Preheating after multifield inflation with nonminimal couplings. III. Dynamical spacetime results
The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters.
Citation DeCross, Matthew P., et al. “Preheating after Multifield Inflation with Nonminimal Couplings. III. Dynamical Spacetime Results.” Physical Review D, vol. 97, no. 2, Jan. 2018. © 2018 American Physical Society
As Published http://dx.doi.org/10.1103/PhysRevD.97.023528
Publisher American Physical Society
Version Final published version
Citable link http://hdl.handle.net/1721.1/113868
Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. PHYSICAL REVIEW D 97, 023528 (2018)
Preheating after multifield inflation with nonminimal couplings. III. Dynamical spacetime results
† ‡ Matthew P. DeCross,1,* David I. Kaiser,1, Anirudh Prabhu,1, ∥ Chanda Prescod-Weinstein,2,§ and Evangelos I. Sfakianakis3, 1Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 2Department of Physics, University of Washington, Seattle, Washington 98195-1560, USA 3Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA
(Received 15 November 2016; published 26 January 2018)
This paper concludes our semianalytic study of preheating in inflationary models comprised of multiple scalar fields coupled nonminimally to gravity. Using the covariant framework of paper I in this series, we extend the rigid-spacetime results of paper II by considering both the expansion of the Universe during preheating, as well as the effect of the coupled metric perturbations on particle production. The adiabatic and isocurvature perturbations are governed by different effective masses that scale differently with the nonminimal couplings and evolve differently in time. The effective mass for the adiabatic modes is dominated by contributions from the coupled metric perturbations immediately after inflation. The metric perturbations contribute an oscillating tachyonic term that enhances an early period of significant particle production for the adiabatic modes, which ceases on a time scale governed by the nonminimal couplings ξI. The effective mass of the isocurvature perturbations, on the other hand, is dominated by contributions from the fields’ potential and from the curvature of the field-space manifold (in the Einstein frame), the balance between which shifts on a time scale governed by ξI. As in papers I and II, we identify distinct behavior depending on whether the nonminimal couplings are small [ξI ≲ Oð1Þ], intermediate [ξI ∼ Oð1 − 10Þ], or large (ξI ≥ 100).
DOI: 10.1103/PhysRevD.97.023528
I. INTRODUCTION with respect to spacetime transformations (xμ → x0μ)as well as invariant under field-space reparametrizations This paper continues the work of Refs. [1,2] by con- (ϕI → ϕ0I). We also demonstrated in Ref. [1] that the strong sidering the early stage of postinflation reheating in models single-field attractor that such models generically obey that involve multiple scalar fields, each with a nonminimal during inflation [29–32] persists through the early stages coupling to gravity. Reheating is a critical epoch in cosmic of reheating, for at least as long as the linearized approxi- history, connecting early-Universe inflation with the suc- mation (in the field and metric perturbations) remains valid. cesses of the standard big-bang scenario. (For reviews of – The attractor makes these models safe from destabilization postinflation reheating, see Refs. [3 8].) Both of the main issues like the ones described in Ref. [34]. features of the models we consider—multiple fields [9–14] – — In Ref. [2], we applied the covariant formalism in the and nonminimal couplings [15 21] are well motivated by rigid-spacetime limit, in which we imagine holding high-energy theory, and encompass such models as Higgs the energy density fixed while sending M →∞.(The – pffiffiffiffiffiffiffiffiffipl inflation [22] (see also [23 28]) and related models with ≡ 1 8π ≃ 2 43 attractorlike solutions [29–33]. reduced Planck mass is given by Mpl = G . × 1018 In Ref. [1] we established a doubly covariant formalism GeV.) In that limit, we may neglect both the expansion with which to study the behavior of field fluctuations and of spacetime during reheating as well as the effects of the metric perturbations to linear order, which is gauge invariant coupled metric perturbations [7]. Then the condensate(s) that had driven inflation oscillate periodically after the end of inflation, which can lead to efficient particle pro- *[email protected]; Now at the Department of Phys- duction via parametric resonance. Within the rigid-space- ics, University of Pennsylvania. time approximation, we studied the highly nonperturbative † [email protected] ‡ transfer of energy from the inflaton condensate into adia- [email protected]; Now at the Department of Physics, batic and isocurvature modes, using the tools of Floquet Stanford University. → 0 §[email protected] theory. In the long-wavelength limit (k ), we identified ∥ [email protected] three distinct regimes, depending on whether the non- Now at NIKHEF and Leiden University. minimal coupling constants ξI are small [ξI ≲ Oð1Þ],
2470-0010=2018=97(2)=023528(18) 023528-1 © 2018 American Physical Society MATTHEW P. DECROSS et al. PHYS. REV. D 97, 023528 (2018) Z ξ ∼ O 1 − 10 ξ ≥ O 100 pffiffiffiffiffiffi 1 intermediate [ ϕ ð Þ], or large [ ϕ ð Þ]. For 4 I ~ μν I J ~ I S ¼ d x −g~ fðϕ ÞR − δ g~ ∂μϕ ∂νϕ − Vðϕ Þ ; both adiabatic and isocurvature modes, the most efficient 2 IJ particle production occurs for strong couplings, ξ ≥ 100, I ð1Þ and approaches self-similar scaling solutions in the limit ξ → ∞. I where tildes indicate Jordan-frame quantities. We perform a In this paper we consider the postinflation dynamics while conformal transformation relaxing the assumption of a rigid spacetime, incorporating both the Hubble expansion and the contributions from the 2 I g~μνðxÞ → gμνðxÞ¼ fðϕ ðxÞÞg~μνðxÞð2Þ coupled metric perturbations. Consistent with earlier studies M2 [35–44], we find that the coupled metric perturbations can pl have important effects on the preheating dynamics. In the to bring the action in the Einstein frame into the form models we consider here, the metric perturbations have a [47,48] particularly strong effect on the behavior of the adiabatic Z modes, dominating their effective mass at early times with ffiffiffiffiffiffi M2 1 4 p− pl − G ϕK μν∂ ϕI∂ ϕJ − ϕI an oscillating, tachyonic contribution. Within the single- S ¼ d x g 2 R 2 IJð Þg μ ν Vð Þ : field attractor, on the other hand, the metric perturbations play a minimal role for the isocurvature modes. Instead, the ð3Þ behavior of the isocurvature modes is governed by the changing ratio of two other contributions to their effective In the Einstein frame, the (curved) field-space manifold mass: the term arising from (gradients of) the potential, and acquires a metric given by the term from the curved field-space manifold. We identify 2 time scales on which these distinct contributions dominate Mpl 3 G ðϕKÞ¼ δ þ f f ; ð4Þ the dynamics, and how those time scales vary with increas- IJ 2fðϕKÞ IJ fðϕKÞ ;I ;J ing ξI. As in our previous studies [1,2], we work to linear I order in the perturbations, reserving for future study such where f;I ¼ ∂f=∂ϕ . (Explicit expressions for the compo- nonlinear effects as backreaction of the created particles on nents of GIJ for our two-field model may be found in the oscillating inflaton condensate [7,37,45,46]. Appendix A of Ref. [1].) The potential in the Einstein frame In Sec. II we briefly introduce our model and the is likewise stretched by the conformal factor: covariant formalism with which to study linearized per- 4 turbations. Section III examines the behavior of the Mpl ~ ∼ 0 VðϕIÞ¼ VðϕIÞ: ð5Þ adiabatic modes for k , and Sec. IV examines the 4f2ðϕIÞ behavior of the isocurvature modes for k ∼ 0. In Sec. V, we consider the contrasting behavior of adiabatic and We expand the scalar fields and spacetime metric to first isocurvature modes for nonzero wave number, studying order in perturbations. Because we are interested in the the evolution of both subhorizon and superhorizon modes. dynamics at the end of inflation, we consider scalar metric In Sec. VI we briefly consider possible observational perturbations around a spatially flat Friedmann-Lemaître- consequences of the efficient preheating dynamics for this Robertson-Walker (FLRW) line element, family of models, and in Sec. VII we comment on potential 2 μ ν implications of our analyses for reheating after Higgs ds ¼ gμνðxÞdx dx inflation. Concluding remarks follow in Sec. VIII. In the − 1 2 2 2 ∂ i Appendix, we consider the time scale over which the ¼ ð þ AÞdt þ að iBÞdx dt 2 i j background dynamics transition from a matter-dominated þ a ½ð1 − 2ψÞδij þ 2∂i∂jE dx dx ; ð6Þ to radiation-dominated equation of state, and compare this cross-over time with the time scales relevant for the where aðtÞ is the scale factor. We also expand the fields, amplification of adiabatic and isocurvature modes, as derived in Secs. III and IV. ϕIðxμÞ¼φIðtÞþδϕIðxμÞ: ð7Þ
