Understanding gravitational particle production in quintessential inflation
UUnderstanding gravitational particle production in quintessential inflation
Jaume de Haro, ∗ Supriya Pan, † and Llibert Arest´e Sal´o
1, 3, ‡ Departament de Matem`atiques, Universitat Polit`ecnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain Department of Mathematics, Presidency University, 86/1 College Street, Kolkata 700073, India TUM Physik-Department, Technische Universit¨at M¨unchen, James-Franck-Str.1, 85748 Garching, Germany
The diagonalization method, introduced by a group of Russian scientists at the beginning ofseventies, is used to compute the energy density of superheavy massive particles produced due toa sudden phase transition from inflation to kination in quintessential inflation models, the modelsunifying inflation with quintessence originally proposed by Peebles-Vilenkin. These superheavyparticles must decay in lighter ones to form a relativistic plasma, whose energy density will eventuallydominate the one of the inflaton field, in order to have a hot universe after inflation. In the presentarticle we show that, in order that the overproduction of Gravitational Waves (GWs) during thisphase transition does not disturb the Big Bang Nucleosynthesis (BBN) success, the decay has to beproduced after the end of the kination regime, obtaining a maximum reheating temperature in theTeV regime.
PACS numbers: 04.20.-q, 98.80.Jk, 98.80.BpKeywords: Particle production; Inflation; Quintessence; Reheating.
1. INTRODUCTION
Understanding the universe’s evolution has been a great mystery to modern cosmology. There are many questionsrelated to different phases of the universe that are still undisclosed even after continuous investigations with differentobservational missions. In particular, its early and late expansions have been a great deal at present time. Lookingat the literature, one can find two popular and well accepted theories, namely the inflation (the early evolution ofthe universe) and the quintessence (the late evolution of the universe). The inflationary paradigm [1–5] is actually anaccelerating phase of the early universe (in the context of standard Big Bang cosmology) that lasted for an extremelytiny time and became able to solve a number of shortcomings associated with the standard Big Bang cosmology,such as the horizon problem, flatness problem and some more. The potentiality of the inflationary theory was soonrecognized due to its ability to explain the origin of inhomogeneities in the universe [6–10]. Such an explanation wasfound to match greatly with the recent observational data from Planck [11]. Thus, it is interesting to note that thetheory that appeared at the beginning of the 80’s is still surviving quite well with the recent observational data. Andmoreover, the theory of inflation is the simplest viable theory that describes almost correctly the early universe inagreement with the recent observations [11]. On the other hand, the explanation for the current universe’s expansioncomes through the introduction of some quintessence field [12]. Thus, inflation and quintessence were thought to betwo different sides of a coin until the concept of the quintessential inflationary theory was introduced by Peebles andVilenkin [13].The idea to unify inflation with quintessence was indeed a novel attempt by Peebles and Vilenkin [13]. The noveltyof their proposal comes through the introduction of a single potential that at early time allows inflation while at latetime we have quintessence. Thus, a unified picture of the universe was effectively proposed connecting the distantearly phase to the present one. Thanks to this proposal, the origin of the scalar field responsible for the currentacceleration of the universe can be determined, and the fine-tunning problems are reduced [14]. Moreover, as wewill see, the models we deal with only depend on two real parameters, which are determined by the observationaldata. So, because of the behavior of the slow-roll regime as an attractor, the dynamics of the model is obtainedwith the value of the scalar field, its derivative and the initial conditions at some moment during the inflation. Thisshows the simplicity of the quintessential inflation, which from our viewpoint is a little bit simplest than the standardquintessence, where a minimum of two fields are needed to depict the evolution of the universe, namely, the inflatonand a quintessence field, and thus, one needs two different potentials and two different initial conditions, one for theinflaton, which has to be fixed during inflation, and another for the quintessence field, whose initial conditions have ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] a r X i v : . [ g r- q c ] J un to