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provided by Digital.CSIC PHYSICAL REVIEW D 79, 063531 (2009) Preheating in the standard model with the Higgs inflaton coupled to gravity
Juan Garcı´a-Bellido,* Daniel G. Figueroa,† and Javier Rubio‡ Instituto de Fı´sica Teo´rica CSIC-UAM, Universidad Auto´noma de Madrid, Cantoblanco 28049 Madrid, Spain (Received 22 January 2009; published 31 March 2009) We study the details of preheating in an inflationary scenario in which the standard model Higgs, strongly nonminimally coupled to gravity, plays the role of the inflaton. We find that the Universe does not reheat immediately through perturbative decays, but rather initiates a complex process in which perturbative and nonperturbative effects are mixed. The Higgs condensate starts oscillating around the minimum of its potential, producing W and Z gauge bosons nonperturbatively, due to violation of the so- called adiabaticity condition. However, during each semioscillation, the created gauge bosons partially decay (perturbatively) into fermions. The decay of the gauge bosons prevents the development of parametric resonance, since bosons cannot accumulate significantly at the beginning. However, the energy transferred to the decay products of the bosons is not enough to reheat the Universe, so after about a hundred oscillations, the resonance effects will eventually dominate over the perturbative decays. Around the same time (or slightly earlier), backreaction from the gauge bosons into the Higgs condensate will also start to be significant. Soon afterwards, the Universe is filled with the remnant condensate of the Higgs and a nonthermal distribution of fermions and bosons (those of the standard model), which redshift as radiation and matter, respectively. We compute the distribution of the energy budget among all the species present at the time of backreaction. From there until thermalization, the evolution of the system is highly nonlinear and nonperturbative, and will require a careful study via numerical simulations.
DOI: 10.1103/PhysRevD.79.063531 PACS numbers: 98.80. k, 12.60.Fr, 98.80.Cq
the theory of reheating is also far from being complete, I. INTRODUCTION since not only the details but even the overall picture Inflation is nowadays a well-established paradigm, con- depend crucially on the different microphysical models. sistent with all the observations, that solves most of the It seems difficult to study the details of reheating in each puzzles of the hot big bang model in a very simple and concrete model without the experimental knowledge of the elegant way. It is able to explain not only the homogeneity strength of the interactions among the inflaton and the and isotropy of the present Universe on large scales, but matter fields. Because of this, most of the work until now also the generation of almost scale invariant primordial has focused on models encoding the different mechanisms perturbations that give rise to the structure formation [1]. that could play a role in the process, with the strength of the However, the naturalness of inflation is directly related to couplings set essentially by hand. The relative importance the origin of the inflaton. Most of the inflationary models of each one of these mechanisms can only be clarified in proposed so far require the introduction of new degrees of light of an underlying particle physics model, able to freedom to drive inflation. The nature of the inflaton is provide us with the couplings among the inflaton and completely unknown, and its role could be played by any matter fields. From this point of view it is very difficult candidate able to imitate a scalar condensate (typically in to single out a given model of inflation, and even more the slow-roll regime), such as a fundamental scalar field, a difficult to understand the details of the reheating process fermionic or vector condensate, or even higher order terms via the experimental access to the couplings. of the curvature invariants. The number of candidates We may be very far away from understanding the micro- motivated by particle physics is as big as the number of physical mechanisms responsible for inflation, but maybe extensions of the standard model (grand unified theories, the natural candidate for being the inflaton was already supersymmetry, extra dimensions, etc.), where it is not there long ago. If we do not want to introduce new dy- very difficult to find a field that could play the role of the namical degrees of freedom in the theory, apart from those inflaton [2]. present in the standard model (SM) of particle physics, and In addition, given a model we must find a graceful exit to if at the same time we require Lorentz and gauge invari- inflation and a mechanism to bring the Universe from a ance, we are left with just one possibility: the Higgs field. cold and empty post-inflationary state to the highly en- Early models of inflation in terms of a Higgs-like scalar 4 tropic and thermal Friedmann Universe [3]. Unfortunately, field h with a quartic self-interaction potential 4 h need an extremely small coupling constant 10 13 [2], and are *[email protected] also nowadays excluded at around 3 by the present ob- †daniel.fi[email protected] servational data [4]. However, the SM-based inflation may ‡[email protected] be rescued from these difficulties by replacing the usual
1550-7998=2009=79(6)=063531(22) 063531-1 Ó 2009 The American Physical Society GARCI´A-BELLIDO, FIGUEROA, AND RUBIO PHYSICAL REVIEW D 79, 063531 (2009) Einstein-Hilbert action by a nonminimal coupling of the the action (1) is not the most general one that can be written Higgs field to the Ricci scalar [5–9], in a scalar-tensor in a nontrivial background. As shown in [9,16] the simul- theory fashion, taneous existence of a reduced bare Planck mass MP and a nonminimal coupling of a symmetry breaking field to the Z ffiffiffiffiffiffiffi 4 p y scalar curvature, SIG ¼ d x g H HR: (1) Z ffiffiffiffiffiffiffi 2 4 p MP y The induced gravity (IG) action is indeed the most natural SHG d x g R þ H HR ; (2) 2 generalization of the standard model in a curved space- time, given the relevance of the nonminimal coupling to avoid the decoupling of the gauge bosons and can give rise gravity for renormalizing the theory [10]. It belongs to a to an inflationary expansion of the Universe together with a group of theories known as scalar-tensor theories, origi- potentially successful reheating. nally introduced by Brans and Dicke [11] to explain the In this paper we initiate the study of the reheating origin of the masses. In those theories not only the active process in the model presented in Ref. [16], in which the masses but also the gravitational constant G are determined symmetry breaking field is the standard model Higgs, by the distribution of matter and energy throughout the strongly nonminimally coupled to gravity and playing the Universe, in a clear connection with the Mach principle. role of the inflaton. The novelty and great advantage of this The interaction that gives rise to the masses should be the model is its connection with a well-known microphysical gravitational one, since gravity couples to all particles, i.e. mechanism, hopefully accessible in the near future accel- to their masses or energies. The gravitational constant G in erator experiments. The measurement of the Higgs mass these theories is replaced by a scalar function that perme- will complete the list of the couplings of the standard ates the whole space-time and interacts with all the ordi- model and, therefore, one should be able to study all the nary matter content, determining how the latter moves details of the reheating mechanism. This makes the model through space and time. Any measurements of an object’s under consideration extremely interesting and, potentially, mass therefore depends on the local value of this new field. predictive. Reheating in the context of scalar-tensor theo- The successful Higgs mechanism lies precisely in the ries has been studied in the Hartree approximation by same direction as Mach’s original idea of producing mass Ref. [17] and perturbatively by Ref. [18], but without the by a gravitational-like interaction. The role of the Higgs Higgs boson playing the role of the inflaton, and therefore field in the standard model is basically to provide the without an explicit coupling of the fundamental scalar field inertial mass of all matter fields through the local sponta- 1 to matter. neous symmetry breaking mechanism. The Higgs boson Studying the details of reheating in this Higgs-inflaton couples to all the particles in the standard model in a very scenario, we have paid special attention to the relative specific way, with a strength proportional to their masses impact of the different mechanisms that can take place. [12], and mediates a scalar gravitational interaction of We have found that the Universe does not reheat immedi- Yukawa type [13,14], between those particles which be- ately through perturbative decays, but rather initiates a come massive as a consequence of the local spontaneous complex process in which perturbative and nonperturbative symmetry breaking. According to the equivalence princi- effects are mixed. The Higgs condensate starts oscillating ple, it seems natural to identify the gravitational and par- around the minimum of its potential, producing Z and W ticle physics approaches to the origin of the masses. From gauge bosons due to violation of the so-called adiabaticity this point of view, the induced gravity action (1) would be condition. During each semioscillation, the nonperturba- an indication of a connection between the Higgs, gravity, tively created gauge bosons decay (perturbatively) into and inertia. Indeed, the action (1) is, at least at the classical fermions. This decay prevents the development of the usual level, just a different representation of Starobinsky’s model parametric resonance, since bosons do not accumulate of inflation [1,15], where inflation is entirely a property of significantly at the beginning. The energy transferred to the gravitational sector. Both representations of the same the decay products of the bosons is not enough to reheat the theory are simply related by a Legendre transformation. Universe within a few oscillations, and therefore the reso- This fact, together with the possibility of having an infla- nance effects will eventually dominate over the perturba- tionary expansion of the Universe, makes the model ex- tive decays. Around the same time, the backreaction from tremely appealing. Unfortunately, the induced gravity the gauge bosons into the Higgs condensate will also start model cannot be accepted as a completely satisfactory to be significant. Soon afterwards, the Universe is filled inflationary scenario, since the gauge bosons acquire a with the remnant condensate of the Higgs and a nonthermal constant mass in the Einstein frame and totally decouple distribution of fermions and bosons (those of the SM), from the Higgs-inflaton field [9], which translates into an which redshift as radiation and matter, respectively. We inefficient reheating of the Universe. Notice, however, that end the paper by computing the distribution of the energy budget among all the species present at the time of back- 1We are not considering here Majorana masses. reaction. From there until thermalization, the evolution of
