Nonlinear Dynamics of Preheating after Multifield Inflation with Nonminimal Couplings
Rachel Nguyen, Jorinde van de Vis, Evangelos I. Sfakianakis, John T. Giblin Jr., David I. Kaiser
NNonlinear Dynamics of Preheating after Multifield Inflationwith Nonminimal Couplings
Rachel Nguyen, ∗ Jorinde van de Vis, † Evangelos I. Sfakianakis,
2, 3, ‡ John T. Giblin, Jr.,
1, 4, § and David I. Kaiser ¶ Department of Physics, Kenyon College, Gambier, Ohio 43022, USA Nikhef, Science Park 105, 1098XG Amsterdam, The Netherlands Lorentz Institute for Theoretical Physics, Leiden University, 2333CA Leiden, The Netherlands CERCA/ISO, Department of Physics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106 Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
We study the post-inflation dynamics of multifield models involving nonminimal couplings us-ing lattice simulations to capture significant nonlinear effects like backreaction and rescattering.We measure the effective equation of state and typical time-scales for the onset of thermalization,which could affect the usual mapping between predictions for primordial perturbation spectra andmeasurements of anisotropies in the cosmic microwave background radiation. For large values ofthe nonminimal coupling constants, we find efficient particle production that gives rise to nearlyinstantaneous preheating. Moreover, the strong single-field attractor behavior that was previouslyidentified persists until the end of preheating, thereby suppressing typical signatures of multifieldmodels. We therefore find that predictions for primordial observables in this class of models retaina close match to the latest observations.
Introduction . Post-inflation reheating plays a criti-cal role in our understanding of the very early Universe(see Ref. [1] for a recent review). By the end of thereheating phase — and before big-bang nucleosynthesis(BBN) can commence [2] — the Universe must achieve aradiation-dominated equation of state and become filledwith (at least) a thermal bath of Standard Model par-ticles at an appropriately high temperature. Althoughthe earliest stages of reheating can be studied within alinearized approximation, some of the most critical pro-cesses arise from nonlinear physics, including backreac-tion and rescattering among the produced particles.In addition to setting appropriate conditions for BBN,the reheating phase plays a critical role in compar-isons between inflationary predictions and recent high-precision measurements of the cosmic microwave back-ground (CMB). In particular, if there were a prolongedperiod after inflation before the Universe attained aradiation-dominated equation of state (EOS), that wouldimpact the mapping between perturbations on observa-tionally relevant length-scales and when those scales firstcrossed outside the Hubble radius during inflation [3–6]. Residual uncertainty on the duration of reheating, N reh , is now comparable to statistical uncertainties inmeasurements of CMB spectral observables. Hence un-derstanding the time-scale N reh is critical for evaluatingobservable predictions from inflationary models.In this Letter we study the nonlinear dynamics of theearly preheating phase of reheating in a well-motivatedclass of models. These models include multiple scalarfields, as typically found in realistic models of high-energyphysics [7, 8]; and each scalar field, φ , has a nonmin-imal coupling to the spacetime Ricci curvature scalar, R , of the form ξφ R . Such nonminimal couplings arequite generic: they are induced by quantum corrections for any self-interacting scalar field in curved spacetime,and they are required for renormalization [9, 10]. More-over, the dimensionless coupling constants, ξ , grow withenergy-scale under renormalization-group flow, with noUV fixed point [11]. Hence they can attain large valuesat inflationary energy scales. Upon transforming to theEinstein frame, such