be fixed at the beginning of radiation era.This enhanced more investigation in order to connect quintessential inflation with the observational data [15–26] andconsequently this particular topic has become a popular area of research. The mechanism of the quintessential inflationmodel is very simple: once the inflationary phase is completed, a reheating mechanism is needed to match inflationwith the hot Big Bang universe [1] because the particles existing before the beginning of this period were completelydiluted at the end of inflation resulting in a very cold universe. The most accepted idea to reheat the universe inthe context of quintessential inflation comes through an abrupt phase transition of the universe from inflation tokination (a regime where all the energy density of the inflation turns into kinetic [27]) where the adiabatic regimeis broken and the particles are produced. The mechanism of particle production is not unique in this context sincea number of distinct mechanism are available and can be used. The first one is the gravitational particle production studied long time ago in [28–33], at the end of the 90’s in [34, 35] and more recently applied to quintessential inflationin [13, 36–38] for massless particles. A second well-known mechanism is the so-called instant preheating introducedin [39] and applied for the first time to inflation in [40] and recently in [38, 41] in the context of α -attractors insupergravity. Other less popular mechanisms are the curvaton reheating applied to quintessence inflation in [42, 43],production of massive particles self-interacting and coupled to gravity [44] and the reheating via production of heavymassive particles conformally coupled to gravity [26, 45–48]. The production of superheavy massive particles is theprimary concern of this work. Our main motivation for using a conformally-coupled scalar field is its simplicity.Alternatively one could use other massive fields but the calculations would be more cumbersome. For instance, theWentzel-Kramers-Brilloui (WKB) solution in equation 2.20 in [49] gets considerably simplified when the scalar fieldis conformally coupled, i.e. ξ = 1 / n s , r ) where n s is the scalar spectral index and r is the ratio of tensor toscalar perturbations, does not enter into the 95% confidence-level of Planck results [11]. However, a simple changein the inflationary piece − quartic to quadratic − can solve this issue (see [22] for a detailed discussion and also see[26]). On the other hand, the reheating mechanism followed in [13] is gravitational production of massless particlesthat results in a reheating temperature of the order of 1 TeV. This reheating temperature is not sufficient to solve theoverproduction of the Gravitational Waves (GWs). As a result the Big Bang Nucleosynthesis process can be hampered.Now, a lower bound for the reheating temperature comes in the following way. Since the radiation-dominated eraoccurs before the Big Bang Nucleosynthesis (BBN) epoch which takes place in the 1 MeV regime [50], the reheatingtemperature should naturally be greater than 1 MeV. But the upper bound of this reheating temperature is dependenton the theory we are concerned with. That means, in some supergravity and superstring theories containing particles(for instance the gravitino or a modulus field) with only gravitational interactions, the thermal production of theserelics and its late time decay may jeopardize the success of the standard BBN [51]. However, this problem can beavoided with the consideration of sufficiently low reheating temperature (of the order of 10 GeV) [52]. Finally, onealso needs to take into account that a viable reheating mechanism should deal with the pretension of the GravitationalWaves (GWs) in the BBN success that must satisfy the observational bounds appearing from the overproduction ofthe gravitational waves [13].Here we also consider a pre-heating due to the gravitational production of superheavy particles at the beginningof kination, where the inflationary and quintessence pieces of the quintessential potential are matched. The heavymassive particles due to this pre-heating will start decaying in lighter ones to form a thermal relativistic plasma. Weuse the well-known Hamiltonian diagonalization method (see [53] for a review) to calculate the energy density of theproduced particles, showing that before the beginning of kination the vacuum polarization effects, which are geometricobjects associated to the creation and annihilation of the so-called quasiparticles [53], are sub-dominant and have norelevant effect in the Friedmann equation. On the contrary, after the abrupt phase transition to kination heavymassive particles are produced and, since their energy density decreases as a − before decaying in lighter particlesand as a − after that, they will eventually dominate the energy density of the inflation whose decrease is as a − , andthus the universe will become reheated. Finally, we show that in our model the overproduction of GWs is compatiblewith the BBN success only when the decay of the superheavy particles is after the end of the kination phase, leadingto a reheating temperature of a few TeVs.As usual we note that in the present manuscript we have worked on the units where (cid:126) = c = 1 and the reducedPlanck’s mass is M pl ≡ √ πG ∼ = 2 . × GeV. We devote a full section on the use of WKB approximation