063531-2 PREHEATING IN THE STANDARD MODEL WITH THE ... PHYSICAL REVIEW D 79, 063531 (2009) the system is highly nonlinear and nonperturbative, and Ricci scalar, H the Higgs field, and the announced non- will require a careful study via numerical simulations, to be minimal coupling constant. As shown in Ref. [16], the described in a future publication. parameters and the self-couplingpffiffiffiffi of the Higgs potential The paper is organized as follows. In Sec. II we present arep relatedffiffiffi by ’ 49 000 . In the unitary gauge, H ¼ the model with the Higgs field nonminimally coupled to h= 2, and neglecting all gauge interactions for the time the scalar curvature, transform it into a new frame where being, the Lagrangian for the Higgs-gravity sector in the the action takes the usual Einstein-Hilbert form, and derive so-called Jordan (J) frame takes the form an approximate inflationary potential. In Sec. III we study Z ffiffiffiffiffiffiffi the effect of the conformal transformation in the matter 4 p 1 SHG þ SSSB d x g fðhÞR g @ h@ h UðhÞ ; sector, including the interaction among the Higgs, vector 2 bosons, and fermions. Section IV is devoted to the analysis (5) of the different reheating mechanisms, both perturbative 2 2 and nonperturbative, that can take place, leaving for Sec. V where fðhÞ¼ðMP þ h Þ=2, and the analysis of the combined effect of parametric reso- nance and perturbative decays. We then study the back- UðhÞ¼ ðh2 v2Þ2 (6) reaction of the produced particles on the Higgs oscillations 4 and the end of preheating in Sec. VI. Finally, the conclu- is the usual Higgs potential of the standard model, with sions are presented in Sec. VII. vacuum expectation value (vev) v ¼ 246 GeV. In order to get rid of the nonminimal coupling to gravity, II. THE STANDARD MODEL HIGGS AS THE we proceed as usual, performing a conformal transforma- INFLATON tion [19] 2 The Glashow-Weinberg-Salam [12] action is divided g ! g~ ¼ g ; (7) into four parts: a fermion sector (F) which includes the kinetic terms for the fermions and their interaction with the such that we obtain the Lagrangian in the so-called gauge bosons; a gauge sector (G), including the kinetic Einstein (E) frame terms for the intermediate bosons as well as the gauge Z pffiffiffiffiffiffiffi fðhÞ E E 4 ~ ~ ~ 2 fixing and Faddeev-Popov terms; a spontaneous symmetry S þ S d x g~ R þ 3g~ r r ln HG SBS 2 breaking sector, with a Higgs potential and the kinetic term @~ h@~ h for the Higgs field including its interaction with the gauge 3 ~ 2 ~ 2 g~ r ln r ln 2 fields; and finally, a Yukawa sector (Y), with the interaction 2 2 among the Higgs and the fermions of the standard model, 1 4 UðhÞ : (8) SSM ¼ SF þ SG þ SSSB þ SY: (3) The simplest versions of this Lagrangian in curved space- The usual Einstein-Hilbert term can then be obtained im- 2 2 time follow the principles of general covariance and local- posing fðhÞ= MP=2, which implies the following ity for both matter and gravitational sectors. To preserve relation between the conformal transformation and the the fundamental features of the original theory in flat Higgs field space-time, one must also require the gauge invariance h2 and other symmetries in flat space-time to hold for the 2 ðhÞ¼1 þ 2 : (9) curved space-time theory. The number of possible terms in MP the action is unbounded even in this case, and some addi- This allows us to write the Lagrangian (8) completely in tional restrictions are needed. A natural requirement could terms of h be renormalizability and simplicity. Following these three Z principles (locality, covariance, and restricted dimension), pffiffiffiffiffiffiffi M2 1 2 þ 6 2h2=M2 SE þ SE d4x g~ P R~ P and the previously motivated requirement of not introduc- HG SSB 2 2 4 ing new dynamical degrees of freedom, the form of the 1 action is fixed, except for the values of some new parame- g~ @ h@ h 4 UðhÞ ; (10) ters to be determined by the physics. This procedure leads to the nonminimal Lagrangian for the standard model in where we have neglected a total derivative that does not the presence of gravity, given by contribute to the equations of motion. As we will be work- S ¼ S þ S ; (4) ing in the Einstein frame from now on, we will skip over SMG SM HG the tilde in all the variables to simplify the notation. where SSM is the standard model part (3) defined above, Notice that the conformal transformation (7) leads to a and SHG is the new Higgs-gravity sector, given by Eq. (2). nonminimal kinetic term for the Higgs field, which can be 1=2 Here MP ¼ð8 GÞ is the reduced Planck mass, R the reduced to a canonical one by making the transformation
063531-3 GARCI´A-BELLIDO, FIGUEROA, AND RUBIO PHYSICAL REVIEW D 79, 063531 (2009) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 2 2 2 2 d þ 6 h =MP 1 þ ð1 þ 6 Þh =MP ¼ 4 ¼ 2 2 2 ; dh ð1 þ h =MPÞ 0.8 (11) 0.6 where is a new scalar field. Doing this, the total action in V the Einstein frame, without taking into account the gauge V0 interactions, is simply 0.4 Z pffiffiffiffiffiffiffi M2 1 SE þ SE d4x g P R g @ @ Vð Þ ; 0.2 HG SSB 2 2
(12) -3 -2 -1 0 1 234 with 1 Vð Þ Uðhð ÞÞ (13) FIG. 1 (color online). Comparative plot of the exact solution 4ð Þ (red solid line) obtained parametrically from Eq. (14), the analytic formula (18) for the potential (blue dashed line), and the potential in terms of the new field . To find the explicit their parametrization (20) (green dotted line). form of the potential in this new variable , we must find the expression of h in terms of . This can be done by integrating Eq. (11), whose general solution is given by exact solution (red solid line) obtained parametrically from pffiffiffi Eq. (14) with the analytic formula (18) (blue dashed line). pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðhÞ¼ 1 þ 6 sinh 1ð 1 þ 6 uÞ Both solutions agree very well in the region of positive M but differ substantially for <0. The conformal trans- P pffiffiffiffiffiffi pffiffiffiffiffiffi u formation is even ill defined in the negative field region. 6 sinh 1 6 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; (14) From Eqs. (9) and (16) we have 1 þ u2 ffiffiffi p h2 where u h=MP. Since 1, we can take 1 þ 6 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ¼ 1 ¼ e 1 ¼ð1 e Þe ; (19) 6 and, using the identity sinh 1x ¼ lnðx þ x2 þ 1Þ for MP 1 2 4 ¼ e ; (16) MP j j 2 pffiffiffiffiffiffiffiffi Vð Þ¼ 2 ð1 e Þ ; (20) 1 4 where ¼ 2=3 and ¼ MP . The field is therefore directly related in this approximation (just in the limit which correctly describes the potential obtained from 1 and far from u ¼ 0) to the conformal transformation in Eq. (14), for the whole field range of interest. In Fig. 1, a very simple way, and the inflationary potential (13) is just this parametrization (green dotted line) is again compared given by to the exact solution (red solid line) obtained from Eq. (14). Around the minimum, the potential (20) can be approxi- Vð Þ¼ 4UðhÞ mated as 4 2 2 MP v 2 1 2 2 ¼ 2 e 1 þ 2 e : (17) Vð Þ¼2M þ Vð Þ; (21) 4 MP 2 2 2 v2 where M ¼ MP=3 is the typical frequency of oscil- Since v Mp, then 1 þ 2 1, and we can safely MP lation and V are some corrections to the quadratic ap- ignore the vev for the evolution during inflation and pre- proximation, which soon become negligible after inflation heating, and simply consider the potential ends; see Sec. IV. Note, nevertheless, that approximations (15) and (16), and therefore parametrization (20), do not M4 Vð Þ¼ P ð1 e Þ2: (18) describe correctly the potential for very small values of the 4 2 field v MP= . As can be read from Eq. (11), for d Notice that the previous potential only partially parame- j j t MP= , we have dh 1, and therefore, there is 4 trizes the original potential (6), since it neglects the region a transition in the potential from (21)toVð Þ 4 . <0, as can be seen in Fig. 1, where we compare the However, as will be shown in Sec. IV C, the transition 063531-4 PREHEATING IN THE STANDARD MODEL WITH THE ... PHYSICAL REVIEW D 79, 063531 (2009) region j j < t is several orders of magnitude smaller than where the functions VðhÞ, UðhÞ, and GðhÞ are given by the nonadiabaticity region j j < [see Eq. (79)], inside a h4 h2 which the concept of particle is not properly defined. UðhÞ¼ ðh2 v2Þ2 þ A ln þ B ; (25) Therefore, from now on we will neglect the change in the 4 128 2 Q2 1 2 2 4 behavior of the potential (from 2 M to 4 ) in this ‘‘small’’ field region, since . See Sec. IV for 1 h2 h2 t a FðhÞ¼ ðM2 þ h2Þþ C ln þ D ; (26) more details. 2 P 384 2 Q2 The analysis of the inflationary potential (20) can be performed either in the Jordan [20,21] or in the Einstein 1 h2 GðhÞ¼1 þ F ln þ E : (27) frame [16] with the same result. The slow-roll parameters 192 2 Q2 can be expressed analytically, in the limit of h2 2 2 MP= v , as a function of , Here A, B, C, D, E, and F are different combinations of the Higgs, gauge, and Yukawa couplings and their logarithms M2 V0ð Þ 2 2 2 [22,25,26], and Q is the normalization scale. Following ¼ P ¼ ; 2 Vð Þ ðe 1Þ2 Ref. [23], for the analysis of inflation we will just consider (22) the explicit form of the combination A, V00ð Þ 2 2ð2 e Þ ¼ M2 ¼ : X X P ð Þ 2 2 V ðe 1Þ A ¼ 3 g4 y4 ; (28) A f The slow-roll regime of inflation will end when ’ 1, A f which corresponds to the field value which is related with the local conformal anomaly. The factor 3 accounts for the polarizations of the gauge bosons, 1 2ffiffiffi end ¼ ln 1 þ p : (23) and a similar factor 4 for the fermions has been taken into 3 account. The new inflationary potential in the Einstein frame for a Note that the slow-roll parameter is then negative, Higgs field with nonminimal coupling 1 and mean ¼ 1 p2ffiffi < 0, so there is a small region of negative end 3 value much greater than the minimum of the classical (mass squared) curvature in the potential just after the end potential is given by [23] of inflation. The effective curvature of the potential will be 1 M2 2 UðhÞ negative until ¼ ln2, which corresponds to the in- ~ P Vð ~Þ¼ 2 : (29) flection point, given by ¼ 0. 