models feature curved field-spacemanifolds [12].Multifield models with nonminimal couplings naturallyyield a plateau-like phase of inflation at large field values,of the sort most favored by recent observations [13]. Dur-ing inflation the fields generically evolve within a single-field attractor, thereby suppressing typical multifield ef-fects that could spoil agreement with observations, suchas large primordial non-Gaussianities and isocurvatureperturbations [14–16].Previous work, which studied the onset of preheat-ing in this class of models semi-analytically, identifiedthree regimes that yielded qualitatively distinct behav-ior: ξ (cid:46) O (1), ∼ O (10), and (cid:38) O (10 ) [17–19]. In thisLetter we significantly expand this work, employing lat-tice simulations to study the complete preheating phase,deep into the nonlinear regime. We restrict attention tocoupled scalar fields, and neglect the production of Stan-dard Model particles such as fermions or gauge fields [20–33]. Nonetheless, we are able to analyze the typical time-scales required for the Universe to achieve a radiation-dominated EOS; for the produced particles to backreacton the inflaton condensate, ultimately draining away itsenergy; and for rescattering among the particles to yielda thermal spectrum. For large couplings, ξ (cid:38) , ofthe sort encountered in Higgs inflation [34], we find veryefficient preheating, typically completing within the firsttwo e -folds after the end of inflation, thereby protect-ing the close match between predictions for primordial a r X i v : . [ h e p - ph ] O c t observables and the latest CMB measurements. Model . In the Jordan frame, the nonminimal cou-pling between the N scalar fields and the spacetimeRicci scalar ˜ R remains explicit in the action throughthe term f ( φ I ) ˜ R . Upon rescaling ˜ g µν ( x ) → g µν ( x ) =Ω ( x )˜ g µν ( x ), with Ω = 2 f ( φ I ) /M , we transform theaction into the Einstein frame. (Here M pl ≡ / √ πG =2 . × GeV is the reduced Planck mass.) TheEinstein-frame potential is stretched by the conformalfactor, V ( φ I ) = ˜ V ( φ I ) / Ω , compared to the Jordan-frame potential ˜ V ( φ I ). Taking canonical scalar fieldsin the Jordan frame, the nonminimal couplings inducea curved field-space manifold in the Einstein frame, withfield-space metric given by G IJ ( φ K ) = [ M / (2 f )] { δ IJ +3 f ,I f ,J /f } [12]. The equation of motion for the fields inthe Einstein frame is then (cid:3) φ I + g µν Γ IJK ∂ µ φ J ∂ ν φ K − G IJ V ,J = 0 , (1)where Γ IJK ( φ L ) is the Christoffel symbol constructedfrom G IJ . We consider an unperturbed, spatially flatFriedmann-Lemaˆıtre-Robertson-Walker (FLRW) space-time metric, so the Einstein field equations yield H ( t ) = ρ total / (3 M ), where ρ total is the total energy density ofthe system, H ( t ) ≡ ˙ a/a , and overdots denote derivativeswith respect to cosmic time.We consider two-field models, φ I = { φ, χ } , with f ( φ I ) = 12 (cid:2) M + ξ φ φ + ξ χ χ (cid:3) , ˜ V ( φ I ) = λ φ φ + g φ χ + λ χ χ . (2)The topography of the Einstein-frame potential gener-ically includes “ridges” and “valleys” along certain di-rections χ/φ = const . For non-fine-tuned parameters,the fields quickly fall to a local minimum (valley) of thepotential, and the background dynamics obey a strong“single-field attractor” [15–17]. For symmetric couplings,with ξ φ = ξ χ and λ φ = g = λ χ , any initial angular mo-tion within field space damps out within a few e -foldsafter the start of inflation, and the system flows towardthe minimum of the potential along a single-field trajec-tory [35]. Within a single-field attractor, the predictionsfor the spectral index n s , the tensor-to-scalar ratio r ,the running α = dn s /d ln k , primoridal non-Gaussianities f NL , and isocurvature perturbations β iso remain consis-tent with the latest observations across large regions ofphase space and parameter space [15–17].Field fluctuations in these models are sensitive to thecurvature of the field-space manifold, which is greatestnear the origin. During preheating, as the inflaton con-densate oscillates through zero, the effective mass for thefluctuations δχ receives quasi-periodic “spikes” propor-tional to a component of the field-space Riemann tensor.In the limit ξ I (cid:29)
