General relativistic effects in preheating

Transcription

1 7 May 1999 Physics Letters B General relativtic effects in preheating Bruce A. Bassett a, David I. Kaer b, Roy Maartens c a Department of Theoretical Physics, Oxford UniÕersity, Oxford OX1 3NP, UK b Lyman Laboratory of Physics, HarÕard UniÕersity, Cambridge, MA 0138, USA c School of Computer Science and Mathematics, Portsmouth UniÕersity, Portsmouth PO1 EG, UK Received 17 February 1999 Editor: L. Alvarez-Gaumé Abstract General relativtic effects in the form of metric perturbations are usually neglected in the preheating era that follows inflation. We argue that in realtic multi-field models these effects are in fact crucial, and the fully coupled system of metric and quantum field fluctuations needs to be considered. Metric perturbations are resonantly amplified, breaking the scale-invariance of the primordial spectrum, and in turn stimulate scalar field resonances via gravitational rescattering. Th non-gravitationally dominated nonlinear growth of gravitational fluctuations may have significant effects on the Doppler peaks in the cosmic background radiation, primordial black hole formation, gravitational waves and nonthermal symmetry restoration. q 1999 Elsevier Science B.V. All rights reserved. PACS: Cq As recent ideas about the end of an inflationary era have shown, post-inflation reheating was one of the most violent and explosive processes occurring in the early universe w1 3 x. The nonequilibrium, nonperturbative, resonant decay of the inflaton was demonstrated in Minkowski spacetime. The first steps toward a gravitationally self-constent treatment, which modeled inflaton decay in an expanding, dynamical background spacetime, revealed that preheating may proceed with qualitative, as well as quantitative, differences from the Minkowski case w,3 x. Yet even these studies neglected an essential feature of gravitational physics: the production and amplification of metric perturbations attending such a sudden transfer of energy from the oscillating inflaton to higher-momentum particles. Other papers began the study of metric perturbations induced durwx 4 a perfect-fluid analys with ing preheating: in Born decay was used, and crucial features of preheatwx 5 these limita- ing were thus not incorporated; in tions are avoided, but the effects of the amplified metric perturbations on the process of preheating itself are not considered. In th Letter, we pursue a more self-constent relativtic treatment of preheating, by studying both the field fluctuations Žresponsible for particle production. and the coupled metric perturbations Ždescrib- ing gravitational fluctuations in the curvature.. In the process, we clarify the question of causality and the amplification of long-wavelength perturbations during preheating. We give qualitative arguments, confirmed by numerical results Žsee also wx. 6, to show r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S

2 ( ) B.A. Bassett et al.rphysics Letters B that metric perturbations typically undergo rapid growth during multi-field preheating, and in turn act as a source and a pump for the growth of field fluctuations Õia graõitational rescattering, ultimately making the preheating process even more efficient than previously realized. Neglecting th general relativtic effect can produce mleading results. Moreover, nonlinear growth of metric perturbations itself of potentially major significance, since it precipitates nonlinear density fluctuations, leading to strong mode-mode coupling effects, and nonlinear deviations from the conformally flat background. Th could affect observable quantities such as the cosmic microwave background Ž CMB. spectrum, complicating the usual predictions from inflationary models. Gravitational wave power could be enhanced by gravitational bremstrahlung, and primordial black hole production could occur without the need for special properties in the power spectrum. We work with the gauge-invariant formalm of wx 7 to study the evolution of scalar perturbations of the metric. ŽThe amplification of gravitational waves at preheating has been considered in wx. 8. For scalar perturbations, in the case of a scalar-field energymomentum tensor and spatially flat background, the perturbed metric in the longitudinal gauge ds sa Ž h. Ž 1qF. dh yž 1yF. dx, where F the gauge-invariant gravitational potential. Realtic reheating models involve at least one field coupled to the inflaton wh. First we consider models with only an inflaton, in order to give a simple illustration of some effects. The gauge-invariant field fluctuations dw and the metric perturbations F obey the coupled Eqs. Ž of wx 7. Defining the rescaled