II. MODEL AND FORMALISM To first order in the perturbations, we may then construct generalizations of the gauge-invariant Mukhanov-Sasaki ϕI We consider models with N real-valued scalar fields variable (see Ref. [1] and references therein): in 3 þ 1 spacetime dimensions. We use uppercase Latin letters to label field-space indices, I;J ¼ 1; 2; …;N; Greek φ_ I μ ν 0 QI ¼ δϕI þ ψ: ð8Þ letters to label spacetime indices, , ¼ , 1, 2, 3; and H lowercase Latin letters to label spatial indices, i, j ¼ 1,2,3. The spacetime metric has signature ð−; þ; þ; þÞ. The To background order, the dynamics are governed by the action in the Jordan frame is given by coupled equations,
023528-2 PREHEATING AFTER …. III. DYNAMICAL … PHYS. REV. D 97, 023528 (2018)
I I IJ Dtφ_ þ 3Hφ_ þ G V;J ¼ 0 ð9Þ creation and annihilation operators and associated mode functions. Whereas in Ref. [1] we worked in terms of and conformal time, dη ¼ dt=aðtÞ, in this paper we will use cosmic time, t, since our numerical routines are efficient at 1 1 I I 2 evolving φ t ;H t , and X t in terms of t. Within H G φ_ Iφ_ J V φI ; ð Þ ð Þ ð Þ ¼ 3 2 2 IJ þ ð Þ Mpl the single-field attractor, the mode functions effectively 1 decouple [1]: _ − G φ_ Iφ_ J H ¼ 2 IJ ; ð10Þ 2 2 Mpl k v̈ þ Hv_ þ þ m2 ðtÞ v ≃ 0; k k a2 eff;ϕ k where overdots denote derivatives with respect to t, and the 2 Hubble parameter is given by HðtÞ¼a=a_ . Covariant k 2 ̈z þ Hz_ þ þ m ðtÞ z ≃ 0: ð15Þ derivatives with respect to the field-space metric are given k k a2 eff;χ k D I ∂ I ΓI K I by JA ¼ JA þ JKA for a field-space vector A , from which we may construct the (covariant) directional deriva- The Hubble-drag term enters as Hv_ k [rather than 3Hv_ k,as I J I tive with respect to cosmic time, DtA ¼ φ_ DJA ¼ in Eq. (11)], because we have scaled the fluctuations, −1 _ I ΓI φ_ J K ΓI φL QI ¼ XI=a ∝ a fv ;z gT. Within the attractor (along A þ JK A , where the Christoffel symbols JKð Þ k k K χ ¼ 0), v is the mode function for perturbations along are constructed from GIJðφ Þ. The gauge-invariant pertur- k bations obey the adiabatic direction and zk is the mode function for perturbations along the isocurvature direction. Their effec- k2 tive masses consist of four distinct contributions [1]: D2 I 3 D I δI MI J 0 t Q þ H tQ þ 2 J þ J Q ¼ ; ð11Þ a 2 2 2 2 2 meff;I ¼ m1;I þ m2;I þ m3;I þ m4;I; ð16Þ where the mass-squared matrix is given by with 1 a3 MI ≡ GIK D D V − RI φ_ Lφ_ M − D φ_ Iφ_ J ð J K Þ LMJ 2 3 t J 2 ≡ GϕK D D Mpla H m1;ϕ ð ϕ KVÞ; 2 ϕ L M ð12Þ m2 ϕ ≡ −R ϕφ_ φ_ ; ; LM 3 1 ϕ a and RI is the Riemann tensor constructed from G φK . 2 ≡ − δ δJ D φ_ Iφ_ LMJ IJð Þ m3 ϕ 2 3 ϕ t J ; 1 2 ; M a I H The term in Eq. (12) proportional to =Mpl arises from the pl coupled metric perturbations. 1 m2 ≡ − R; 17 As in Refs. [1,2], we consider a two-field model, 4;ϕ 6 ð Þ ϕI ¼fϕ; χgT, with nonminimal couplings (in the Jordan frame) given by where R is the spacetime Ricci curvature scalar; compa- 2 rable expressions follow for the contributions to meff;χ.We 1 2 I 2 2 2 m fðϕ Þ¼ ½M þ ξϕϕ þ ξχχ ð13Þ note that 1;I arises from (covariant gradients of) the 2 pl 2 potential, m2;I from the curvature of the field-space mani- 2 2 and Jordan-frame potential fold, m3;I from the coupled metric perturbations, and m4;I from the expanding spacetime. The form of each of these λϕ g λχ ~ ϕI ϕ4 ϕ2χ2 χ4 contributions differs from the case of minimally coupled Vð Þ¼ 4 þ 2 þ 4 : ð14Þ 2 fields. The term m2;I, in particular, has no analog for models with minimally coupled fields and canonical kinetic The Einstein-frame potential VðϕIÞ develops ridges and ξ ≠ ξ terms, and can play important roles in the dynamics both valleys for the generic case in which ϕ χ and/or – λ ≠ ≠ λ during and after inflation [1,2,29 32,34]. Meanwhile, ϕ g χ, and the background dynamics exhibit strong within the single-field attractor along χ ¼ 0, the energy – single-field attractor behavior [1,29 32]. Given our covar- densities for adiabatic and isocurvature perturbations take iant framework, we may always perform a field-space the form [1] rotation such that the attractor lies along the direction χ ¼ 0 in field space, without loss of generality. Within such an 1 2 ρðϕÞ _ 2 k 2 2 O χ2 attractor, the field-space metric becomes effectively diago- ¼ jvkj þ 2 þ m ϕ jvkj þ ð Þ; k 2 a eff; nal, with Gϕχ ¼ OðχÞ ∼ 0 [1]. μ μ 1 2 We rescale the perturbations, QIðx Þ → XIðx Þ=aðtÞ, ρðχÞ _ 2 k 2 2 O χ2 k ¼ jzkj þ 2 þ meff;χ jzkj þ ð Þ; ð18Þ and quantize, XI → Xˆ I, expanding Xˆ ϕ and Xˆ χ in sets of 2 a
023528-3 MATTHEW P. DECROSS et al. PHYS. REV. D 97, 023528 (2018)
pffiffiffiffiffi ρðϕÞ ρðχÞ ~ λ ξ 0 FIG. 1. The evolution of the energy densities k (blue) and k (gold) as functions of t ¼ ϕMplt= ϕ for k ¼ . The parameter 2 3 ratios are chosen as ξχ =ξϕ ¼ 0.8, g=λϕ ¼ 1, and λχ=λϕ ¼ 1.25. Top, left to right: ξϕ ¼ 1, 10; bottom, left to right: ξϕ ¼ 10 ; 10 .
ϕ χ again keeping in mind that Q ∼ vk=aðtÞ and Q ∼ zk=aðtÞ. that the growth within the broad-resonance regime that We measure particle production with respect to the instan- we analyzed in Ref. [2] is robust, even when we relax 0 taneous adiabatic vacuum, j ðtendÞi, which minimizes the the assumption of a rigid spacetime. In the next two ρðIÞ sections, we analyze semianalytically the behavior shown energy densities k at the end of inflation [7]. For evolution within an attractor along the direction in Fig. 1. χ ¼ 0, the Hubble scale at the end of inflation satisfies sffiffiffiffiffiffiffiffiffiffi III. RESULTS: ADIABATIC MODES λϕ Hðt Þ ≃ 0.4 M ; ð19Þ Within the single-field attractor, there are two significant end 2 pl 2 12ξϕ components of the effective mass meff;ϕ for the adiabatic 2 perturbations during preheating: m1;ϕ from the potential and the background field’s frequency of oscillation at the 2 and m3 ϕ from the metric perturbations. Within the single- end of inflation satisfies ω >Hfor all ξϕ ≥ 0, with ω=H ≃ ; field attractor, m2 ∼ O χχ_ ∼ 0, and m2 −R=6 ∼ 1.5 for ξϕ ≤ 0.1 and ω=H ≃ 4 for ξϕ ≥ 10 [1]. In particular, 2;ϕ ð Þ 4;ϕ ¼ 2 both the period of oscillation T ¼ 2π=ω and the Hubble Oð1ÞH remains much smaller than the other terms [1]. −1 2 2 time H scale as ξϕ for large ξϕ. Hence we may capture In order to distinguish the effects from m1;ϕ and m3;ϕ and the relevant dynamics by rescaling our time variable ξ ρðϕÞ pffiffiffiffiffi examine their dependence on time and I, we compute k 2 t → t~ ≡ λϕM t=ξϕ ∝ Hðt Þt. For the remainder of this pl end first by neglecting m3;ϕ and compare with the results paper, we will set t˜ 0, where t˜ is the time at which 2 end ¼ end computed using the full effective mass, m ϕ. The results inflation ends and preheating begins. eff; for the k ¼ 0 mode and different values of ξϕ are shown in ρðϕÞ ρðχÞ 0 In Fig. 1 we show the growth of k and k for k ¼ as Fig. 2, from which two main results become clear. First, the 2 we increase the nonminimal couplings ξI. In particular, we contribution from the coupled metric perturbations, m3;ϕ, ξχ 0 8ξϕ g λϕ λχ 1 25λϕ select ¼ . , ¼ , and ¼ . , such that all of dominates during times immediately after the end of O 1 the relevant ratios among couplings are ð Þ, and hence no inflation, yielding a significant, oscillating tachyonic con- softly broken symmetry exists [2]. As in Refs. [1,2],we 2 tribution to meff;ϕ that drives significant particle production. identify three distinct regimes, governed by the magnitude 2 of ξ : inefficient growth of either set of perturbations for The contribution from the potential, m1;ϕ, comes to I 2 ξI ≤ Oð1Þ in the limit k → 0; modest growth of adiabatic dominate meff;ϕ at later times, such that the energy density ξ ∼ O 10 2 2 modes for I ð Þ; and rapid growth for both adiabatic computed with either m ϕ or m1 ϕ alone share the same ξ ≥ O 100 eff; ; and isocurvature modes for I ð Þ. Moreover, the behavior after a characteristic time scale. balance shifts between adiabatic and isocurvature modes We start by examining the time when the metric ξ between the intermediate- and large- I regimes, with perturbations dominate the growth rate of the adiabatic ρðχÞ ξ → ∞ ξ ξ ≥ 10 especially rapid growth of k for I . Hence we find modes, as a function of ϕ, focusing on ϕ . The
023528-4 PREHEATING AFTER …. III. DYNAMICAL … PHYS. REV. D 97, 023528 (2018)
pffiffiffiffiffi ρðϕÞ 0 ~ λ ξ FIG. 2. Left: energy density for the adiabatic perturbations k for k ¼ as a function of t ¼ ϕMplt= ϕ, computed using the full effective mass (black, upper solid line) and by neglecting the coupled metric perturbations (blue, lower solid line). The red dotted curve 2 2 2 is a shifted version of the blue curve, to help indicate the time scales on which m1;ϕ or m3;ϕ dominate meff;ϕ. Right: rescaled adiabatic ~ 2 ξ2 2 ~ 2 ~ 2 effective mass meff;ϕ ¼ ϕmeff;ϕ (back dotted) and its two main components, m1;ϕ (blue) from the potential, and m3;ϕ (red) from the 2 3 metric perturbations. The nonminimal coupling is ξϕ ¼ 1; 10; 10 ; 10 (top to bottom).