1, these scale as R χφφχ ∝ ξ φ . Theselarge “spikes” lead to sharp violations of the adiabatic condition for those modes, driving efficient particle pro-duction [17–19, 36].Within the single-field attractor, the amplitude of pri-mordial perturbations scales as [ λ φ /ξ φ ] / [15]. Presentconstraints on the tensor-to-scalar ratio therefore require λ φ /ξ φ ≤ . × − . We fix λ φ /ξ φ = 10 − and con-sider various values for ξ χ /ξ φ , λ χ /λ φ , and g/λ φ . Weconsider two typical cases: (A) ξ χ = 0 . ξ φ , g = λ φ , and λ χ = 1 . λ φ ; and (B) ξ χ = ξ φ , λ φ = g = λ χ . Forthe “generic” case (A) the single-field attractor lies along χ = 0, while we are free to choose the same attractor di-rection for the symmetric case (B). Once the ratios ofcouplings are fixed, the dynamics of the system changeas we vary ξ φ across (cid:46) O (1) , ∼ O (10), and (cid:38) O (10 ). Results . We employ a modified version of
GABE (Gridand Bubble Evolver) [37] to evolve the fields and thebackground, according to Eq. (1) and the Friedmannequation. Whereas the original software was used tosimulate nonminimally coupled degrees of freedom [38],we have modified the code significantly to allow for acurved field-space metric in both the dynamics of thefields as well as the initial conditions. We start the simu-lations when inflation ends, defined by (cid:15) ( t init ) = 1 where (cid:15) ≡ − ˙ H/H ; the Hubble scale at this time is H end . Weuse a grid with N = 256 points and a comoving boxsize L = π/H end so that the longest wavelength in ourspectra corresponds to k = H end /
2. We match the two-point correlation functions of φ ( t init , x ) and χ ( t init , x )to corresponding distributions for quantized field fluc-tuations. Fourier modes of the quantized fluctuationsevolving during inflation within the single-field attrac-tor may be parameterized as δφ Ik = √G II v Ik ( τ ) /a ( τ ) (nosum on I ), where dτ ≡ dt/a ( t ) is conformal time [17].Near the end of inflation, we use the Wentzel-Kramers-Brillouin (WKB) approximation to estimate amplitudes | v Ik ( τ init ) | = [2Ω ( I ) ( k, τ init )] − / , where Ω I ) ( τ ) = k + a ( τ ) m ,I ( τ ). The effective masses m ,I include dis-tinct contributions from the curvature of the potentialand from the curvature of the field-space manifold, andare analyzed in detail in Refs. [17–19]. (Here we neglectcontributions from coupled metric perturbations.) Theinitial spectra of the fields are subject to a window func-tion that suppresses high-momentum modes above someUV suppression scale, k UV = 50 H end .Figs. 1 and 2 show results for Case A with ξ φ = 10 , (cid:104) φ (cid:105) on the lattice. Backreaction of pro-duced particles — which is absent in linearized analyses— becomes significant beginning around 2 . e -folds afterthe end of inflation for ξ φ = 10. For ξ φ = 100 backre-action is strong enough to completely drain the inflatoncondensate within the first 2 e -folds. Fig. 2 shows theevolution of the peak values of the spatial averages (cid:104) φ (cid:105) Linear ξ ϕ = ξ ϕ = ξ ϕ = ξ ϕ = ��� ��� ��� ��� ��� ��� ��� ��� - ������������ � - ������� ������ � 〈 ϕ 〉 FIG. 1. The evolution of the inflaton condensate (in units of M pl ) versus e -folds N after the end of inflation for Case Awith ξ φ = 10 , (cid:104) φ (cid:105) on thelattice (red, black). 