PHYSICAL REVIEW D 99, 083519 (2019)

Preheating after Higgs inflation: Self-resonance and gauge production

† Evangelos I. Sfakianakis1,2,* and Jorinde van de Vis1, 1Nikhef, Science Park 105, 1098XG Amsterdam, The Netherlands 2Lorentz Institute for Theoretical , Leiden University, 2333CA Leiden, The Netherlands

(Received 6 December 2018; published 18 April 2019)

We perform an extensive analysis of linear fluctuations during preheating in Higgs inflation in the Einstein frame, where the fields are minimally coupled to , but the -space metric is nontrivial. The self-resonance of the Higgs and the Higgsed gauge are governed by effective masses that scale differently with the nonminimal couplings and evolve differently in time. Coupled metric perturbations enhance Higgs self-resonance and make it possible for Higgs inflation to preheat solely through this channel. For ξ ≳ 100 the total energy of the Higgs- condensate can be transferred to Higgs within 3 e-folds after the end of inflation. For smaller values of the nonminimal coupling preheating takes longer, completely shutting off at around ξ ≃ 30. The production of gauge bosons is dominated by the gauge boson mass and the field-space curvature. For large values of the nonminimal coupling ξ ≳ 1000,it is possible for the Higgs condensate to transfer the entirety of its energy into gauge fields within one oscillation. For smaller values of the nonminimal coupling gauge bosons decay very quickly into , thereby shutting off Bose enhancement. Estimates of non-Abelian indicate that they will not suppress preheating into gauge bosons for ξ ≳ 1000.

DOI: 10.1103/PhysRevD.99.083519

I. INTRODUCTION fixed point under renormalization group flow, at least below the Planck scale [10]. The inherent ambiguity in the running While the discovery of the at CERN [1] of the Higgs self-coupling λ at high energies, due to our solidified our understanding of the (SM), its incomplete knowledge of possible new physics between the behavior in the early Universe, above the electroweak TeV and inflationary scales, leads to an ambiguity in the symmetry-breaking scale, remains unclear. An intriguing exact value of the required nonminimal coupling [17–19]. possibility is the identification of the Higgs boson with the λ O 0 01 While simple estimates like infl ¼ ð . Þ lead to the scalar field(s) necessary for driving inflation, the rapid ξ O 104 λ acceleration phase of the Universe required to solve both requirement of ¼ ð Þ, smaller values of can allow for the horizon and flatness problems, as well as seeding much smaller nonminimal couplings. We will remain primordial fluctuations necessary for structure formation agnostic about the exact running of the Standard Model [2–4]. couplings at high energies and instead explore a broad 1 10 ≲ ξ ≲ 104 The original attempt to use the Higgs or a Higgs-like parameter range covering . sector to drive inflation resulted in an inconsistently large A basic feature of inflationary models with nonminimal amplitude of fluctuations [5], because of the value of the couplings is that they provide universal predictions for the Higgs self-coupling λ in the Standard Model. However, the spectral observables ns and r, largely independent of the introduction of a nonminimal coupling between the Higgs exact model parameters and initial conditions [23,24]. field and the Ricci scalar can remedy this [6]. Such non- These observables fall in line with the Starobinsky model α 2 minimal couplings are not only generic, since they arise as [25] as well as with the large family of attractors [26]. necessary renormalization counterterms for scalar fields in – curved spacetime [7 16], but they also grow without a UV 1For inflation on a flat plateau one should consider ξ ≳ 440 (e.g., Ref. [20]). In models of hilltop or inflection-point inflation, smaller values of ξ are possible, although UV corrections are *[email protected] expected to be larger. A discussion of the effects of UV † [email protected] corrections (as well as initial conditions) in Higgs inflation can be found, e.g., in Refs. [21,22]. In order to provide as Published by the American Physical Society under the terms of complete a treatment of Higgs inflation as possible without the Creative Commons Attribution 4.0 International license. referring to specific unknown physics, we choose to consider a Further distribution of this work must maintain attribution to broad range of nonminimal couplings that go below ξ ≈ 400. the author(s) and the published article’s title, journal citation, 2See Ref. [27] for a way to alter the predictions of α-attractor and DOI. Funded by SCOAP3. models through multifield effects.

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Even after the latest Planck release [28], these models, of the Standard Model, the unitarity cutoff scale has been 1–2 O 1 2 – which predict ns ¼ =N and r ¼ ð =NÞ, continue to extensively studied [19,65 68] (see also Ref. [69] for a be compatible with the data for modes that exit the horizon recent review). For large values of the Higgs vacuum ≃ 55 expectation value (VEV) (like the ones appearing during at N e-folds before the end of inflation. pffiffiffi ξ While inflation provides a robust framework for com- inflation) the appropriate unitarity cutoff scale is Mpl= , puting the evolution of the Universe and the generation of while for small values of the Higgs VEV it must be – ξ ≡ 8π −1=2 fluctuations [3,29 35], the transition from an inflating substituted by Mpl= , where Mpl ð GÞ is the universe to a radiation bath (as required for big bang reduced Planck mass. nucleosynthesis [36–38]), known as reheating, remains a In Sec. II of this work, we introduce a simplified model weakly constrained era in the cosmic evolution. Despite the of a complex Higgs field coupled to an Abelian gauge field. difficulty of directly observing reheating due to the very Section III describes the generalization of this model to the short length scales involved, knowledge of how the full electroweak sector of the Standard Model. In Sec. IV equation of state of the Universe transitioned from w ≃ we study the self-resonance of Higgs modes. Section V −1 to w ¼ 1=3 is crucial, since it affects how one relates the deals with the evolution of the gauge fields during and after observed cosmic microwave background (CMB) modes to Higgs inflation. At the end of this section we also address the time during inflation when they exited the horizon the unitarity scale. The decays and scattering processes that [39–46]. This becomes increasingly relevant as new data involve the produced Higgs and gauge bosons are shrink the experimental bounds on primordial observables. described in Sec. VI, and observational consequences The transfer of energy from the inflaton [which carries are described in Sec. VII. Concluding remarks follow in (almost) the entirety of the energy density of the Universe Sec. VIII. during inflation] to radiation degrees of freedom (d.o.f.) can occur either through perturbative decays or through non- II. ABELIAN MODEL AND FORMALISM perturbative processes. The latter case, denoted as preheat- ing, includes parametric and tachyonic resonances (see We build on the formalism of Ref. [70] for the evolution Ref. [47] for a review). The end state of any (p)reheating of nonminimally coupled multifield models, as it was scenario must be a universe filled with SM and dark applied in Refs. [23,71,72] during inflation and in (DM) particles, or at least intermediary particles that decay Refs. [61–63] during preheating. The electroweak sector into the SM and DM sectors. Preheating therefore has the consists of a complex Higgs doublet, expressed using four potential to address other long-standing challenges in real-valued scalar fields in 3 þ 1 spacetime dimensions: cosmological theory (such as generating the observed   -antibaryon asymmetry [48–52]) or leave behind 1 φ þ h þ iθ Φ ¼ pffiffiffi ; ð1Þ cosmological relics (such as cosmological magnetic fields 2 ϕ3 þ iϕ4 [53,54] or primordial black holes [55–57]). Higgs inflation provides a unique opportunity to study where φ is the background value of the Higgs field, h the transition from inflation to radiation domination, since denotes the Higgs fluctuations, and θ; ϕ3, and ϕ4 are the the couplings of the Higgs inflaton to the rest of the SM are 2 1 Goldstone modes. We also add the SUð ÞL and Uð ÞY known. Detailed analyses of reheating in Higgs inflation gauge sectors. We will start by closely examining an were first performed in Refs. [58,59] and extended in Abelian simplified model of the full electroweak sector, Ref. [60] using lattice simulations. However, as discussed consisting of the complex scalar field later in Refs. [61–63] and independently in Ref. [64], multifield models of inflation with nonminimal couplings 1 Φ ¼ pffiffiffi ðφ þ h þ iθÞð2Þ to gravity can exhibit more efficient preheating behavior 2 than previously thought, due to the contribution of the field- space structure to the effective mass of the fluctuations. and a Uð1Þ gauge field only. The full equations of the Furthermore, it was shown in Refs. [61–63] that, in Higgsed electroweak sector are given in Sec. III, where we nonminimally coupled models, preheating efficiency can also discuss their relation to the Abelian simplified model. be vastly different for different values of the nonminimal In order to connect our notation to that of Ref. [70],we coupling, even if these values lead to otherwise identical identify ϕ1 ¼ φ þ h and ϕ2 ¼ θ. We will start by deriving predictions for CMB observables. We will thus perform a the equations of motion for general ϕI fields for notational detailed study of preheating in Higgs inflation, extending simplicity. We use uppercase latin letters to label field- the results of Refs. [58,59,61–64], in order to distinguish space indices, I;J ¼ 1, 2, 3, 4 (or just I;J ¼ 1, 2 in the between Higgs inflation models with different values of the Abelian case); greek letters to label spacetime indices, μ, nonminimal coupling. ν ¼ 0, 1, 2, 3; and lowercase latin letters to label spatial Because of the appeal of Higgs inflation as an economi- indices, i, j ¼ 1, 2, 3. The spacetime metric has signa- cal model of realizing inflation within the content ture ð−; þ; þ; þÞ.

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We first consider Uð1Þ symmetry with the corresponding as in Refs. [70,71]. The potential in the Jordan frame is the gauge field Bμ. The Lagrangian in the Jordan frame is usual Standard Model Higgs potential given by   Z  1 2 pffiffiffiffiffiffi V˜ ðΦ; Φ†Þ¼λ jΦj2 − v2 ≃ λjΦj4; ð10Þ 4 † ˜ μν ˜ † ˜ 4 SJ ¼ d x −g˜ fðΦ; Φ ÞR − g˜ ð∇μΦÞ ∇νΦ  1 246 − ˜μρ ˜νσ − ˜ Φ Φ† where the Higgs v ¼ GeV 4 g g FμνFρσ Vð ; Þ : ð3Þ can be safely neglected at field values that arise during inflation and preheating. Hence, the Higgs potential can be ˜ The covariant derivative ∇μ is given by adequately modeled by a pure quartic term. For the sake of readability, we will drop the arguments of ˜ ˜ G ∇μ ¼ Dμ þ ieBμ; ð4Þ , V, and f from now on. Varying the action with respect to the scalar fields ϕI, the corresponding equation of motion ˜ where Dμ is a covariant derivative with respect to the for ϕI is ˜ spacetime metric gμν and e is the coupling constant. The    corresponding field strength tensor3 is M4 □ϕI μνΓI ∂ ϕJ∂ ϕK GIJ pl 2 2 − þ g JK μ ν þ e B ;J V;J ˜ ˜ 4ξf Fμν ¼ DμBν − DνBμ: ð5Þ M2 pl IJ μ † μ † þ ie f; G ð−B Φ DμΦ þ B ðDμΦ ÞΦÞ By performing a conformal transformation 2f2 J    2 1 1 † − 2 GIJ − μΦ† Φ μΦ† Φ g˜μν x → gμν x f Φ; Φ g˜μν x ; 6 ieM B ;J Dμ þ Dμ B ;J ð Þ ð Þ¼ 2 ð Þ ð Þ ð Þ pl 2f 2f Mpl    † 1 μ 1 μ † − Φ ; Dμ B Φ þ B ðDμΦ ÞΦ; ¼ 0: ð11Þ the action in the Einstein frame becomes J 2f 2f J Z  2  pffiffiffiffiffiffi M 1 4 − pl − μν G Φ Φ† ϕI ϕJ We work to first order in fluctuations, in both the scalar S ¼ d x g 2 R g 2 IJð ; ÞDμ Dν fields and spacetime metric. The gauge fields have no 2 Mpl † † background component, and thus we only treat them þ ððieBμΦÞ ðieBνΦÞþieð−BμΦ DνΦ 2fðΦ; Φ†Þ as first-order perturbations. We consider scalar metric   1 perturbations around a spatially flat Friedmann-Lemaître- Φ† Φ − Φ Φ† − μρ νσ Robertson-Walker metric, þ BνðDμ Þ ÞÞ Vð ; Þ 4 g g FμνFρσ ; 2 μ ν ð7Þ ds ¼ gμνðxÞdx dx − 1 2 2 2 ∂ B i with ¼ ð þ AÞdt þ að i Þdx dt 2 i j þ a ½ð1 − 2ψÞδij þ 2∂i∂jEdx dx ; ð12Þ M4 Φ Φ† pl ˜ Φ Φ† Vð ; Þ¼ 2 † Vð ; Þ; ð8Þ 4f ðΦ; Φ Þ where aðtÞ is the scale factor. We may always choose a coordinate transformation and eliminate two of the four and scalar metric functions that appear in Eq. (12).Weworkin the longitudinal gauge, where BðxÞ¼EðxÞ¼0.Further- 2 † Mpl more, in the absence of anisotropic pressure perturbations, the G ðΦ; Φ Þ¼ ψ IJ 2fðΦ; Φ†Þ remaining two functions are equal, AðxÞ¼ ðxÞ.   We also expand the fields, 3 δ Φ Φ† Φ Φ† × IJ þ Φ Φ† fð ; Þ;I fð ; Þ;J ; fð ; Þ ϕIðxμÞ¼φIðtÞþδϕIðxμÞ: ð13Þ ð9Þ Note that for Higgs inflation only ϕ1 has a background 2 3 value, φðtÞ, whereas the background value of ϕ is zero. The tensor Fμν is defined with lower indices. In that case, it does not matter whether partial or covariant derivatives are used. We may then construct generalizations of the However, when working with F˜ μν it does matter, since the metric Mukhanov-Sasaki variable that are invariant with respect does not commute with partial derivatives. So, F˜ μν is given by to spacetime gauge transformations up to first order in the ˜ μν μρ νσ μρ νσ ˜ ˜ ˜ μ ν ˜ ν μ F ¼ g˜ g˜ Fρσ ¼ g˜ g˜ ðDρBσ − DσBρÞ¼D B − D B . perturbations (see Ref. [61] and references therein):

