Note on Reheating in G-inflation
Hossein Bazrafshan Moghaddam, Robert Brandenberger, Jun'ichi Yokoyama
NNote on Reheating in G-inflation
Hossein Bazrafshan Moghaddam, ∗ Robert Brandenberger, † and Jun’ichi Yokoyama ‡ Department of Physics, McGill University, Montreal, QC, H3A 2T8, Canada Physics Department, McGill University, Montreal, QC, H3A 2T8, Canada, andInstitute for Theoretical Studies, ETH Z¨urich, CH-8092 Z¨urich, Switzerland Research Center for the Early Universe (RESCEU),Graduate School of Science, The University of Tokyo, Tokyo 113-0033, Japan,Department of Physics, Graduate School of Science,The University of Tokyo, Tokyo, 113-0033, JapanKavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU),UTIAS, WPI, The University of Tokyo, Kashiwa, Chiba, 277-8568, Japan (Dated: December 6, 2016)We study particle production at the end of inflation in kinetically driven G-inflation model andshow that, in spite of the fact that there are no inflaton oscillations and hence no parametricresonance instabilities, the production of matter particles due to a coupling to the evolving inflatonfield can be more efficient than pure gravitational Parker particle production.
I. INTRODUCTION
Reheating , namely the transition from the period of in-flation [1], during which the energy-momentum tensor isdominated by the coherent inflaton field, to the radiationphase of Standard Big Bang cosmology, is an importantaspect of inflationary cosmology. Without such an energytransfer, inflation would produce a cold empty universeand would not be a viable early universe scenario. On theother hand, there will inevitably be gravitational parti-cle production of any non-conformal field which lives inthe space-time of an inflationary universe [2] (see [3] fora classic review of quantum field theory in curved space-time). The energy density produced by this mechanismby the end of the inflationary phase will be of the or-der H , where H is the Hubble expansion rate duringinflation. Note that this energy scale is parametricallysuppressed compared to the energy density ρ I during in-flation: H ρ I ∼ H m pl , (1)where m pl is the Planck mass. This ratio is boundedto be smaller than 10 − based on the upper bound onthe strength of gravitational radiation produced duringinflation [4].In simple scalar field models of inflation, based on aslowly rolling scalar field with canonical kinetic term inthe action, there is a much more efficient energy trans-fer mechanism which can reheat the universe. In thepresence of any coupling between the matter field (heremodelled as a scalar field χ ) and the inflaton field φ thereis a parametric resonance instability which causes χ field ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] fluctuations to grow exponentially during the phase af-ter inflation when φ oscillates coherently about its groundstate value [5, 6]. Although this process does not directlyproduce a thermal state of matter particles, it efficientlytransfers the energy density from the inflaton to matter,typically in a period which is short compared to a Hubbleexpansion time. This process is now called preheating [7–9] (see [10] for recent reviews). It produces a state afterinflation in which the matter energy density ρ m ∼ ρ I (2)after inflation is not suppressed compared to the energydensity ρ I during inflation, in contrast to what is ob-tained (1) if only gravitational particle production is op-erative.Simple slow-roll inflation based on an action with acanonical kinetic term is at the moment consistent withthe data we have. In fact, the scenario made a numberof successful predictions (spatial flatness, slight red tilt[11] to the spectrum of cosmological perturbations, etc.)[12–14] . The scenario also predicts a red tilt in thespectrum of gravitational waves [16].There are alternatives to the inflationary paradigm ofearly universe cosmology (see e.g. [17] for reviews). Oneof these alternatives, String Gas Cosmology [18] (see also[19]), while consistent with all current observations ofscalar cosmological perturbations [20], predicts a slightblue tilt in the spectrum of gravitational waves [21]. Inthe context of inflationary cosmology with vacuum ini-tial conditions and with matter obeying the