〈ϕ〉〈χ〉 ϕ rms χ rms ��� ��� ��� ��� ��� ��� ��� ����� - � �� - � �� - � ����� � - ������� ������ � 〈 ϕ 〉 � 〈 χ 〉 � ϕ � � � � χ � � � FIG. 2. Lattice evolution of various fields (in units of M pl )versus e -folds N after the end of inflation for Case A with ξ φ = 10 (solid) and ξ φ = 100 (dotted): peak values of thespatial averages (cid:104) φ (cid:105) (blue) and (cid:104) χ (cid:105) (black); and values of thefluctuations φ rms (green) and χ rms (red). and (cid:104) χ (cid:105) as well as the growth of fluctuations, character-ized by φ rms ≡ (cid:112) (cid:104) φ (cid:105) − (cid:104) φ (cid:105) and χ rms ≡ (cid:112) (cid:104) χ (cid:105) − (cid:104) χ (cid:105) .(Growth of field fluctuations corresponds to particle pro-duction [1].) We have confirmed that the early growth of δφ and δχ fluctuations in our lattice simulations closelymatches the behavior calculated via Floquet analysis inRef. [18]. Beginning around 2.6 e -folds, nonlinear rescat-tering among the δχ fluctuations drives rapid growth ofthe δφ fluctuations for ξ φ = 10. For ξ φ = 100 the sameeffect occurs within the first e -fold. Backreaction andrescattering generally become significant at distinct timesas one varies couplings [39].The dynamics of the δφ and δχ fluctuations vary withcoupling ξ φ , as shown in Fig. 3. For ξ φ = 1 ,
10 para-metric resonance due to the contribution from the po-tential to m ,χ leads to a slow growth of δχ fluctua-tions; these eventually rescatter, leading to the growth � � � ������������� � - ������� ������ � χ � � � / ϕ � � � FIG. 3. The ratio χ rms /φ rms versus e -folds N after the endof inflation, for Case A with ξ φ = 1 , , , , , , , of δφ fluctuations and lowering the χ rms /φ rms ratio. For ξ φ ≥
40 the “Ricci spike” [17, 36] leads to a fast growthof δχ fluctuations. This is seen in Fig. 3 as an earlyrise of the χ rms /φ rms ratio. When χ rms grows enoughit rescatters with δφ fluctuations, eventually leading to χ rms /φ rms ∼
1. The case of ξ φ = 25 is the most interest-ing, since it displays several distinct phases. The initialgrowth occurs due to adiabaticity violation caused bythe Ricci spike. After 1 . e -folds the height of the Riccispike has redshifted, making it comparable to the poten-tial contribution to the effective mass, thereby shuttingoff particle production [17]. When the Ricci spike red-shifts even more, around 2 . e -folds, a second stage ofparametric resonance commences, due to the potentialterm alone. Subsequently, rescattering enhances the δφ fluctuations, lowering the χ rms /φ rms ratio. The situationis qualitatively similar for the symmetric case (B) [39].The rapid growth of fluctuations yields an efficienttransfer of energy from the inflaton condensate into ra-diative degrees of freedom. Within the single-field at-tractor, we may approximate the energy density in theinflaton condensate as [17] ρ bg (cid:39) G φφ (cid:104) ˙ φ (cid:105) + λ φ M (cid:104) φ (cid:105) M + ξ φ (cid:104) φ (cid:105) ) , (3)where we evaluate G φφ with φ → (cid:104) φ (cid:105) and χ ∼
0. Fig. 4shows that across Cases A and B the fraction of energydensity in the inflaton condensate falls sharply within thefirst few e -folds after the end of inflation; for ξ φ ≥ N = 1 . e -folds.The rapid transfer of energy to radiative degrees offreedom is similarly reflected in Fig. 5, which shows theevolution of the EOS, w = p total /ρ total , where ρ total and p total are the total energy density and pressure for thesystem, respectively. In this case, the system approaches ( A ) ξ ϕ = ( A ) ξ ϕ = ( A ) ξ ϕ = ( B ) ξ ϕ = ( B ) ξ ϕ = ( B ) ξ ϕ = ��� ��� ��� ��� ��� ��������������������� � - ������� ������ �� - ρ �� ρ ��� FIG. 4. The fraction of energy density that has left the in-flaton condensate versus e -folds N after the end of inflationfor the generic case (A) and the symmetric case (B) with ξ φ = 1 , , w = 1 / ξ φ ∼ O (1), becausein that regime the Einstein-frame potential for the infla-ton approximates a quartic form, so that even the con-densate’s oscillations correspond to w (cid:39) / ξ φ increases, the Einstein-frame potential for φ approachesa quadratic form, for which the condensate’s oscillationsbehave like w (cid:39) ξ φ = 100, we find a transientphase with a stiff EOS, w > /