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φ_ I attractor [23,61,71], the background field motion proceeds QI ¼ δϕI þ ψ: ð14Þ H along a straight single-field trajectory φðtÞ. GIJ and MIJ are then diagonal at background order, so the equations of The background equation of motion for φI is unchanged motion for the first-order fluctuations h and θ do not mix: with respect to models with multiple scalar fields and no   k2 gauge bosons, D2Qh þ 3HD Qh þ þ Mh Qh ¼ 0; t t a2 h D φ_ I 3 φ_ I GIJ 0   t þ H þ V;J ¼ ; ð15Þ 2 D2 θ 3 D θ k Mθ θ and t Q þ H tQ þ 2 þ θ Q   a 1 1 2    2 M 1 H ¼ G φ_ Iφ_ J þ VðφIÞ ; pl θθ μ μ μ 3 2 2 IJ − e G 2B ∂μφ þðDμB Þφ þ 2fB φDμ Mpl 2f 2f 1 _ − G φ_ Iφ_ J ¼ 0; ð20Þ H ¼ 2 2 IJ ; ð16Þ Mpl where where overdots denote derivatives with respect to t, and the _ φ_ Hubble parameter is given by HðtÞ¼a=a. Covariant Qh ¼ h þ Ψ;Qθ ¼ θ: ð21Þ derivatives with respect to the field-space metric are given H D I ∂ I ΓI K 4 I by JA ¼ JA þ JKA for a field-space vector A , We see that only the Higgs fluctuations, generated along the from which we may construct the (covariant) directional direction of background motion, are coupled to the metric derivative with respect to cosmic time, perturbations Ψ. In the language of Refs. [61–63], the I Higgs fluctuations correspond to adiabatic modes. D AI ¼ φ_ JD AI ¼ A_ þ ΓI φ_ JAK; ð17Þ t J JK The equations are simplified if we replace QI → XI=aðtÞ ΓI φL where the Christoffel symbols JKð Þ are constructed and use covariant derivatives with respect to conformal K τ from GIJðφ Þ. time instead of cosmic time. We multiply the equations by We now specify our analysis to the case of a complex a3 and obtain Higgs field with background φðtÞ and fluctuations hðt; x⃗Þ    1 and θðt; x⃗Þ as in Eq. (2). The equation of motion for the 2 h 2 2 h h h Dτ X þ k þ a M h − RG h X ¼ 0; ð22Þ I 6 gauge-invariant fluctuation Q is identical to the case    – without the presence of a gauge field [61 63,70],upto 2 θ 2 2 θ 1 θ θ Dτ X k a M θ − RG θ X terms that mix θ and Bμ: þ þ 6     2 M2 θ 2 1 2 I I k I I J pl IJ d 3 Mpl θθ 0 μ 0 _ D Q þ 3HD Q þ δ þ M Q − e G − ea G 2B φ_ þðDμB Þφ − B φf ¼ 0; t t a2 J J 2f dϕJ 2    f f 1 μ μ μ ð23Þ × 2B ∂μφ þðDμB Þφ þ 2fB φDμ ¼ 0; 2f where R is the spacetime Ricci curvature. ð18Þ The variation of the action with respect to the gauge field δS S → S δBμ gives where we define the mass-squared matrix by þ δBμ   3 1 a M2 e2 M2 MI ≡GIK D D V −RI φ_ Lφ_ M − D φ_ Iφ_ ; νμ pl † μ pl μν † † J ð J K Þ LMJ 2 3 t J DνF − Φ ΦB þie g ðΦ ∂νΦ−ð∂νΦ ÞΦÞ¼0: Mpla H f 2f ð19Þ ð24Þ

RI 5 and LMJ is the Riemann tensor constructed from the Since there is no background value for the gauge field, the G φK field-space metric IJð Þ. The term in Eq. (19) propor- first-order perturbation equation is 1 2 tional to =Mpl arises from the coupled metric perturbations ’ 5 through expanding Einstein s field equations to linear order There has been a growing interest in inflation models where and using Eq. (14). It hence vanishes in the limit of an gauge fields acquire a nontrivial background value during → ∞ 2 infinitely rigid spacetime MPl . In the single-field inflation. While this is not possible for Abelian fields, SUð Þ gauge fields can have nontrivial vacuum configurations during inflation, leading to interesting phenomenology, like the violation 4Examples of field-space vectors include AI ¼ δϕI and of the Lyth bound [73] and tensor non-Gaussianity [74], while I _ I A ¼ φ . providing ns and r in agreement with CMB observations.

083519-4 PREHEATING AFTER HIGGS INFLATION: SELF-RESONANCE … PHYS. REV. D 99, 083519 (2019)   2 2 2 2 φ_ _ _ νμ Mple 2 μ Mpl μν ̈ _ k f kikjBj;k DνF − φ B þ e g ðθ∂νφ − φ∂νθÞ¼0; B k þ HB k þ B k þ 2 − þ H 2 2 2 2 i; i; 2 i; φ 2 M a f f a f 2 pl 2φ2 k þ 2f e ð25Þ 2 2 Mple 2 þ φ B k ¼ 0; ð31Þ 2f i; where we used Eq. (2) and we stress again that Fνμ is defined using covariant derivatives. which is somewhat simplified in conformal time, Until now we have worked in full generality and not   chosen a gauge. Hence, we are in principle working with ∂ 2 2 ∂τφ ∂τf ∂τa kikj τBj;k ∂τ B k þ k B k þ 2 − þ 2 2 more d.o.f. than needed. We will distinguish two frequently i; i; φ 2 M a f a 2 pl 2φ2 used gauges: unitary and Coulomb gauge. The equation of k þ 2f e motion of Xh is unaffected by the gauge choice. 2 2 2 Mple 2 þ a φ B k ¼ 0: ð32Þ 2f i; A. Unitary gauge θ In the unitary gauge θ ¼ 0 ¼ X . Equation (23) thus We now distinguish between transverse (Bk ) and longi- L becomes a constraint equation, tudinal (Bk ) modes:

  ⃗ ϵˆL L ϵˆþ þ ϵˆ− − 2φ_ _ Bk ¼ kBk þ k Bk þ kBk; ð33Þ μ − f 0 DμB ¼ φ þ B : ð26Þ f with

L The equations of motion for the gauge fields are ik · ϵˆk ¼jkj; k · ϵˆk ¼ 0: ð34Þ rewritten as The equations of motion for the transverse and longitudinal 1 ffiffiffiffiffiffi M2 e2 modes become ffiffiffiffiffiffi ∂ p− νρ μσ − pl φ2 μ 0 p− νð gg g FρσÞ 2 B ¼ : ð27Þ   g f M2 e2 2 2 2 pl 2 ∂τ B þ k þ a φ B ¼ 0; k 2f k   Separating the time and space components, the equation for 2 ∂τφ ∂τf ∂τa k B0 becomes 2 L L ∂τ B þ 2 − þ 2 2 ∂τB k φ 2 M a k f a 2 pl 2φ2 k þ 2f e 2 2   1 Mple 2 2 2 − ∂ ∂ B0 − ∂0B φ B0 0: 28 2 2 Mple 2 2 ið i iÞþ 2 ¼ ð Þ þ k þ a φ BL ¼ 0: ð35Þ a f 2f k

Performing the analysisR in Fourier space, with the con- d3k −ik·x vention f x 3 2 fke , we derive an algebraic ð Þ¼ ð2πÞ = B. Coulomb gauge equation for B0;k, i In the Coulomb gauge (∂iB ¼ 0), the Goldstone mode θ remains an explicit dynamical d.o.f., and thus the relevant _ ikiBi;k equations of motion are B0 k ¼ 2 2 2 : ð29Þ ; a M e    k2 þ pl φ2 1 2f D2 θ 2 2 Mθ − Gθ θ τ X þ k þ a θ 6 R θ X 2   The equation of motion for the spatial components Bi is 1 3 Mpl θθ _ _ þ ea G 2B0φ_ þðB0 þ 3HB0Þφ − φfB0 ¼ 0; 2f f _ 1 1 a _ ̈ _ 2 2 2 2 ð∂ B0 − B Þ − ðB − ∂ B0Þþ ð∂ B − ∂ ∂ B Þ 1 M e M a3 i i a2 i i a4 j i i j j − ∂2 pl φ2 − pl θφ_ − φθ_ 0 2 i B0 þ 2 B0 e 2 ð Þ¼ ; 1 M2 e2 a f f − pl φ2 0 _ 1 1 2 Bi ¼ : ð30Þ a _ ̈ _ 2 a 2f ð∂ B0 − B Þ − ðB − ∂ B0Þþ ∂ B a3 i i a2 i i a4 j i 1 M2 e2 M2 Using the constraint (26), going to Fourier space, and − pl φ2 − e pl φ∂ θ 0 2 Bi 2 i ¼ : ð36Þ multiplying by a2, the equation of motion becomes a 2f a 2f

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Going to Fourier space, we can solve for B0;k in terms of L k Bk ¼ θk: ð42Þ θk, similarly to the situation in the unitary gauge, eφ

M2 It is a straightforward algebraic exercise to show that by pl _ _ e 2 ðθkφ − φθkÞ L f using Eq. (42), the equations of motion for Bk and θk can B0 k ¼ 2 2 : ð37Þ ; 2 M e k pl 2 be transformed into each other, providing a useful check for 2 þ 2 φ a f our derivation. During preheating, when the background inflaton field By plugging the longitudinal mode into the last equation of oscillates, the unitary gauge becomes ill defined at the Eq. (36) and demanding that it is zero, we get the additional times where φðtÞ¼0, as can be seen for example in the constraint transformation relation of Eq. (42). We will compute gauge 2 boson production during preheating by solving the corre- _ Mpl sponding linearized equations of motion in the Coulomb HB0 þ B0 ¼ eφθ: ð38Þ 2f gauge, which is always well defined. The evolution of the background Higgs field φðtÞ will be numerically computed Substituting the two previous relations into the equation of in an expanding background according to Eqs. (15) and motion for Xθ gives (16), neglecting backreaction from fluctuations and produced particles. 4 φ ∂ φ − ∂τf φ ∂τa φ 2 θ 2 Mpl θθ ð τ 2f þ a Þ θ Dτ X − 2e G 2 2 2 DτX 2 2 M e φ C. Single-field attractor and parameter choices 4f k pl 2 þ 2   a  f For Higgs inflation, the function fðΦ; Φ†Þ is given by [6] 1 2 2 Mθ − Gθ þ k þ a θ 6R θ 2 † Mpl †  fðΦ; Φ Þ¼ þ ξΦ Φ: ð43Þ 4 ∂τf ∂τa ∂τa 2 M ð∂τφ − φ þ φÞð∂τφ þ φÞ 2 pl Gθθ 2φ2 2 2f a a þ e 2 a þ 2 2 4 2 M e f k pl φ2 For typical values of Higgs inflation λ ¼ Oð0.01Þ, and a2 þ 2f correspondingly ξ ∼ 104. If we consider a different renorm- ∂τf ∂τa  ∂τφφð∂τφ − φ þ φÞ λ 2 2f a Γθ θ 0 alization group flow for the self-coupling through the þ 2 2 θ X ¼ : ð39Þ 2 M e h k pl 2 introduction of unknown physics before the inflationary 2 þ φ a 2f scale, λ will become smaller or larger at inflationary energies. As we will show below, since the combination We must demand that physical observables are identical λ=ξ2 is fixed by the amplitude of the scalar power spectrum, θ in the two gauges, and derive a relation between k in the a larger or smaller value of λ during inflation will lead to a L h Coulomb gauge and Bk in the unitary gauge. Xk and Bk are correspondingly larger or smaller value of the nonminimal already identical in the two gauges. The longitudinal coupling ξ. We will consider values of ξ in the range 6 component of the electric field is given by 10 ≤ ξ ≤ 104. The inflationary predictions for the scalar and tensor modes for nonminimally coupled models with L _ L Ek ¼ Bk − kB0;k: ð40Þ ξ ≥ 10 fall into the large-ξ single-field attractor regime, as described, e.g., in Ref. [23]. This results in very simple In the unitary and Coulomb gauge we get expressions for the scalar spectral index ns, the tensor-to- scalar ratio r, and the running of the spectral index α as a 2 2 M e _ L function of the number of e-folds at horizon crossing N L pl 2 Bk Unitary∶ Ek ¼ φ 2 2 ; 2 2 M e f k pl φ2 2 3 12 a2 þ 2f ≃ 1 − − ≃ ns 2 ;r2 ; 2 N N N M e θ φ_ − φθ_   ∶ L − pl k k Coulomb Ek ¼ k 2 2 : ð41Þ dn 2 3 2 2 M e s f k pl 2 α ¼ ≃− 1 þ : ð44Þ 2 þ 2 φ 2 a f d ln k N N