2 F ðhÞ h¼hð ~Þ During the slow-roll regime H ¼ p ffiffi V1=2, which, eval- SR 3 The corresponding slow-roll parameters are uated at N ¼ 60 e-folds, is approximately given by H60 ’ M 4M4 h2 2 4 M2 A 2 2 , where M defines the natural inflationary energy scale of P P ~ ¼ 2 4 1 þ 2 ¼ 2 þ 2 ; this model as well as the frequency of oscillations (21) 3 h hI 3 h 64 (30) during reheating. At the end of inflation H ¼ p ffiffi V1=2 or, 2 end 2 4MP 2 M ~ ¼ 2 ; equivalently, Hend ’ 3 H60 ¼ 3 . 3 h The radiative corrections for a model containing a single 2 2 2 64 MP scalar field h nonminimally coupled to gravity were ge- with hI ¼ A . Using the WMAP þ BAO þ SN con- nerically calculated in Ref. [22]. The specific radiative straint [27] at the 2 confidence level gives a value for the corrections for the model under consideration, estimated spectral index [23] in Ref. [16], were recently reviewed in Ref. [23]. In what follows until the end of this section, we will summarize the 0:934 063531-5 GARCI´A-BELLIDO, FIGUEROA, AND RUBIO PHYSICAL REVIEW D 79, 063531 (2009) obtain the Higgs self-coupling at such scale. This differs clear: It changes all dimensional quantities (lengths, from the analysis performed in [23], where the coupling masses, etc.) in every point of the space-time, leaving their constants were evaluated at the electroweak scale instead ratios unchanged. of at the characteristic energy scale of inflation, M. At that In this section we will apply the previous prescription to scale the coupling constants for the gauge groups SUð2ÞL the different sectors of the action (3). Consider, for in- 2 2 and Uð1ÞY are roughlypffiffiffi equal, g1 g2 0:3 and stance, the spontaneous symmetry breaking sector, respon- 2 2 cos W ¼ sin W 1= 2. The total anomalous scaling sible for the masses of the intermediate gauge bosons W constant at that scale [see Eq. (28)] in terms of g2 and and Z, cos , W Z ffiffiffiffiffiffiffi 4 p 2 þ 1 2 6 g4 1 SSSB d x g mW W W þ mZZ Z : (40) A 2 1 þ cos 4 y4 ; (35) 2 8 2 W f In the standard model, the masses of the SUð2Þ bosons are together with the bounds obtained from the WMAP þ due to a spontaneous symmetry breaking mechanism, real- BAO þ SN 2 c.l. constraints [27] [see Eq. (32)], give ized by the constant vev of the Higgs field, and therefore us the following range for the self-coupling of the Higgs are constant. In our case the Higgs field evolves with time, field at the scale M, giving rise to variable effective masses for the gauge 0:424 < ðMÞ < 0:482: (36) bosons This range can be propagated back to obtain a value at the g2h mW M2 mW ¼ ;mZ ¼ ; (41) scale Z through the renormalization group equations, 2 cos W 2 1 1 3 M ð Þ¼ ðMÞþ log : (37) with W the Weinberg angle defined as W ¼ 4 2 2 1 tan ðg1=g2Þ, and g1 and g2 are the coupling constants Note that we have neglected the effect of the gauge and corresponding to Uð1ÞY and SUð2ÞL at the scale M, where Yukawa couplings, since the complete solution of the the relevant physical processes during preheating will take renormalization group equations is out of the scope of place. As mentioned before, numerically this correspond to 2 2 2 this preliminary study. If we integrate this equation and a value g1p ffiffiffi g2 0:30, which implies sin W ¼ 2 take into account the present observational bounds [28], we cos W 1= 2. From now on we will use these values obtain a Higgs mass in the range for numerical estimations. In agreement with the above prescription for transforming masses and fields, the action 114:5 GeV III. THE STANDARD MODEL MATTER SECTOR W Z W~ ; Z~ ; IN THE EINSTEIN FRAME 2 2 2 j j 2 The length scales are conventionally defined in such a 2 mW g2MPð1 e Þ 2 m~W m~W ¼ 2 ¼ ; m~Z ¼ 2 : way that elementary particle masses are the same for all 4 cos W times and in all places. This implies, for instance, that if (43) under a conformal transformation the Lagrangian of a free particle transforms as The same can be applied to the interactions between fer- Z Z mions and gauge bosons. Let us consider, for instance, L L~ m ~ 1P ¼ mds ! 1P ¼ ds; (39) Z pffiffiffiffiffiffiffi S ¼ S þ S d4x g m F NC CC the mass should be accordingly redefined as m~ to g g g express it in the new system of units. The previous argu- 2ffiffiffi þ 2ffiffiffi þ 2 p W J þ p W J þ Z JZ ; (44) ment applies also for classical fields [29]. The rescaling of 2 2 cos W all fields (including the metric tensor) with an arbitrary þ space-time dependent factor , taken with a proper con- where J dL uL, J u L dL are the charged cur- formal weight for each field, will leave the physics unaf- rents carrying the information about the couplings of the fected. The physical interpretation of this symmetry is W to the standard model fermions, and 063531-6 PREHEATING IN THE STANDARD MODEL WITH THE ... PHYSICAL REVIEW D 79, 063531 (2009) 1 1 2sin2 2 3g3M W E 3g2m~W 2 p j j 1=2 J u L uL dL dL u L uL ¼ ¼ ð1 e Þ Z 2 2 3 W 16 32 1=2 sin2 3 W 3cos W þ dL dL (45) ¼ E (53) 3 2LIPS Z j j 1=2 is the neutral current with the information of the couplings where m~W , m~Z /ð1 e Þ are the dynamical of the Z boson. In the Einstein frame the action (44) masses in the Einstein frame; see Eq. (43). preserves its form, Z pffiffiffiffiffiffiffi E E E 4 IV. REHEATING IN THE STANDARD MODEL SF ¼ SNC þ SCC d x g~ OF PARTICLE PHYSICS g g g 2ffiffiffi ~ þ ~ 2ffiffiffi ~ ~þ 2 ~ ~ In this section we will analyze the different mechanisms p W J þ p W J þ Z JZ ; (46) 2 2 cos W that could give rise to efficient reheating of the Universe, both perturbative and nonperturbative. The natural mecha- as long as we redefine the currents as nism, given the strength of the interactions of the Higgs boson with the standard model particles, would be a per- J J ~ Z ~ turbative reheating process right after the end of the slow- J Z ; J ; (47) 3 3 roll regime. However, as we will see, perturbative reheat- ing is not efficient enough, and nonperturbative effects which is equivalent to redefining the Dirac fields must be taken into account. Given the form of the potential d u (20), very different (p)reheating mechanisms could, in d~ ; u~ : (48) 3=2 3=2 principle, take place, from a tachyonic production in the region between the end of inflation and the inflection point Finally, concerning the Yukawa sector, we have, for a given [31–33], an instant preheating mechanism [34], and a para- family of the quark sector, metric resonance effect around the minimum of the poten- tial [35–37]. Therefore, it will be crucial to disentangle the Z pffiffiffiffiffiffiffi 4 contributions of each mechanism and quantify their rela- SY d x gfmddd þ muuu g; (49) tive importance. In order to study the different reheating mechanisms it where d and u stand for down- and up-type quarks, re- will be useful to have an approximate expression for the spectively. The effective masses in the Jordan frame, mf ¼ evolution of the inflaton. As mentioned before, we can y h pfffiffi , become expand potential (20) for the range of interest, as 2 Vð Þ¼1M2 2 þ Vð Þ; (54) y M 2 f ffiffiffiffiffiffiP j j 1=2 m~ f p ð1 e Þ (50) 2 2 2 2 with M ¼ MP=3 the typical frequency of oscillation. The first terms of the corrections V to the quadratic in the Einstein frame. potential are given explicitly by On the other hand, the total decay widths, summing over all the allowed decay channels in the standard model of the Vð Þ¼ j j3 þ 4 þ Oðj j5Þ; (55) W and Z bosons into any pair of fermions and over all the 3 4 pffiffiffi polarizations of the gauge bosons, are given, respectively, 2 2 with ¼ MP= 6 and ¼ 7 =27 . The Klein- by [30] Gordon equation for the inflaton, 2 2 0 3g2mW g2mZ € þ 3H _ þ V ð Þ¼0; (56) þ W ¼ W ¼ ; Z ¼ 2 LIPS; 16 8 cos W can then be written, for a power-law evolution a / tp,as (51) t2 € þ 3pt _ þ t2M2½1 þ M2ð Þ ¼ 0; (57) where LIPS denotes the Lorentz invariant phase-space where we have neglected the Higgs’ interactions with other factors fields, because they are proportional to the number density 7 11 49 of particles of a given species and therefore they will be LIPS sin2 þ sin4 : (52) 4 3 W 9 W negligible (see Sec. VI) during the first oscillations of the Higgs field. The backreaction of other particles into the The decay rates will preserve their functional form in the dynamics of the Higgs will only be relevant once their Einstein frame, since they are only changed through the occupation numbers have grown sufficiently. The nonlin- conformal transformation of the masses, i.e. ear terms of the Higgs’ self-interaction, described by 063531-7 GARCI´A-BELLIDO, FIGUEROA, AND RUBIO PHYSICAL REVIEW D 79, 063531 (2009) 2 2 O 3 end M j jþ þ ð Þ; (58) ðtÞ¼ sinðMtÞ: (66) Mt will also be negligible from the very beginning of reheat- Rewriting the previous equation in terms of the number of ing, j M2ð Þj 1, as we will justify a posteriori. Thus, times the inflaton crosses zero, j ¼ðMtÞ= , or equiva- neglecting such a term in the effective equation of , the lently in terms of the number of oscillations N ¼ j=2, then general solution can be expressed as ðtÞ end sinð2 NÞ¼ end sinð jÞ XðjÞ sinð jÞ: 1 2 N j ðtÞ¼ ½AJþ ðMtÞþBJ ðMtÞ ; (59) ðMtÞ (67) with A and B constants depending on the initial conditions Therefore, the Higgs condensate oscillates with a decreas- (end of inflation), and J ðxÞ Bessel functions of order , ing amplitude XðjÞ/1=j. We can obtain an upper bound, with ¼ð3p 1Þ=2. For a reasonable power index, j j < 0:122=N, on the amplitude of the Higgs field p>1=3—for matter p ¼ 2=3, while for radiation after N oscillations, which in terms of the correction of p ¼ 1=2—the second term in the right-hand side of M2 to 1 [see Eq. (57)] implies j M2j < 0:122, 0.0615, or Eq. (59) diverges in the limit Mt ! 0 and therefore should 0.0244, after the first N ¼ 1, 2, and 5 oscillations, respec- be discarded on physical grounds. The physical solution is tively. Thus, from the very beginning, the effective poten- then simply given by tial of the Higgs field tends very rapidly to that of a harmonic oscillator, which justifies a posteriori the ap- ðtÞ¼AðMtÞ ðð3p 1Þ=2ÞJ ðMtÞ; (60) ðð3p 1Þ=2Þ proximation j M2j 1 used in the derivation of which, making use of the large argument expansion (Mt Eqs. (59) and (67). 