Null EnergyCondition (NEC), one always obtains a red tilt .However, it was pointed out in [22] (see also [23]), thatby introducing Galileon type terms (in particular kinetic But see [15] for a different view. This is different than the scalar spectrum for which either a redor a blue tilt can be obtained, although the simplest slow-rollmodels of inflation also predict a red tilt of the scalar spectrum. a r X i v : . [ h e p - t h ] D ec terms) in the action of a scalar field φ , it is possible to ob-tain an inflationary model in which matter violates theNEC and hence a blue tensor tilt is possible . Thismodel is called G-inflation . In this model, inflation isdriven by the kinetic term in the action which at earlytimes has the “wrong” sign and hence can lead to the vi-olation of the NEC. Nevertheless thanks to the Galileon-type terms, the stability of fluctuations is maintainedeven in the presence of NEC violation contrary to thecase of k-inflation [25]. Inflation terminates at a scalarfield value above which the sign of the kinetic term re-verts back to its canonical form. This leads to a transitionfrom an inflationary phase to a standard kinetic-drivenphase with equation of state w = 1, where w is the ratioof pressure to energy density. The energy density in φ then decreases as a ( t ) − , where a ( t ) is the cosmologicalscale factor.Since there is no phase during which φ ( t ) oscillatesthere is no possibility of preheating. As discussed in [22],the production of regular matter particles after Galileoninflation is still possible by the gravitational Parker par-ticle production mechanism. However, the resulting mat-ter energy density will be suppressed as in (1). The ques-tion we ask in this note is whether in the presence of acoupling between matter and the inflaton there is nev-ertheless some non-gravitational channel which transfersenergy to matter faster than what can be achieved bygravitational effects.In the following we point out that there is indeed achannel for direct particle production, and we derive con-ditions on the coupling constant for which this directchannel is more efficient than Parker particle production . Our analysis is based on the general framework setout in [5].We begin with a brief review of G-inflation, move on toa discussion of the particle production mechanism we use,before presenting the calculations applied to our model.We work in natural units in which the speed of light,Planck’s constant and Boltzmann’s constant are set to 1. II. G-INFLATION
The original G-inflation [22] is based on a scalar field φ minimally coupled to gravity with an action L = K ( φ, X ) − G ( φ, X ) (cid:3) φ , (3)where X is the standard kinetic operator X = 12 ∂ µ φ∂ µ φ , (4) It is still possible to distinguish String Gas Cosmology from G-Inflation by considering non-Gaussianities or consistency rela-tions [24]. A similar channel is operative in the “emergent Galileon” sce-nario of [26] - see [27]. Particle-induced particle production hasalso recently been studied in a bouncing cosmology in [28]. and K and G are general functions of φ and X . See [29]for its generalized version. The special property of thisclass of Lagrangians is that the resulting equations of mo-tion contain no higher derivative terms than second order[30]. In the case K = K ( X ) and G ( φ, X ) ∝ X the actionhas an extra shift symmetry (“Galilean symmetry”) andthese Lagrangians were introduced and studied in [31].The model of kinetically driven G-inflation [22] is basedon choosing K ( φ, X ) = − A ( φ ) X + δK , (5)with A ( φ ) = tanh (cid:2) λ ( φ e − φ ) (cid:3) , (6)and G ( φ, X ) = ˜ g ( φ ) X = ˜ gX . (7)Here λ and ˜ g are coupling constants and δK includeshigher order terms in X which are important during in-flation. After the sign of the linear kinetic term in theaction is flipped at φ = φ e , they soon become negligibleand do not affect our analysis of reheating.We consider homogeneous and isotropic cosmologicalsolutions resulting from this action. As shown in [22],for φ < φ e there are inflationary trajectories for whichthe quasi-exponential expansion of space is driven by thewrong-sign kinetic term. Inflation ends at φ = φ e , and for φ > φ e the background becomes that of a kinetic-drivenphase with w = 1, a ( t ) ∼ t / and˙ φ ( t ) ∼ t . (8)We call this stage the kination regime of the model. Sincethe energy density in φ decays so rapidly, eventually thekination regime will end and regular radiation and matterwill begin to dominate. The energy density at which thistransition happens determines the reheating temperature of the Universe.Knowledge of the reheating temperature is importantfor various post-inflationary processes such as baryogene-sis or the possibility of production of topological defects.It may also be possible to directly probe the physics of thephase between the initial thermal stage and the hot BigBang phase with precision observations (see e.g. [32]).Regular matter and radiation are produced by gravi-tational particle production. However, if this is the onlymechanism, then the reheating temperature will be lowas it is suppressed by (1). In the following we will assumethat there is a direct coupling between matter (describedby a free massless scalar field χ ) and the inflaton field φ .We consider two possible couplings. The first is of theform L I = 12 g ˙ φχ , (9)where g is a dimensionless coupling constant. Note thatwe have chosen a derivative coupling of φ with χ to pre-serve the invariance of the interaction Lagrangian undershifting of the value of φ (which is part of the Galileansymmetry. The disadvantage of this coupling is that itviolates the symmetry φ → − φ . The second couplingobeys this symmetry but involves non-renormalizable in-teractions. It is L I = − M − ˙ φ χ , (10)where M is a new mass scale which is expected to besmaller than the Planck mass. These couplings open upnon-gravitational channels for the production of χ parti-cles. In the following we will study the conditions underwhich these direct production channels are more efficientthan the gravitational particle production channel. III. INFLATON-DRIVEN PARTICLEPRODUCTION
Assuming that the Lagrangian for the matter field χ has canonical kinetic term, then the Lagrangian for χ isthat of a free scalar field with a time dependent mass,the time dependence being given by the interaction La-grangians (9) or (10). Each Fourier mode χ k of χ evolvesindependently, the equation of motion is¨ χ k + 3 H ˙ χ k + (cid:18) k a − g ˙ φ (cid:19) χ k = 0 . (11)or ¨ χ k + 3 H ˙ χ k + (cid:18) k a + M − ˙ φ (cid:19) χ k = 0 , (12)depending on the form of the interaction Lagrangian.The effects of the expansion of space can be pulled outby rescaling the field X k ≡ a − χ k . (13)Then, in terms of conformal time τ (which is related tophysical time t by dt = a ( t ) dτ ), the equation of motionbecomes X (cid:48)(cid:48) k + (cid:18) k − g ˙ φa − a (cid:48)(cid:48) a (cid:19) X k = 0 , (14)or X (cid:48)(cid:48) k + (cid:18) k + M − ˙ φ a − a (cid:48)(cid:48) a (cid:19) X k = 0 , (15)where a prime denotes a derivative with respect to τ .The qualitative features of the equations of motion (14)or (15) are well known from the theory of cosmologicalperturbations (see e.g. [33] for an in-depth review and[34] for a brief overview): In the absence of the inter-action term, X k will oscillate on sub-Hubble scales, i.e.scales for which k > a (cid:48)(cid:48) a ∼ H , (16) whereas the mode function X k is squeezed on super-Hubble scales, i.e. X k ∼ a . (17)Following [5], we will treat the effects of the interactionterm in leading order Born approximation, i.e. we write X ≡ X + X , (18)(here and in the following we will drop the subscript k )where X is the solution of the equation in the absenceof interactions, i.e. a solution of X (cid:48)(cid:48) + (cid:18) k − a (cid:48)(cid:48) a (cid:19) X = 0 , (19)solving the initial conditions of the problem, and X isthe solution of the inhomogeneous equation X (cid:48)(cid:48) + (cid:18) k − a (cid:48)(cid:48) a (cid:19) X = g ˙ φa X (20)or X (cid:48)(cid:48) + (cid:18) k − a (cid:48)(cid:48) a (cid:19) X = − M − ˙ φ a X (21)(with vanishing initial conditions) obtained by taking theinteraction term in (14) or (15) to the right hand sideof the equation and replacing X by the “unperturbed”solution X .The inhomogeneous equation (20) (or (21)) can besolved by the Green’s function method X ( τ ) = (cid:90) ττ i dτ (cid:48) G ( τ, τ (cid:48) ) g a ( τ (cid:48) ) ˙ φ ( τ (cid:48) ) X ( τ (cid:48) ) , (22)or X ( τ ) = − (cid:90) ττ i dτ (cid:48) G ( τ, τ (cid:48) ) M − a ( τ (cid:48) ) ˙ φ ( τ (cid:48) ) X ( τ (cid:48) ) , (23)where the Green’s function G ( τ, τ (cid:48) ) is determined interms of the two fundamental solutions u and u of thehomogeneous equation via G ( τ, τ (cid:48) ) = W − (cid:0) u ( τ ) u ( τ (cid:48) ) − u ( τ ) u ( τ (cid:48) ) (cid:1) , (24)where W is the Wronskian W = u ( τ ) u (cid:48) ( τ ) − u ( τ ) u (cid:48) ( τ ) . (25)In the above, the time τ i is the time when the initialconditions are imposed. In our case it is the end of theperiod of inflation.The condition that direct particle production is moreefficient than gravitational particle production is X ( τ ) > X ( τ ) (26)at some time τ > τ i before the time when the kineticphase would be terminated by gravitational particle pro-duction alone. IV. ANALYSIS