fields Y'aF and X'adw, these become, in momentum space: XX 1 X XX Y q k y kw Y sk w X, Ž 1. k k k XX 3 X XX Xk q k qa Vwwy kw y H Xksw Y k, X 1 X Yks kwx k, 3 X where V w the potential, Hsa ra and k s 8p M y pl. Eq. Ž. 3 a constraint, showing that if there explosive growth in the field fluctuations X k the heart of preheating then th automatically constrains the gravitational fluctuations Yk to follow, and Õice Õersa. Th equation essentially says that if you shake the right-hand side of Einstein s field equations Gmnsk T mn, then unavoidably you are simultaneously shaking the left hand side. Clearly, neglecting the metric perturbations Yk can be seri- ously mleading under many conditions, since it tantamount to ignoring the perturbed Einstein equations. Th may be reasonable in a slow-roll inflationary regime, but generally not in an oscillatory regime. A clear illustration provided by super-hubble 1 modes in the simplest model, Vs m w.ifdw) the field fluctuation calculated by neglecting metric perturbations, then for k 0, X )' adw) satfies Eq. Ž. with gravitational fluctuations eliminated, i.e. XX 1 X ) ) X q maq kw y H X s0. Th equation then of precely the same form as the background Klein-Gordon equation, so that dw ) Awfw sinžbh 3. ržbh 3. 0, where w0 and b are constants. The approximation ares from using the time-averaged scale factor a A h, and improves in accuracy as h increases. When metric perturbations are incorporated, the long-wavelength solution given in general by wsee w7 x, Eq. Ž 6.57.x dwa w X a y Ha dh. We find that at reheating dw 3 Ž 3. y1 f 5 w0cos bh, to lowest order in h. Gravitational rescattering produces a non-decaying term in the field fluctuations, which dominates the rapidly decaying fluctuation dw) calculated by neglecting gravitational fluctuations. Relating in the usual way w1,3x the field fluctuations to the density of particles produced per mode, we find that th non-decaying solution indicates nonzero particle production. If gravitational perturbations are neglected, no particle production found for th model. GraÕitational rescattering dramatically alters the eõolution of the matter field fluctuations eõen in th simplest of all models, especially on super-hubble-radius scales. Since preheating concerns primarily the behavior of such matter field fluctuations after inflation, it thus crucial to study the coupled metric-perturbation field-fluctuation system. Now we consider the amplification of gravitational fluctuations. The constrained system of equa-

3 86 ( ) B.A. Bassett et al.rphysics Letters B Ž.Ž. tions Eqs. 1 3 has one degree of freedom, reflected in the decoupled equation XX XX X X 1 X k k k Y y w rw Y y kw yk Y s0. To avoid the periodic singularities when w X s0, Nambu and Taruya w5x employ the rescaled Mukhanov variable Žyn.rŽ nq1. X Qsa Xq w rh Y, where n the index in the power-law potential V 1 n s m w0 wrw 0. Using the time-averaged forms of a and w, they find that, to leading order in t y1 Ah 3rŽ1y n. Aa y3rž nq1., Q q 1qk gt tt 4Ž ny. r3 y Ž 4rt. sint Qs0, Ž 4. where g a constant. Th equation has Mathieu XX form, y qwa y q sin xx k ys0, with time-depen- dent Ak and q, so that modes can be drawn through instability bands by the expansion of the universe, if they were not already there initially w5,9 x. Note that the resonance parameters scale as y3rž nq1. 4Ž ny. rž nq1. y1 qrak Aa b qk a, where b constant. If metric fluctuations are neglected, then qrak Aa y3. Therefore, for n-8, ex- pansion less effectiõe in ending resonance when metric fluctuations are incorporated. Further dcussion of single-field models given in w10,11 x, which confirm the analytical conclusions of wx 5 : in the n s 1 Ž quadratic potential. model, there no resonance in the long-wavelength limit, whereas large, resonant growth found for other models, such as the massless, quartically-coupled case Žn s.. However, as pointed out above, the single-field case completely inadequate as a model of reheating, and we turn now to consider the multi-field case. The rapid growth of metric perturbations in the multi-field case Ž see Fig. 1. will produce a backreac- 3 Fig. 1. Metric perturbation evolution in -field preheating, with q' gw1 t0 rm s8=10. The main graph shows the ks0 mode, the y3 y5 inset the ks0 mode. Initially, mt s100 Žwell into the small amplitude phase: w Ž t. rmt s3=10. and F Ž t. s10, F Ž t k 0 k 0 s0. The ks0 mode becomes nonlinear at mt; 150 after less than 10 inflaton oscillations Ž i.e. well before the end of preheating. and continues growing without bound. Two strong resonance bands are evident, with different Floquet indices Ž given by the slopes of the dotted lines., for 140FmtF160 and 50FmtF300.