2 2 ≲ maximum value (positive or negative) of m3;ϕ may be well perturbations become unimportant when maxðm3;ϕÞ 1 2 fit numerically by 3 maxðm1;ϕÞ. Numerical evaluation of the relevant functions ξ ≥ 10 ffiffiffi for several values of pϕ ffiffiffiffiffi pinpoints this crossover as 2 p ≃ 2 6 ϕ̈ occurring at t˜ ∼ 9.5 ξϕ. Upon using Eq. (19) and the maxðm3;ϕÞ j maxj; ð20Þ pcrossffiffiffiffiffi ~ λ ξ ∼ relationpffiffiffiffiffi t ¼ ϕMplt= ϕ, this is equivalent to Hendtcross 1 1 ξ ≡ where overdots denote derivatives with respect to t˜. We can . ϕ, where Hend HðtendÞ is the value of the Hubble see from the right panels of Fig. 2 that the coupled metric parameter at the end of inflation. The exact constant of
023528-5 MATTHEW P. DECROSS et al. PHYS. REV. D 97, 023528 (2018)
pffiffiffiffiffi ~ 2 ξ2 2 δ ξ ϕ ξ 1 10 102 103 104 105 FIG. 3. Left: rescaled mass m1;ϕ ¼ ϕm1;ϕ as a function of ¼ ϕ =Mpl, for ϕ ¼ ; ; ; ; ; (color coded in a ξ 1 ~ 2 δ rainbow scale with the bottom red curve corresponding to ϕ ¼ ). Right: maximum value of m1;ϕ (red dotted line) and the value of peak −1=4 for which it occurs (blue dots) as a function of ξϕ, along with the fitting curve δpeak ¼ 0.4ξϕ (blue solid line). proportionality is not particularly important; the point is As shown on the right-hand side of Fig. 3, the quantity −1 4 that the growth of adiabatic modes for ξϕ ≥ 10 is domi- δ δ ≃ 0 4ξ = peak is well fit numerically by the curve peak . ϕ nated by the effects of the coupled metric perturbationspffiffiffiffiffi at across 9 orders of magnitude. The amplitude of oscillation, early times, up to around H t ∼ O 1 ξϕ. δ end cross ð Þ max, will redshift due to Hubble friction, having begun at δ 0 8 ξ ≥ 1 As we can see from the blue curves in the left-hand maxðtendÞ¼ . for ϕ [1]; at early times after tend, the panels of Fig. 2, the times t
023528-6 PREHEATING AFTER …. III. DYNAMICAL … PHYS. REV. D 97, 023528 (2018)
~ 2 ξ2 2 FIG. 4. Left: rescaled effective mass for the isocurvature modes, meff;χ ¼ ϕmeff;χ (black dotted line) along with thep contributionsffiffiffiffiffi from ~ 2 ~ 2 ξ 10 ~ λ ξ the potential, m1;χ (blue), and from the curved field-space manifold, m2;χ (red), for ϕ ¼ , as functions of t ¼ ϕMplt= ϕ. Right: 2 2 2 quantities meff;χ (black dotted line), m1;χ (blue), and m2;χ (red) rescaled by the Hubble parameter.
2 ξ 10 2 Figure 4 shows the evolution of meff;χ for ϕ ¼ , along m2;χ 2 ∝ ð1 − εÞξϕ; ð26Þ with its two main components within the single-field hHi2 2 2 attractor, m1;χ from the potential and m2;χ from the curved 2 2 field-space manifold. We see that m2;χ dominates for early indicating that the contribution to meff;χ that arises from the 2 2 2 times and m1;χ dominates at later times. Furthermore, the curved field-space manifold, m2;χ, scales with H ðtÞ after 2 2 maximum amplitude of m2;χ is proportional to H ðtÞ. This the end of inflation. proportionality may be easily understood. Within the On the other hand, the contribution arising from the 2 2 single-field attractor along χ ¼ 0 [1], potential, m1;χ, grows compared to H ðtÞ, and thus domi- 2 ξ 10 2 nates meff;χ at late times. For ϕ ¼ , the terms m1;χ and 2 1 2 2 ˜ RG ϕ˙ m2 χ become approximately equal at t ≃ 80, while for m2;χ ¼ ϕϕ ; ð22Þ ; 2 ˜≳ 150 2 t , the term m1;χ clearly dominates. At these late 2 where R is the field-space Ricci scalar, not to be confused times, meff;χ oscillates quasisinusoidally and can drive 2 with the space-time Ricci scalar R. The peaks of m2;χ occur parametric resonance, depending sensitively on wave when ϕ vanishes and ϕ˙ reaches its maxima. Using the number and couplings; this accounts for the late-time ξ ρðχÞ ξ 10 expressions in Appendix A of Ref. [1], for large I and growth of k for ϕ ¼ in Fig. 1. ϕ 0 2 2 ¼ we may easily derive The evolution of m1;χ=H does not admit a simple 2 2 solution, contrary to the adiabatic case m1 ϕ=H , for the R ∝ 6 1 − ε ξ2 ; max ð Þ ϕ; ð23Þ 2 following reason. In the case of m1;ϕ the peak value occurs at ϕ 0, which drastically reduces the dependence of the ε ξ − ξ ξ ¼ where ¼ð ϕ χÞ= ϕ is the ellipticity of the field-space peak on parameters, since all terms involving powers of ϕ χ 0 potential. Furthermore, for motion along ¼ (within the vanish identically. In the case of m2 , the peak occurs at ϕ 0 2 1;χ single-field attractor) and with ¼ (where m2;χ peaks), ϕ ϕ 2 some nonzero max. Hence not only do higher powers of we find Gϕϕ ¼ 1. Thus the peak of the contribution m2 χ has 2 ϕ ; not vanish at the maximum of m1;χ,but,as ðtÞ redshifts, a very simple expression: 2 ξ2 ϕ2 terms such as Mpl þ ϕ change their character over time. 2 2 For ξϕϕ =M ¼ Oð1Þ, which is true immediately after maxðm2 Þ ∝ 3ð1 − εÞξ2 ϕ˙2: ð24Þ pl 2;χ ϕ 2 ξ2 ϕ2 ≈ ξ2 ϕ2 inflation, Mpl þ ϕ ϕ . However, as the amplitude 2 2 2 ˙2 of ϕ redshifts, the two terms in M þ ξ ϕ become After inflation the kinetic energy density (Gϕϕϕ =2) and the pl ϕ 2 ξ2 ϕ2 → 2 → ∞ potential energy density [Vðϕ; χÞ] attain equal maximum comparable. Ultimately Mpl þ ϕ Mpl as t . 2 2 values but oscillate out of phase. Averaged over a period of Thus we can show that m1;χ=H grows, but the nature the inflaton oscillation, the Hubble parameter is of the growth does not admit a simple expression. In order to compare the two contributions m2 and m2 , 1 1;χ 2;χ hHi¼pffiffiffi jϕ˙ j; ð25Þ we may consider the former in two distinct regions: 6 max Mpl 2 OðξϕÞ; δ ¼ Oð1Þ ˙ m1;χ where jϕ j is the field’s maximum velocity within one ¼ −1 2 ð27Þ max H2 O ξ2 δ O ξ = oscillation. Then one finds ð ϕÞ; ¼ ð ϕ Þ;
023528-7 MATTHEW P. DECROSS et al. PHYS. REV. D 97, 023528 (2018)
2 2 FIG. 5. The effective mass of the isocurvature perturbations, meff;χ =H (black-dotted line), shown with the contributions from the 2 2 2 2 ξ 102 ξ 103 potential, m1;χ=Hpffiffiffiffiffi(blue), and from the curved field-space manifold, m2;χ=H (red), for ϕ ¼ (left) and ϕ ¼ (right), as ~ λ ξ functions of t ¼ ϕMplt= ϕ. pffiffiffiffiffi pffiffiffiffiffi δ ξ ϕ δ O 1 ξ 2 where ðtÞ¼ ϕ ðtÞ=Mpl. We use ¼ ð = ϕÞ as a maxima in meff;χ at early times, and is in keeping with point at which the amplitude of the background field has Eqs. (26) and (27).] Using Figs. 1 and 6, we infer that the ðχÞ 2 2 redshifted significantly, since for this field value the term growth of ρ ceases when the ratio falls to m2 χ=m1 χ ≃15, 2 ξ ϕ2 k ; ; Mpl þ ϕ changes its behavior. We can immediately see which occurs at around one-sixth the crossover time at 2 2 2 2 2 that m2 χ=H ∝ ξ dominates initially, but will ultimately ˜ ∼ 800 6 ∼ 130 ; ϕ which m2;χ ¼ m1;χ, corresponding to t = for ξ become subdominant. We may further analyze the ϕ ξϕ ¼ 100. Since the relevant transition times scale as 2 2 ˜ dependence of the interplay between m1;χ and m2;χ. The t ∼ ξϕ, we expect the period of rapid, initial growth in δ ˜ 3 function ðtÞ asymptotes to one universal behavior in the the case ξϕ ¼ 10 to cease at t˜ ∼ 8000=6 ∼ 1300, which limit ξϕ ≫ 1 [1]. We noted above that the amplitude of indeed matches the behavior shown in Fig. 1. the inflaton decays as δ ∝ 1.5t˜−1=2. This suggests that the We may understand the transition out of the initial period 2 2 ξ crossover time between when m2;χ and m1;χ dominates of rapid growth for large I as the change from the broad- to ˜ ∼ ξ 2 ≃ 2 ˜ ≃ 80 the narrow-resonance regime [49]. At early times, the should scale as t ϕ. We found that m1;χ m2;χ at t 2 2 isocurvature mode zkðtÞ (with k ¼ 0) oscillates multiple for ξϕ ¼ 10. Figure 5 shows that m ≃ m at t˜ ≃ 800 for 1;χ 2;χ times for each oscillation of the inflaton field, as shown on ξ 102 ˜ ≃ 8000 ξ 103 ϕ ¼ and at t for ϕ ¼ , confirming that the the right-hand side of Fig. 6. These early oscillations drive a crossover time scales as t˜ ∼ ξϕ. 2 broad-resonance amplification during times when m2;χ As we increase the nonminimal couplings from ξ ∼ 2 I dominates m . At later times, however, the frequency O 1–10 to ξ ≥ O 100 , we find a dramatic change in the eff;χ ð Þ I ð Þ ’ behavior of the isocurvature modes in the long-wavelength of oscillation for zkðtÞ asymptotes toward the inflaton s χ oscillation frequency, which shifts the growth of z t from ξ 100 ρð Þ kð Þ limit. For ϕ ¼ , for example, k exhibits strong initial broad resonance toward the less-efficient narrow-resonance growth (as shown in Fig. 1), which ceases around t˜ ≃ 130. 2 regime. Although narrow-band resonances will drive As shown in Fig. 6, meff;χ is dominated at early times by exponential amplification in the rigid-spacetime limit, 2 2 2 ≃ 100 ξ m2;χ, with m2;χ=m1;χ ¼ ϕ. [This comes from com- generically they become considerably less effective in an paring the heights of the global maxima and the local expanding universe [49,50]. Figure 6(b) indicates that zkðtÞ
pffiffiffiffiffi 2 2 ξ 100 ˜ λ ξ FIG. 6. Left: effective mass for the isocurvature perturbations, meff;χ=H , for ϕ ¼ as a function of t ¼ ϕMplt= ϕ. The time interval of 0 < t<˜ 300 is folded into the interval 0 < t<˜ 100 as follows: 0 < t<˜ 100 (blue), 100 < t<˜ 200 (red), and 200 < t<˜ 300 (green). Right: isocurvature mode zkðtÞ for k ¼ 0, folded into the same time interval as the left panel. The green curve is shifted upwards for clarity. The black-dotted vertical lines correspond to the zero crossings of the background field ϕðtÞ in the interval 0 < t<˜ 100. The brown lines show the transition between exponential growth and slow red-shifting, which occurs around t˜ ∼ 130.