3, which likely arises be-cause typical momenta for the fluctuations are compa-rable to m eff ,I , and the contributions to ρ total and p total from kinetic and spatial-gradient terms are weighted bycomponents of G IJ , which are significant for ξ φ (cid:29)
1. Atlater times, as m eff ,I →
0, the system relaxes to a gasof massless particles with w = 1 /
3. Across a wide rangeof couplings for this family of models, we therefore findthat the Universe rapidly achieves a radiation-dominatedEOS within N rad ∼ − . e -folds after the end of infla-tion. Preheating in α -attractor models with α = O (1), incontrast, can lead to a prolonged period with w (cid:39) δφ and δχ for Case A with ξ φ = 10. Although the spec-tra are dominated at early times by increased power indistinct resonance bands, by later times rescattering hasflattened out the distributions for both δφ and δχ . By N therm = 2 . e -folds after the end of inflation, both fieldshave attained a spectrum consistent with a thermal dis-tribution, | δφ Ik | ∝ [ k (exp[ k/T ] − − , at a temperature T reh ∼ O ( H end ). We find comparable behavior across ( A ) ξ ϕ = ( A ) ξ ϕ = ( A ) ξ ϕ = ( B ) ξ ϕ = ( B ) ξ ϕ = ( B ) ξ ϕ = ��� ��� ��� ��� ��� ��� ������������������ � - ������� ������ � 〈 � 〉 FIG. 5. The averaged effective equation of state (cid:104) w (cid:105) for ξ φ =1 , ,
100 and the two representative cases, “generic” (A) andsymmetric (B). � �� �� ���������� � �� � � / � ��� | δ χ � | � � | δ ϕ � | � FIG. 6. Spectra for the fluctuations δφ (dashed) and δχ (solid) versus k/H end , where k is comoving wavenumber, forCase A with ξ φ = 10 at N (cid:39) , . , . , . , . e -folds afterthe end of inflation (purple, orange, blue, red, green respec-tively). The black-dotted curve shows a thermal spectrum. Cases A and B for ξ φ ≥ N ad = min[ N bg , N therm ], where N bg isthe time by which super-Hubble coherence of the infla-ton condensate is lost, indicated by φ rms > (cid:104) φ (cid:105) . Anysignificant turning of the system within the field spacebetween the end of inflation and N ad could amplify non-Gaussianities and isocurvature perturbations, therebythreatening the close agreement between predictions inthese models and measurements of the CMB [41–44].In Fig. 7, we plot ω/H across cases of interest, where ω = | ω I | is the covariant turn-rate [45]. Even as the Hub-ble rate falls over time, we nonetheless find ω/H < . N ad , indicating minimal turning of the systemwithin field space.Our late-time results were unchanged as we variedthe initial UV suppression scale k UV = bH end between ( A ) ξ ϕ = ( A ) ξ ϕ = ( A ) ξ ϕ = ( B ) ξ ϕ = ( B ) ξ ϕ = ( B ) ξ ϕ = ��� ��� ��� ��� ��� ��� ��� - � - � - � - �� � - ������� ������ � � �� �� ω / � FIG. 7. The quantity | ω I | /H versus e -folds N after the end ofinflation for the generic case (A) and the symmetric case (B)and for ξ φ = 1 , , ω I is the covariant turn-rate[14]. Each curve is shown up to N ad = min[ N bg , N therm ]. b = 25, 50, and 100, and the number of grid-points be-tween 128 , 256 and 512 . We discuss this and relatednumerical convergence tests in Ref. [39]. Conclusions . Multifield models of inflation with non-minimal couplings generically yield predictions for pri-mordial observables in close agreement with the latestobservations, deriving from the strong single-field attrac-tor behavior of these models [15–17]. Throughout thecases we have examined and across parameter space, wefind that this single-field attractor behavior remains ro-bust until the system reaches the adiabatic limit afterinflation, with no significant turning in field space evenin the midst of strongly nonlinear dynamics.Preheating in this class of models is efficient, drainingthe energy density from the inflaton condensate within N bg (cid:46) . e -folds in the limit of strong couplings, ξ I ∼ N rad (cid:46) .