4 ( ) B.A. Bassett et al.rphysics Letters B tion on the background quantities a and w w1 x. Similarly, the amplified field fluctuations will grow to be of the same order as the tree-level terms, such as w, and hence will damp the inflaton s oscillations, ending the parametric resonance. Let us consider an effective single-field model. We may then estimate these two time-scales, expecting the initial preheating phase to end at h sminh, h 4 end m f, where ² : ² : F hm s1 and dw hf sw. In a resonance m k h band, F ksfk h e, where Fk oscillates and mk the Floquet exponent. From Eq. Ž. 3, and working in the saddle-point approximation, we may estimate ² F : s Ž p. HdkF < < k y3 3 4 X ² : ;k w dw r 4m, k max where mk max the maximum Floquet exponent. For single-inflaton models of chaotic inflation with polynomial potentials, the slow-roll conditions are violated, and the inflaton begins to oscillate at w s 0 am pl, with a;0.3. In models with a massive infla- ton, we may further approximate < w X < ; a mm. The field fluctuations saturate their Ž linear-theory. upper ² : limit at dw ; w. Combining these yields ² : 4 F h m ; 16p a m rm k max. The specific spec- trum of fluctuations, governed by the values of m k, depends on details of the potential. For modes subject to a parametric resonance, m k max ;a m, whereas modes subject to a negative-coupling instability may have m ; Om w1 3,13 x k max. Thus first-order analy- s reveals that gravitational backreaction will become relevant at around the same time as the backreaction of the nonlinearly-coupled field fluctuations. Another important sue to demonstrate how super-hubble modes may be amplified causally. In inflation, modes which had been deep within the Hubble radius during inflation become amplified and stretched to super-hubble scales w7,14 x. Reheating was believed not to be able to affect these super-hubble scales. In preheating, however, the coherence of the inflaton condensate immediately after inflation does allow for super-hubble dynamics. At first, such behavior might appear to violate causality. Th not the case; consider the following: Ž. 1 the field equations are relativtic, hence causality automatically built into the solutions; Ž. causality concerns space-like related events, and does not translate into direct constraints on individual modes pl in Fourier space w15 x; Ž. 3 explicit calculation of the unequal-time two-point correlation function reveals that no mass or energy being transported superluwx 6 minally by these super-hubble resonances. See for th calculation and further dcussion. Similar conclusions regarding the possibility for the causal amplification of super-hubble modes at preheating have been reached in w10 x. The main point we wh to emphasize that causality restricts the shape of the spectrum of amplified modes, but not directly the wavelengths that can be amplified. 1 Preheating in single-field models typically restricted to the narrow-resonance regime, and we may expect much larger effects in models with multiple scalar fields coupled to the oscillating inflaton, because the resonance parameter, q, may be much greater than unity w3,6 x. The dynamics of such couw16x and lead to enhanced particle production w17 x. The pled-oscillator systems are in general chaotic multi-field generalization of Eqs. Ž. 1 Ž. 3 can be given as w18,6x 3HFq Ž kra. q3h F 1 P i i i i i PP P Ž dwi. q3hž dwi. qž k ra. dwi sy ks wž dw. yfw qv dw, Ž 5. s4fw yv FyS V dw, Ž 6. i i ij j 1 ḞqHFs kswdw, i i 7 dropping the mode label k, using proper time t, and writing V s E VrEw. Eq. Ž. i i 7 shows that resonant amplification of field fluctuations accompanied by similar behavior of gravitational fluctuations. In the single-field case, any resonant growth occurs with the same charactertics for metric and field fluctuations, and stability bands are thus also the same. In the multi-field case, th simple relation broken, and the stability band structure much more complicated. The non-inflaton fields w Ž i) 1. i grow rapidly under resonance, while the inflaton w1 strongly damped. Thus metric fluctuations F grow more quickly than any of the field fluctuations dw i 1 The convenient approximation sometimes used in preheating, that the dtribution of amplified modes falls as a spike, džky k. resonance, violates causality, since th requires that the field fluctuation contain correlations on all length scales, even for modes with kra4 H.

