy Rasanen. Higgs inflation in the Palatini Emil Mottola. Cosmological dark energy: formulation with kinetic terms for the metric, Prospects for a dynamical theory. New J. arXiv: 1811.09514. Phys., 9:11, 2007. [209] T. Matsumura et al. LiteBIRD: Mission Overview and Focal Plane Layout. J. Low. [217] A. M. Polyakov. Decay of Vacuum Energy. Temp. Phys., 184(3-4):824–831, 2016. Nucl. Phys., B834:316–329, 2010. [210] Florian Bauer and Durmus A. Demir. Higgs- [218] C. Wetterich. Graviton fluctuations erase the Palatini Inflation and Unitarity. Phys. Lett., cosmological constant. Phys. Lett., B773:6– B698:425–429, 2011. 19, 2017.