1) of fractional Bessel functions [38], can be approximated Note that if we neglect the presence of other fields and by a cosinusoidal function consider the Higgs condensate as a free field only damped sffiffiffiffi by the expansion rate, then we can easily estimate the 2 number of semioscillations before the amplitude of the ðtÞ A ðMtÞ ð3p=2Þ cosðMt ð3p=2Þð =2ÞÞ: (61) field becomes smaller than the transition value t Mp= , defined in Sec. II.Forj j < t, the Higgs potential The normalization constant A can be fixed if we consider will not be approximated anymore by Eq. (54), but rather 4 that the oscillatory behavior starts just at the end ofp in-ffiffiffi by a quartic form ð =4Þ . This will happen when 1 end flation, i.e. ðt ¼ 0Þ¼ end ¼ Mp logð1 þ 2= 3Þ Xðj Þ= < 1, which implies t t jt (23). In this case, pffiffiffi logð1 þ 2= 3Þ 4 1=2ð3p 1Þ 1 j jt Oð10 Þ: (68) A ¼ end2 ð2ð3p 1ÞÞ!; (62) where we have made use of the limit of the Bessel func- Therefore, if before that moment, the Higgs field has not tions when Mt 1, yet efficiently transferred its energy into other fields, then after jt semioscillations, the transition in the behavior of ðMtÞ1=2ð3p 1Þ the potential will also imply a change in the expansion rate, J ðMtÞ : (63) 1=2ð3p 1Þ 1=2ð3p 1Þ 1 from matterlike, characteristic of quadratic potentials, to 2 ð2 ð3p 1ÞÞ! radiationlike, characteristic of quartic potentials. The energy and pressure densities associated with the general solution (60) are given, after averaging over several A. Perturbative decay of the Higgs field oscillations, by A natural reheating mechanism, given the strength of the 1 2 1 2 2 h2 _ þ 2M i interactions of the Higgs boson with the standard model particles, would be a perturbative decay process of the 1M2X2½hcos2ðMt 3 p=4Þi þ hsin2ðMt 3 p=4Þi 2 inflaton quanta into the standard model particles right after 1 2 2 ¼ 2M X ; (64) the end of the slow-roll regime. As we saw, soon after the end of inflation the effective potential for the Higgs field can be approximated by a simple quadratic potential (21). p h1 _ 2 1M2 2i 2 2 In this approximation, the masses of fermions and gauge 1 2 2 2 2 2M X ½hcos ðMt 3 p=4Þi hsin ðMt 3 p=4Þi bosons [see Eqs. (43) and (50)] are simply given by ¼ 0; (65) j j 1=2 j j 1=2 m ’ y M ;m’ g M ; f f 2 M P W 2 4 M P with XðtÞ/ðMtÞ ð3p=2Þ. Since the averaged pressure is P P 1=2 negligible, p 0, then aðtÞ/tp with p 2=3. Using g2 j j mZ ’ MP: (69) this fact, the physical solution is finally expressed as cos W 4 MP 063531-8 PREHEATING IN THE STANDARD MODEL WITH THE ... PHYSICAL REVIEW D 79, 063531 (2009) In order to have a perturbative decay, two conditions must that allows the decay to be possible, which prevents con- be fulfilled: dition (2) from being fulfilled. The decay width is much (1) There should be enough phase space in the final smaller than the Hubble rate for a huge number of oscil- states for the Higgs field to decay, i.e. M>2mf, lations. Looking at Eq. (71), we realize that the Higgs 12 mA, which will only happen when the amplitude of condensate should oscillate j 10 times before the de- the Higgs field becomes smaller than a certain criti- cay rate into electrons overtakes the Hubble rate. cal value c. In particular, for a decay into gauge One can check that the previous conclusions also hold bosons and/or fermions, in the light of Eq. (69), one for the rest of the fermions of the standard model. When needs there is phase space for the Higgs to decay into a given sffiffiffiffi species, the decay rate does not catch up with the expan- 1 sion rate and, vice versa, if the decay rate of a given species & M; (70) c g2 2 overtakes the expansion rate, there is no phase space for the decay to happen.2 Therefore, during a large number of where g ¼ g2, g2= cos w for the W and Z bosons, oscillations, the Higgs field is not allowed to decay pertur- respectively, and g ¼ yf for the fermions. When batively in any of the standard model fields. Moreover, compared to the initial amplitude (23) of the Higgs before any of those decay channels is opened, many other 2 field at the end of inflation, c= end ’ 1=ð3g Þ,we interesting (nonperturbative effects) will take place, as we see that this critical value is much smaller than end will describe in detail in the next sections. for the gauge bosons and the top quark, of the same order for the bottom and charm quarks, and even B. Tachyonic preheating and nonadiabatic particle greater for the rest of the quarks and the SM leptons. production at the inflection point g2 (2) The Higgs decay rate 8 M has to be greater 2 1 M 2 end 2 As we pointed out in Sec. II, the effective square mass of than the rate of expansion H ¼ 2 ð Þ ð Þ , 3MP 6 MP j the Higgs field is negative just after the end of inflation where we have used Eq. (64). Such a condition, > and will be so till the inflection point. When this happens H, can be translated into the following inequality: spinoidal instability takes place [31–33] and long wave- lengths quantum fluctuations k, with momenta k 063531-9 GARCI´A-BELLIDO, FIGUEROA, AND RUBIO PHYSICAL REVIEW D 79, 063531 (2009) can still consider an initial spectrum of quantum vacuum frequency of oscillation of the gauge boson is much higher fluctuations even at the inflection point. For simplicity, all than the one of the Higgs field . This implies that during our analytical estimations have been done ignoring this most of the time the effective masses of the intermediate period of tachyonic instability, taking as initial conditions boson are changing adiabatically and an adiabatic invariant at the end of inflation the amplitudepffiffiffi of the Higgs conden- can be defined: the number of particles. However, for MP values of very close to zero, the adiabaticity conditions sate end logð1 þ 2= 3Þ (23) and quantum vacuum fluctuations. m~_ m~_ Another physical effect before the Higgs condensate W Z 2 1; 2 1 (78) reaches the bottom of the potential for the first time will m~W m~Z be the particle production in the inflection point, due to the are violated. In such a case, there will be an inequivalence violation of the adiabaticity condition, between the vacua before and after the passage of 2 through the minimum of the potential, which can be inter- j!_ kj j!kj; (73) preted as particle production [35,36]. In terms of the field where the frequency of oscillation of the fluctuations is , the violation of (78) corresponds to the region a & 2 2 00 !kðtÞ¼k þ V ð Þ. Differentiating this and rewriting the & a, 3 adiabaticity condition as !_ k!k !k, we find that only j _ ðtÞj2 1=3 those modes within the band ¼ ; (79) a 2 000 g MP V ð Þ _ k3 (74) 2 where, from now on, g ¼ g2, g2= cos W for the W or Z bosons. Only outside this region, the notion of particle 2 are amplified. At the end of inflation H_ ¼ H which makes sense and an adiabatic invariant can be defined. implies _ V1=2ð Þ. Extrapolating the previous for- Taking this into account and approximating the velocity mula to the inflection point (ip), we get of the field around zero as _ ðjÞ M end ¼ MXðjÞ [see sffiffiffiffiffiffi j pffiffiffi Eq. (67)], the general expressions (79) can be approxi- V0 3M _ ip ¼ ; (75) mated as 4 4 2 2 1=3 M j endj which indeed seems to be a very good approximation if we aðjÞ¼ 2 2 2 j g MP compare it with the result of a numerical solution beyond the slow-roll regime. Inserting (75) into Eq. (74) we get 1=3 j1=3 ¼ pffiffiffi XðjÞ: (80) 2=3 000 3 3 logð1 þ 2= 3Þ g V ð Þ _ ip H k3 ¼ 60 ; (76) ip 2 2 Note that the previous regions are indeed very narrow which corresponds to a maximum excited wave number at compared to the amplitude of the oscillating Higgs, a 10 2j1=3XðjÞ. Therefore, the particle production that takes the inflection point given by kip < 0:4 M. Again, here the time of production is so brief that the occupation number of place in that region happens within a very short period of ¼ modes within the band is not significantly enhanced. We time as compared to the inflatons’ oscillation period T will have to wait until the next stage of consecutive oscil- 2 =M, lations of the Higgs field around zero for a significant 2 t ðjÞ a 10 2j1=3M 1 T: (81) production of particles. a j _ j C. Instant preheating Notice, indeed, that different values of do not appreci- ably change the above conclusions about the smallness of During each oscillation of the Higgs field , the rest of the nonadiabatic regions. Given the weak dependence of the quantum fields that couple to it will oscillate many t / j1=3, many semioscillations ( 103) will pass before times. Consider for instance the interaction of the Higgs the fraction of time spent in the nonadiabatic zone will field with the Z bosons around the minimum of the poten- increase from 1% to 10%, as compared with the period of tial. In this region the associated action (42) can be ap- oscillations. This holds also independently of the species, proximated by a trilinear interaction where the masses of W or Z bosons. the W and Z bosons (43) are given by Moreover, despite the smallness of a as compared to g2M g2M the amplitude XðjÞ, it is important to note that the field m~ 2 ’ 2 P j j; m~2 ’ 2 P j j; (77) W 4 Z 4 cos2 range corresponding to the region of nonadiabaticity still is W much greater than those critical regions defined in Sec. II. which are much greater than the inflaton mass M for the In particular, let us recall that there is a field value, t main part of the oscillation of . As a result, the typical MP= , below which there is a transition of the effective 063531-10 PREHEATING IN THE STANDARD MODEL WITH THE ... PHYSICAL REVIEW D 79, 063531 (2009) potential from a quadratic to a quartic behavior. However, þ 1 1 nkðj Þ¼ðTk ðjÞ 1Þþð2Tk ðjÞ 1Þnkðj Þ this is well inside the region of nonadiabaticity, t a, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 as we emphasized before. Moreover, there is also an inter- þ 2 cos j Tk ðjÞðTk ðjÞ 1Þ val of Higgs field values, j j < c, for which the Higgs qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi perturbative decay into W, Z and top quarks can occur [see nkðj Þðnkðj Þþ1Þ; (86) Eq. (70) in Sec. IVA], which nevertheless is also much smaller than the nonadiabaticity interval, c a. where j are some accumulated phases at each scattering, We will now discuss the nonperturbative creation of which we will discuss later on in Sec. V since they will not particles in the nonadiabatic region. This production is play any role in the following discussion of this section. formally equivalent to the quantum mechanical problem The inverse of the transmission probability for the jth of a particle scattering in a periodic potential. In the case scattering can be expressed as [39] under consideration the equations of motion for the fluc- 1 2 2 0 2 tuations of each gauge field with a given polarization will Tk ðjÞ¼1 þ ½Aið xj ÞAi ð xj Þ 00 2 2 2 be given by Wk þðk =a þ m~WÞWk ¼ 0, and the corre- þ Bið x2ÞBi0ð x2Þ 2; (87) sponding one for the Z fluctuations. Expanding Eq. (67) j j around the jth zero at time t ¼ j, the evolution equation 1=3 j with xj K=ðq=jÞ and AiðzÞ, BiðzÞ the Airy functions. of the fluctuations can be approximated as Note that we have used the Wronskian normalization, 0 0 1 AiðzÞBi ðzÞ BiðzÞAi ðzÞ¼ . k2 g2M j sinðMðt t ÞÞj Consider the situation n ðj Þ 1, which is certainly W00 þ þ 2 p end j W ¼ 0; (82) k k a2 4 j k true in the first scattering j ¼ 1, or can happen for j>1 if the previously produced gauge bosons have fully decayed into fermions. In such a case, k2 g2M j sinðMðt t ÞÞj n ðjþÞ T 1ðjÞ 1; (88) Z00 þ þ 2 p end j Z ¼ 0: (83) k k k a2 4 j cos2 k W where we have retained only the first term of Eq. (86). This corresponds to the spontaneous particle creation