023528-8 PREHEATING AFTER …. III. DYNAMICAL … PHYS. REV. D 97, 023528 (2018)
pffiffiffiffiffi 1 Δ˜ ˜ λ ξ ϕ FIG. 7. Change in the effective local frequencies, = ti, as a function of t ¼ ϕMplt= ϕ, for the background inflaton field ðtÞ 2 3 (blue) and for the isocurvature mode zkðtÞ (with k ¼ 0, red), for different values of ξϕ: ξϕ ¼ 10 (left) and ξϕ ¼ 10 (right).
– 2 transitions from 2 3 oscillations per inflaton oscillation Another way to identify the importance of m2;χ in driving toward one oscillation per inflaton oscillation around the early amplification of isocurvature modes is to calculate ˜ ∼ 130 χ t , at the same time that the exponential growth of ρð Þ 2 2 k by neglecting either m1;χ or m2;χ. The results are shown zkðtÞ ceases. ξ 100 We confirm the behavior, and the scaling with t˜,in in Fig. 8 for ϕ ¼ . If there were no contributions from — 2 0 Fig. 7. For these plots, we calculate all zero crossings of the potential that is, if we set m1;χ ¼ , which is akin to ˜ 0 ϕðtÞ and of zkðtÞ (for k ¼ 0). For each field, we denote Δti setting g ¼ so that there were no direct coupling between as the difference in time between the ith and (i þ 1)th zero ϕ and χ—one would still find the same initial burst of crossings. The inverse of this quantity, 1=Δt˜ , may be ρðχÞ i growth in k , stemming entirely from the curved field- considered a local measure of frequency, which acquires space manifold (which itself arises from the nonminimal ϕ different values for ðtÞ and zkðtÞ. Figure 7 shows the couplings upon transforming to the Einstein frame). On the evolution of these local frequencies for the background 2 other hand, if one neglected the effects of m2;χ and only field and the isocurvature mode. As expected, the fre- considered the role of m2 , then one would find little initial quency of ϕðtÞ does not show significant variation, whereas 1;χ growth. The nonminimal couplings thus open a distinct the local frequency of zkðtÞ displays significant dispersion, 2 “channel” for ϕ and χ to interact, via their shared coupling due to the strongly anharmonic oscillation of meff;χ. 2 to the spacetime Ricci scalar, R, in the Jordan frame Because of the structure of m χ, the local frequency 2 eff; [51,52]. In the limit of large couplings, ξI ≥ Oð10 Þ, this for zkðtÞ is comprised of both short- and long-duration channel can drive broad resonances. components. Whereas at early times, the effective local We conclude this section by considering the case of ϕ frequency for zkðtÞ is considerably greater than for ðtÞ, the ξϕ ≤ 1, which departs strongly from both the intermediate- 2 two become comparable around t˜ ∼ 130 for ξϕ ¼ 10 , and and large-ξI regimes. From Fig. 9 we note that the 3 2 around t˜ ∼ 1300 for ξϕ ¼ 10 , in keeping with our expect- contribution to meff;χ arising from the curved field- ations from the scaling arguments given above. 2 space manifold, m2;χ, becomes subdominant after the first
ρðχÞ FIG.p 8.ffiffiffiffiffi The energy density of isocurvature modes k versus ~ t ¼ λϕMplt=ξϕ for k ¼ 0 and ξϕ ¼ 100, calculated based on FIG. 9. The rescaled effective mass of the isocurvature pertur- 2 2 2 2 bations m~ ξϕm (black-dotted line) as a function of m2;χ alone (orange, upper curve), on m1;χ alone (green, lower pffiffiffiffiffi eff;χ ¼ eff;χ 2 ˜ λ ξ ξ 1 ~ 2 curve), and on meff;χ (yellow, middle curve), for the same ratios of t ¼ ϕMplt= ϕ for ϕ ¼ . Also shown are m1;χ (blue) and ~ 2 couplings as in Fig. 1. m2;χ (red).
023528-9 MATTHEW P. DECROSS et al. PHYS. REV. D 97, 023528 (2018) pffiffiffiffiffi oscillation of the background field. In keeping with our which scales (as we found in Sec. III)ast˜ ∼ ξϕ. By the ξ analysis of the small- I regime in the rigid-spacetime time the wavelengths of subhorizon modes redshift to approximation [2], we find that the effects of the curved superhorizon scales, the tachyonic amplification typically ξ ≤ O 1 field-space manifold remain modest for I ð Þ, making has ceased. little change from the minimally coupled case. The behavior of the isocurvature modes is quite different. 2 In Ref. [2] we found that for ξI ≥ Oð10 Þ, the Floquet V. RESULTS: NONZERO WAVE NUMBER charts for the isocurvature modes show dense bands of instability regions, with little k dependence across a wide In Secs. III and IV we analyzed the behavior of super- range of k. Moreover, as discussed in Sec. IV, the growth of horizon modes with k=aH ≪ 1, focusing on the limit long-wavelength isocurvature perturbations occurs in the k → 0. In this section we briefly examine the behavior broad-resonance regime at early times, when the modes of adiabatic and isocurvature modes with nonzero wave z t oscillate more rapidly than the inflaton ϕ t . The number, for modes with wave number comparable to the kð Þ ð Þ effective frequency for modes z t increases with larger k, Hubble radius (k=aH ∼ 1) as well as for modes whose kð Þ and the broad-resonance instabilities identified in Ref. [2] wave number begins well within the Hubble radius remain robust in an expanding spacetime. As shown in (k=aH ≫ 1) at the start of preheating. We restrict attention ρðχÞ to the large-ξI regime, in which both adiabatic and Fig. 10, the spectrum of k therefore remains nearly flat isocurvature modes with k ¼ 0 are amplified efficiently. for superhorizon and subhorizon modes. ρðϕÞ ρðχÞ Figure 10 shows the behavior of k and k for values 0 ≤ ≤ 20 of k within the range k=½aðtendÞHðtendÞ . For the VI. OBSERVATIONAL CONSEQUENCES adiabatic perturbations, we find a sharp distinction between In general for multifield models, the growth of subhorizon and superhorizon modes: only modes with long-wavelength isocurvature perturbations can affect the k=aH ≤ 1 at the start of preheating become amplified. evolution of superhorizon adiabatic modes [10–14,41–43, This behavior is in accord with our findings in Ref. [2]: for 53–56]. Such transfers of power from isocurvature to ξ ≥ Oð102Þ, the dominant resonance band for adiabatic I adiabatic perturbations can affect CMB observables, such modes occurs for k → 0, while resonance bands for larger k as the spectral index n or primordial non-Gaussianity f . shrink to the narrow-resonance regime (and hence do not s NL In some cases, this process can produce significant cor- remain very effective in an expanding universe). Moreover, rections from the preheating phase to values of observables as we found in Sec. III, in a dynamical spacetime the calculated during inflation. (See the discussion in Sec. 7 of dominant amplification for adiabatic modes at early times Ref. [7], and references therein.) stems from effects of the coupled metric perturbations, Although we have found very efficient growth of which drive tachyonic growth with m2 < 0. Such eff;ϕ isocurvature modes on superhorizon scales during preheat- tachyonic growth will only be effective for modes with ξ ≥ O 102 2 2 ing for I ð Þ, such growth need not pose a direct ðk=aÞ pffiffiffiffiffi ˜ λ ξ ξ 100 0 FIG. 10. Left: energy density for adiabatic modes versus t ¼ ϕMplt= ϕ with ϕ ¼ and k=½aðtendÞHðtendÞ ¼ , 1, 5, 10, 20 (top to bottom: black-dashed line, red, brown, green, and blue, respectively). Right: energy density for isocurvature modes with ξϕ ¼ 100 0 ξ ξ 0 8 and k=½aðtendÞHðtendÞ ¼ , 2, 10, 20 (top to bottom: blue, red, green, and black, respectively). For both plots we use χ = ϕ ¼ . , λχ=λϕ ¼ 1.25, and g=λϕ ¼ 1. 