5 88 ( ) B.A. Bassett et al.rphysics Letters B due to the quadratic products wdw. i i In models which include a substantial broad resonance regime for the coupled field fluctuations, resonance parameters are much larger than in the single-field case, typically in the range q;10 y10 6, rather than qfož.wx 1, so the associated Floquet indices are much larger w,3 x. These qualitative remarks are illustrated in Fig. 1 and amplified in the extensive simulations in wx 6, aring from integrating Eqs. Ž.Ž. 5 7 for the potential 1 1 Vs m w q gw w, which describes decay of the 1 1 massive inflaton w1 into the boson field w. The graph shows the resonant amplification and nonlinear growth of gravitational fluctuations on both super- and sub-hubble scales. Also clear the validity of the Floquet index as a characterizer of growth at strong resonance in an expanding universe Žthe slopes in the resonance bands are very nearly constant.. The expansion has the effect of pulling modes through the resonance bands leading to phases of explosive growth and quiescence. Note that even with the expansion of the universe included, the Floquet index a useful concept at broad resonance the growth of F k almost exactly exponential in the resonance bands, as evident in Fig. 1. A further crucial point, with wide-ranging implications, that the non-gravitationally dominated evolution of perturbations during preheating breaks the scale-invariance of the primordial spectrum, as evident from comparing the evolution of the ks0, 0 modes in Fig. 1 wx 6. The numerical simulations in wx 6 reveal that inclusion of the coupled metric perturbations enhances the strength of the resonances in dw i, compared with when F neglected. Th can be understood by the fact that the gravitational field has negative specific heat. For th reason, F does not act as a parasite or competitor with the dwi for the energy of the oscillating inflaton, but rather can serve as a source or pump for field fluctuations growth. Preheating in the narrow resonance and broad resonance regimes shows another important qualitative difference: broad-resonance preheating in an expanding universe proceeds stochastically wx 3. The phases of the amplified field modes are virtually uncorrelated between each moment when the oscillating inflaton passes through zero. Both the field modes and the metric perturbations will have stochastic driving terms. It has been shown that such stochastic terms in general remove all stability bands, so that all modes kg0 grow subject to a parametric resonance, with, in general, larger charactertic exw19,17 x. Note that once the fluctuations have become ponents than in the simple periodic case strongly nonlinear, the linearized perturbation equations are no longer valid, and mode-mode coupling, aring due to convolutions which are ignored in the linearized Eqs. Ž.Ž. 5 7, must be included wx 3. Nonlinear gravitational fluctuations could produce various observable signatures, in particular on the CMB. Firstly, if the preheating phase followed by a second round of inflationary expansion, then subhubble amplified modes would get stretched into observationally-interesting scales. Such double-inflation typical for most realtic scenarios based on supergravity or supersymmetry w0 x. Secondly, acoustic Doppler peaks at lg 100 could also be affected. These are a key prediction of most inflationary models. In contrast, defect models often predict no secondary peaks and a shift in the position of the first peak, because the metric perturbations are produced randomly, receiving out of phase kicks due to the mode-mode coupling inherent in the nonw1 x. In the multi-field, broad-resonance case, stochas- linearity of defect models tic amplification leads to nonlinear metric fluctuations, producing mode-mode coupling which drives the unequal-time correlation function D 0 for < h y h < ) h c, the coherence time of the system. As h 0 we get d-correlated Ž white. noe w17 x c, mem- ory of initial conditions Ž and hence coherence. lost, and the field evolution mimics the actiõe, incoherent evolution of defects. The crucial question whether the nonlinear mode-mode coupling survives local interactions and persts up to nucleosynthes and photon decoupling. If so, there could be significant limits placed on inflationary reheating by observed element abundances and small-scale CMB anotropies. By smearing out the Doppler peaks, surviving nonlinear mode-mode coupling would greatly reduce the effectiveness of small-angle CMB observations for differentiating inflationary scenarios from defects models. The implied nonlinearity would lead to hybrid CMB anotropies passive and typically inflationary on large angular scales, active and defect-like on smaller scales.