Since EL should not depend on the gauge, we can use these The values for the spectral observables given in Eq. (44) expressions to solve for BL in terms of θ. We obtain correspond to single-field background motion. Multifield nonminimally coupled models of inflation at large ξ show a 6The gauge field being studied is not the Uð1Þ of the very strong single-field attractor behavior. The strength of electromagnetic sector. However, we will use the more familiar the attractor was analyzed in Ref. [71] for the case of an nomenclature found in . SOðNÞ-symmetric model, similar to Higgs inflation

083519-6 PREHEATING AFTER HIGGS INFLATION: SELF-RESONANCE … PHYS. REV. D 99, 083519 (2019)  Z pffiffiffiffiffiffi without gauge fields. The more general case of two-field 4 −˜ Φ Φ† ˜ − ˜μν ∇˜ Φ †∇˜ Φ inflation with generic potential parameters was studied in SJ ¼ d x g fð ; ÞR g ð μ Þ ν  Refs. [61,72], where it was shown that the single-field 1 1 attractor becomes stronger for larger ξ and that it persists − ˜μρ ˜νσ − ˜μρ ˜νσ − ˜ Φ Φ† 4 g g BμνBρσ 4 g g Aμν · Aρσ Vð ; Þ ; not only during inflation but also during the (p)reheating era. For generic initial conditions, the isocurvature fraction ð49Þ β iso is exponentially small for random potentials, while for β O 10−5 a symmetric potential iso ¼ ð Þ, as was shown in with the Higgs doublet Ref. [72]. As discussed in Sec. VA, during inflation the   gauge bosons are very massive compared to the Hubble 1 ϕ3 þ iϕ4 Φ ¼ pffiffiffi : ð50Þ scale, making the single-field attractor behavior of Higgs 2 φ þ h þ iθ inflation stronger than the one described in Ref. [71] for the ˜ scalar symmetric case. Hence, the use of a single-field The covariant derivative ∇μ is given by motion φðtÞ for the background is well justified during and 1 1 after Higgs inflation. ˜ ˜ 0 ∇μ Dμ ig YBμ ig Aμ · τ; 51 The dimensionless power spectrum of the (scalar) ¼ þ 2 þ 2 ð Þ density perturbations is measured to be where Y is the generator of hypercharge Uð1Þ and Bμ is the −9 As ≃ 2 × 10 : ð45Þ corresponding gauge field. The Higgs doublet has hyper- þ1. We have also introduced the vector notation 55 Using the tensor-to-scalar ratio from Eq. (44) with N ¼ ≃ 3 3 10−3 yields r . × , and hence the tensor power spectrum Aμ ≡ ðA1;μ;A2;μ;A3;μÞ; τ ≡ ðτ1; τ2; τ3Þ: ð52Þ becomes The Aμ are the gauge fields corresponding to SUð2Þ and τ P 2 2 i T H ≃ 6 6 10−12 are the Pauli matrices. The corresponding field-strength 2 ¼ π2 2 ¼ r × As . × : ð46Þ MPl MPl tensors are Given that the Hubble scale during inflation is approx- Bμν ¼ ∂μBν − ∂μBν; imately [6] X3 λ A μν ¼ ∂μA ν − ∂νA μ − g ϵ A μA ν: ð53Þ H2 ≃ M2 ; ð47Þ a; a; a; abc b; c; infl 12ξ2 Pl b;c¼1 † the Higgs self-coupling and nonminimal coupling must Defining the fields Wμ, Wμ, Zμ, and Aμ as obey the relation A1;μ − iA2;μ λ Wμ ¼ pffiffiffi Aμ ¼ sin θ A3 μ þ cos θ Bμ ≃ 5 10−10 2 W ; W ξ2 × : ð48Þ † A1;μ þ iA2;μ Wμ ¼ pffiffiffi Zμ ¼ cos θ A3 μ − sin θ Bμ; ð54Þ We keep the value of the Hubble scale fixed and determine 2 W ; W the value of λ that corresponds to each ξ through Eq. (48). with III. ELECTROWEAK SECTOR 0 e ¼ g sin θW ¼ g cos θW; ð55Þ We now consider the full SUð2Þ × Uð1Þ gauge sym- metry, as it exists in the electroweak sector of the SM. The the components of the covariant derivative of Φ are Lagrangian in the Jordan frame is given by given by

2θ ! ˜ g cos W igffiffi 1 Dμðϕ3 þ iϕ4ÞþiðeAμ þ 2 θ ZμÞðϕ3 þ iϕ4Þþp Wμðφ þ h þ iθÞ ˜ cos W 2 ∇μΦ ¼ pffiffiffi : ð56Þ ˜ ig igffiffi † 2 Dμðφ þ h þ iθÞ − Zμðφ þ h þ iθÞþp Wμðϕ3 þ iϕ4Þ 2 cos θW 2

The structure of the equations is almost identical to the one studied in the earlier parts of this work, where we focused on the Abelian case. For θ, the Goldstone mode that becomes the longitudinal polarization of the Z boson, we substitute

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g The Aμ does not couple to the Higgs: 2eBν → − Zν ð57Þ cos θW νμ DνF ¼ 0: ð64Þ in our Abelian equation and obtain    1 The Z boson obeys D2 θ 2 2 Mθ − Gθ θ τ X þ k þa θ 6R θ X   M2 2 M2 1 νμ − pl g φ2 μ 3 pl g θθ 0 μ 0 _ DνF 2 Z þa G 2Z φ_ þðDμZ Þφ− Z φf ¼ 0: Z 2f 4cos θ 2f 2cosθ f W W 2 Mpl g μν ð58Þ − g ðθ∂νφ − φ∂νθÞ¼0; ð65Þ 2f 2 cos θW

The Goldstone bosons ϕ3 and ϕ4 become the longitudinal modes of the W bosons. Making the substitutions θ → ϕ3 and correspondingly for the W bosons and θ → ϕ4 in the Abelian equation and 2 2 2 νμ Mpl g 2 μ Mpl ig μν g † g † DνF − φ W − pffiffiffi g ððϕ3 þ iϕ4Þ∂νφ 2eBν → i pffiffiffi ðWν − WνÞ; 2eBν → pffiffiffi ðWν þ WνÞ; W 2 2 2 2 2 2 2 f f − φ∂νðϕ3 þ iϕ4ÞÞ ¼ 0: ð66Þ ð59Þ we obtain    A. Unitary gauge 2 ϕ 2 2 ϕ 1 ϕ ϕ Dτ X 3 þ k þ a M 3 ϕ − RG 3 ϕ X 3 In the unitary gauge, the equations of motion for the 3 6 3 three Goldstone d.o.f. θ and ϕ3; ϕ4 give the constraints 2  3 Mpl ig ϕ ϕ 0 †0 − a pffiffiffi G 3 3 2ðW − W Þφ_  _  _ 2f 2 2 μ 2φ_ f 0 μ 2φ_ f 0  DμZ ¼ − þ Z ;DμW ¼ − þ W : 1 φ f φ f μ †μ 0 †0 _ þðDμðW − W ÞÞφ − ðW − W Þφf ¼ 0; ð60Þ f ð67Þ    μ μ 2 ϕ 2 2 ϕ 1 ϕ ϕ The equations of motion for Z and W are Dτ X 4 þ k þ a M 4 ϕ − RG 4 ϕ X 4 4 6 4  2 M2 2 3 Mpl g ϕ ϕ 0 †0 νμ pl g 2 μ − a pffiffiffi G 4 4 2ðW þ W Þφ_ DνF − φ Z ¼ 0; 2f 2 2 Z 2f 4cos2θ  W 1 2 2 μ †μ 0 †0 _ νμ Mpl g 2 μ þðDμðW þ W ÞÞφ − ðW þ W Þφf ¼ 0: ð61Þ DνF − φ W ¼ 0: ð68Þ f W 2f 2

At quadratic order, the field-strength term for the These equations are identical to the equations in the electroweak case is no more complicated than the Abelian case, albeit with different couplings. The equations Abelian case; it simply contains more fields, for the longitudinal and transverse modes are thus given by 1 1 1   † μν μν μν 2 2 L − F F − F μνF − FμνF ; 62 M gauge ¼ Wμν W Z Z ð Þ 2 2 2 pl g 2 2 4 4 ∂τ Z þ k þ a φ Z ¼ 0; k 2f 4cos2θ k  W with ∂ φ ∂ ∂ 2 2 L τ τf τa k L ∂τ Z þ 2 − þ 2 2 ∂τZ k M a 2 k φ 2f a 2 pl g 2 k 2 φ ∂ − ∂ ∂ − ∂ þ 2f 4cos θ FWμν ¼ μWν νWμ;FZμν ¼ μZν νZμ;   W 2 2 ∂ − ∂ Mpl g Fμν ¼ μAν νAμ: ð63Þ 2 2 φ2 ZL 0 þ k þ a 2 2 k ¼ ; ð69Þ f 4cos θW Comparing Eqs. (56) and (62) with the Abelian case, we can easily find the equations of motion for the gauge fields. and

083519-8 PREHEATING AFTER HIGGS INFLATION: SELF-RESONANCE … PHYS. REV. D 99, 083519 (2019)   2 2 M2 M g pl g φθ_ − θφ_ ∂2 2 2 pl φ2 0 2 2 θ ð Þ τ Wk þ k þ a Wk ¼ ; f cos W 2 2 Z0 ¼ 2 ; f ;k 2 M 2   k pl g φ2 2 2 þ 2 4 2θ ∂τφ ∂τf ∂τa k a f cos W ∂2 L 2 − ∂ L τ Wk þ þ 2 2 τWk M2 φ 2 2 M a 2 2 pl igffiffi _ _ f a pl g φ p ðφðϕ3 þ iϕ4Þ − ðϕ3 þ iϕ4Þφ_ Þ k þ 2f 2 2f 2 2 W0 ¼ 2 : ð72Þ   ;k 2 M 2 2 2 k pl g 2 M g 2 þ φ þ k2 þ a2 pl φ2 WL ¼ 0; ð70Þ a 2f 2 2f 2 k From the decoupling of the longitudinal modes from the where W denotes the polarization of the field W (so the equations for the corresponding transverse ones we get the W W† does not distinguish or ). constraints B. Coulomb gauge 2 The Coulomb gauge for the two types of bosons, Z and _ Mpl g HZ0 þ Z0 ¼ − φθ; 2 2 θ W , is defined through the conditions f cos W 2 _ Mpl ig ∂ Zi ¼ 0; ∂ Wi ¼ 0: ð71Þ HW0 þ W0 ¼ − pffiffiffi φðϕ3 þ iϕ4Þ: ð73Þ i i 2f 2 2

In Fourier space, we can express Z0;k in terms of θ and W0;k θ in terms of ϕ3 and ϕ4: Substituting into the equation for X gives