023528-10 PREHEATING AFTER …. III. DYNAMICAL … PHYS. REV. D 97, 023528 (2018) k2 important result follows from the fact that the longitudinal- R_ ¼ 2ωS þ O ; c a2H2 polarization vectors for the gauge bosons scale with 2 momentum whereas the transverse-polarization states do S˙ β S O k not. Thus the amplitudes for high-energy scattering proc- ¼ H þ 2 2 ; ð28Þ a H esses are dominated by the longitudinal polarizations of the vector bosons. And the longitudinal polarizations, in turn, ω ’ where ðtÞ is the turn rate of the fields trajectory in field are related to the Goldstone bosons via spontaneous β 2 space, and ðtÞ encodes the effects of meff;χ. For motion symmetry breaking. Thus one may calculate such quantities within a single-field attractor the turn rate vanishes iden- as the amplitude for WμWμ scattering or Wμ-Higgs scatter- tically, ω ¼ 0, and hence the gauge-invariant curvature ing in terms of the simpler interactions among the perturbation Rc remains conserved in the long-wavelength Goldstone and Higgs scalars [71]. limit. We found in Ref. [1] that the single-field attractor The Goldstone boson equivalence theorem stipulates that behavior persists in these models throughout preheating as amplitudes for the high-energy scattering of vector gauge well as during inflation. Thus, at least at the level of a bosons are identical to corresponding amplitudes involving linearized analysis, we do not anticipate significant effects the Goldstone and Higgs scalars, up to corrections of order pffiffiffi pffiffiffi ρðχÞ O m = s , where m is the mass of the W boson and s on primordial observables from the amplification of k ð W Þ W after the end of inflation. is the center-of-mass energy for a given process [71]. Nonlinear effects in such models could certainly During and after inflation, the Higgs field has a (time- become important after the end of inflation, such as the dependent) vacuum expectation value, hðtÞ ∼ hϕðtÞi, since formation of oscillons, which could affect the evolution of the background field ϕ is displaced from the minimum of the effective equation of state and thereby the expansion its potential. Therefore the electroweak symmetry is broken ðχÞ during and after inflation, and should remain broken until history [45,57–60]. Moreover, the rapid growth of ρ in k the Higgs condensate dissolves at the end of reheating. (See these models suggests that the majority of the inflaton’s also Refs. [72–75].) In the broken-symmetric phase, energy could be transferred to isocurvature modes within a m ¼ gh=˜ 2, where g˜ ∼ Oð1Þ is the electroweak gauge single background oscillation [61]. Recent lattice simula- W coupling. During reheating, we then have m ∼ h, and the tions of related models that yield near-instantaneous pre- pffiffiffiW equivalence theorem holds for energies s >h. heating have identified qualitative differences from linear At the start of preheating, the value of the vacuum analyses, such as significant rescattering between the infla- expectation value is given by the dynamics of the back- ton and produced particles [62–64]. Such inherently non- ground field ϕ t : linear effects are beyond the scope of the present analysis. ð Þ M h ∼ hϕi ≲ pffiffiffiffiffipl : ð29Þ VII. POTENTIAL IMPLICATIONS ξϕ FOR HIGGS INFLATION During reheating, the amplitude of the Higgs condensate The strong resonances that we have identified in multi- decays due to the expansion of the Universe (even in the ξ ≫ 1 field models with I have potential implications for linearized approximation), and also due to the transfer of reheating after Higgs inflation [22]. Most studies of Higgs energy to fluctuations (which requires a nonlinear analysis). inflation have adopted the unitary gauge, in which the For the scalar sector, meanwhile, the strong-coupling scale Higgs sector reduces to a single scalar degree of freedom. Λ – in the Einstein frame, E, as calculated in Refs. [76p78]ffiffiffiffiffi, However, it is well known that the unitary gauge suffers satisfies ðh=ΛEÞ < 1 for all values 0 023528-11 MATTHEW P. DECROSS et al. PHYS. REV. D 97, 023528 (2018) For best-fit values of the masses of the Higgs boson and the transition to a radiation-dominated equation of state after top quark, the running of the Higgs self-coupling λϕ yields the end of inflation. λ μ ≃ 5 0 10−3 μ 6 ϕð Þ . × at the energy scale ¼ Hend ¼ × The successes of big-bang nucleosynthesis require more 12 10 GeV [79]. Equation (30) then suggests that ξϕ ≃ than just a radiation-dominated equation of state; the 3 Universe must also become filled with a plasma of 3.3 × 10 at the start of preheating. (If the top-quark mass is Standard Model particles some time after inflation ends greater than the current best-fit value, then one expects −3 3 [7]. The authors of Ref. [63] studied the generation of a λϕðμÞ < 5.0 × 10 and hence ξϕ < 3.3 × 10 at μ ¼ charged plasma from instantaneous preheating. In that 6 1012 × GeV [80,81].) Given the efficient amplification scenario, the Universe becomes filled with hypercharge ξ ∼ 103 of isocurvature modes in such models with ϕ ,we gauge bosons after inflation, which scatter into Standard may expect preheating after Higgs inflation to conclude Model particles. In the case of Higgs inflation, we may within a few oscillations of the inflaton. expect an inverse process to unfold. The nonzero value of h ¼hϕi induces a mass for the The energy density of the Universe at the end of inflation Goldstone bosons χ. We may estimate the mass scale due to ρ ∼ 2 2 is MplHend, while the momentum of each Higgs mode electroweak symmetry breaking at the start of preheating as can be taken to be comparable to the Hubble scale at the end of inflation, E ∼ k=a ∼ H . Thus the particle λ 2 end end ϕMpl ρ ∼ 2 2 ∼ λ 2 ∼ density scales as n ¼ =E M Hend, while the velocity mvev ϕh : ð32Þ pl ξϕ for radiation modes is simply v ∼ c. The speed of the transition from a Higgs boson bath into a plasma depends Given Eq. (31), we then find on the ratio of the scattering rate to the Hubble expansion rate, 2 mvev ∼ 100ξ 2 ϕ: ð33Þ Γ nσv Hend ¼ : ð36Þ H H Meanwhile, in Eq. (26) we calculated the ratio of the The scattering rate of Higgs particles into gauge bosons or Hubble scale to the spike in m χ due to the nontrivial eff; fermions proceeds through vertices like the ones shown in field-space manifold, m2 χ, and found ; Fig. 11, and scales as 2 2 2 m2;χ 2 α α ∼ ξϕ; ð34Þ σ ∼ Y ∼ Y H2 2 ð37Þ s Hend ξ ∼ 103 ≫ so for ϕ , we expect m2;χ mvev at the start of for the case of scattering into hypercharge Uð1Þ bosons. preheating, since Scattering into fermions has a similar cross section, and thus we may estimate for Eq. (36), m2 ξ 2;χ ∼ ϕ 2 100 : ð35Þ Γ M 2 mvev ∼ 2 pl ∼ 109 2 ≫ 1 aY aY : ð38Þ H Hend Even in the presence of electroweak symmetry breaking during the preheating epoch, the features that are most We therefore expect that scattering of the Higgs bosons into responsible for rapid particle production—namely, the the rest of the Standard Model particles should occur 2 spikes in m2;χ which drive violations of the adiabatic almost instantaneously. Thermalization would proceed condition—dominate the dynamics of the scalar sector. through similar interactions, and should also be efficient. Significant backreaction from created particles will halt After the electroweak phase transition (which presumably the background field’s oscillations, with ϕ settling near ϕ ∼ 0, leaving a universe filled with scalar Higgs bosons. A (See also Refs. [61,72–75].) We found in Ref. [1] that the f effective equation of state after inflation tends toward ∼ 1 3 wavg = within several efolds after the end of inflation for ξI ≥ 100, based on analysis of the background field’s dynamics. Since the Higgs potential is dominated by the A f quartic term after the end of inflation, the Higgs bosons produced during preheating should behave as nearly FIG. 11. Scattering vertices that allow the transfer of energy massless radiation modes (especially once ϕ ∼ 0), quicken- from the Higgs bosons to the rest of the Standard Model, → 1 3 ing the rate at which wavg = . Thus we expect a rapid specifically into gauge fields (left) and fermions (right). 023528-12 PREHEATING AFTER …. III. DYNAMICAL … PHYS. REV. D 97, 023528 (2018) would occur at much lower energies than either inflation or Addressing the late-stage thermalization following the reheating), the Higgs field would acquire a nonzero vacuum initial phase of preheating remains beyond the scope of this expectation value of v ≈ 246 GeV. Then any remaining paper, and deserves further study. One interesting question massive Higgs particles would decay into Standard relevant to such further study concerns the behavior of the Λ ∼ ξ Model particles, following the experimentally known decay unitarity cutoff scale itself. The estimate E Mpl= ϕ rates [82,83]. comes from analyzing perturbative processes involving a In sum, for the case of reheating after Higgs inflation, few particles in both the incoming and outgoing states, such 3 with ξϕ ≳ 10 , we may expect that the efficient production as 2 → 2 scattering [76–78]. But preheating involves the of isocurvature modes—which, in this case, correspond to rapid production of many particles in the final state, with ∼ 2μ ≫ 1 Goldstone modes within the Standard Model Higgs doublet number densities nk exp½ kt . Even for perturba- and therefore behave as the longitudinal modes of the W tive analyses, if the final state involves n ≫ 1 bosons, then — ΛðnÞ and Z bosons when electroweak symmetry is broken phase-space factors generically raise pert above what one would lead to the rapid break-up of the Higgs condensate would calculate for 2 → 2 scattering, with ΛðnÞ typically into