We now apply the formalism of the previous section toour specific Galileon inflation model. We are interestedin super-Hubble modes for which the k term in the equa-tion of motion (11) can be neglected. The fundamentalsolutions are then u ( τ ) = (cid:18) ττ i (cid:19) / (27) u ( τ ) = (cid:18) ττ i (cid:19) / ln (cid:18) ττ i (cid:19) , and hence the Wronskian becomes W = 1 τ i , (28)and the Green’s function is G ( τ, τ (cid:48) ) = (cid:0) τ τ (cid:48) (cid:1) / ln (cid:18) τ (cid:48) τ (cid:19) . (29)The contribution X ( τ ) induced by the direct couplingbetween φ and χ thus becomes X ( τ ) = g (cid:90) ττ i dτ (cid:48) (cid:0) τ τ (cid:48) (cid:1) / ln (cid:18) τ (cid:48) τ (cid:19) ˙ φ ( τ (cid:48) ) a ( τ (cid:48) ) X ( τ (cid:48) ) , (30)or X ( τ ) = − M (cid:90) ττ i dτ (cid:48) ( τ τ (cid:48) ) / ln (cid:18) τ (cid:48) τ (cid:19) ˙ φ ( τ (cid:48) ) a ( τ (cid:48) ) X ( τ (cid:48) ) , (31)For X ( τ ) we can take the dominant solution of the ho-mogeneous equation X ( τ (cid:48) ) = X ( τ i ) (cid:18) τ (cid:48) τ i (cid:19) / ln (cid:18) τ (cid:48) τ i (cid:19) . (32)Making use of the scaling (8) of ˙ φ and after a couple oflines of algebra we obtain the approximate result (keepingonly the contribution from the upper integration limit) X ( τ ) (cid:39) g ˙ φ i ( τ i ) τ X ( τ i ) . (33)or X ( τ ) (cid:39) − M − ˙ φ i ( τ i ) τ i (cid:18) ττ i (cid:19) / X ( τ i )ln (cid:18) ττ i (cid:19) . (34)If we take the initial time τ i to correspond to the end ofinflation, we have ˙ φ ( τ i ) (cid:39) H ( τ i ) m pl , (35)where H ( τ i ) is the value of H at the end of inflation. Inthis case X ( τ ) ∼ g m pl τ i (cid:18) ττ i (cid:19) / , (36) or X ( τ ) (cid:39) − (cid:16) m pl M (cid:17) X ( τ ) . (37)The criterion (26) for direct particle production todominate over gravitational particle production then be-comes (up to logarithmic factors) g > H ( τ i ) m pl (cid:18) t i t (cid:19) . (38)or (cid:16) m pl M (cid:17) > . (39)Note that for the second interaction term, particle pro-duction via direct interactions dominates within oneHubble expansion time (the time interval after which thecontribution from the lower integration end can be ne-glected), provided that M < m pl , a condition which hasto be satisfied if we are to trust the effective field justifi-cation of the interaction term.Once X ( τ ) starts to dominate over X ( τ ), the Bornapproximation ceases to be valid. At that point, the cou-pling term in the equation of motion for X will becomethe dominant one, and an approximation to (14) (we willfirst focus on the case of the first interaction term) whichis self-consistent for long wavelength modes (for whichthe k term in the equation is negligible) is X (cid:48)(cid:48) − g ˙ φa X = 0 . (40)An approximate solution of this equation is X ( τ ) = A ( τ ) e ˜ f ( ττi ) / τ i (41)with ˜ f ≡ (cid:0) g ˙ φ ( τ i ) (cid:1) / . (42)Inserting this ansatz (41 and 42) into (40) we find anequation for the amplitude A ( τ ) A (cid:48)(cid:48) + 32 ˜ f τ − / τ − / i A (cid:48) −
316 ˜ f τ − / τ − / i A = 0 , (43)which both for ˜ f τ i (cid:28) f τ i (cid:29) X which becomes important once˜ f τ i (cid:18) ττ i (cid:19) / > , (44)which in terms of physical time is tt i > ˜ f − τ − i . (45)In the above we are implicitly assuming that ˜ f τ i <
1. If˜ f τ i > f τ i <
1, the we see that once thetime t is larger than the one given by (45), the energytransfer from the inflaton to matter is exponentially fastand will immediately drain all of the energy from theinflaton. Hence, the “reheating time” t RH is t RH ∼ t i ( ˜ f τ i ) − , (46)and since the energy density between t i and t RH de-creases as a ( t ) − ∼ t − we have ρ ( t RH ) ∼ ρ ( t i )( ˜ f τ i ) . (47)Making use of ρ ( t i ) = H ( t i ) m pl (up to a numerical fac-tor) and ρ ( t RH ) ∼ T RH we finally obtain the reheatingtemperature T RH to be T RH ∼ ˜ f τ i ( H ( t i ) m pl ) / (48)which is larger than the reheating temperature H ( t i )which would be obtained if only gravitational particleproduction were effective, provided that˜ f τ i > (cid:18) H ( t i ) m pl (cid:19) / . (49)In the case of the second coupling, the conclusions aresimilar. Once the coupling term in the equation of motiondominates over the expansion term, the equation can beapproximated as (changing the sign of the coupling term) X (cid:48)(cid:48) − M − ˙ φ a X = 0 . (50)Since ˙ φ ( τ ) a ( τ ) = ˙ φ ( τ i ) (cid:16) τ i τ (cid:17) , (51)the equation has power law solutions with an exponent∆ given by ∆ = 12 (cid:2) ± (cid:112) R (cid:3) , (52)where R ≡ m pl M . (53)We see that if M (cid:28) m pl , then the power of the dominantsolution is ∆ (cid:29) χ within one Hub-ble expansion time. Hence, the reheating temperature isgiven by the energy density at the end of inflation, i.e. T RH ∼ ( H ( t i ) m pl ) / . (54) V. CONCLUSIONS AND DISCUSSION