∂ φ ∂ ∂    2 M4 φ2ð τ − τf þ τaÞ 1 D2 θ − 2 g pl Gθθ φ 2f a D θ 2 2 Mθ − Gθ τ X 2 2 2 τX þ k þ a θ R θ 2 M 2 6 4cos θW 4f k pl g 2 2 þ 2 2 φ a f 4cos θW   2 4 2 ∂τφ ∂τf ∂τa 2 ∂τφ ∂τf ∂τa ∂τφ ∂τa M φ ð∂τφÞð − þ Þ φ ð − þ Þð þ Þ g pl Gθθ 2φ2 2 φ 2f a Γθ 2 φ 2f a φ a θ 0 þ a þ 2 θ þ 2 X ¼ ; ð74Þ 2 2 2 M 2 h 2 M 2 4cos θW 4f k pl g 2 k pl g 2 2 þ 2 2 φ 2 þ 2 2 φ a f 4cos θW a f 4cos θW and substituting into the equation for Xϕ3 gives    2 4 φ2 ∂τφ − ∂τf ∂τa g Mpl ð φ 2f þ a Þ 1 D2 ϕ3 − 2 Gϕ3ϕ3 D ϕ3 2 2 Mϕ3 − Gϕ3 τ X 2 2 τX þ k þ a ϕ3 R ϕ3 4 4 2 M 2 6 f k pl g φ2 a2 þ 2f 2   2 4 φ2 ∂ φ ∂τφ − ∂τf ∂τa φ2 ∂τφ − ∂τf ∂τa ∂τφ ∂τa g Mpl ð τ Þð φ 2f þ a Þ ϕ ð φ 2f þ a Þð φ þ a Þ Gϕ3ϕ3 2φ2 2 Γ 3 2 ϕ3 0 þ 2 a þ 2 ϕ þ 2 X ¼ ; ð75Þ 4 4 2 M 2 h 3 2 M 2 f k pl g φ2 k pl g φ2 a2 þ 2f 2 a2 þ 2f 2 and likewise    2 4 φ2 ∂τφ − ∂τf ∂τa g Mpl ð φ 2f þ a Þ 1 D2 ϕ4 − 2 Gϕ4ϕ4 D ϕ4 2 2 Mϕ4 − Gϕ4 τ X 2 2 τX þ k þ a ϕ4 R ϕ4 4 4 2 M 2 6 f k pl g φ2 a2 þ 2f 2   2 4 φ2 ∂ φ ∂τφ − ∂τf ∂τa φ2 ∂τφ − ∂τf ∂τa ∂τφ ∂τa g Mpl ð τ Þð φ 2f þ a Þ ϕ ð φ 2f þ a Þð φ þ a Þ Gϕ4ϕ4 2φ2 2 Γ 4 2 ϕ4 0 þ 2 a þ 2 ϕ þ 2 X ¼ : ð76Þ 4 4 2 M 2 h 4 2 M 2 f k pl g φ2 k pl g φ2 a2 þ 2f 2 a2 þ 2f 2 The equations of motion of the transverse modes of the Z and W are   1 M2 g2 ̈ _ 2 Pl φ2 0 Zk þ HZk þ 2 k þ Zk ¼ ; ð77Þ a 2f 4 cos θW   1 M2 g2 Ẅ þ HW_ þ k2 þ Pl φ2 W ¼ 0: ð78Þ k k a2 2f 2 k

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IV. HIGGS SELF-RESONANCE ω2 ̈ _ hðk; tÞ 0 vk þ Hvk þ 2 vk ¼ ; ð88Þ We now focus on the Higgs fluctuations, neglecting the a effects of Goldstone modes and gauge fields. In our linear analysis the Higgs fluctuations do not couple to the gauge where the frequency is defined in Eq. (80). We examine the two dominant terms of the effective field. The equation of motion for the rescaled fluctuations 2 Xh xμ ≡ a t Qh xμ mass: the one arising from the potential (m1;h) and the one ð Þ ð Þ ð Þ is 2 arising from the coupled metric perturbations (m3;h). The D2 h ω2 τ h 0 τ Xk þ hðk; ÞXk ¼ ; ð79Þ latter is often overlooked in studies of preheating, perhaps because it is vastly subdominant during inflation. It arises where the effective frequency is defined as by combining the equation of motion for δϕ and the metric perturbation ψ, defined through Eq. (12), in conjunction ω2 τ 2 hðk; Þ k 2 with the definition of the Mukhanov-Sasaki variables, 2 ¼ 2 þ meff;h: ð80Þ given in Eq. (14). a a 2 The expression for m1;h is For notational simplicity and connection to earlier work – [61 63] we define the various contributions to the effective λM4 φ2ðξφ2ð12ξM2 −2ξð6ξþ1Þφ2 þM2 Þþ3M4 Þ m2 Pl Pl Pl Pl mass of the Higgs fluctuations, 1;h ¼ ξφ2 2 2 ξ 6ξ 1 φ2 2 2 ð þMPlÞ ð ð þ Þ þMPlÞ 4 4 2 2 2 1 2 2 2 2 λM λM ðφ þ18M Þ m ≡ Mh − R ¼ m þ m þ m þ m ; ð81Þ ≃− Pl Pl Pl eff;h h 6 1;h 2;h 3;h 4;h 3ξ3φ2 þ 18ξ4φ4 ; ð89Þ

Mh where h was defined in Eq. (19) and where we used ξ ≫ 1 in expressions such as ð6ξ þ 1Þ ≃ 6ξ. Furthermore, since we are initially interested in studying 2 Ghh D D m1;h ¼ ð φ φVÞ; ð82Þ the behavior during inflation where analytic progress can be made, we use ξφ2 ≫ M2 as an approximation. As we 2 −Rh φ_ Lφ_ M Pl m2;h ¼ LMh ; ð83Þ will see, this works reasonably well even close to the end of   inflation. We normalize the effective mass by the Hubble 1 3 2 − D a φ_ 2G scale, m3;h ¼ 2 3 t hh ; ð84Þ MPla H m2 12 2 ξφ2 12ξ 2 −2ξ 6ξ 1 φ2 2 3 4 1 1;h MPlð ð MPl ð þ Þ þMPlÞþ MPlÞ 2 2 2 ¼ 2 2 2 2 m4 ¼ − R ¼ðϵ − 2ÞH : ð85Þ H ðtÞ φ ðξð6ξþ1Þφ þM Þ ;h 6   Pl 4M2 4M4 M6 ≃− Pl þ Pl þO Pl : ð90Þ For the case of fluctuations along the straight background ξφ2 ξ2φ4 ξ3φ6x trajectory (like Higgs fluctuations), the Riemann contribu- 2 tion m2;h vanishes identically. As described in Ref. [75] and We can use the single-field slow-roll results further utilized in Ref. [61], the mode functions can be decomposed using the vielbeins of the field-space metric. 3 ξφ2 1 φ2 In the single-field attractor—which exists in the non- −N ¼ þ þ Oðlog½φ=M Þ; ð91Þ 4 M2 8 M2 Pl minimally coupled models that include Higgs inflation Pl Pl — both during [23] and after inflation [61] the decomposi- 2 h where we went beyond lowest order in ξφ and we measure tion of Xk into creation and annihilation operators is trivial, the number of e-folds from the end of inflation, meaning Z 7 d3k that negative values correspond to the inflationary era. ˆ h h ˆ ik·x h ˆ † −ik·x X ¼ ½vke1ake þ v e1ake ; ð86Þ This leads to ð2πÞ3=2 k pffiffiffiffiffiffiffi   m2 3 9 1 eh Ghh 1;h ≃ O where 1 ¼ . Since the vielbeins obey the parallel 2 þ 2 þ 3 : ð92Þ h H ðtÞ N 4N N transport equation Dτe1 ¼ 0, the equation of motion for the mode function vk becomes ffiffiffi 2 p If we minimize m1 as a function of δ ¼ ξφ, the field ∂2 ω2 τ 0 amplitude that minimizes the mass is τ vk þ hðk; Þvk ¼ : ð87Þ

We solve the equation in cosmic rather than conformal 7We neglected the contributions coming from the lower end of time, which is better suited for computations after inflation, the integral leading to Eq. (91).

083519-10 PREHEATING AFTER HIGGS INFLATION: SELF-RESONANCE … PHYS. REV. D 99, 083519 (2019)   pffiffiffi M 0.82λM4 2λM4 δ ¼ 2M þ O Pl ; ð93Þ φ_ 2 ≃ Pl ≃ Pl : 97 min Pl ξ end 24ξ3 75ξ3 ð Þ

≃−1 5 2 2 ≃−11 or, equivalently, Nmin . . For the minimization we Numerically, we get m3;h=H ðtÞ at the end of used the full expression for the effective mass and only took inflation, in rough agreement with the approximate expres- the Taylor expansion for large ξ at the end. We can see that sions given above. 2 for ξ ≫ 1 the minimum of m1 is independent of ξ and thus The numerical results for ξ ¼ 10 are shown in Fig. 1, occurs at the same value of δ (which will also be the same along with the approximate analytical expressions that we value of N) in the approximation of Eq. (91). In general, the derived. We only show the ξ ¼ 10 case, since all cases with 2 2 function m1;hðNÞ=H shows no appreciable difference for higher values of the nonminimal coupling exhibit visually different values of ξ ≫ 1 during inflation. This can be identical results. After the end of inflation the two dominant easily seen by substituting Eq. (91) into Eq. (90). As shown components of the effective mass of the Higgs fluctuations in Ref. [62], this behavior persists during the time of evolve differently for different values of ξ. In Ref. [62] the 2 coherent inflaton oscillations. behavior of m1;h was analyzed in the static universe The mass component arising from the metric fluctua- approximation. It was shown that for ξ ≳ 100 the effective 2 tions is mass component m1;h quickly approaches a uniform shape regardless of the value of ξ. The consequence of this is that ðξð6ξ þ 1Þφ2 þ M2 Þφ_ ðHðtÞðϵðtÞþ3Þφ_ þ 2φ̈Þ m2 − Pl the Floquet chart for the inflaton self-resonance also 3;h ¼ ξφ2 2 2 HðtÞð þ MPlÞ approaches a common form for ξ ≳ 100. This can be seen 2 2 18φ_ in the left panel of Fig. 2, where m1;h is very similar ≃− ; 3 φ2 ð94Þ between ξ ¼ 100 and ξ ¼ 10 , but different for ξ ¼ 10. The coupled metric fluctuations component of the effective where the last approximation holds during inflation. Using mass has a similar shape for ξ ¼ 100 and ξ ¼ 103, but for the slow-roll expression for φ_ we get that during inflation ξ ¼ 10 it is significantly less pronounced, as seen in the 2 right panel of Fig. 2. m3 9 ;h ≃− : ð95Þ H2ðtÞ 2N2 A. Superhorizon evolution and thermalization 2 An important notion when dealing with (p)reheating is This contribution is clearly subdominant to m1 , and hence ;h the transfer of energy from the inflaton condensate to the it can be safely neglected during inflation. However, jm2 j 3;h radiation d.o.f. Naively, one must compute all the power grows near the end of inflation, since it is proportional to concentrated in the wave numbers that are excited above φ_ 2 , which at the end of inflation is given by the vacuum state (meaning that they are different than the adiabatic vacuum at any time) and compare that to the λM2 φ4 λM2 φ2 φ_ 2 GφφV Pl ≃ Pl : 96 energy density stored in the condensate. However, when end ¼ ¼ 4 6ξ2φ2 ξφ2 2 24ξ2 ð Þ ð þ þ MPlÞ dealing with inflationary perturbations, one must keep in mind that computations should refer to modes, whose It has been numerically shown in Ref. [61] that the field pffiffiffi length scales are relevant to the dynamics being studied. ξφ ≃ 0 8 value at the end of inflation is end . MPl, leading to For curvature perturbations, the use of a finite box was

2 2 FIG. 1. The components of the effective mass of the Higgs fluctuations m1;h and m3;h rescaled by the Hubble scale. The blue curves show the numerical curves for ξ ¼ 10 and the red dashed curves show the approximate analytic expressions of Eqs. (92) and (95), respectively.