5, while rescatteringyields a rapid onset of thermalization within N therm (cid:46) φ and χ to Standard Modelparticles. Acknowledgements . RN received support from a ClareBooth Luce Undergraduate Research Award, Grant ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected][1] M. A. Amin, M. P. Hertzberg, D. I. Kaiser, andJ. Karouby, “Nonperturbative Dynamics Of ReheatingAfter Inflation: A Review,” Int. J. Mod. Phys. D24 ,1530003 (2015), arXiv:1410.3808 [hep-ph].[2] R. H. Cyburt, B. D. Fields, K. A. Olive, and T.-H.Yeh, “Big Bang Nucleosynthesis: Present status,” Rev.Mod. Phys. , 015004 (2016), arXiv:1505.01076 [astro-ph.CO].[3] P. Adshead, R. Easther, J. Pritchard, and A. Loeb,“Inflation and the scale dependent spectral index:prospects and strategies,” JCAP , 021 (2011),arXiv:1007.3748 [astro-ph.CO].[4] L. Dai, M. Kamionkowski, and J. Wang, “ReheatingConstraints to Inflationary Models,” Phys. Rev. Lett. , 041302 (2014), arXiv:1404.6704 [astro-ph.CO].[5] V. Vennin, K. Koyama, and D. Wands, “Encyclopædiacurvatonis,” JCAP , 008 (2015), arXiv:1507.07575[astro-ph.CO].[6] K. D. Lozanov and M. A. Amin, “Equation of State andDuration to Radiation Domination after Inflation,” Phys.Rev. Lett. , 061301 (2017), arXiv:1608.01213 [astro-ph.CO].[7] A. Mazumdar and J. Rocher, “Particle physics modelsof inflation and curvaton scenarios,” Phys. Rept. ,85–215 (2011), arXiv:1001.0993 [hep-ph].[8] D. Baumann and L. McAllister, Inflation and String The-ory (Cambridge University Press, Cambridge, 2015).[9] C. G. Callan, Jr., S. R. Coleman, and R. Jackiw, “Anew improved energy-momentum tensor,” Annals Phys. , 42–73 (1970).[10] T. S. Bunch, P. Panangaden, and L. Parker, “On renor-malization of λφ field theory in curved space-time,” J.Phys. A13 , 901–918 (1980).[11] S. D. Odintsov, “Renormalization Group, Effective Ac-tion and Grand Unification Theories in Curved Space-time,” Fortsch. Phys. , 621–641 (1991).[12] D. I. Kaiser, “Conformal Transformations with Mul-tiple Scalar Fields,” Phys. Rev. D81 , 084044 (2010),arXiv:1003.1159 [gr-qc].[13] Y. Akrami and
Planck Collaboration , “Planck 2018 re-sults. X. Constraints on inflation,” arXiv:1807.06211[astro-ph.CO].[14] D. I. Kaiser, E. A. Mazenc, and E. I. Sfakianakis, “Pri-mordial bispectrum from multifield inflation with non-minimal couplings,” Phys. Rev. D , 064004 (2013),arXiv:1210.7487 [astro-ph.CO].[15] D. I. Kaiser and E. I. Sfakianakis, “Multifield Inflation af-ter Planck: The Case for Nonminimal Couplings,” Phys.Rev. Lett. , 011302 (2014), arXiv:1304.0363 [astro-ph.CO].