We have derived the condition under which direct par-ticle production in G-inflation dominates over gravita-tional particle production. The discussion also applies tok-inflation [25]. We consider two possible interaction La-grangians, namely (9) and (10). We first study the onsetof matter particle production from the direct couplingusing the Born approximation. We find that for both in-teraction terms we consider the direct particle productionchannel eventually dominates. This happens within oneHubble expansion time for the coupling (10), whereas inthe case of (9) the time when direct particle productionstarts to dominate depends on the coupling constant g .Once direct particle production begins to dominateover gravitational particle production we must use a dif-ferent approximation scheme to solve the equation of mo-tion. We can now neglect the squeezing term in the equa-tion of motion. We provide solutions of the resultingapproximate equations of motion and show that once di-rect particle production begins to dominate, the energytransfer from the inflaton to the matter fields will be al-most instantaneous. This allows us to estimate the valueof the reheating temperature, the temperature of matteronce the inflaton field has lost most of its energy densityto particle production. In the case of the second interac-tion term (10), the reheating temperature is given by theenergy density at the end of inflation, in the case of thefirst interaction term (9), it is reduced by a factor whichinvolves the interaction coupling constant g .In the present work we have studied the productionof particles which correspond to an entropy fluctuationdirection. There is the danger (see e.g. [35] for aninitial study and [36] for recent work) that the inducedentropy fluctuations might induce too large curvatureperturbations, as it does in the model studied in [37]. Astudy of this question will be the focus of future work. Acknowledgements
One of us (RB) wishes to thank the Institute for The-oretical Studies of the ETH Z¨urich for kind hospitalityduring the 2015/2016 academic year. RB acknowledgesfinancial support from Dr. Max R¨ossler, the “WalterHaefner Foundation” and the “ETH Zurich Foundation”,and from a Simons Foundation fellowship. The researchis also supported in part by funds from NSERC and theCanada Research Chair program. HBM also wishes tothank the Institute for Theoretical Studies of the ETHZ¨urich for kind hospitality while this work was being pre-pared. HBM is supported in part by an Iranian MSRTfellowship. This work was partially supported by OpenPartnership Joint Projects Grant of JSPS.
Appendix
In this Appendix we compare our calculation of par-ticle production in G-inflation with what is obtained us-ing more standard methods of quantum field theory incurved space-time (see e.g. [2, 38, 39]). Recall the equa-tion of motion for our canonical field X . In the case ofthe second coupling which we consider in the main textthis equation is X (cid:48)(cid:48) k + (cid:18) k + M − ˙ φ a − a (cid:48)(cid:48) a (cid:19) X k = 0 . (55)We will compare the energy density in produced particlesin the initial stages of particle production (before back-reaction becomes important).Starting from vacuum initial conditions, we can ob-tain the energy density from the Bogoliubov coefficients β k which describe how the solution of (55) that initiallycorresponds to the vacuum can at late times τ be decom-posed into positive and negative frequency modes (seee.g. [3] for a textbook treatment): ρ ( τ ) = 12 π a ( τ ) (cid:90) ∞ | β k | k dk . (56)The Bogoliubov coefficient β k is given by β k = i k (cid:90) ∞−∞ e − ikτ V ( τ ) dτ , (57)where in the case of the equation (55) V ( τ ) = (cid:32) m pl M − (cid:33) τ (58) (for times in the kination phase). The first term on theright hand side of this equation represents particle pro-duction via the particle interactions, whereas the secondterm corresponds to gravitational particle production.From this equation it is already clear that for M (cid:28) m pl particle interactions will dominate the energy transferfrom the inflaton field to matter.Following [37], the expression (56) for the energy den-sity can be rewritten as ρ ( τ ) = − π a (cid:90) τ −∞ dτ dτ ln( µ | ( τ − τ ) | ) V (cid:48) ( τ ) V (cid:48) ( τ )(59)where µ is a regularization scale which has been intro-duced to remove the ultraviolet divergence. 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