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2 2 FIG. 2. The ratio of the components of the effective mass of the Higgs fluctuations m1;h (left) and m3;h (right) rescaled by the Hubble scale at the end of inflation. The blue, red dashed, and green dotted curves correspond to ξ ¼ 10; 102; 103, respectively. described in Ref. [76]. For preheating, since thermalization added to the local background energy density. We will skip proceeds through particle interactions, the relevant length this last step, as their contribution is subdominant com- scales are those that allow for particle interactions, meaning pared to the energy density stored in the inflaton con- subhorizon scales or short wavelengths. densate. In Fig. 3 we see the evolution of the comoving The parametric excitation of long-wavelength modes has Hubble radius, which shrinks during inflation and grows been extensively studied [77–86]. It has been demonstrated afterwards. We also see that different values of ξ lead to that the coupled metric fluctuations lead to an enhancement different post-inflationary evolutions, which is expected, of (in particular) long-wavelength modes [80–84], which is since the effective equation of state of the background larger than the one computed using a rigid background. dynamics after inflation depends strongly on ξ, as shown in Furthermore, the amplification of long-wavelength modes, Ref. [61]. More specifically, large nonminimal couplings even on super-Hubble scales, does not violate causality, as ξ ≳ 100 lead to a prolonged period of matter-domination- discussed, e.g., in Refs. [80,82–84,86]. Intuitively, the like expansion, which can last for several e-folds in the inflaton condensate has a super-Hubble correlation length absence of backreaction. As we will see in the next and can thus consistently affect super-Hubble modes. sections, the majority of the parametric resonance effects While UV modes encounter the complication of possibly occur for N ≲ 3 e-folds, placing the entirety of the being excited for wave numbers that exceed the unitarity reheating dynamics inside the matter-dominated back- bound (this does not occur for Higgs modes), the IR modes ground era for large values of ξ. In order to consistently have a different conceptual difficulty: since thermalization take into account the relevant wave numbers, we use an occurs when particles interact and exchange energy, in adaptive code that only sums up the contributions of modes order to lead to a thermal distribution modes that are that are inside the horizon at the point in time when we superhorizon are “frozen in” and hence cannot take part in compute the energy density of the Higgs field fluctuations. such processes.8 Hence, it is normal to only consider modes that have large enough physical wave numbers that place them inside the horizon at the instant in time that we are considering. Modes that have longer wavelengths are frozen outside the horizon and do not contribute to the thermalization process. They should be summed over and

8Generically, in multifield models one would not expect the curvature perturbations to remain “frozen in” when stretched outside the Hubble radius, since multifield interactions can generate nonadiabatic pressure, which in turn will source changes in the gauge-invariant curvature perturbations on arbitrarily long length scales. However, in models like Higgs inflation that feature strong single-field attractor dynamics during inflation, the non- FIG. 3. The size of the comoving Hubble radius during and adiabatic pressure effectively vanishes and the long-wavelength 2 3 4 modes remain “frozen in,” akin to the expected behavior in simple after inflation for ξ ¼ 10; 10 ; 10 ; 10 (black, blue, red, and single-field models. Details on the single-field attractor in such green, respectively). The curves are normalized to unity at the end models can be found in Refs. [23,61,71]. of inflation.

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B. Preheating We see in the right panels of Fig. 4 that at early times (before the end of inflation) the energy density in Higgs We now move to the computation of the energy density −4 in the Higgs particles that are produced during preheating. modes (indicated by the solid blue line) decays as a A detailed analysis was performed in Ref. [63]. However, (indicated by the dotted line), in keeping with the expect- all computations were initialized at the end of inflation, ation for modes in the BD state. However, approximately thereby neglecting the amplification of long-wavelength one e-fold before the end of inflation, the evolution of the −4 modes during the last e-folds of inflation. We initialize all energy density in Higgs modes departs from a , because computations at 4.5 e-folds before the end of inflation, in the low-k modes are enhanced with respect to the BD 2 0 order to ensure that all relevant modes are well described by spectrum. This enhancement occurs because meff;h < ,an the Bunch-Davies (BD) vacuum solution early tachyonic amplification phase driven largely by the effect of coupled metric perturbations. An immediate 1 ffiffiffiffiffi −ikτ consequence of this fact is that one would underestimate vk;h ≃ p e : ð98Þ 2k the true amount of growth by starting the computation in a BD-like vacuum state at the end of inflation.

FIG. 4. Left: The effective mass squared (black-dotted curve), along with the contributions from the potential (blue curve) and the coupled metric perturbations (red curve). Right: The energy density in the background Higgs condensate (orange curve) and the Higgs fluctuations (blue curve) for ξ ¼ 10; 102; 103 (top to bottom). The green curve shows 10% of the background energy density, which is −4 used as a proxy for the limit of our linear analysis. The orange-dashed curve is ρ0a , corresponding to the redshifting of the background 4 energy density during radiation-dominated expansion. The energy density is plotted in units of MPl.

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The right panels of Fig. 4 present the results for the the energy density of the background condensate redshifts energy transfer into Higgs particles for ξ ¼ 10; 102; 103. as a−3 ¼ e−3N. For small values of ξ, however, the Universe Preheating is completed when the energy density in the passes briefly through the background (average) equation Higgs fluctuations (blue line) becomes equal to the energy of state w ¼ 0 and after the first e-fold approaches w ≃ 1=3. density of the background field (orange line). However, the Figure 5 shows the evolution of the energy density in Higgs linear analysis is expected to break down when the energy modes for the marginal case of ξ ¼ 30. We see that the density of the Higgs fluctuations becomes comparable to fluctuation energy density in the Higgs modes would that of the inflaton field. As an indicator of the validity of always be smaller than the background if the background the linear theory, which neglects backreaction of the excited evolved with w ≃ 0, as indicated by the orange dashed line. modes onto the background, the green line shows 10% of However, the fact that the background energy density the energy density of the inflation field. redshifts faster (w ≃ 1=3) allows for complete preheating. For all values of ξ studied, the system exhibits an Simply put, nonminimal couplings in the “intermediate” amplification of inflaton (Higgs) fluctuations. This is regime of ξ ¼ Oð10Þ exhibit a shorter period of tachyonic- mainly caused by the periodic negative contribution of parametric amplification, while at the same time following 2 2 ρ ∼ −4N m3;h to the effective mass squared meff;h, which is plotted in a background evolution of ϕ e . the left panels of Fig. 4. This is the term arising from We distinguish two time points relevant for preheating: considering the effect of the coupled metric perturbations at Nreh is the time at which the energy density in the linear linear order. As shown in Ref. [63] and further reiterated in fluctuations equals the background energy density (which 2 we take as the time of complete preheating), and N is the Fig. 4, the amplification driven by m3;h lasts longer for br larger values of ξ. Specifically, the time at which the time at which the energy density in the linear fluctuations pffiffiffi ∼ ξ −1 equals 10% of the background energy density (which is the tachyonic resonance regime stops scales as t Hend,as shown in Ref. [63]. However, for ξ > 100 the differences point at which backreaction effects may become impor- are irrelevant (in the simplified linear treatment), since the tant). We have numerically found that the self-resonance of Universe will have preheated already by N ≃ 3 e-folds. the Higgs field becomes insufficient to preheat the Universe at ξ < 30. In particular, the results for N ξ can be fitted Hence, for ξ > 100 the self-resonance of the Higgs field rehð Þ by a simple analytical function, as shown in Fig. 6: leads to predictions for the duration of preheating that are almost independent of the exact value of ξ. 21 After the tachyonic resonance has ended (and if preheat- N ðξÞ ≃ þ 3; ð99Þ reh ξ 1 0.016ξ ing has not completed yet), the modes undergo parametric ð þ Þ resonance, driven by the oscillating effective mass term for ξ ≳ 30, where complete preheating is possible, at least 2 ≃ 0 m1;h. However, for very long-wavelength modes k the ξ 100 in the linear approximation that we used. For > , Nreh Floquet exponent vanishes [62] and the amplification is becomes largely independent of ξ, as expected from the polynomial in time rather than exponential, and hence results of Fig. 4. significantly weaker. As shown in Ref. [62], the maximum As a final note, we must say that the results were Floquet exponent in the static universe approximation is insensitive to the exact value of the maximum wave number μ ≈ 0 3 k;maxT . , where T is the background period. Using the relation ω=H ≃ 4, which was derived in Ref. [61] for ω ¼ 2π=T, the maximum Floquet exponent is expressed as μk ∼ 0.5H. Hence, the Floquet exponent is too small to lead to an efficient amplification of Higgs fluctuations in an expanding universe. Thus the early-time tachyonic reso- nance driven by the coupled metric fluctuation is crucial for preheating the Universe through Higgs particle production. For ξ ¼ 10 the situation is significantly different. Both tachyonic resonance (due to the coupled metric fluctuations 2 encoded in m3;h) as well as parametric resonance (due to 2 the potential term m1;h) become inefficient earlier, leading to a slower growth of the fluctuations and the energy FIG. 5. The energy density in the background Higgs condensate density that they carry and an incomplete preheating. (orange curve) and the Higgs fluctuations (blue curve) for the However, for smaller values of the nonminimal coupling marginal case of ξ ¼ 30 (top to bottom). The green line shows ξ ¼ Oð10Þ one must take into account another important 10% of the background energy density, which is used as a proxy feature, namely, the evolution of the background. As shown for the limit of our linear analysis. The orange-dashed curve is −4 in Ref. [61], larger values of ξ put the Universe into a ρ0a , corresponding to the redshifting of the background energy prolonged matter-dominated state (w ¼ 0). This means that density during radiation-dominated expansion.

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where e is the Uð1Þ gauge coupling. These equations are of the form   ∂ ∂2 I ∂ I ω2 τ I 0 τ Bk þ ∂τ logðbIÞ τBk þ I ðk; ÞBk ¼ ; ð102Þ

ω2 with I denoting either L or polarization, and bI and I are given by   22 −1 τ 1 k f bLðk; Þ¼ þ 2 2 2φ2 ; FIG. 6. The number of e-folds after inflation when the energy MPla e density in the Higgs fluctuations equals the background energy 2 2 2 2 M 2 2 density N (blue solid curve) or 10% of the background ω ðk; τÞ¼k þ a Pl e φ ; reh L 2f energy density Nbr (black dashed curve). M2 b k; τ 1; ω2 k; τ k2 a2 Pl e2φ2: ð Þ¼ ð Þ¼ þ 2 ð103Þ considered. This is due to the fact that the small (but f O subhorizon) wave numbers k ¼ ðHendÞ are exponentially amplified and dominate the fluctuation energy density After integrating by parts, we rewrite the quadratic action in shortly after the end of inflation. Hence, we do not need Fourier space as to implement any scheme to subtract the vacuum contri- Z bution from large-k modes, since it is vastly subdominant SI ¼ dτL ðτÞ for any reasonable UV cutoff. I Z Z   1 1 3 I 2 2 I 2 ¼ dτ d kb ðk; τÞ j∂τB j − ω ðk; τÞjB j ; V. GAUGE/ PRODUCTION I 2 k 2 I k A. Evolution during inflation and initial conditions ð104Þ for preheating We use the equations of motion derived in the Abelian and follow the same procedure as the one model in the unitary gauge, in order to study the evolution appearing in Ref. [87]. This is the standard method used to of gauge fields during inflation. The unitary gauge is well quantize models with noncanonical kinetic terms, which defined in this period, since φðtÞ does not vanish. The include nonminimally coupled models in the Einstein ;L values of Bk at the end of inflation serve as initial frame. The canonical momentum is conditions for preheating. Accurate knowledge of the δL τ spectrum of gauge fields at the end of inflation is essential, π τ ð Þ ∂ I τ I;kð Þ¼ I ¼ bI τBkð Þ; ð105Þ especially when initializing lattice simulations which are δð∂τB−kðτÞÞ increasingly expensive to start deeper within inflation. During preheating, the unitary gauge is not well defined ˆ I and the commutator relation of the operator BkðτÞ is at moments when φðtÞ¼0, so we use the Coulomb gauge. In order to determine the initial condition for θk, we will L 1 use Eq. (42), which relates B in the unitary gauge to θk in ˆ I τ ∂ ˆ J τ δIJδ k q k ½Bkð Þ; τBqð Þ ¼ i τ ð þ Þ: ð106Þ the Coulomb gauge. bIðk; Þ The equations of motion for the longitudinal and trans- ˆ I verse modes in the unitary gauge in conformal time τ are We decompose the field operator BkðτÞ in terms of creation   and annihilation operators, ∂ φ ∂ ∂ 2 2 L τ τf τa k L ∂τ B þ 2 − þ 2 2 ∂τB k M a k ˆ I I I I† I φ 2f a 2 Pl 2 2 BkðτÞ¼aˆ ku ðτÞþaˆ u ðτÞ; ð107Þ k þ 2 e φ k −k k   f M2 2 2 Pl 2φ2 L 0 I τ þ k þ a e Bk ¼ ; ð100Þ where the mode function ukð Þ satisfies the same equation 2f ˆ I of motion as the field operator BkðτÞ [Eq. (100)]. ∂τω   As long as the adiabaticity condition j 2 j ≪ 1 holds M2 e2 ω 2 2 2 pl 2 ∂τ Bk þ k þ a φ Bk ¼ 0; ð101Þ [87], the modes can be described by the WKB approxi- 2f mation,

