PHYSICAL REVIEW D 99, 083519 (2019)
Preheating after Higgs inflation: Self-resonance and gauge boson production
† Evangelos I. Sfakianakis1,2,* and Jorinde van de Vis1, 1Nikhef, Science Park 105, 1098XG Amsterdam, The Netherlands 2Lorentz Institute for Theoretical Physics, 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 gravity, but the field-space metric is nontrivial. The self-resonance of the Higgs and the Higgsed gauge bosons 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-inflaton condensate can be transferred to Higgs particles 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 fermions, thereby shutting off Bose enhancement. Estimates of non-Abelian interactions 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 Higgs boson at CERN [1] of the Higgs self-coupling λ at high energies, due to our solidified our understanding of the Standard Model (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 =N Þ, 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 matter 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 baryon-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 particle 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 vacuum expectation value 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∂jE dx 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 electromagnetism. 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- charge 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 photon 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/GOLDSTONE BOSON 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 quantization 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