of W and Notice that around the zeros of the inflaton the sinusoidal Z bosons each time the Higgs crosses zero and, therefore, behavior j sinðMðt tjÞÞj can be very well approximated tells us about the number of particles of these species that by j sinðMðt tjÞÞj Mjt tjj , which allows us to are created in each zero crossing. The momenta distribu- rewrite Eqs. (82) and (83) as Schro¨dinger-like equations, tion is shown in Fig. 2. In particular, the total number of q 00 W 2 0.035 Wk j jWk ¼ K Wk; j (84) q 0.030 Z00 Z j jZ ¼ K2Z ; k j k k 0.025 where primes denote derivatives with respect to the re- 0.020 scaled time ¼ Mt, K is the rescaled momentum K k , j aM k and n 0.015 0.010 2 2 2 2 g2 end Mp 3g2 end qW ¼ cos WqZ ¼ ¼ 4 M 4 0.005 (85) 0.000 0.0 0.5 1.0 1.5 2.0 2.5 3.0 are the usual resonance parameters [36]. Each time the x inflaton crosses zero can therefore be interpreted as the quantum mechanical scattering problem of a particle cross- FIG. 2 (color online). Spectral distributions (88) for the gauge bosons created in a single zero crossing through the first term of ing an inverted triangular potential. Let T and R ¼ 1 T, Eq. (86), calculated after j ¼ 1, 2, 5, and 10 oscillations (from for either W or Z, be the transmission and reflection left to right). The horizontal axis represents x k=Mq1=3,so probabilities for a single scattering in this periodic trian- x ¼ 1 is the typical width of the band of momenta of particles gular barrier. The number of particles just after the jth created at the first scattering. For later times, the distributions þ scattering, nkðj Þ, in terms of the previous number of broaden out to greater momenta, since the argument of Eq. (88), 1=3 particles nkðj Þ just before that scattering, can be written xj, behaves as / j . The typical momenta of the distribution as [35,36] agree with the one calculated in Sec. IV C. 063531-11 GARCI´A-BELLIDO, FIGUEROA, AND RUBIO PHYSICAL REVIEW D 79, 063531 (2009) produced particles of a given species with a given polar- ~2 2 ~2 2 ~2 2 ~2 2 2ðkF þ m~FÞ¼kW þ m~W; 2ðkF þ m~FÞ¼kZ þ m~Z: ization, just after exiting the nonadiabatic region around (90) the jth zero crossing, can be obtained as In terms of the field , the previous equations can be Z 1 1 q rewritten as þ 2 1 I 3 nðj Þ¼ 2 3 dkk ½Tk ðjÞ 1 ¼ M ; (89) 2 aj 0 2j k~2 1 k~2 g2 F ¼ W þ 2 1; (91) R m~2 y2 M2 ðe j j 1Þ 4 I 1 2 0 2 2 0 2 2 2 F F P with ¼ 0 ½Aið x ÞAi ð x ÞþBið x ÞBi ð x Þ x dx 0:0046 and q the resonant parameters given by Eq. (85). Thus, the only difference between the number of W and Z k~2 1 k~2 g2 bosons produced is simply encoded in the different reso- F ¼ Z þ 2 1; (92) 2 2 2 j j 2 2 2 2 m~F yF M ðe 1Þ 4cos W nance parameter, qW / g2 and qZ / g2=cos W, respec- P tively. Notice that the effect of the nonperturbative production of these particles is proportional to the coupling for the W and Z fields, respectively. Note that the relativ- square. If the couplings of the Higgs field to the gauge istic or nonrelativistic nature of a given particle is some- 1 thing intrinsic to the particle and should not depend on the bosons were not so large (g2 0:5, cos W 1:4), then their production would be very suppressed. Strictly speak- conformal frame. As expected, the transitions Z ! tt , ing, the previous analysis is just valid for gauge bosons. W ! tb are not allowed in the Einstein frame. For the ~2 2 The production of fermions through this mechanism is rest of the quarks kF m~F, which implies that all the different and more involved than for bosons. Never- fermions produced in the decay of the W and Z bosons are theless, if the effect is, as expected, proportional to the clearly relativistic, as was the case in the Jordan frame. The Yukawa coupling squared [40], then only the top quark total number density of gauge bosons nðjþÞ present just production would be non-negligible. after the jth crossing will decay exponentially fast until the After the passage through the minimum of the effective next crossing, due to the perturbative decay into fermions. potential, the number of W and Z particles remains (al- Therefore the total number density just previous to the ðj þ most) constant while their masses grow when the field 1Þth zero crossing, nððj þ 1Þ Þ, is given by increases. The W and Z bosons tend to decay into fermions R E 1 E tjþ1 dt in a time t h i , where are given by Eq. (53), t W;Z j W;Z nððj þ 1Þ Þ¼nðjþÞe j ¼ nðjþÞe h ijT=2: (93) while h ij represents a time average between the jth and the ðj þ 1Þth scatterings. Given the time dependence of the The number of fermions produced between those two field (67), the typical time of decay turns out to be t ’ scatterings, nFðjÞ, is simply given by 0:64j1=2M 1 for the Z bosons and a bit bigger, t ’ 1=2 1 þ h ZijT=2 1:55j M , for the W bosons, as was expected. This nFðjÞ¼2 3 ½nZðj Þð1 e Þ implies that in a semiperiod, T=2 ¼ M 1, the nonpertur- þ 2n ðjþÞð1 e h W ijT=2Þ ; (94) batively produced gauge bosons at the jth scattering decay W significantly before the next scattering takes place, at least for the first scatterings. However, as the amplitude of the where the factor 2 3 takes into account that each gauge Higgs field decreases with time due to the expansion of the boson can have one out of three polarizations and decay into two fermions, while the extra factor 2 in front of nW Universe, the probability of decay of the gauge bosons [see þ Eq. (53)] becomes smaller and smaller as time goes by. accounts for both the W and W decays. The averaged This explains the j1=2 behavior, which essentially means value of the decay widths in the previous expressions can that after a certain number of oscillations, the number of be estimated [see Eq. (53)] as produced fermions through the perturbative decays per g 3 M LIPS 2 P ffiffiffi j j 1=2 semioscillation will eventually become negligible. h Z!allij ¼ p hð1 e Þ ij In the Jordan frame the standard model presents its usual cos W 16 form and the fermions produced in the decay of the Z and 2 Z FðjÞ; (95) W bosons are mainly relativistic. Since both momenta and T masses transform in the same way under a change of conformal frame, if a gauge boson is allowed to decay into a pair of fermions in the Jordan frame, it will also be 3cos3 2 h i ¼ W h i W FðjÞ; (96) able to decay in the Einstein frame. Therefore, the relation W!all j 2LIPS Z!all j T between the typical momenta and masses of those fermions and gauge bosons (W, Z) in the conformally transformed where T ¼ 2 =M is the typical oscillation period and we frame is simply given by have defined 063531-12 PREHEATING IN THE STANDARD MODEL WITH THE ... PHYSICAL REVIEW D 79, 063531 (2009) j j 1=2 FðjÞ hð1 e Þ ij Fð1Þ 6½ nZð1ÞEF ð1Þþ2 nWð1ÞEF ð1Þ Z W Z ðjþ1Þ j sinx=xj 1=2 1 1 1 2ffiffiffi 2 2 ¼ dx 1 1 þ p ¼ " M end 2 Fð1Þ; (101) j 3 2 1 with 0:3423 pffiffiffi : (97) j 33=2 2ð2 þ cos 3 ÞIg3 " W 2 3 10 5 3=4: Note that the last approximated equality is simply a (good) 1=2 1=2 8 ð endÞ fit to FðjÞ for all j. The constants , are just numerical Z W (102) factors depending of the parameters of the model and the decaying species, The energy density of the inflaton, evaluated at the maxi- pffiffiffi mum amplitude of the first semioscillation, is given (64)by g 3 3 1=2 ¼ 2 LIPS 14:23 ð1=4Þ; Z 1=2 1 2 2 cos W 16 (98) 2 2 ð1Þ M end : (103) 3cos3 2 3 W 5:91 ð1=4Þ: W 2LIPS Z Therefore, the ratio between the energy density of the Mt : ð Þ ð Þ fermions and of the inflaton, at that moment, 1 5 ,is Using the notation EFZ j and EFW j for the mean energy of the fermions produced between tj and tjþ1, Fð1Þ "Fð1Þ 5 3=4 ð1Þ ¼ 2 2 10 ; (104) from the decay of Z or W bosons, respectively, then we find ð1Þ ð2=3Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 which means that initially, e.g. for ¼ 0:4, only 0:004% EF ðjÞ h kF þ mFij hkFij hmZij Z 2 of the inflaton’s energy has been transferred to the fermi- g 2 FðjÞM ; (99) ons. Thus, the so-called instant preheating mechanism [34] 1=2 p 4 cos W becomes frustrated here, because in order to make it work efficiently, the couplings of the theory must be really fine- qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 tuned, in such a way that a significant fraction of the energy E ðjÞ h k2 þ m2 i hk i hm i FW F F j F j 2 W j of the inflaton was transferred (in the first semioscillation) g to the decay products of the bosons to which the inflaton is 2 FðjÞM ; (100) 4 1=2 p coupled. Moreover, in the instant preheating scenario, the produced fermions must be nonrelativistic, while the ef- where we have used the fact that the produced fermions are fective behavior of the background inflaton should be relativistic [see Eqs. (91) and (92)], while the gauge bosons effectively mimicking that of relativistic matter (like e.g. are nonrelativistic [see Eq. (130) in Sec. V]. in 4 models). Only in this case, it would be guaranteed Let us work now under the following hypothesis: We that the remnant energy of the inflaton would decay faster will consider that the perturbative decay of the gauge than that of the fermions, thanks to the extra suppression bosons into fermions is sufficiently effective, such that factor 1=a due to the expansion of the Universe. If the the gauge bosons do not accumulate significantly. This inflaton would effectively behave as nonrelativistic matter amounts to neglecting initially a potential effect of para- and the produced fermions were relativistic, the energy of metric resonance. This is of course a rough approximation, the inflaton could again overtake that of the fermions very which is only valid for the first oscillations, where soon, because now the fermion’s energy would decrease e FðjÞ 1. Numerically, after j ¼ 1, 2, 10, 15, and 20 faster than that of the background. That is, precisely, the zero crossings, the 99.5%, 98.5%, 94.2%, 87.4%, 81.9%, situation we have in the scenario under discussion. Even if 77.4%, respectively, of the produced Z particles have de- we had found that ð1Þ Oð1Þ, the relativistic nature of the cayed into fermions (and a similar though smaller fraction fermions and the nonrelativistic effective behavior of the of the W bosons). This implies that there will always be a Higgs oscillations would have prevented the Universe from remnant of the gauge bosons produced at each scattering, instantaneously reheating at that point. that will not decay in one semiperiod of the inflaton’s One could hope that after a certain number of oscilla- oscillation. Let us neglect this for the time being, therefore tions, let us say jp, that ratio would grow to a value ðjpÞ ignoring the possibility of having parametric resonance, Oð1Þ. The successively produced fermions, generated each and estimate the energy transferred simply through the semioscillation through the perturbative decay of the (non- perturbative decay into fermions, during the first perturbatively