083519-15 SFAKIANAKIS and VAN DE VIS PHYS. REV. D 99, 083519 (2019)  Z  αI 1 We will mainly focus on the longitudinal mode, since it will I ffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 u ðτÞ¼p exp −i dτ ωIðk;τ Þ k 2 b ðk;τÞ ω ðk;τÞ be amplified most efficiently during preheating. I I  Z  During inflation we can rewrite the equations of motion βI 1 − τ þpffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiexp þi dτ0ω ðk;τ0Þ : using x ¼ k as the time variable. If we further make use 2 I τ −1 φ bIðk;τÞ ωIðk;τÞ of the de Sitter approximation ( ¼ =aH) and take ðtÞ as a constant, the equations of motion become9 ð108Þ   2 τ 2 xc The behavior is different for modes with jk j >xc (early ∂ B þ 1 þ B ¼ 0; ð114Þ x k x2 k times/short wavelengths) and jkτj , does not apply for nonminimally coupled models of inflation with large ξ unless one takes a very ≪ 1 M2 a2e2φ2 weakly coupled gauge field e , making such a value b ðk; τÞ → Pl : ð111Þ very different to gauge couplings found in the SM. L k22f For the longitudinal mode BL the presence of a first- Putting everything together, the mode function for jkτj > derivative term is important for x

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Eq. (115) can be solved exactly using Hankel functions, resulting in rffiffiffiffiffi ffiffiffiffiffiffiffiffi p π ð1Þ π π uðτÞ¼ −kτ H ð−kτÞeiz2þi4; ð118Þ k 4k z qffiffiffiffiffiffiffiffiffiffiffiffi 1 2 where z ¼ 4 − xc, as described, e.g., in Ref. [87]. The analysis of the transverse modes is essentially identical to the minimally coupled case of Ref. [87]. Since they will not be significantly amplified during preheating, we will not discuss them further. FIG. 7. The evolution of the longitudinal gauge field mode ξ 103 during inflation for ¼ . The solid lines correspond to the 2. Initial conditions in the Coulomb gauge 1 10 102 103 104 numerical solution for k=Hend ¼ ; ; ; ; (blue, green, black, orange, and brown, respectively), along with the approxi- Having explored in detail the behavior of the longi- mate solutions of Eqs. (112) and (117) (red-dotted lines). The tudinal gauge fields during inflation, we focus on their form vertical lines correspond to the points kjτj¼xc, where the close to the end of inflation and the start of the (p)reheating matching between the two asymptotic regimes is performed. era. Since we are interested in the details of the vacuum (as For the brown curve this does not occur during inflation. will be evident later), we compare the adiabatic vacuum during inflation [given in Eq. (108)] to the approximate τ background trajectory discussed in Sec. II C, since the analytic expressions derived for uLðk; Þ, as well as to the orthogonal direction(s)—described equally well through numerically derived values. It is a straightforward exercise τ ω τ the scalar d.o.f. θ or through the longitudinal polarization of to expand bLðk; Þ and Lðk; Þ in the two limiting cases of τ the gauge boson BL—are very massive, making the back- kj j to see that the WKB expression given in Eq. (108) with α 1 β 0 ground single-field trajectory a stable one.10 It is interesting ¼ and ¼ matches Eqs. (112) and (117) in the to note that, due to the relation between ξ and λ arising from appropriate limits. the normalization of the power spectrum, the gauge fields Figure 8 shows the comparison of the WKB solution of become less massive for larger ξ, meaning that the ratio of Eq. (108), the approximate expressions of Eqs. (112) and the gauge field mass to the Hubble scale becomes smaller. (A8), as well as the numerical results from the modes Hence, the single-field attractor, at least in the linearized shown in Fig. 7. We see an excellent agreement between all τ ∼ analysis, becomes weaker for larger ξ. This is opposite to three, with the exception of the modes around kj j xc, the case of a scalar-only multifield model with a non- where the approximate expressions fail, since they were τ ≫ τ ≪ symmetric potential, where the attractor strength increases derived using the limits kj j xc or kj j xc. We must τ −1 with ξ, as shown in Ref. [61]. While for the SM the gauge also note that we used the approximation ¼ =aH for couplings are large enough to make the gauge field much the analytically derived expressions, and hence we expect heavier than the Hubble scale, one can construct more some discrepancy close to the end of inflation. This general inflationary models involving a Higgs-like field agreement has a significant physical meaning: since the and the associated gauge sector. In this case weakly coupled adiabatic vacuum follows the evolution of the mode gauge sectors might leave observational imprints through functions, there is no particle production during inflation. oscillations of the background during inflation. A search We can thus begin our numerical computations at the end of for “primordial clocks” [89,90] in these models is beyond inflation, unlike the case of Higgs self-resonance where we the scope of the present work because they do not arise in needed to initialize our simulations several e-folds before SM Higgs inflation, but they could provide a useful tool for the end of inflation in order to capture nontrivial dynamics exploring gauge-field phenomena in broader classes of that took place during the last stages of inflation itself. nonminimally coupled inflation. The initial conditions in the Coulomb gauge can be The transverse modes are significantly easier to analyze, easily read off from the unitary gauge solutions using Eq. (42). It is interesting to note that there is no ξ-dependent since Eq. (114) makes clear that the B are conformally θ L coupled at early times and will become massive (and thus term in the relation of k to Bk. The initial conditions that we will use for the computations in the Coulomb gauge are be suppressed) for x

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FIG. 8. The mode-function amplitude (left) and frequency (right) for N ¼ 0,2,4e-folds before the end of inflation (blue, green, and black, respectively). Solid lines correspond to the WKB expression of Eq. (108) and red-dotted lines correspond to the approximate solutions for x ≫ xc and x ≪ xc. The dots show the full numerical results.

Before we conclude the analysis of the gauge-field and Xθ ¼ aðtÞ · θ. We normalize the scale factor as a ≡ 1 at evolution during inflation, let us focus on the case of the end of inflation. The effective mass of the Goldstone kjτj ≫ xc, where the initial conditions for preheating are mode θ in the absence of gauge fields is ϕ τ e in 1 θkðτ Þ ≈ pffiffiffiffiffi ; ð121Þ 2 θ 2 2 2 2 in m ≡ M θ − R ¼ m þ m þ m þ m ; ð125Þ xc 2k eff;θ 6 1;θ 2;θ 3;θ 4;θ   ik θ_ τ ≈ θ τ with kð inÞ kð inÞ × τ : ð122Þ að inÞ 2 Gθθ D D It is reassuring that for large wave numbers the coupling m1;θ ¼ ð θ θVÞ; ð126Þ constant e drops out of the initial conditions for the θ field ∝ (since xc e), and hence the decoupling limit is trivially 2 −Rθ φ_ 2 m2;θ ¼ hhθ ; ð127Þ obtained. For kjτj

B. Preheating The numerical solution of Eq. (123) was obtained in cosmic rather than conformal time, since this is more We start by rewriting Eq. (39) in a somewhat more convenient for numerical simulations after the end of compact way, inflation. The computations were initialized at the end of    ˜ 2 inflation, according to Eqs. (119) and (120). 2 θ mB θ 2 2 2 Dτ X − ∂τ log 1 þ DτX þ k þ a m We can follow the quantization method described in k2 eff;θ     Ref. [61] and utilized in Sec. IV for the study of the Higgs ∂ φ ∂ ∂ ˜ 2 2 τ τa τf mB θ self-resonance, þ m˜ þ þ − ∂τ log 1 þ X ¼ 0; B φ a 2f k2 Z 3 ð123Þ ˆ θ d k θ ik·x θ † −ik·x X ¼ ½z e2aˆ ke þ z e2aˆ ke ; ð130Þ ð2πÞ3=2 k k where we defined the gauge-field mass pffiffiffiffiffiffiffi where eθ ¼ Gθθ. Using the vielbein decomposition, the M2 2 ˜ 2 ≡ 2φ2 pl 2 covariant derivatives are effectively substituted by partial mB e a ; ð124Þ 2f ones,

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∂2 − ∂ 1 ˜ 2 2 ∂ τ zk τ logð þ mB=k Þ · τzk where hHðtÞi is a time-averaged version of the Hubble  scale over the early oscillatory behavior. The range of 2 2 2 2 1 k a m m˜ ∂τ excited wave numbers is given by the relation þ þ eff;θ þ B þ 2 log  sffiffiffiffiffiffiffiffi   2 ˜ 2 k2 ≲ a2m2 ; 2 f mB 2;θjmax ð137Þ × m˜ ∂τ log 1 þ z ¼ 0: ð131Þ B M2 k2 k Pl 2 ˜ 2 assuming that the spike of m2;θ dominates over mB near In order to eliminate the first-derivative term we can use the φðtÞ¼0. Each subsequent inflaton zero crossing affects a 2 ∝ 2 ∝ rescaled variable z˜k, defined as smaller range of wave numbers, since m2;θ H ρ ∝ −3 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi infl: a , where we assumed w ¼ for the averaged 2 2 ∝ −1 ˜ background evolution. Altogether kmax a , and hence the 1 mB ˜ ≡ ˜ zk ¼ þ 2 zk T · zk; ð132Þ maximum excited wave number shrinks for every sub- k sequent inflaton oscillation. The maximum comoving leading to wave number after the first inflaton zero crossing, where aðtÞ ≈ 1,is 2 2 ∂τ z˜ þ ω z˜ ¼ 0; ð133Þ k z k 2 O 10 ξ2 2 O 1 λ 2 kmax ¼ ð Þ Hend ¼ ð Þ MPl; ð138Þ where where we used Eq. (47) and H ≈ 0.5H . This is in  sffiffiffiffiffiffiffiffi end infl agreement with Ref. [64]. We focus primarily on the first 2 2 2 2 2 1 2 2f 2 ω ¼ k þ a m þ m˜ þ ∂τ log m˜ ∂τ logðT Þ inflaton zero crossing, since the produced gauge bosons will z eff;θ B 2 B M2 Pl decay into fermions between two subsequent background 2 2 2 2 ∂τ ðm˜ Þ 3 ð∂τm˜ Þ zero crossings, and hence Bose enhancement is lost. This þ B − B ; ð134Þ 2k2T2 4 k4T4 was shown in Refs. [58,59] and will be discussed in detail in Sec. VI. ˜ 2 2 2 where mB is larger than m1;θ and m4;θ. As discussed The second dominant component of the gauge-field ˜ 2 extensively in Refs. [61–63] for the case of a purely scalar effective frequency squared is mB, which scales simply as multifield model with large nonminimal couplings to gravity, the field-space manifold is asymptotically flat m˜ 2 =a2 M2 e2 1 ξ 1010 B Pl φ2 O 1 O 1 ; 139 for large field values and exhibits a curvature “spike” at 2 ¼ 2 2 ¼ ð Þ λ ¼ ð Þ ξ ð Þ Hend f Hend the origin φðtÞ ≃ 0. This “Riemann spike” is exhibited in 2 the effective mass of the isocurvature modes meff;θ, more where the λ − ξ relation given in Eq. (48) was used in the 2 ξ 103 specifically in the m2;θ component, which is subdominant last step. We can see that for ¼ the maxima of the two ˜ 2 3 for all times away from the zero crossings of the back- contributions mB and m2;θ are similar, as shown in Fig. 9. ground value of the inflaton field φðtÞ. We will not Computing the energy density transferred from the reproduce the entirety of the Floquet structure of this inflaton condensate into the gauge-field modes requires model, both because we do not wish to repeat the analysis more attention than the corresponding computation of of Ref. [62], and because (as we will see in the subsequent Sec. IV for the Higgs self-resonance. In the case of section) the first zero crossing of φðtÞ is the only relevant Higgs self-resonance, the range of excited wave numbers h ∼ one for preheating through gauge modes. is kmax H. A naive computation of the energy density in In order to estimate the maximum excited wave number the local adiabatic (WKB) vacuum for the same modes k , we consider the following approximation containing ρ ∼ 4 ∼ 4 max gives BD kmax H , which is 10 orders of magnitude only the dominant terms: smaller than the background energy density.11 In that case, we do not need to subtract this unphysical vacuum ω2 ≡ 2 2 2 ˜ 2 z;approx k þ a m2;θ þ mB; ð135Þ contribution from the energy density of the Higgs modes, since the energy density in the parametrically amplified ˜ 2 where mB dominates over all subsequent terms in Eq. (134) modes is exponentially larger. for large k. Figure 9 shows the three contributions to 2 3 4 ω for ξ 10 ; 10 . As shown in Ref. [63], the scaling 11 z;approx ¼ Any computation that does not involve vacuum subtraction, of the spike in the effective mass is including lattice simulations (such as in Refs. [53,54]), deals with classical quantities and computes the energy density of the m2 j vacuum modes as if they were physical. Such a computation 2;θ max O 10 ξ2 2 ¼ ð Þ ; ð136Þ is valid as long as the unphysical energy density of the vacuum hHðtÞi modes is vastly subdominant.