radiation modes, followed by a fast transition to a pert growing linearly with n in the limit n ≫ 1 [84]. Moreover, primordial plasma of Standard Model particles. In such a the production of particles during preheating involves scenario, the reheatpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi temperature after inflation could be as ∼ ∼ 10−3 inherently nonperturbative processes. The cross section high as Treh HendMpl Mpl. for nonperturbative processes that yield n-particle final Such a high reheat temperature may pose a challenge for states, σ1→n, has a very different scaling with energy than a more thorough investigation of reheating after Higgs the corresponding perturbative cross sections for few → few inflation, beyond the linearized analysis of the preheating scattering [85]. If these considerations were to yield a cutoff phase we have pursued here. Typical momenta for quanta in scale considerably different than ΛE, then ratios like a blackbody distribution scale as k ∼ 3T. So if the Universe Eq. (39) would need to be revisited. Such questions remain achieves thermal equilibrium at a temperature as high as the topic of further research. ∼ 10−3 Treh Mpl after inflation, then typical scatterings among particles would involve momentum exchanges of VIII. CONCLUSIONS order k ∼ 3T . Meanwhile, in the scenario we have reh Building on Refs. [1,2], we have studied the preheating described here, the inflaton condensate would likely dis- pffiffiffiffiffi dynamics of adiabatic and isocurvature perturbations in sipate quickly, falling from h ∼ M = ξϕ to h 023528-13 MATTHEW P. DECROSS et al. PHYS. REV. D 97, 023528 (2018) to the Einstein frame). In contrast to the adiabatic modes, APPENDIX: COMPARING RELEVANT the amplification of isocurvature modes is efficient for TIME SCALES subhorizon and superhorizon scales alike, and their In Sec. III, we found that the effective mass for the growth rate for ξI ≥ Oð100Þ exceeds that of the super- 2 adiabatic modes, meff;ϕ, is dominated by two contributions: horizon adiabatic modes. The duration of efficient, 2 broad-resonance amplification for the isocurvature modes m1;ϕ, arising from gradients of the Einstein-frame potential, ˜ ∼ ξ 2 scales as t ϕ. and m3;ϕ, arising from the coupled metric perturbations. In 2 The amplification of both adiabatic and isocurvature particular, the effects of m3;ϕ dominate the dynamics at modes becomes more efficient as the nonminimal cou- 2 early times, while m1 ϕ dominates at later times. In terms of plings become large, ξ ≥ O 100 , underscoring the role of ; I ð Þ the rescaled time coordinate, the nonminimal couplings in driving the dynamics. The adiabatic modes grow primarily due to their coupling sffiffiffiffiffi ξ λϕ to metric perturbations; that coupling increases as I ~ ≡ t 2 Mplt; ðA1Þ increases, which is most clear in the Jordan frame, given ξϕ the coupling fðϕIÞR in the action. Likewise, the isocurva- ture modes grow primarily due to the nontrivial field-space we may label the cross-over time between these two ϕ ϕ manifold, which is likewise a manifestation of the under- regimes as t˜ð Þ . In Sec. III we found that t˜ð Þ scales as lying nonminimal couplings, and whose curvature cross cross increases with ξ . The curvature of the field space manifold, qffiffiffiffiffi I ðϕÞ ˜ ∼ ξϕ RILMJ, allows for efficient transfer of energy from the tcross ðA2Þ inflaton condensate to isocurvature perturbations, even in the absence of a direct coupling (within the potential) in the limit ξϕ ≫ 1, for evolution within a single-field between the various fields. attractor along the direction χ ∼ 0. Within the scope of our linearized analysis, we do not In Sec. IV, meanwhile, we found that the effective mass expect the rapid amplification of superhorizon isocurvature 2 for the isocurvature modes, meff;χ, is dominated by a modes to affect inflationary predictions for primordial 2 different pair of contributions: m1;χ, arising from gradients observables (such as the spectral index or non- 2 Gaussianity), because the single-field attractor prohibits of the Einstein-frame potential, and m2;χ, arising from the isocurvature modes from sourcing a change in the gauge- nontrivial field-space manifold. In the case of the isocur- 2 invariant curvature perturbation on long wavelengths. vature perturbations, the effects of m2;χ dominate at early Whether such behavior persists beyond linear order in times. In terms of t˜, we found the cross-over time between the perturbations remains the subject of further study. In the the two regimes for the isocurvature modes to scale as meantime, the rapid amplification of isocurvature modes suggests that reheating in models like Higgs inflation, with ˜ðχÞ ∼ ξ 3 tcross ϕ ðA3Þ ξϕ ∼ 10 , could be especially efficient. in the limit ξϕ ≫ 1, again for evolution within a single-field ACKNOWLEDGMENTS attractor along the direction χ ∼ 0. ˜ðϕÞ ˜ðχÞ It is a pleasure to thank Mustafa Amin, Bruce Bassett, We may compare the scaling of tcross and tcross with that Jolyon Bloomfield, Peter Fisher, Tom Giblin, Alan Guth, of a third time scale, namely, the cross-over time in the Mark Hertzberg, and Johanna Karouby for helpful dis- equation of state for the background dynamics, wavg, cussions. We also thank the anonymous referee for helpful between a matter-dominated and radiation-dominated ∼ 0 comments and suggestions. We would like to acknowledge phase. We found in Ref. [1] that wavg at the start of → 1 3 support from the Center for Theoretical Physics at MIT. preheating in this family of models, and that wavg = This work is supported by the U.S. Department of Energy within several efolds after the end of inflation, though the ∼ 1 3 ξ under grant Contract No. DE-SC0012567. M. P. D. and number of efolds until wavg = grew with ϕ. (See Fig. 9 A. P. were also supported in part by MIT’s Undergraduate ˜ðwÞ in Ref. [1].) Here we estimate tcross, the cross-over time Research Opportunities Program (UROP). C. P. W thanks between matter-dominated and radiation-dominated evolu- the University of Washington College of Arts & Sciences tion of the background dynamics. for financial support. She also gratefully acknowledges Within a single-field attractor along χ ∼ 0, the Einstein- support from the MIT Dr. Martin Luther King, Jr. Visiting frame potential simplifies to Professors and Scholars program and its director Edmund 4 4 4 4 Bertschinger. E. I. S gratefully acknowledges support λϕM ϕ λϕM δ from a Fortner Fellowship at the University of Illinois at V ϕ; χ ≃ pl pl ; ð Þ 4 2 ξ ϕ2 2 ¼ 4ξ2 1 δ2 2 ðA4Þ Urbana-Champaign. ½Mpl þ ϕ ϕ ½ þ 023528-14 PREHEATING AFTER …. III. DYNAMICAL … PHYS. REV. D 97, 023528 (2018) pffiffiffiffiffi δ ≡ ξ ϕ 2 3ξ2 ϕ2 2 2 where, as usual, we define ϕ =Mpl. Inflation Mpl ϕ 1 1 þ δ þ 6ξϕδ Gϕϕ ¼ 1 þ ¼ : ends at δ ≲ 1, and the amplitude of ϕ continues to fall 2f f 1 þ δ2 1 þ δ2 as preheating proceeds. For δ < 1, we may therefore ðA13Þ approximate δ ∼ O 1 λ 4 At the start of preheating, we have ð Þ, and hence ϕMpl 4 G ∼ ξ δ2 ≫ 1 ξ ≫ 1 G V ≃ δ : ϕϕ ϕ for ϕ . However, ϕϕ falls during 4ξ2 ðA5Þ ϕ preheating as the amplitude of ϕðtÞ falls, eventually reach- 2 ing Gϕϕ → 1 for 6ξϕδ ≪ 1. We may then approximate the At early times during preheating, the time-averaged equa- ðwÞ cross-over time for the background equation of state, t˜ , tion of state obeys w ≃ 0, and hence the energy density cross avg 6ξ δ2 ðwÞ ≃ 1 scales as as the time when ϕ maxðtcrossÞ ,or 2 3N=2 6ξϕα ≃ e ; A14 −3N ð Þ ρ ≃ ρ0e ; ðA6Þ upon using Eq. (A8). Comparing with Eq. (A11), we then where N is the number of efolds since the end of inflation. find We may estimate 24 ~ ðwÞ tcross ≃ pffiffiffi ξϕ: ðA15Þ 4 3 λϕM ρ ≃ V ¼ pl δ4 ; ðA7Þ max 4ξ2 max ˜ðwÞ ϕ Hence we find the scaling tcross ∼ ξϕ, akin to the large-ξϕ pffiffiffiffiffi scaling we had found for the cross-over among dominant δ ξ ϕ where max ¼ ϕ max=Mpl is the rescaledpffiffiffiffiffi amplitude contributions to the effective mass of the isocurvature ’ α ≡ ξ ϕ ˜ðχÞ of the field s oscillations. If we label ϕ 0=Mpl, perturbations, tcross. ϕ ϕ ˜ðwÞ ˜ðχÞ where 0 is the initial amplitude of ðtÞ at the start Though the similarity in scaling between tcross and tcross of preheating, then from Eqs. (A6) and (A7) we may is interesting, we note that wavg is calculated to back- estimate ground-order only, and does not take into account the contributions to the total energy density from particles δ ≃ α −3N=4 maxðtÞ e : ðA8Þ produced during preheating. Hence wavg, as we have calculated it numerically in Ref. [1] and estimated its Next