[16] K. Schutz, E. I. Sfakianakis, and D. I. Kaiser, “Mul-tifield inflation after Planck: Isocurvature modes fromnonminimal couplings,” Phys. Rev. D , 064044 (2014),arXiv:1310.8285 [astro-ph.CO].[17] M. P. DeCross, D. I. Kaiser, A. Prabhu, C. Prescod-Weinstein, and E. I. Sfakianakis, “Preheating after mul- tifield inflation with nonminimal couplings. I. Covariantformalism and attractor behavior,” Phys. Rev. D ,023526 (2018).[18] M. P. DeCross, D. I. Kaiser, A. Prabhu, C. Prescod-Weinstein, and E. I. Sfakianakis, “Preheating after mul-tifield inflation with nonminimal couplings. II. Resonancestructure,” Phys. Rev. D , 023527 (2018).[19] M. P. DeCross, D. I. Kaiser, A. Prabhu, C. Prescod-Weinstein, and E. I. Sfakianakis, “Preheating after mul-tifield inflation with nonminimal couplings. III. Dynam-ical spacetime results,” Phys. Rev. D , 023528 (2018).[20] P. B. Greene and L. Kofman, “Preheating of fermions,”Phys. Lett. B448 , 6–12 (1999), arXiv:hep-ph/9807339[hep-ph].[21] P. B. Greene and L. Kofman, “Theory of fermionic pre-heating,” Phys. Rev.
D62 , 123516 (2000), arXiv:hep-ph/0003018 [hep-ph].[22] M. Peloso and L. Sorbo, “Preheating of massive fermionsafter inflation: Analytical results,” JHEP , 016 (2000),arXiv:hep-ph/0003045 [hep-ph].[23] S. Tsujikawa, B. A. Bassett, and F. Viniegra, “Multifieldfermionic preheating,” JHEP , 019 (2000), arXiv:hep-ph/0006354 [hep-ph].[24] A.-C. Davis, K. Dimopoulos, T. Prokopec, andO. Tornkvist, “Primordial spectrum of gauge fieldsfrom inflation,” Phys. Lett. B501 , 165–172 (2001),arXiv:astro-ph/0007214 [astro-ph].[25] J. Garcia-Bellido, M. Garcia-Perez, and A. Gonzalez-Arroyo, “Chern-Simons production during preheating inhybrid inflation models,” Phys. Rev.
D69 , 023504 (2004),arXiv:hep-ph/0304285 [hep-ph].[26] F. Bezrukov, D. Gorbunov, and M. Shaposhnikov, “Oninitial conditions for the Hot Big Bang,” JCAP ,029 (2009), arXiv:0812.3622 [hep-ph].[27] J. Garcia-Bellido, D. G. Figueroa, and J. Rubio, “Pre-heating in the Standard Model with the Higgs-Inflatoncoupled to gravity,” Phys. Rev.
D79 , 063531 (2009),arXiv:0812.4624 [hep-ph].[28] J.-F. Dufaux, D. G. Figueroa, and J. Garcia-Bellido,“Gravitational Waves from Abelian Gauge Fields andCosmic Strings at Preheating,” Phys. Rev.
D82 , 083518(2010), arXiv:1006.0217 [astro-ph.CO].[29] R. Allahverdi, A. Ferrantelli, J. Garcia-Bellido, andA. Mazumdar, “Non-perturbative production of matterand rapid thermalization after MSSM inflation,” Phys.Rev.
D83 , 123507 (2011), arXiv:1103.2123 [hep-ph].[30] J. T. Deskins, J. T. Giblin, and R. R. Caldwell, “GaugeField Preheating at the End of Inflation,” Phys. Rev.