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FIG. 9. Dominant components of the effective frequency squared for ξ ¼ 103 (left) and ξ ¼ 104 (right). Color coding is as follows: ˜ 2 2 2 2 2 mB=a (red), m2;θ (blue), and k =a (black) for the maximum excited wave number kmax. The orange-dotted curve shows the scaling a−3.

For the case of gauge fields the maximum wave number Both the particle number and the energy density can be L up to which modes can be excited is given in Eq. (138). The computed equally well using the field θk or Bk, since the vacuum energy density in these modes, naively computed, only moment for which the longitudinal gauge fields are ρ ∼ 4 ∼ λ2 4 φ 0 is BD kmax MPl. The total energy density in the not defined is when ðtÞ¼ . At this instant we cannot ρ 3 2 2 ρ ρ ∼ inflaton field is infl ¼ H MPl, leading to BD= infl define the particle number either way, since there is no λξ2 ∼ 10−10ξ4. This is much greater than unity for large well-defined adiabatic vacuum. Figure 10 shows the values of the nonminimal coupling. We thus need to evolution of the particle number density for a few values remove the unphysical vacuum contribution to the energy of the comoving wave number after the first few inflaton density using the adiabatic subtraction scheme [9]. In this zero crossings, neglecting the effect of particle decays, as scheme we compare the wave function of the gauge fields described in Sec. VI. The left panel of Fig. 11 shows the to the instantaneous adiabatic vacuum, computed in the particle number density per k mode for ξ ¼ 10; 102; 103 WKB approximation, isolating the particle number for each after the first inflaton zero crossing. The condition of wave number k. The particle number corresponding to a Eq. (138) for the maximum excited wave number kmax is mode vk is given by evident.   At this point, it is worth performing a simple estimate of ω jv_ j2 1 n ¼ k k þjv j2 − : ð140Þ the energy density that can be transferred to the gauge-field k 2 ω2 k 2 φ 0 k modes away from the first point ðtÞ¼ , A drawback of this method is that the particle number is only well defined when the adiabaticity condition holds, Z ω_ =ω2 ≪ 1, and thus we cannot define the particle number 3 k k ρ d k ω ∼ ˜ 3 “ ” φ 0 12 ¼ 3 nk k hnimBkmax in the vicinity of the Riemann spike when ðtÞ¼ . ð2πÞ The energy density is easily computed through the particle   105 number as ∼ ffiffiffi λ3=2 3 ∼ 4 10−15ξ5=2 hni p Hend ð M Þ hniM ; ð142Þ Z ξ Pl Pl d3k ρL;θ ¼ n ω : ð141Þ ð2πÞ3 k k where hni is the average occupation number. The back- ρ 3 2 2 ∼ ground inflaton energy density is infl ¼ H MPl 12 10−11 2 ξ ≳ 103 Reference [64] computed the particle number, working in the MPl, and hence for the transfer of energy Jordan frame, and arrived at similar results. The energy of is enough to completely drain the inflaton condensate within the gauge fields was subsequently computed using the value one zero crossing of φðtÞ if we take the particle number “ ” of the gauge-field mass directly on the Riemann spike. We shown in Fig. 11 into account. The right panel of Fig. 11 refrain from using m2 j as an indicator of the gauge-field 2;θ max shows the ratio of the energy density in gauge fields to the mass, since the particle number is not a well-defined quantity there. For ξ ≈ 103, the two contributions to the gauge-field mass background energy density of the inflaton after the first zero 2 ˜ 2 crossing. Obviously, values of ρ =ρ > 1 are not m2;θ and mB are comparable, as shown in Fig. 9, which does not gauge infl hold for other values of ξ. physical but signal the possibility of complete preheating.

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FIG. 10. The particle number density for k=Hend ¼ 1; 150; 550; 2600; 28000 (blue, black, green, red, and purple, respectively). From left to right: ξ ¼ 102; 103; 104. If a colored curve is missing from a panel, the corresponding wave number is not excited.

C. Unitarity scale cutoff 1 5 104 kUV;2 ffiffiffi ∼ × So far we have computed the excitation of gauge-field ¼ p 2 : ð144Þ kmax ξ λ ξ modes of arbitrary wave number k

FIG. 11. Left: The particle number density after the first inflaton zero crossing for ξ ¼ 102; 103; 104 (blue, orange, and green, respectively) Right: The ratio of the energy density in gauge fields to the background inflaton energy density as a function of the nonminimal coupling ξ after the first zero crossing. We see that for ξ ≳ 103 gauge boson production can preheat the Universe after one background inflaton zero crossing, and hence it is much more efficient than Higgs self-resonance.

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≳ ξ 1000 ≪ 2 since kUV;1 kmaxð ¼ Þ, as can be seen from Fig. 11. keeping the relation mh mgauge valid at all times. If instead we place the UV cutoff at kUV;2, the gauge fields Hence, the Higgs field cannot decay into gauge bosons, do not carry enough energy to completely preheat the as long as the background Higgs condensate follows the Universe after one inflaton zero crossing, regardless of the evolution that is derived neglecting backreaction. value of the nonminimal coupling ξ. We thus conclude that The decays of Higgs bosons to fermions deserve closer preheating into gauge fields is very sensitive to unknown attention, due to the fact that small Yukawa couplings for UV physics, since the majority of the energy density is some fermions (like and ) can make carried by high-k modes, whose number density in a UV- them much lighter than the Higgs particles, and hence complete model can be much different than the one kinematically open the decay channel. Furthermore, fer- computed here. It is worth noting that the excitation of mion masses do not have a Riemann component, and hence Higgs fluctuations occurs entirely below the unitarity scale, when φðtÞ crosses zero fermions become instantaneously and hence it is not UV sensitive. We will not consider any massless, making the decay even easier. A similar analysis UV cutoff for the remainder of this work, unless explicitly of kinematical blocking of perturbative decays during stated. reheating was performed in Ref. [91], where the Higgs field was a light spectator field during inflation rather than VI. SCATTERING, DECAY, AND BACKREACTION playing the role of the inflaton itself. We will compute each component of the Higgs field mh;1 So far we have computed the parametric excitation of and mh;3 separately. We begin with the potential contribution particles, either Higgs or gauge bosons, from the oscillating Higgs condensate during preheating. With the exception of λ 2 δ2 δ2 12ξ − 12ξδ2 3 λ 2 MPl ð ð Þþ Þ 2 2 m 1 ¼ ∼ M ∼ H ; the brief discussion in Sec. IVA, the interactions of the h; ξ ð1 þ δ2Þ2ð1 þ 6ξδ2Þ2 3ξ2 Pl end resulting particles have been completely ignored. However, as discussed in Refs. [58,59], certain types of decays of the ð146Þ produced particles can suppress Bose enhancement and δ ξφ2 δ ≃ 0 8 thus effectively shut off preheating. We will discuss in turn where ¼ and . at the end of inflation, as – (A) the decay of Higgs particles into gauge bosons and discussed in Refs. [61 63]. The value in Eq. (146) holds at the start of preheating and until the crossover time fermions, pffiffiffi ∼ ξ −1 (B) the scattering of Higgs particles into gauge bosons tcross Hend. The expression for tcross was derived in and fermions, Ref. [63].Forttcross the Higgs particle mass mh;1 decreases slowly with Higgs bosons, and time. For ξ ¼ 10; 102; 103 the crossover time occurs at ≃ 1 5 2 5 3 2 (E) possible effects arising from non-Abelian inter- Ncross . ; . ; . , respectively. actions of the produced . Figure 12 shows the ratio of the to the Higgs Any of the above-mentioned processes can suppress or shut mass at the end of inflation as a function of the Yukawa off the resonances. Due to their inherent differences, we coupling, for different values of the nonminimal coupling. will explore them separately. We see that the decay is kinematically possible for small Yukawa couplings. Furthermore, the decay channel is less A. Higgs decay constrained for later times and larger nonminimal coupling. The perturbative decay rate of Higgs particles to fermions is In the Standard Model, Higgs particles can decay into pairs of fermions or gauge bosons. The fermion masses are given by 2 2 φ2 y 2 yf Γ f m ¼ ; ð145Þ ¼ mh: ð147Þ f 2 2f 8π while the gauge boson masses were extensively studied in Figure 12 shows the ratio Γ=H, which must be greater than Sec. VB. For now, it is enough to consider the part of the unity in order for the decay to be efficient. It is clear that, in the parameter range where the decay is kinematically gauge-field mass analogous to mf in Eq. (145) with the Yukawa coupling substituted by the gauge coupling. allowed, it is very inefficient. This can be intuitively ∼ We start with the process of a Higgs particle decaying understood since mh H, mf is proportional to yf and Γ 2 Γ into two gauge bosons. In order for this to be kinematically is proportional to yf, and hence =H is suppressed by an 2 allowed, the following relation must hold: mh > mgauge.It extra factor of the Yukawa coupling compared to mf=mh. ≪ 2 is straightforward to see that mh mgauge, at least for This conclusion does not change, even if one considers the φðtÞ ≠ 0. When φðtÞ¼0, the Riemann contribution to the short increase in the mass of the Higgs modes due to the “ ” 2 2 gauge-field mass (the Riemann spike ) dominates, coupled metric fluctuations term m3;h. Even though m3;h

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FIG. 12. Left: Fermion to Higgs mass ratio as a function of the Yukawa coupling for ξ ¼ 10; 100; 103 (blue, red, and green, respectively) at N ¼ 0, 1.5, 3 (solid, dashed, and dotted curves, respectively). Right: Ratio of the decay rate to the Hubble rate. The color coding is the same.

ρ ρ has a large positive spike, its duration is too small to allow n ≈ h ≤ infl ; ð149Þ for a significant decay of Higgs particles into fermions. mh mh Before we conclude this section, we will make one ρ ρ where h ¼ infl at the point of complete preheating. The further note regarding the evolution of fermion masses. cross section is Equation (145) shows that fermions become massless when φðtÞ¼0. The distinction between computing the fermion y4 σ ≈ e : mass during reheating by either using an averaged quantity 8π 2 ð150Þ mh for the Higgs VEVor by using the full time dependence was explored in Ref. [91]. In order to explore possible effects of Putting everything together, we arrive at the time dependence of the fermion mass, we focus on the 2 3 Γ 1 ρ 3 M H case of ξ ¼ 10 and choose a large Yukawa coupling ≤ y4 tot ¼ y4 Pl : ð151Þ H e 8π m3H e 8π m3 yw ¼ 1, since that provides the largest decay rate to h h Hubble scale ratio Γ=H ≃ 10, as shown in Fig. 12. The It is easy to see that Γ=H ≪ 1 since y4 ≃ 10−24, time per oscillation that m

The Higgs particles are heavy (mh >H) and have small wave numbers (k=a ≲ H), and hence will be non- relativistic. We will take v ¼ c ≡ 1 as an upper limit. The FIG. 13. Loop diagrams that contribute to the scattering of number density of Higgs particles is approximately Higgs bosons to gluon pairs.