6 ( ) B.A. Bassett et al.rphysics Letters B Another possible observational consequence ares from gravitational bremsstrahlung in scattering of the large scalar perturbations wx 8. Combined with our analys of causality above, the extra gravitational wave power due to the previously neglected metric fluctuations implies that the probability of detection in instruments such as LIGO may be higher than previously estimated, at least for preheating following chaotic inflation. Nonlinear gravitational fluctuations could also create the large density contrast necessary to produce primordial black holes Ž PBHs. w x, without the need for a blue spectrum or a large peak in the power spectrum at some k. PBH limits may then be able to constrain the nature of preheating and the associated amplification of metric perturbations. Finally, amplified metric perturbations affect non-thermal symmetry restoration at preheating w3 x, by altering the inflaton s effective potential via addition of the terms Žsee Eq. Ž of wx. 7 : ² : DVsa V dw q av² dwf :. ww The first term was originally considered in w3 x, while the second due to the direct Ž gravitational. coupling between dw and F. It of the same perturbative order as the first term, and msed if gravitational fluctuations are neglected in preheating. Acknowledgements Thanks to Fabrizio Tamburini for very valuable asstance with the numerics, and to Juan Garcia-Bellido, Andrew Liddle, Fabrizio Tamburini, Robert Brandenberger, Fabio Finelli, Marco Bruni and David Wands for dcussions. DK receives partial support from NSF grant PHY References w wx 1 L. Kofman, A. Linde, A. Starobinsky, Phys. Rev. Lett. 73 Ž ; Y. Shtanov, J. Traschen, R. Brandenberger, Phys. Rev. D 51 Ž ; D. Boyanovsky et al., Phys. Rev. D 51 Ž ; 5 Ž ; M. Yoshimura, Prog. Theo. Phys. 94 Ž ; H. Fujaki et al., Phys. Rev. D 53 Ž ; earlier perturbative studies of resonance included J. Traschen, R.H. Brandenberger, Phys. Rev. D 4 Ž ; A. Dolgov, D. Kirilova, Sov. J. Nuc. Phys. 51 Ž wx D. Kaer, Phys. Rev. D 53 Ž ; S.Y. Khlebnikov, I.I. Tkachev, Phys. Rev. Lett. 77 Ž ; 79 Ž ; Phys. Lett. B 390 Ž ; T. Prokopec, T.G. Roos, Phys. Rev. D 55 Ž ; D. Boyanovsky et al., Phys. Rev. D 56 Ž ; I. Zlatev, G. Huey, P.J. Steinhardt, Phys. Rev. D 57 Ž ; B.A. Bassett, S. Liberati, Phys. Rev. D 58 Ž wx 3 L. Kofman, A. Linde, A. Starobinsky, Phys. Rev. D 56 Ž wx 4 H. Kodama, T. Hamazaki, Prog. Theo. Phys. 96 Ž , 113. wx 5 Y. Nambu, A. Taruya, Prog. Theo. Phys. 97 Ž ; Phys. Lett. B 48 Ž wx 6 B.A. Bassett, F. Tamburini, D.I. Kaer, R. Maartens, hepphr ; WWWrUsersrBruceBassettrreheatingr wx 7 V.F. Mukhanov, H.A. Feldman, R.H. Brandenberger, Phys. Rep. 15 Ž wx 8 S.Y. Khlebnikov, I.I. Tkachev, Phys. Rev. D 56 Ž ; B.A. Bassett, Phys. Rev. D 56 Ž wx 9 H.E. Kandrup, astro-phr w10x F. Finelli, R.H. Brandenberger, Phys. Rev. Lett. 8 Ž w11x M. Parry, R. Easther, Phys. Rev. D 59 Ž w1x V. Mukhanov, L. Abramo, R. Brandenberger, Phys. Rev. Lett. 78 Ž ; Phys. Rev. D 56 Ž w13x P.B. Greene, T. Prokopec, T.G. Roos, Phys. Rev. D 58 Ž w14x A.D. Linde, Particle Physics and Inflationary Cosmology Harwood, Chur, 1990; E.W. Kolb, M.S. Turner, The Early Universe, Addon-Wesley, Redwood City, CA, 1990; A. Liddle, D. Lyth, Phys. Rep. 31 Ž w15x J. Robinson, B.D. Wandelt, Phys. Rev. D 53 Ž ; N. Turok, Phys. Rev. D 54 Ž ; Phys. Rev. Lett. 77 Ž w16x N.J. Cornh, J.J. Levin, Phys. Rev. D 53 Ž ; R. Easther, K. Maeda, Class. Quantum Grav. 16, in press, 1999 Ž gr-qcr w17x B.A. Bassett, F. Tamburini, Phys. Rev. Lett. 81 Ž w18x L. Kofman, D. Pogosyan, Phys. Lett. B 14 Ž w19x V. Zanchin et al., Phys. Rev. D 57 Ž ; B.A. Bassett, Phys. Rev. D 58 Ž w0x J.A. Adams, G.G. Ross, S. Sarkar, Nucl. Phys. B 503 Ž ; R. Jeannerot, Phys. Rev. D 53 Ž ; D.H. Lyth, E.D. Stewart, Phys. Rev. Lett. 75 Ž w1x A. Albrecht et al., Phys. Rev. Lett. 76 Ž ; J. Magueijo et al., Phys. Rev. Lett. 76 Ž ; Phys. Rev. D 54 Ž wx J. Bullock, J. Primack, Phys. Rev. D 55 Ž w3x L. Kofman, A. Linde, A. Starobinsky, Phys. Rev. Lett. 76 Ž ; I.I. Tkachev, Phys. Lett. B 376 Ž