we may estimate cross-over time here, is independent of the production of (say) χ particles. Rather, we may understand the similar 2 χ 1 λϕM ˜ðwÞ ˜ð Þ ξ 2 ≃ pl δ4 scaling of tcross and tcross with ϕ by looking more closely at H 3 2 Vmax ¼ 12ξ2 maxðtÞ; ðA9Þ Mpl ϕ several relevant quantities. ˜ðwÞ The behavior of wavg, and hence of tcross, depends on the with which we may write ϕI σ_ 2 2 ratio Vð Þ= , while the behavior of meff;χ, and hence of sffiffiffiffiffi ˜ðχÞ 2 2 t , depends on the ratio m1 χ=m2 χ. Within a single-field M λ cross ; ; ≃ plffiffiffi ϕδ2 χ ∼ 0 dN ¼ Hdt p 2 maxðtÞdt; ðA10Þ attractor along , we have, from Eqs. (22) and (A12), 2 3 ξϕ 1 2 Rσ_ 2 m2;χ ¼ 2 ; ðA16Þ or, upon using Eqs. (A1) and (A8), ffiffiffi where R is the Ricci scalar for the field-space manifold. In p 2 2 3 the regime ξ ≫ 1, δ ≪ 1, and ξϕδ ∼ Oð1Þ, along χ ∼ 0, e3N=2 ≃ α2t:~ ðA11Þ I 4 we have 12ξ ξ As we saw in Sec. III C of Ref. [1], the evolution of the R ≃ ϕ χ 2 1 6ξ δ2 2 ; ðA17Þ background-order equation of state depends on the shifting Mpl½ þ ϕ balance between the kinetic energy, σ_ 2, and the potential, VðϕIÞ. Within the single-field attractor, we have upon using the expressions in Appendix A of Ref. [1].In that same regime, we find σ_ 2 ≡ G φ_ Iφ_ J → G ϕ˙2 IJ ϕϕ ; ðA12Þ 2 4 6jΛϕjM δ 2 χK pl m G DχD V ≃ ; 1;χ ¼ ð K Þ ξ 1 6ξ δ2 ðA18Þ where ϕ ð þ ϕ Þ 023528-15 MATTHEW P. DECROSS et al. PHYS. REV. D 97, 023528 (2018) where Λϕ ≡ λϕξχ − gξϕ. Then in the regime of interest, we For the case of weakly broken symmetries, with λϕ ∼ g find and ξϕ ∼ ξχ, of the sort we examined throughout Secs. III and IV, we can expect jΛϕj=ξχ ∼ λϕ. Then we find that the 2 Λ 4 4 m1;χ j ϕjMpl δ 2 ratios in Eqs. (A19) and (A20) scale quite similarly with ∼ 1 6ξ δ ; 2 ξ ξ2 σ_ 2 ð þ ϕ Þ ðA19Þ m2;χ χ ϕ parameters, especially around the critical transition period 2 ˜ðχÞ when 6ξϕδ ∼ 1. The cross-over time tcross depends on the while for this same regime, we find ˜ðwÞ ratio in Eq. (A19), while the cross-over time tcross depends 4 4 on the ratio in Eq. (A20), and hence we may understand V λϕM δ ∼ pl : why these two distinct cross-over times scale in a compa- σ_ 2 ξ2 σ_ 2 ðA20Þ ϕ rable way with ξϕ. [1] M. P. DeCross, D. I. Kaiser, A. Prabhu, C. Prescod- 13, 901 (1980); T. S. Bunch and P. Panangaden, On Weinstein, and E. I. Sfakianakis, Preheating after multifield renormalization of λϕ4 field theory in curved space-time, inflation with nonminimal couplings. I. Covariant formal- II, J. Phys. A 13, 919 (1980). ism and attractor behavior, Phys. Rev. D 97, 023526 (2018). [17] N. D. Birrell and P. C. W. Davies, Quantum Fields in [2] M. P. DeCross, D. I. Kaiser, A. Prabhu, C. Prescod- Curved Space (Cambridge University Press, New York, Weinstein, and E. I. Sfakianakis, preceding paper, Preheat- 1982). ing after multifield inflation with nonminimal couplings. II. [18] I. L. Buchbinder, S. D. Odintsov, and I. L. Shapiro, Effective Resonance structure, Phys. Rev. D 97, 023527 (2018). Action in Quantum Gravity (Taylor and Francis, New York, [3] A. H. Guth and D. I. Kaiser, Inflationary cosmology: 1992). Exploring the Universe from the smallest to the largest [19] L. E. Parker and D. J. Toms, Quantum Field Theory in scales, Science 307, 884 (2005). Curved Spacetime (Cambridge University Press, New York, [4] B. A. Bassett, S. Tsujikawa, and D. Wands, Inflation 2009). dynamics and reheating, Rev. Mod. Phys. 78, 537 (2006). [20] S. D. Odintsov, Renormalization group, effective action, and [5] R. Allahverdi, R. Brandenberger, F.-Y. Cyr-Racine, and A. Grand Unification Theories in curved spacetime, Fortschr. Mazumdar, Reheating in inflationary cosmology: Theory Phys. 39, 621 (1991). and applications, Annu. Rev. Nucl. Part. Sci. 60, 27 (2010). [21] T. Markkanen and A. Tranberg, A simple method for one- [6] A. V. Frolov, Non-linear dynamics and primordial curvature loop renormalization in curved spacetime, J. Cosmol. perturbations from preheating, Classical Quantum Gravity Astropart. Phys. 08 (2013) 045. 27, 124006 (2010). [22] F. L. Bezrukov and M. E. Shaposhnikov, The Standard [7] M. A. Amin, M. P. Hertzberg, D. I. Kaiser, and J. Karouby, Model Higgs boson as the inflaton, Phys. Lett. B 659, Nonperturbative dynamics of reheating after inflation: 703 (2008). A review, Int. J. Mod. Phys. D 24, 1530003 (2015). [23] T. Futamase and K. Maeda, Chaotic inflationary scenario of [8] M. A. Amin and D. Baumann, From wires to cosmology, the Universe with a nonminimally coupled “inflaton” field, J. Cosmol. Astropart. Phys. 02 (2016) 045. Phys. Rev. D 39, 399 (1989). [9] D. H. Lyth and A. Riotto, Particle physics models of [24] D. S. Salopek, J. R. Bond, and J. M. Bardeen, Designing inflation and the cosmological density perturbation, Phys. density fluctuation spectra in inflation, Phys. Rev. D 40, Rep. 314, 1 (1999). 1753 (1989). [10] D. Wands, Multiple field inflation, Lect. Notes Phys. 738, [25] R. Fakir, S. Habib, and W. G. Unruh, Cosmological density 275 (2008). perturbations with modified gravity, Astrophys. J. 394, 396 [11] A. Mazumdar and J. Rocher, Particle physics models of (1992); R. Fakir and W. G. Unruh, Improvement on cos- inflation and curvaton scenarios, Phys. Rep. 497, 85 (2011). mological chaotic inflation through nonminimal coupling, [12] J. Martin, C. Ringeval, and V. Vennin, Encyclopedia Phys. Rev. D 41, 1783 (1990). inflationaris, Phys. Dark Universe 5–6, 75 (2014). [26] N. Makino and M. Sasaki, The density perturbation in the [13] V. Vennin, K. Koyama, and D. Wands, Encylopedia chaotic inflation with nonminimal coupling, Prog. Theor. Curvatonis, J. Cosmol. Astropart. Phys. 11 (2015) 008. Phys. 86, 103 (1991). [14] J.-O. Gong, Multi-field inflation and cosmological pertur- [27] D. I. Kaiser, Primordial spectral indices from generalized bations, Int. J. Mod. Phys. D 26, 1740003 (2017). Einstein theories, Phys. Rev. D 52, 4295 (1995). [15] C. G. Callan, Jr., S. R. Coleman, and R. Jackiw, A new [28] D. I. Kaiser, Constraints in the context of induced gravity improved energy-momentum tensor, Ann. Phys. (N.Y.) 59, inflation, Phys. Rev. D 49, 6347 (1994). 42 (1970). [29] D. I. Kaiser, E. A. Mazenc, and E. I. Sfakianakis, Primordial [16] T. S. Bunch, P. Panangaden, and L. Parker, On renormal- bispectrum from multifield inflation with nonminimal cou- ization of λϕ4 field theory in curved space-time, I, J. Phys. A plings, Phys. Rev. D 87, 064004 (2013). 023528-16 PREHEATING AFTER …. III. DYNAMICAL … PHYS. REV. D 97, 023528 (2018) [30] R. N. Greenwood, D. I. Kaiser, and E. I. Sfakianakis, [46] D. G. Figueroa and F. Torrenti, Parametric resonance in the Multifield dynamics of Higgs inflation, Phys. Rev. D 87, early universe: A fitting analysis, J. Cosmol. Astropart. 064021 (2013). Phys. 02 (2017) 001. [31] D. I. Kaiser and E. I. Sfakianakis, Multifield Inflation after [47] D. I. Kaiser, Conformal transformations with multiple scalar Planck: The Case for Nonminimal Couplings, Phys. Rev. fields, Phys. Rev. D 81, 084044 (2010). Lett. 112, 011302 (2014). [48] H. Abedi and A. M. Abbassi, Gravitational constant in [32] K. Schutz, E. I. Sfakianakis, and D. I. Kaiser, Multifield multiple field gravity, J. Cosmol. Astropart. Phys. 05 (2015) inflation after Planck: Isocurvature modes from nonminimal 026. couplings, Phys. Rev. D 89, 064044 (2014). [49] L. Kofman, A. Linde, and A. Starobinsky, Towards the [33] R. Kallosh and A. Linde, Non-minimal inflationary attrac- theory of reheating after inflation, Phys. Rev. D 56, 3258 tors, J. Cosmol. Astropart. Phys. 10 (2013) 033; Multi-field (1997). conformal cosmological attractors, J. Cosmol. Astropart. [50] D. I. Kaiser, Postinflation reheating in an expanding uni- Phys. 12 (2013) 006; R. Kallosh, A. Linde, and D. Roest, verse, Phys. Rev. D 53, 1776 (1996). Universal Attractor for Inflation at Strong Coupling, Phys. [51] B. A. Bassett and S. Liberati, Geometric reheating after Rev. Lett. 112, 011303 (2014); R. Kallosh, A. Linde, and D. inflation, Phys. Rev. D 58, 021302 (1998). Roest, Superconformal inflationary α-attractors, J. High [52] S. Tsujikawa, K. Maeda, and T. Torii, Resonant particle Energy Phys. 11 (2013) 198; M. Galante, R. Kallosh, A. production with nonminimally coupled scalar fields in Linde, and D. Roest, Unity of Cosmological Attractors, preheating after inflation, Phys. Rev. D 60, 063515 Phys. Rev. Lett. 114, 141302 (2015); R. Kallosh and A. (1999); Preheating with nonminimally coupled scalar fields Linde, Planck, LHC, and α-attractors, Phys. Rev. D 91, in higher-curvature inflation models, Phys. Rev. D 60, 083528 (2015); J. J. M. Carrasco, R. Kallosh, and A. Linde, 123505 (1999); Preheating of the nonminimally coupled Cosmological attractors and initial conditions for inflation, inflaton field, Phys. Rev. D 61, 103501 (2000). Phys. Rev. D 92, 063519 (2015). [53] C. Gordon, D. Wands, B. A. Bassett, and R. Maartens, [34] S. Renaux-Petel and K. Turzyński, On reaching the adia- Adiabatic and entropy perturbations from inflation, Phys. batic limit in multi-field inflation, J. Cosmol. Astropart. Rev. D 63, 023506 (2000). Phys. 06 (2015) 010; Geometrical Destabilization of [54] S. Groot Nibbelink and B. J. W. van Tent, Density pertur- Inflation, Phys. Rev. Lett. 117, 141301 (2016). bations arising from multiple field slow-roll inflation, arXiv: [35] A. Taruya and Y. Nambu, Cosmological perturbation with hep-ph/0011325; Scalar perturbations during multiple-field two scalar fields in reheating after inflation, Phys. Lett. B slow-roll inflation, Classical Quantum Gravity 19, 613 428, 37 (1998). (2002). [36] B. A. Bassett, D. I. Kaiser, and R. Maartens, General rela- [55] J. Elliston, D. J. Mulryne, D. Seery, and R. Tavakol, tivistic effects in preheating, Phys. Lett. B 455, 84 (1999). Evolution of fNL to the adiabatic limit, J. Cosmol. Astro- [37] B. A. Bassett, F. Tamburini, D. I. Kaiser, and R. Maartens, part. Phys. 11 (2011) 005. Metric preheating and limitations of linearized gravity, [56] G. Leung, E. R. M. Tarrant, C. T. Byrnes, and E. J. Nucl. Phys. B561, 188 (1999). Copeland, Reheating, multifield inflation and the fate of [38] B. A. Bassett, C. Gordon, R. Maartens, and D. I. Kaiser, the primordial observables, J. Cosmol. Astropart. Phys. 09 Restoring the sting to metric preheating, Phys. Rev. D 61, (2012) 008. 061302 (2000). [57] M. Gleiser, Pseudostable bubbles, Phys. Rev. D 49, 2978 [39] F. Finelli and R. Brandenberger, Parametric amplification of (1994); E. J. Copeland, M. Gleiser, and H.-R. Mueller, metric fluctuations during reheating in two field models, Oscillons: Resonant configurations during bubble collapse, Phys. Rev. D 62, 083502 (2000). Phys. Rev. D 52, 1920 (1995). [40] S. Tsujikawa and B. A. Bassett, A new twist to preheating, [58] M. A. Amin, R. Easther, and H. Finkel, Inflaton fragmen- Phys. Rev. D 62, 043510 (2000); When can preheating tation and oscillon formation in three dimensions, J. Cosmol. affect the CMB?, Phys. Lett. B 536, 9 (2002). Astropart. Phys. 12 (2010) 001; M. A. Amin, R. Easther, H. [41] A. Chambers and A. Rajantie, Lattice Calculation of Finkel, R. Flauger, and H. P. Hertzberg, Oscillons after Non-Gaussianity from Preheating, Phys. Rev. Lett. 100, Inflation, Phys. Rev. Lett. 108, 241302 (2012). 041302 (2008). [59] S.-Y. Zhou, E. J. Copeland, R. Easther, H. Finkel, Z.-G. [42] J. R. Bond, A. V. Frolov, Z. Huang, and L. Kofman, Mou, and P. M. Saffin, Gravitational waves from oscillon Non-Gaussian Spikes from Chaotic Billiards in Inflation preheating, J. High Energy Phys. 10 (2013) 026; S.-Y. Zhou, Preheating, Phys. Rev. Lett. 103, 071301 (2009). Gravitational waves from Affleck-Dine condensate frag- [43] L. Bethke, D. G. Figueroa, and A. Rajantie, Anisotropies in mentation, J. Cosmol. Astropart. Phys. 06 (2015) 033. the Gravitational Wave Background from Preheating, Phys. [60] K. D. Lozanov and M. A. Amin, End of inflation, oscillons, Rev. Lett. 111, 011301 (2013). and matter-antimatter asymmetry, Phys. Rev. D 90, 083528 [44] H. B. Moghaddam, R. H. Brandenberger, Y.-F-. Cai, and (2014). E. G. M. Ferreira, Parametric resonance of entropy [61] Y. Ema, R. Jinno, K. Mukaida, and K. Nakayama, Violent perturbations in massless preheating, Int. J. Mod. Phys. D preheating in inflation with nonminimal coupling, J. Cos- 24, 1550082 (2015). mol. Astropart. Phys. 02 (2017) 045. [45] K. D. Lozanov and M. A. Amin, Equation of State and [62] P. Adshead, J. T. Giblin, T. R. Scully, and E. I. Sfakianakis, Duration to Radiation Domination after Inflation, Phys. Rev. Gauge-preheating and the end of axion inflation, J. Cosmol. Lett. 119, 061301 (2017). Astropart. Phys. 12 (2015) 034. 023528-17 MATTHEW P. DECROSS et al. PHYS. REV. D 97, 023528 (2018) [63] P. Adshead, J. T. Giblin, T. R. Scully, and E. I. Sfakianakis, [74] J.-F. Dufaux, D. G. Figueroa, and J. García-Bellido, Magnetogenesis from axion inflation, J. Cosmol. Astropart. Gravitational waves from Abelian gauge fields and cosmic Phys. 10 (2016) 039. strings at preheating, Phys. Rev. D 82, 083518 (2010). [64] H. L. Child, J. T. Giblin, Jr, R. H. Ribeiro, and D. Seery, [75] J. García-Bellido, J. Rubio, and M. Shaposhnikov, Higgs- Preheating with Non-Minimal Kinetic Terms, Phys. Rev. dilaton cosmology: Are there extra relativistic species?, Lett. 111, 051301 (2013). Phys. Lett. B 718, 507 (2012). [65] S. Weinberg, The Quantum Theory of Fields, Modern [76] F. Bezrukov, A. Magnin, M. Shaposhnikov, and S. Applications (Cambridge University Press, New York, Sibiryakov, Higgs inflation: Consistency and generalisa- 1996), Vol. 2. tions, J. High Energy Phys. 01 (2011) 016. [66] A. O. Barvinsky, A. Yu. Kamenshchik, C. Kiefer, A. A. [77] F. Bezrukov, G. K. Karananas, J. Rubio, and M. Starobinsky, and C. F. Steinwachs, Asymptotic freedom in Shaposhnikov, Higgs-dilaton cosmology: An effective field inflationary cosmology with a nonminimally coupled Higgs theory approach, Phys. Rev. D 87, 096001 (2013). field, J. Cosmol. Astropart. Phys. 12 (2009) 003; Higgs [78] F. Bezrukov, The Higgs field as an inflaton, Classical boson, renormalization group, and naturalness in cosmol- Quantum Gravity 30, 214001 (2013). ogy, Eur. Phys. J. C 72, 2219 (2012). [79] F. Bezrukov and M. Shaposhnikov, Standard Model Higgs [67] C. P. Burgess, H. M. Lee, and M. Trott, Power-counting and boson mass from inflation: Two loop analysis, J. High the validity of the classical approximation during inflation, Energy Phys. 07 (2009) 089. ξ J. High Energy Phys. 09 (2009) 103; Comment on Higgs [80] K. Allison, Higgs -inflation for the 125-126 FeV Higgs: inflation and naturalness, J. High Energy Phys. 07 (2010) A two-loop analysis, J. High Energy Phys. 02 (2014) 040. 007. [81] F. Bezrukov and M. Shaposhnikov, Why should we care [68] M. P. Hertzberg, On inflation with nonminimal coupling, about the top quark Yukawa coupling?, J. Exp. Theor. Phys. J. High Energy Phys. 11 (2010) 023. 120, 335 (2015). [69] S. Mooij and M. Postma, Goldstone bosons with a [82] C. Patrignani, Review of particle physics, Chin. Phys. C 40, dynamical Higgs field, J. Cosmol. Astropart. Phys. 09 100001 (2016). (2011) 006. [83] It is also possible that the electroweak symmetry was never [70] D. I. Kaiser and A. T. Todhunter, Primordial perturbations thermally restored, in which case perturbative decays of the Higgs into Standard Model particles would be occurring from multifield inflation with nonminimal couplings, Phys. from the onset of reheating. Rev. D 81, 124037 (2010). [84] D. A. Dicus and H.-J. He, Scales of fermion mass generation [71] J. M. Cornwall, D. N. Levin, and G. Tiktopoulos, Derivation and electroweak symmetry breaking, Phys. Rev. D 71, of gauge invariance from high-energy unitarity bounds on 093009 (2005); A. Belyaev, A. C. A. Oliveira, R. Rosenfeld, the S matrix, Phys. Rev. D 10, 1145 (1974); B. W. Lee, C. and M. C. Thomas, Multi-Higgs and vector boson produc- Quigg, and H. B. Thacker, Weak interactions at very high tion beyond the Standard Model, J. High Energy Phys. 05 energies: The role of the Higgs-boson mass, Phys. Rev. D (2013) 005. 16, 1519 (1977); M. S. Chanowitz and M. K. Gaillard, The ’ ’ [85] M. B. Voloshin, Strong high-energy scattering in theories TeV physics of strongly interacting W s and Z s, Nucl. Phys. with weak coupling, Phys. Rev. D 43, 1726 (1991);D.T. B261, 379 (1985); K. Riesselmann, Higgs physics and the Son, Semiclassical approach for multiparticle production equivalence theorem, arXiv:hep-ph/9504321. in scalar theories, Nucl. Phys. B477, 378 (1996);F.L. [72] F. Bezrukov, D. Gorbunov, and M. Shaposhnikov, On initial Bezrukov, M. V. Libanov, and S. V. Troitsky, Oð4Þ sym- conditions for the hot big bang, J. Cosmol. Astropart. Phys. metric singular solutions and multiparticle cross sections in 06 (2009) 029. ϕ4 theory at three level, Mod. Phys. Lett. A 10, 2135 (1995); [73] J. García-Bellido, D. G. Figueroa, and J. Rubio, Preheating F. Bezrukov, Use of singular classical solutions for calcu- in the Standard Model with the Higgs-inflaton coupled to lation of multiparticle cross sections in field theory, Theor. gravity, Phys. Rev. D 79, 063531 (2009). Math. Phys. 115, 647 (1998). 023528-18