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3 2 hence there is no significant particle decay. We will thus Γ g W ¼ 16π mW; ð152Þ neglect this altogether. However, after the particle production has taken place at   2 φ 0, the particle number is a constant, if one neglects g2 7 11 49 ¼ Γ ¼ m − sin2 θ þ sin4 θ ; 2 ∼ φ2 Z 2 2 Z 2 3 W 9 W decays, and the particle mass grows sharply as mW;Z . 8π cos θW We rewrite the equation for the energy density in the gauge ð153Þ sector as

Z 3 R where the decay widths are obtained by summing over all − t Γ 0 0 L;θ d k ðt Þdt ρ ¼ n0e t0 ω : ð155Þ allowed decay channels into SM fermions. The decay of the ð2πÞ3 k Z boson to a pair of Higgs particles proceeds similarly. Using the gauge boson mass given in Eq. (139), we see that Figure 14 shows the energy density per particle number of a Γ =H ≫ 1, and hence the produced gauge-boson pop- W;Z random excited k mode as ρ ≃ nðtÞmA, with A denoting any ulation is depleted within a Hubble time, or between two gauge field. We see that decays into fermions completely consecutive inflaton background zero crossings φðtÞ¼0. deplete the produced gauge-boson population within far There are two issues that need to be addressed: the possible less than a period of background oscillations. Hence, in decay of particles during their production close to the order for the energy transfer to be able to preheat the Riemann spike at φðtÞ¼0 and the decay away from Universe, the energy density in the gauge fields must be φ 0 ˜ 2 2φ2 2 2 ðtÞ¼ , when the mB ¼ e ðMPl= fÞ component domi- equal to the energy density in the inflaton condensate as nates the gauge-field mass. In both cases, we will approxi- soon as the adiabaticity condition is restored. The fact that mate the total decay of the particle number as the particle decays during the “Riemann spike” are insuf- R ficient to suppress gauge boson production shows that this − t Γðt0Þdt0 is indeed possible. nðtÞ¼n0e t0 ; ð154Þ where ΓðtÞ is defined through Eqs. (152) and (153) by D. Gauge scattering considering the time-dependent mass of the gauge bosons. Instead of decaying into fermions, gauge bosons can also We will focus only on the cases of ξ ¼ 103 and ξ ¼ 104. scatter into Higgs particles or fermion-antifermion pairs. During the spike, the particle number is not a well-defined We will estimate the rate of the Higgs scattering to Higgs quantity, since an adiabatic vacuum cannot be constructed bosons. The scattering rate is Γ ¼ nσv, where we take due to the violation of the adiabaticity condition. We will v ¼ c and however compute the exponential decay factor of Eq. (154) HM2 pffiffiffi as an estimate of possible particle decays. We choose the ≃ Pl ≃ 2 ξ10−5 m HMPl ; ð156Þ limits of integration to correspond to the times for which mA adiabaticity is violated, and hence particle production 2 occurs. This is also the time at whichR the Riemann spike α 1 t σ ≃ ≃ pffiffiffi ; ð157Þ − Γðt0Þdt0 λ 2 is pronounced. For all cases we get e t0 > 0.5, and s ð MPlÞ

FIG. 14. Energy density per mode (in arbitrary units) with (blue) and without (red) considering particle decays for ξ ¼ 103 (left) and ξ ¼ 104 (right). The time is rescaled by the period of background oscillations.

083519-24 PREHEATING AFTER HIGGS INFLATION: SELF-RESONANCE … PHYS. REV. D 99, 083519 (2019) where we computed the number density using the condition to an extended momentum distribution. Particles with such of complete preheating and we took the Mandelstam high momenta are energetic enough to scatter off the Higgs ≃ 2 variable s kmax. Altogether, condensate and fragment it, thereby shutting off any further Γ 105 parametric resonance. In the case of Higgs inflation the ≃ gauge fields produced do not survive long before decaying 3 2 ; ð158Þ H ξ = into fermions, due to their large masses. Hence, this is where we used the relation between λ and ξ given in unlikely to be an issue in the present case. Eq. (48). Since Γ=H ≲ 1 for ξ ≳ 103, gauge field scatter- ings are not important. This is different from other cases of VII. OBSERVATIONAL CONSEQUENCES preheating into gauge bosons, such as that in Ref. [54] Observing reheating is difficult due to the inherently small where gauge boson scattering is extremely efficient. The length scales involved. However, there are two important difference is that in the present case the number density is quantities that can be used to connect reheating to particle not large, but the average energy carried by each gauge physics processes or CMB observables: the reheat temperature boson is, due to the large range of excited wave numbers. Treh, and the number of e-folds of an early matter-dominated epoch in the expansion history of the Universe Nmatter. E. Non-Abelian effects The reheat temperature is computed using the Hubble 1 ρ ρ Since we are using an Abelian Uð Þ gauge field as a scale at the instant when infl ¼ rad as proxy for preheating into SM W and Z bosons, we must π2 estimate the possible non-Abelian effects. As long as the 3M2 H2 ¼ ρ ¼ T4 ; ð161Þ linear analysis holds, the electroweak sector can be decom- Pl 30g reh posed into three almost identical Abelian copies. A where g ¼ 106.75 is the number of relativistic d.o.f. at high numerical example of the relation between an SUð2Þ gauge energies. For instantaneous reheating from gauge field field and its three Abelian copies at low field values was production, which happens for ξ ≳ 1000, the Hubble scale shown in Ref. [92]. is H ≃ H .Forξ ≲ 1000 preheating proceeds through However, once the gauge-field modes become suffi- end Higgs self-resonance, leading to a smaller value of the ciently populated, their true non-Abelian nature cannot energy density, as shown in Fig. 4. The monotonic increase be neglected. The relevant term in the non-Abelian of the reheat temperature T as a function of the non- Lagrangian is reh minimal coupling ξ is shown in Fig. 15. It must be noted that 1 bμ d cν e Eq. (161) assumes the immediate transition to a thermal state L ⊂− f f A AμA Aν; ð159Þ non-Abelian 4 abc ade after preheating has ended. For the case of Higgs self- resonance, this will occur through efficient scattering of where fabc and fade are SUð2Þ structure constants. In the Higgs bosons to the rest of the SM. For the case of equation of motion for the gauge field strength Ai, this term 2 j instantaneous preheating to gauge fields, the situation is in the Lagrangian will induce a term of the form g AjA Ai, which has the form of an effective non-Abelian mass term. more complicated. In that case the number density of gauge Using a Hartree-type approximation we can define the bosons is not exponentially large, as is usually the case in non-Abelian contribution to the gauge-field mass squared preheating. On the contrary, the transfer of energy to gauge 2 2 fields is done primarily through the production of fewer as m ∼ g hAAi. We estimate hAAi through the pffiffiffi non-Abelian ∼ λ ρ ≃ 2 2 high-momentum modes kmax MPl. A fraction of the energy density of the gauge fields as mAhA i. Taking as 2 2 produced W and Z bosons will decay to , while a maximum value ρ ¼ ρ ¼ H M , we estimate infl Pl another fraction will decay into and antiquarks that 2 ≃ 10−10ξ 2 hA i MPl: ð160Þ will eventually hadronize. The approach to thermal equi- librium will thus be more complicated. We leave the study of In order for the non-Abelian mass contribution to suppress 2 the thermalization process for future work and we use particle production, it must dominate over mθ;2. However, Eq. (161) as an estimate of the reheat temperature, under 2 ≃ ξ2 2 ≃ ξ210−12 2 we know that mθ;2 H MPl, meaning that for the assumption of efficient thermalization. ξ ≳ 103 the “Riemann spike” dominates over the possible However, a high reheat temperature may pose a chal- non-Abelian mass contribution. Hence, we expect the lenge for any computation that goes beyond the linearized explosive transfer of energy from the inflaton to the gauge analysis that we presented, due to possible conflicts with fields to persist even in the full SUð2Þ × Uð1Þ sector. the unitarity scale. Since thermalization of the reheating A further phenomenon that has been observed during products will result in a blackbody spectrum, we can take ∼ 3 simulations of preheating of a non-Abelian Higgsed sector the typical momentum involved to be k Treh, which is was described in Ref. [93]. There, the decay of the Higgs thus the typical momentum exchange in particle scatterings condensate through resonant decay of electroweak bosons inside the plasma. Since complete reheating means that the was simulated. Non-Abelian gauge boson interactions led inflaton condensate will have completely decayed, the

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VIII. CONCLUSIONS Higgs inflation is an appealing way to realize inflation within the particle content of the Standard Model, by coupling the Higgs field nonminimally to the gravity sector with a large value of the nonminimal coupling. We analyzed the non- perturbative decay of the Higgs condensate into Higgs bosons and electroweak gauge fields, finding distinct behavior for different ranges of values of the nonminimal coupling ξ. The self-resonance of the Higgs field leads to preheating ≃ 4 after Nreh e-folds for values of the nonminimal cou- pling ξ ≳ 30. For large values ξ > 100 the inflaton can FIG. 15. Reheat temperature in units of MPl as a function of the transfer all of its energy into nonrelativistic Higgs modes nonminimal coupling ξ. The discontinuity at ξ ≃ 103 occurs due ≈ 3 within Nreh , independent of the exact value of the to the instantaneous preheating to gauge fields. The light red nonminimal coupling. The dominant contribution to the region represents the uncertainty of the exact threshold of parametric excitation of Higgs modes is the effect of instantaneous preheating to gauge fields. The black-dotted line ≲ 3ξ coupled metric fluctuations. In order to accurately capture corresponds to the unitarity scale constraint Treh MPl= . The blue-dashed line shows the reheat temperature due entirely to the amplitude of the Higgs wave function, the computation Higgs self-resonance, assuming gauge boson production above must be initiated before the end of inflation. the unitarity scale is suppressed due to unknown UV physics. The excitation of gauge bosons is much more dramatic, reminiscent of the purely scalar case of preheating in multifield inflation with nonminimal couplings [61–63]. ≡ ξ unitarity scale is kUV;2 MPl= . The typical particle Gauge fields are excited after the first zero crossing of the 3 pffiffiffi momenta are below the unitarity scale for T

083519-26 PREHEATING AFTER HIGGS INFLATION: SELF-RESONANCE … PHYS. REV. D 99, 083519 (2019) black hole production in critical Higgs inflation has been We require c2 ¼ 0 in order for the phases to match the −ikτ proposed in Ref. [98]. It was however shown that large Bunch-Davies form e in the past. We must also set c1 curvature fluctuations in Higgs inflation are hard or impos- such that the norm matches Eq. (112) for kjτj¼xc, sible to produce without violating observational limits on the pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi tensor-to-scalar ratio and the running of the spectral tilt 1πℑ 1−4 2 1ℜðð1− 1−4x2Þ logð x ÞÞ τ ≃ 4 ð xcÞ 2 c xc [99,100], unless one postulates a large running of the juLðk; Þj jc1je e : ðA3Þ nonminimal coupling ξ that is not found in the renormaliza- ≫ 1 tion group flow of the Standard Model nonminimally If we work in the regime xpc ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi, which is true for Higgs 2 coupled to gravity. Recent studies of Higgs inflation inflation, we approximate 1 − 4xc ≃ i2xc, and the above involving nonminimal couplings in the Palatini formulation expression simplifies to of gravity [101,102] can also have different preheating   phenomenology. Exploring the preheating phenomenology 1=2 1πx x of these models is interesting and can be performed using the ju ðk; τÞj ≃ jc1je2 c ; ðA4Þ L x techniques applied here. Such analyses can provide unique c methods to probe the Higgs potential at energy scales that are τ out of reach for the LHC and any future accelerator. and hence equating this to Eq. (112) for kj j¼xc reveals the value of the integration constant c1, ACKNOWLEDGMENTS 1 −1πx We are indebted to David Kaiser and Marieke Postma for jc1j¼e 2 c pffiffiffiffiffi : ðA5Þ invaluable discussions and comments on the manuscript. 2k The authors gratefully acknowledge support from the Dutch Organisation for Scientific Research (NWO). Altogether, the evolution of the mode function is   APPENDIX: Gauge field modes 1=2 1 kjτj − τ ffiffiffiffiffi τ ixc τ during inflation for jkτj < xc uLðk; Þ¼p ðkj jÞ ;kj j

1 conditions for preheating computations are 1 1−ν ðν−1Þ 1 1−ν L τ −1 4ð Þ 2 τ 2ð Þ u ðk; Þ¼c1ð Þ xc ðk Þ 1F1     1 τ 1=2 1 1 1 k2τ2 τ ffiffiffiffiffi k in × − ν;1− ν; uLðk; inÞ¼p ; ðA8Þ 4 4 2 2 2k xc xc 1 1 −ν−1 1 ðνþ1Þ 2ð Þ ðνþ1Þ   þ c2ð−1Þ4 xc ðkτÞ2 1F1   1 1 2 2 u_ k; τ B k; τ − ix 1 1 1 k τ Lð inÞ¼ Lð inÞ τ τ 2 c ν ; ν 1; að inÞ in × 4 þ 4 2 þ 2 ; ðA1Þ   xc 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ≃ B ðk; τ ÞHðτ Þ − ix ðA9Þ 2 L in in 2 c where ν ¼ 1 − 4xc and c1, c2 are integration constants. This is a rather cumbersome formula that does not provide a τ ≫ 1 lot of insight. By Taylor expanding it for values of kjτj ≪ for wave numbers such that jk inj

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