produced) W and Z bosons, could perhaps oscillations. accumulate a sufficient amount of energy that could finally In particular, the energy density of those fermions pro- equal that of the Higgs condensate. This does not seem duced after the first scattering, averaged over the first totally unreasonable because the total energy stored in the semioscillation between Mt ¼ and Mt ¼ 2 , will be Higgs decreases with the expansion of the Universe as 063531-13 GARCI´A-BELLIDO, FIGUEROA, AND RUBIO PHYSICAL REVIEW D 79, 063531 (2009) 2 / 1=j [see Eq. (64)], such that the total amount of question of fermionic preheating at each zero crossing energy that we would require to transfer to the fermions deserves more investigation, and we will address it in a would be less and less. Moreover, the number of fermions future publication. would only increase as time goes on, so one keeps adding The numerical values of the ratio "ðjÞ after e.g. j ¼ 1,2, energy each semioscillation in the form of newly produced 5, 10, 15, and 20 semioscillations, e.g. for ¼ 0:4, are, fermions. Thus, these two effects would contribute to the respectively, "ðjÞ½ 105 3:90, 5.97, 11.82, 37.26, 65.04, increment of the ratio of the energy between the fermions and 97.59. For different values of , these numbers do not and the Higgs. On the other hand, the relativistic nature of change significantly. Thus, we see that the transferred the fermions and the decreasepffiffiffi of their production rate with energy from the Higgs field to the fermions through the the expansion, as / 1= j, would tend to decrease such a gauge bosons is generically a very slowly growing func- ratio. Therefore, one must put together all these competing tion. After 20 crossings the transferred energy is still only effects in order to obtain the evolution in time of the energy 0:03% of the Higgs energy at that time. Therefore, we transferred from the Higgs to the fermions. To do this, we clearly see that this successive instant preheating mecha- will assume, for simplicity as well as to try to make this nism is not efficient enough as to rapidly reheat the mechanism more efficient, that since the gauge bosons do Universe. If we consider that the former formalism (107) not accumulate significantly, only the first term of Eq. (86) is valid up to an arbitrary number of oscillations, then we should be considered, such that W and Z bosons are only can estimate the number jp of semioscillations required to produced through spontaneous creation at each zero achieve "ðjÞ Oð1Þ. Equating "ðjÞ to 1 in Eq. (107), we 4 crossing. obtain jp Oð10 Þ. However, much earlier than that, para- In this case the averaged energy density of the fermions metric resonance effects should be considered; see Sec. V. produced between tj and tjþ1 will be given by In other words, notice that we have neglected the pres- ence of those Z and W bosons that did not decay into ðjÞ 6½ n ðjÞð1 e ZFðjÞÞE ðjÞ F Z FZ fermions in each semioscillation. The occupation number þ ð Þð W FðjÞÞ ð Þ of the bosons produced at the bottom of the potential is not 2 nW j 1 e EFW j simply generated by the first term of Eq. (86), but rather by 1 1 FðjÞ 2 2 the rest of the terms in Eq. (86), which indeed give rise to ¼ " M end 2 ðjÞ; (105) 2 j the phenomena of resonant production of bosons. Taking where " is given by Eq. (102) and we have defined this into account will have very interesting consequences. The number density of the bosonic species will grow ð1 þ 2cos3 e ð W ZÞFðjÞÞ W ZFðjÞ exponentially fast and thus will also transfer energy into ðjÞ 1 3 e : ð1 þ 2cos WÞ the fermions exponentially fast. We must therefore develop (106) a mixed formalism that takes into account the two com- peting effects: that of parametric resonant production of Then the ratio between the energy of the fermions to the bosons versus the effect of their perturbative decay into Higgs condensate at the jth zero crossing finally reads fermions. It is crucial to note that while the perturbative decay does not transfer enough energy (as we have just ðjÞ ðj þ 1Þ2 Xj i 5=3 "ðjÞ F " 2 FðiÞ ðiÞ seen), the fact that those bosons disappear will have very ðjÞ j i¼1 j important consequences for the development of the reso- ðj þ 1Þ2 nant effect. In particular, the resonance will not become ¼ "GðjÞ 2 ; (107) effective at the beginning of the oscillations of the inflaton j right after inflation, as usually assumed, but only after the where inflaton has already performed a significant number of oscillations. Xj i 5=3 GðjÞ FðiÞ ðiÞ : (108) i¼1 j V. COMBINED PREHEATING: MIXED PARAMETRIC RESONANCE AND Note that the strength of the effect, i.e. the amplitude of 3 PERTURBATIVE DECAYS "ðjÞ, is modulated by the gauge couplings through " / g2 so, even in this case in which the SM gauge couplings of Let us now analyze how the occupation number of the Z 2 the Higgs to the vector bosons are quite big (g2 0:3), that and W bosons grows if we consider the effect of all the does not help transfer sufficient energy initially. As men- terms in Eq. (86). The production of gauge bosons will also tioned before, if we could apply this formalism to the occur, as before, in a very short interval of time (81) when production of fermions at each Higgs’ zero crossing, by the Higgs condensate crosses around zero, then violating substituting the gauge couplings with the Yukawa ones, we the adiabaticity conditions (80), j j < a. In the large would obtain an even more ridiculous production of parti- occupation limit nk 1, the first term in Eq. (86) can be cles (except perhaps for the top quarks). Of course, the neglected, and therefore the spectral number density of the 063531-14 PREHEATING IN THE STANDARD MODEL WITH THE ... PHYSICAL REVIEW D 79, 063531 (2009) produced gauge bosons just after the jth scattering is given and the average index over an oscillation, obtained as by Z 1 2 1 þ 1 ðavÞ 1 nkðj Þ ðð2Tk ðjÞ 1Þ k ¼ kð Þd ¼ logð2Tk Þ: (114) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 0 2 2 cos T 1ðjÞðT 1ðjÞ 1ÞÞn ðj Þ; (109) j k k k All these possibilities are shown in Fig. 3, as a function of þ 1=3 which indicates the spectral number density nkðj Þ just x K=ðq=jÞ , where q are the resonant parameters (85) after the jth scattering, in terms of the spectral number for the Z and W bosons, while xj is the natural argument of density nkðj Þ just before such scattering. Since the inter- the transmission probability scattering functions (87). val between successive scatterings is M t ¼ , we can As explained in Ref. [36], when j jþ1 j , naturally define a growth (Floquet) index kðjÞ as [35,36] the effect of resonance will be chaotic, since then the phases are essentially random at each scattering. For in- þ 2 kðjÞM t 2 kðjÞ nkðj Þ nkðj Þe ¼ nkðj Þe : (110) stance, using the effective frequencies of the fluctuations Comparing formulas, we obtain (82) of the W field, these phases can be estimated, for the relevant range of momenta, as follows: 1 1 kðjÞ logðð2Tk ðjÞ 1Þ Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 tjþ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 j ¼ dt K þ m~W 1 1 2 cos T ðjÞðT ðjÞ 1ÞÞ: (111) tj j k k pffiffiffiffiffiffi g2 3 2 1=2 The j are some accumulated phases at the jth scattering, pffiffiffiffi FðjÞ Oð10 Þj ; (115) which can indeed play a very important role since they can 2 enhance ( cos < 0) or decrease ( cos > 0) the effect of j j where FðjÞ was defined in Eq. (97) and we have neglected production of particles at each scattering. 2 2 Depending on the phases, we can consider the K versus m~W in the second equality, since, as will be following cases: the typical behavior of the Floquet index, justified later (130), the produced bosons are nonrelativis- for cos ¼ 0, tic. Comparing the above formula with , we see that the end of the stochastic behavior will occur after ð2 5Þ 1 3 ðtypÞ ¼ logð2T 1 1Þ; (112) 10 zero crossings, depending on . For the case of the Z k 2 k boson the previous estimation of the end of the stochastic 1 O the maximum index, achieved for cos ¼ 1, given by resonance is modified by a factor ðcos WÞ ð1Þ, and qffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi thus the result is essentially unaffected. Therefore, since 1 ðmaxÞ ¼ logð T 1 þ T 1 1Þ; (113) for the first thousand oscillations of the Higgs, the accu- k k k mulated phases of the fluctuations of the gauge bosons will 0.20 0.8 0.15 0.6 k k 0.10 n 0.4 0.05 0.2 0.00 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 x x FIG. 3 (color online). Left panel: The Floquet index for a given polarization of the W and Z bosons as a function of the variable xj ¼ k=k ðjÞ. Here we show the maximum (solid red line), the average (short dashed green line), and the typical (long dashed blue þ þ line) indices. Right panel: The initial spectral distribution nkð1 Þ (lower blue curve) and the Gaussian approximation nðj 2Þ (127) for different j’s greater than 2 (rest of the curves), describing the resonant behavior. The approximation is so good that it is hard to distinguish it from the real curve, presenting small deviations just on the tail. The horizontal axis is x ¼ k=k ð1Þ and the curves correspond to different j’s. It is clearly distinguishable that only the range x<1 [k 063531-15 GARCI´A-BELLIDO, FIGUEROA, AND RUBIO PHYSICAL REVIEW D 79, 063531 (2009) be chaotic, we will average out the phases and work with tively. Since k ðjÞ is the natural scale for the momenta of ðavÞ the problem, its order of magnitude should coincide simply k . On the other hand, the perturbative decay of the pro- with the one obtained via the Heisenberg uncertainty prin- duced vector bosons occurs precisely just between two ciple [see Eq. (81)], as is indeed the case since successive Higgs zero crossings, nððj þ 1Þ Þ¼nðjþÞ 1=3 expð FðjÞÞ, where FðjÞ is given by Eq. (97) and ¼ 1 j k ð1Þ kðjÞ ajð taÞ k ðjÞ: (123) Z, W [see Eq. (98)]. Taking into account Eqs. (110) and 21=3 (111), we can express the number of gauge bosons just after the ðj þ 1Þth scattering in terms of the number just Notice that the typical momenta range will be redshifted after the previous one, because of the expansion of the Universe, and even the comoving typical moment k is not a static quantity but þ 2 kðjþ1Þ nkððj þ 1Þ Þ¼nkððj þ 1Þ Þe rather depends on j. þ FðjÞ 2 kðjþ1Þ ¼ nkðj Þe e : (116) Let us now obtain the total number density of created particles. Just after the jth scattering, this will be given by Applied recursively, this formula allows us to obtain the P Z occupation number for each species and polarization, just þ 1 f j 1 FðiÞg 2 þ nðj 2Þ¼ e i¼2 dkk n ð1 Þ after the ðj þ 1Þth scattering in terms of the initial abun- 2 2a3 k þ j dances nkð1 Þ, P f2 j ðiÞg e i¼1 k Xj þ þ 3 P Z nkððj þ 1Þ Þ¼nkð1 Þ exp FðiÞ k ð1Þ f j 1 FðiÞg 3 i¼1 2 2 2 i¼1 ¼ M 3 e duu Ci ð u Þ 2aj Xj Yj exp 2 kði þ 1Þ : (117) 2 2 2 2=3 i¼1 ð2 