D88 , 063530 (2013), arXiv:1305.7226 [astro-ph.CO].[31] P. Adshead and E. I. Sfakianakis, “Fermion produc-tion during and after axion inflation,” JCAP , 021(2015), arXiv:1508.00891 [hep-ph].[32] P. Adshead, J. T. Giblin, and Z. J. Weiner, “Non- Abelian gauge preheating,” Phys. Rev.
D96 , 123512(2017), arXiv:1708.02944 [hep-ph].[33] E. I. Sfakianakis and J. van de Vis, “Preheating af-ter Higgs Inflation: Self-Resonance and Gauge bo-son production,” Phys. Rev.
D99 , 083519 (2019),arXiv:1810.01304 [hep-ph].[34] F. Bezrukov and M. Shaposhnikov, “The Standard ModelHiggs boson as the inflaton,” Physics Letters B , 703–706 (2008), arXiv:0710.3755 [hep-th].[35] R. N. Greenwood, D. I. Kaiser, and E. I. Sfakianakis,“Multifield dynamics of Higgs inflation,” Phys. Rev. D , 064021 (2013), arXiv:1210.8190 [hep-ph].[36] Y. Ema, R. Jinno, K. Mukaida, and K. Nakayama, “Vio-lent Preheating in Inflation with Nonminimal Coupling,”JCAP , 045 (2017), arXiv:1609.05209 [hep-ph].[37] http://cosmo.kenyon.edu/gabe.html.[38] H. L. Child, J. T. Giblin, Jr, R. H. Ribeiro, and D. Seery,“Preheating with Non-Minimal Kinetic Terms,” Phys.Rev. Lett. , 051301 (2013), arXiv:1305.0561 [astro-ph.CO].[39] R. Nguyen, J. van de Vis, E. I. Sfakianakis, J. T. Giblin,Jr., and D. I. Kaiser, “Preheating after multifield infla-tion with nonminimal couplings: Lattice Simulations,”in preparation.[40] O. Iarygina, E. I. Sfakianakis, D.-G. Wang, andA. Ach´ucarro, “Universality and scaling in multi-field α -attractor preheating,” JCAP , 027 (2019),arXiv:1810.02804 [astro-ph.CO].[41] J. Elliston, D. J. Mulryne, D. Seery, and R. Tavakol,“Evolution of f NL to the adiabatic limit,” JCAP ,005 (2011), arXiv:1106.2153 [astro-ph.CO].[42] J. Elliston, S. Orani, and D. J. Mulryne, “Gen-eral analytic predictions of two-field inflation and per-turbative reheating,” Phys. Rev. D89 , 103532 (2014),arXiv:1402.4800 [astro-ph.CO].[43] J. Meyers and E. R. M. Tarrant, “Perturbative Reheat-ing After Multiple-Field Inflation: The Impact on Pri-mordial Observables,” Phys. Rev.
D89 , 063535 (2014),arXiv:1311.3972 [astro-ph.CO].[44] S. Renaux-Petel and K. Turzynski, “On reaching the adi-abatic limit in multi-field inflation,” JCAP , 010(2015), arXiv:1405.6195 [astro-ph.CO].[45] The covariant turn-rate ω I ≡ D t ˆ σ I , where ˆ σ I ≡ ˙ ϕ I / ˙ σ , D t A I = ˙ ϕ J D J A I is the covariant directional deriva-tive, and ˙ σ ≡ G IJ ˙ ϕ I ˙ ϕ J [14]. One may show that ω = | ω I | = [ V ,K V ,L ( ˙ σ G KL − ˙ ϕ K ˙ ϕ L )] / / ˙ σ . To cal-culate ω during preheating, we evaluate the spatiallyhomogeneous fields via spatial averages on the lattice, { ϕ I , ˙ ϕ I } → {(cid:104) φ I (cid:105) , (cid:104) ˙ φ I (cid:105)} , and replace the term ˙ σ2