Ci ð u =l Þþ1Þ; (124) l¼2 The initial abundances are, of course, only generated 3 2 through Eq. (88), and are given by where k ð1Þ / g (122) should be evaluated with g ¼ g2 or g = cos , and ¼ or , respectively, for W or Z n ð1þÞ¼T 1ð1Þ 1 2Ci2ð x Þ; (118) 2 W W Z k k 1 bosons. This formula encodes the usual resonant behavior where we have defined the function discovered in the 1990s, see Refs. [35,36], in which it was implicitly assumed that the produced bosons did not decay 2 0 2 2 0 2 Ci ðxjÞ¼Aið xj ÞAi ð xj ÞþBið xj ÞBi ð xj Þ; (119) between successive inflaton zero crossings. However, as we saw in Sec. IV C, the bosons produced each time the j1=3k Higgs condensate crosses zero significantly decay before xj : (120) the next scattering. Therefore, we had to correct our for- Mq1=3a j mulas for this effect. Fortunately, this was easily done, Again, we used xj in light of Eq. (84), as the natural since the resonant growth occurs in a steplike form instan- argument of the expression of the transmission probability taneously [within a time ta; see Eq. (81)] when the Higgs (87). Note that here the species are only distinguished condensate crosses around zero, while the perturbative through the resonance parameters in Eq. (85). decay of the produced vector bosons occurs during the Normalizing the scale factor at the first zero crossing as time just between two successive Higgs zero crossings. Thus, the occupation number just before the ðj þ 1Þth a1 ¼ 1, we can simply write the evolution of the scale 2=3 scattering, in terms of the occupation number just after factor as aj ¼ j . Thus, the behavior of xj with the the jth scattering, has been corrected by the factor number of zero crossings goes as / j 1=3. Then, we can P f j 1 ð Þg define a typical momentum of the problem, k ðjÞ, related in exp i¼1 F i , which accounts for the accumulated effect of the perturbative decays up to the jth scattering. a very simple way to the resonance parameters qZ, qW (85), as The combined effect of the nonperturbative parametric resonance at the nonadiabatic regions at the bottom of the k potential, together with the perturbative decay along the x ¼ ) k ðjÞ k ð1Þj1=3; (121) j 1=3 j k ð1Þ adiabatic zone during the rest of the semioscillation, gives rise to a new phenomenology, as we will immediately see. where Therefore, to emphasize the difference from the usual 2 1=3 parametric resonance or instant preheating-like mecha- 1=3 2g end k ð1Þ q M M; (122) nisms, we will call this effect combined preheating. 4 Expanding the combination of Airy functions (119), it is with g ¼ g2 and g2= cos W for W and Z bosons, respec- possible to write 063531-16 PREHEATING IN THE STANDARD MODEL WITH THE ... PHYSICAL REVIEW D 79, 063531 (2009) 2 Du2=j2=3 2 2 2 2=3 Bu2=l2=3 Ci ðxjÞ Ce and ð2 Ci ð u =l Þþ1Þ Ae ; (125) 1=3 where u j xj and ð2=3Þ2 12 ð2=3Þ ð16=3Þ2 2 1 C ¼ ;D¼ ;A¼ 2 þ 2 2C; B ¼ 1 : (126) 2ð1=3Þ 2ð2=3Þ 32=3 ð1=3Þ 32=3 3ð1=3Þ ð2=3Þ A Substituting Eqs. (125) and (126) into Eq. (124), we then obtain F ðj 1Þ 3 Z P þ 3 e k ð1Þ j 1 2 Du2 Bð j i 2=3Þu2 nðj 2Þ M A C duu e e i¼2 2j2 ffiffiffiffi p j F ðj 1Þ 3 X 3=2 3 e k ð1Þ j 1 2=3 ¼ M 2 A C D þ B i ; (127) 2j 4 i¼2 where we have used a ¼ j2=3, performed the resulting k ð1Þ j k P : (129) Gaussian integral, and defined p j 2=3 1=2 ðD þ B i¼2 i Þ Xj In the right panel of Fig. 4, one can easily observe this F ðjÞ FðiÞ; (128) i¼1 behavior: The value of the momentum at which the distri- bution peaks slightlyP moves to smaller values, according to for simplicity. Notice that the resonant behavior is now k /ðD þ B j i 2=3Þ 1=2. Thus, it is noticeable that the j 1 p i¼2 encoded in the factor A , which, for sufficiently large j, typical momentum k of the resonant fluctuations is always F ðj 1Þ will finally overtake the decaying factor e , since of order k ð1Þ, independently of how many oscillations the 2 A>2. Taking also into account the factor 1=j due to the Higgs performs. The reason for this is that the parametric expansion of the Universe, the first result we can read from resonance effect builds up initially from the spectral here is that only for those values of j for which ðj 1Þ þ distribution nkð1 Þ, which only filters k & k ð1Þ; see logA 2 logj> F ðj 1Þ, the resonant effect will Eq. (117) and Fig. 4. dominate over both the perturbative decay and the expan- 2 2 On the other hand, the ratio kp=hmij for both W and Z sion rate. bosons, between the typical momenta produced around Note thatP inside the integral (127), the function zero and the average masses in every oscillation, is shown 2 ðDþB j i 2=3Þu2 u Pe i¼2 has a maximum at a value up ðD þ in Fig. 4. In particular, it is easy to estimate the evolution in j 2=3 1=2 B i¼2 i Þ , which implies that the typical (comov- time of such a ratio, in terms of the resonant parameters ing) excited momentum is (85), as 0.05 0.0020 0.04 0.0015 0.03 2 j m k 0.0010 n 2 2 k k 0.02 0.0005 0.01 0.0000 0.00 0 20 40 60 80 0.0 0.5 1.0 1.5 2.0 j x FIG. 4 (color online). Left panel: The ratio k2=hm2i between the typical momenta produced around zero and the average mass in every oscillation for the W (dashed blue line) and Z bosons (solid red line) as a function of the number of oscillations. This ratio is significantly smaller than 1 for all crossings,P which allows us to consider the produced gauge bosons as nonrelativisitic. Right panel: j 2 þ 2 kðjÞ 2 Successive spectral distributions k nkð1 Þe k¼2 , at different j’s, including the volume factor k . One can see the predicted (129) slow displacement of the maxima of the distribution. The x axis is given in terms of x ¼ k=ðk ð1ÞÞ. 063531-17 GARCI´A-BELLIDO, FIGUEROA, AND RUBIO PHYSICAL REVIEW D 79, 063531 (2009) ðk =a Þ2 q2=3 bosons produced at the bottom of the potential are always p j ¼ P hmi2 j 2=3 2 3 2 nonrelativistic. This justifies a posteriori the calculation of j ðD þ B k¼2 k Þg ð2 Þ FðjÞ 1 the energy of the fermions as EFðjÞ 2 hmZ;Wij; see 1 1 / P 1 8 j; Eq. (99) in Sec. IV C. Extrapolating Eq. (130) to the case 2=3 1=3 j 2=3 g j ðD þ B k¼2 k Þ of fermions, we realize that the produced particles at each zero crossing would be mainly relativistic, due to the (130) smallness of the Yukawa couplings, with the exception of Taking into account that the previous ratio (130)isa the top quarks. decreasing function with j, as well as its dependence on The energy density transferred to the fermions between the gauge couplings g 2=3, we can conclude that the vector the jth and the ðj þ 1Þth scatterings will be ð Þ¼ ½ð ZFðjÞÞ ð þÞ ð Þþ ð W FðjÞÞ ð þÞ ð Þ F j 6 1 e nZ j EFZ j 2 1 e nW j EFW j pffiffiffiffi 3 Xj ð Þ ~ k 3=2 2 2 j 1 2=3 ZFðjÞ Z ðj 1Þ ¼ 2 M endA C 2 D þ B l FðjÞðð1 e Þe 2 4 j l¼2 3 W FðjÞ W ðj 1Þ þ 2 cos Wð1 e Þe Þ; (131) where we have used the energy of the fermions (99) and as usual, as if the produced bosons would not decay defined a momentum scale independent of the gauge cou- perturbatively during each semioscillation. plings, common to both bosonic species, as In order to estimate the time in which the perturbative decays stop blocking the parametric resonance effect, we k ð1Þ k : can evaluate numerically when theP expression e ðj 1Þ 2=3 (132) g 2 j ðiÞ becomes subdominant versus e i¼2 k . In particular, The gauge coupling dependence is indeed incorporated in we can evaluate the ratios, for either W or Z, P the definition of the parameter 2 j ðiÞ i¼2 k ; (136) 3g3 1=2 2 ðj 1Þ ~ 2 ; (133) 3 5=2 2 ðcos WÞ ð endÞ for the fastest growing mode k ¼ kp (129), and find the number of semioscillations j for which the previous ratio ~ / 3 R which modulates again the strength of the effect as g2. becomes greater than 1, 1. We find j 62 for the W The total energy density transferred into the fermions R bosons and jR 360 for the Z bosons. The fact that para- will be metric resonance becomes important much earlier for W’s than for Z’s is not a surprise, since their decay rates (53) Xj i 8=3 FðjÞ¼ FðiÞ ; (134) differ by a factor Z= W 2:4, which simply means that i¼1 j there are many more W bosons surviving per semioscilla- tion than Z bosons. Therefore, the combined preheating of and the ratio of such an energy to that of the inflaton is the W bosons is driven into the parametric-like behavior much faster, while the evolution of the Z bosons is much ðjÞ 2 2ðj þ 1Þ2 Xj i 8=3 F 2 more affected by the perturbative decays, delaying (or even "FðjÞ ¼ 2 2 FðiÞ ; (135) M end i¼1 j completely preventing) the development of parametric resonance. Obviously, after a dozen oscillations, the trans- with FðiÞ given by Eq. (131). Here we can clearly see fer of energy from the inflaton to the gauge bosons will be the two competing effects: That of the perturbative decay completely dominated by the channel into the W bosons, of the bosons, given by the factors of the form ð1 since by that moment they will be fully resonant while the ð Þ ð Þ e F j Þe F j 1 , tends to decrease the rate of production Z bosons will still be severely affected by their perturbative of bosons and fermions, while the factors e2 k encoded in decay. the form of the Gaussian approximation describe the reso- Finally, to conclude this section and achieve an overall nant effect due to the accumulation of previously produced complete picture of all the details, let us also estimate the bosons and fermions. Initially, the perturbative decay will transfer of energy from the inflaton to the gauge bosons. In prevent the resonance from being effective. However, after particular, the total energy transferred to them just after the a certain number of oscillations (a number that we will jth scattering, BðjÞ, is given simply by estimate next), the resonant effect will overtake the pertur- þ þ bative decays and parametric resonance will be developed BðjÞ¼3ðnZðj ÞhmZij þ 2nWðj ÞhmWijÞ; (137) 063531-18 PREHEATING IN THE STANDARD MODEL WITH THE ... PHYSICAL REVIEW D 79, 063531 (2009) TABLE I. Number of semioscillations of the Higgs required, tion of the produced gauge fields into the homogeneous as a function of , for an efficient transfer of energy from the Higgs condensate will become significant, and it will have inflaton to the gauge fields and/or to the fermions. to be taken into account. 0.2 0.4 0.6 0.8 1.0 ðBÞ jeff 74 64 60 57 55 VI. BACKREACTION ðFÞ jeff 79 69 64 61 59 Let us now calculate the backreaction from the W and Z bosons into the Higgs condensate. Neglecting the vectorial nature of the bosons, the effective equation of the Higgs condensate can be written, in the Hartree approximation, as where we have used the fact that the gauge bosons are nonrelativistic and have three polarizations. Therefore, the 1 € þ 3H _ r2 ratio of the energy of the gauge bosons to the energy of the a3 inflaton can be expressed as g2M2 1 @ þ M2 þ p ð1 e j jÞh’2i ¼ 0; (139) 4 @ BðjÞ "BðjÞ pffiffiffiffi where there is a ’ field for each polarization of each gauge 1 2 1 2 k~ 3FðjÞAj 1C boson species, such that g2 ¼ g2 and g2 ¼ g2=cos2 ,as ¼ j þ P 2 2 W 2 j 2=3 3=2 usual, for the W and Z bosons, respectively. From here, 2 cos W 4j ðD þ B i Þ i¼2 performing the derivative, one obtains for the effective ZF ðjÞ 3 W F ðjÞ ðe þ 2 cos We Þ; (138) Higgs frequency where we have used Eq. (127), and ~, defined in Eq. (133), 2 g Mp again modulates the amplitude of this growing function. !2 ¼ M2 þ e j jh’2i; (140) 4 j j Using Eqs. (135) and (138), we can estimate the time in which the energy of the inflaton would finally be trans- where the second term in the r.h.s. should be summed over ferred efficiently to the fermions or the bosons. Defining polarizations and species. For the fraction of time of each that moment, respectively, like "FðjeffÞ 1 and "BðjeffÞ semioscillation, during which the Higgs frequency evolves 1, one obtains the numbers in Table I. Note that the bosons adiabatically, we can use the correlation function receive the transfer of energy from the inflaton before the fermions, since by the time that parametric resonance 3 2 0 h’ ’ 0 i¼ð2 Þ j’ j ðk k Þ; (141) overtakes the perturbative decay, the fraction of bosons k k k decaying (per semioscillation) into fermions is very small with ’ ðtÞ expressed as and, therefore, the fraction of newly added fermions is less k and less important, while the amount of produced bosons is R R ðtÞ t 0 ðtÞ t 0 3=2 k i !kdt k þi !kdt more and more prominent. Note also that the number of a ’kðtÞ¼pffiffiffiffiffiffiffiffiffiffiffiffiffi e 0 þ pffiffiffiffiffiffiffiffiffiffiffiffiffi e 0 : oscillations jeff required for an efficient transfer of energy 2!ðkÞ 2!ðkÞ depends on the parameter , although the overall order of (142) magnitude does not change appreciably. Unfortunately, as we will see in the next subsection, Thus, one can compute the expectation value of the bo- before reaching the stage in which F;B 1, the backreac- sonic fields (components), Z Z 2 R 1 1 dkk 1 t 0 2 2 2 2 i2 !dt þarg kþarg k h’ i 2 3 dkk j’kj ¼ 2 3 þj kj þ Ref k ke g 2 a 2 a !k 2 pffiffiffi Z 1 2 1 2 X pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 3 dkk nk 1 þ cos h!ij þ arg k þ arg k ; (143) 2 a gMp 1 e j j M j pffiffiffi where, to obtain the last expression we have used X pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 n’ 2 gMpffiffiffi j j 2 2 h’ i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ A cos h!ij ; !k ¼ p 1 e , j kj ¼j kj 1 ¼ nk, and gM j j M R 2 P p 1 e j ð 0Þ 0 ¼ð Þ n h i h i ¼ M R !k t dt =M j¼1 ! j, with ! j (144) tjþ1 0 ð 0Þ tj dt ! t . Following [36], since we do not know the R 2 3 1 2 accumulated phases of k and k, we will write with A<1 and n’ ¼ð2 a Þ dkk nk. 063531-19 GARCI´A-BELLIDO, FIGUEROA, AND RUBIO PHYSICAL REVIEW D 79, 063531 (2009) From here, one can define the effective frequency of the Backreaction ,ðnZðjÞ= cos W þ 2nWðjÞÞ Higgs condensate as pffiffiffi 2 j ðt Þjð j ðt ÞjÞ1=2M2 P * j j ; (146) gn ½1 þ A cosð2 h!i Þ 3 g !2 M2 þ pffiffiffi ’ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM j j : (145) 2 2 j j e2 j j e j j pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where we have expanded e2 j j e j j ð j jÞ1=2, The backreaction of the gauge boson fields over the Higgs which is certainly accurate after a couple of dozens of 1 field will be non-negligible when the last term in the r.h.s. oscillations, since j ðtnÞj / j . Substituting the averaged of the previous expression becomes of the order of M2.In value per semioscillation RðtÞ!h ðtÞij, we will then terms of the number densities of the Z and W bosons, i.e. end 1 2 end take j ðtjÞj ! j ð 0 sinðxÞÞ ¼ j . Using summing the contribution over polarizations and species of the analytical expressions for the occupation numbers t ¼ all the fields that backreact, this will happen at a time j (127), we can translate the above condition into the follow- j =M, ing one: pffiffiffi pffiffiffi Aðj 1Þ 32 6 3=2 logð1 þ 2= 3Þ3=2 ðe Z ðj 1Þ=cos3 þ 2e W ðj 1ÞÞ P : (147) W 1=2 j 3=2 1=2 2 3 7=2 3 j ðD þ B i¼2 i Þ 3 g2 Ck Thus, if we find numerically the number of semioscilla- VII. CONCLUSIONS tions of the Higgs, jbackr, for which j>jbackr fulfills the above condition, then we know the moment at which back- We have studied the different stages of reheating after reaction of the bosonic fields becomes significant, tbackr inflation in a model where the role of the inflaton is played jbackr=M. Note that the above condition depends on by the Higgs field of the standard model of particle physics both in the left- and right-hand sides. In particular, the with a nonminimal coupling to gravity. Inflation in this dependence is rather weak in the constant of the right-hand model takes place at the grand unified theory scale, along side of the inequality, since it goes as 1=4, while in the left- the lines of the Starobinsky model of inflation since a hand side, it enters through the exponentials so it can conformal transformation makes these two models indis- change the number nbackr in a more significant manner. tinguishable from the point of view of inflation. The usual Taking values of between 0.2 and 1.0, we obtain the difficulty with large self-couplings of the Higgs is tamed numbers in Table II. here by the inclusion of a large nonminimal coupling to We clearly see that backreaction seems to become im- gravity, 105, which nevertheless does not leave any portant at a time slightly earlier than that at which we were signature at low (electroweak) scales due to the fact that the expecting the Higgs to have efficiently transferred its Higgs field acquires a vacuum expectation value and does energy to the bosons and fermions. This means that our not evolve at present, while the local space-time curvature analytical estimates of these transfers were biased, and a is negligible. careful numerical study of the process is required. Beyond The advantage of this model of inflation for the study of backreaction, the strength of the resonance very quickly reheating after inflation is that all the couplings of the decreases due to the increased frequency of oscillations of Higgs inflaton to matter fields are known at the electroweak the Higgs. Eventually, the broad resonance driving the scale, and can be extrapolated to the GUT scale using the production of gauge bosons, and thus their decay into renormalization group equations, and therefore one can SM particles, becomes a narrow resonance and finally study in detail the process of reheating the Universe, shuts off. From then on, the inflaton will oscillate like a without having to impose ad hoc assumptions about their matter field, while the produced particle will redshift as values. The surprise is that the process becomes more radiation, its effect on the expansion becoming negligible complicated than expected, and a series of subsequent after a few hundred oscillations. stages take place, where essentially all different types of particle production mechanisms at preheating occur. Moreover, since the standard model couplings of the Higgs to gauge and matter fields are non-negligible, nor TABLE II. Number of semioscillations of the Higgs required, are their couplings among themselves, the process of non- as a function of , for the backreaction of the gauge fields into perturbative decay via parametric resonance is mixed with the Higgs background to become significant. the usual perturbative decays of the decay products, which 0.2 0.4 0.6 0.8 1.0 complicates things significantly. j 67 57 52 50 48 Inflation ends at values of the Higgs field of order the backr Planck scale and goes through a brief stage of tachyonic 063531-20 PREHEATING IN THE STANDARD MODEL WITH THE ... PHYSICAL REVIEW D 79, 063531 (2009) preheating soon after the end of inflation. The passage is so gauge fields decay perturbatively until their energy is trans- short that particle production is not significant at that stage. ferred to SM particles. Since the stage after backreaction is The same occurs with the production at the inflection point. very nonlinear and nonperturbative, it cannot be solved Finally, the Higgs-inflaton field starts oscillating around analytically and we have to resort to numerical studies in the minimum of its potential with a curvature scale of order the lattice. We leave the description of our numerical 1013 GeV. At this stage, particle production occurs when- studies to a future publication. ever the Higgs passes through zero, creating mostly vector gauge bosons W and Z. These gauge bosons acquire a large ACKNOWLEDGMENTS mass while the Higgs increases towards a maximum am- plitude, and start to decay into all standard model leptons We would like to thank Geneva University, SISSA- and quarks within half a Higgs oscillation, rapidly deplet- Trieste, and MPI-Munich for hospitality during the devel- ing the occupation numbers of gauge bosons, like in instant opment of parts of this research. D. G. F. is supported by preheating. However, the fraction of energy of the Higgs FPU Contract No. AP2005-1092 and J. R. by a I3P that goes into SM particles is still very small compared Contract. We also acknowledge financial support from with the energy in the oscillations, and therefore the non- the Madrid Regional Government (CAM) under the pro- perturbative decay is slow. This implies that a relatively gram HEPHACOS P-ESP-00346, and the Spanish large number of oscillations take place before a significant Research Ministry (MEC) under Contract No. FPA2006- amount of energy is transferred to the gauge bosons and 05807. The authors participate in the Consolider-Ingenio fermions. 2010 CPAN (CSD2007-00042) and PAU (CSD2007- The amplitude of Higgs boson oscillations decreases as 00060), as well as in the European Union 6th Framework the Universe expands in a matterlike dominated stage with Marie Curie Network ‘‘UniverseNet’’ under Contract zero pressure. Eventually, this amplitude is small enough No. MRTN-CT-2006-035863. that the gauge boson masses are not large enough for Note added.—Upon completion of this paper, we re- inducing a quick decay of the gauge bosons, and they start ceived through the arXiv the preprint of Bezrukov et al. to build up their occupation numbers very rapidly via [44], where they also study preheating in the MSM. parametric amplification. The question of whether this Although the formalism is common to both, our conclu- effect can give rise to the production of a significant sions are somewhat different from those of Ref. [44]. We gravitational wave background potentially observable to- find that the Higgs decay into gauge bosons is significantly day remains to be addressed. Several papers have recently faster, and that backreaction occurs much before thermal- studied such an issue in the chaotic and hybrid models of ization. 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