2. CREATION OF SUPERHEAVY PARTICLES CONFORMALLY COUPLED TO GRAVITY
In this section we shall describe the superheavy particles creation conformally coupled to gravity. Before that werefer to Appendix A (diaginalization method) and Appendix B (WKB approximation and its use in particle creation)which will be used throughout this work extensively. We begin this section with the consideration of the modelsbelonging to the category of quintessential inflation with an abrupt phase transition from the end of inflation to thebeginning of kination, as exactly in the Peebles-Vilenkin model [13], where some of the higher order derivatives of ω k ( τ ) are discontinuous, which is essential for an efficient production of superheavy particles. Otherwise, if we hada smooth transition, the production of particles would be exponentially suppressed [45] and its energy density wouldbe abnormally small. Therefore, it would never dominate those of the background, which means that the universewould never be reheated. In this way, the two quintessential inflationary models considered in this work are theimprovements of the well known Peebles-Vilenkin model as follows:1. The first quintessential inflationary model that we consider is, V ( ϕ ) = m (cid:16) ϕ − M pl + M (cid:17) for ϕ ≤ − M pl , m M ( ϕ + M pl ) + M for ϕ ≥ − M pl . (1)2. The second quintessential inflationary model in this work is, V ( ϕ ) = (cid:40) m ( ϕ + M ) for ϕ ≤ m M ϕ + M for ϕ ≥ . (2)While to understand the behavior of the above two modified potentials, we plot them in Fig. 1 [for eqn. (1)] and Fig.2 [for eqn. (2)] in two different scales in order to exactly show the abrupt phase transition. The left panels of bothFig. 1 and Fig. 2 are drawn in higher scale while the right panels of Fig. 1 and Fig. 2 are for smaller scales. Let usnote that while drawing the plots we have used the derived values of other parameters, namely, m and M , shown inthe latter part of this section. - - - φ M pl - - - - - V ( φ ) M pl4 - - - φ M pl × - × - × - × - × - V ( φ ) M pl4 FIG. 1: The figure depicts the evolution of the first improved version of the Peeble-Vilenkin potential of eqn. (1), in twodifferent scales, using the values derived in this section.
The inflation’s mass m is obtained from the power spectrum of the curvature fluctuation in co-moving coordinateswhen the pivot scale leaves the Hubble radius [57], given by P ζ ∼ = H ∗ π M pl (cid:15) ∗ ∼ × − , where (cid:15) = M pl (cid:16) V ϕ V (cid:17) is a slowroll parameter and the “star” ( ∗ ) attached to any quantity means that the quantity is evaluated when the pivot scaleleaves the Hubble radius. For the first potential one has (cid:15) ∗ = M pl ϕ ∗ ( ϕ ∗ − M pl ) ∼ = M pl ϕ ∗ , where we have used that − ϕ ∗ (cid:29) M pl .In the same way η ∗ = M pl V ϕϕ V ∼ = M pl ϕ ∗ , and since the spectral index is given by 1 − n s = 6 (cid:15) ∗ − η ∗ one gets (cid:15) ∗ ∼ = − n s .Finally, since at the time of the inflation the energy density is dominated by the potential term, using the Friedmannequation H ∗ = V ( ϕ ∗ )3 M pl one has m ∼ × − π (1 − n s ) M pl . (3) - - φ M pl - - - - - V ( φ ) M pl4 - - φ M pl × - × - × - × - × - V ( φ ) M pl4 FIG. 2: The figure depicts the evolution of the first improved version of the Peeble-Vilenkin potential of eqn. (2), in twodifferent scales, using the values derived in this section.
Thus, since recent observations constrain the value of the spectral index to be n s = 0 . ± .
006 [11], hence,taking its central value one can evaluate m ∼ = 5 × − M pl . The other parameter M is a very small mass comparedto the reduced Planck’s mass M pl , whose numerical value is determined so that at the present time the ratio of theenergy density of the inflaton field ϕ to the critical energy density is approximately around 0 . ρ ϕ, / (3 H M pl ) ∼ = 0 .
7, where the sub-index 0 means “at present time” and ρ ϕ = ˙ ϕ / V ( ϕ ) is the energy densityof the inflaton field. Numerical calculations performed in [58] show that the value of M depends on the reheatingtemperature and for a reheating temperature of the order of 100 TeV, which is the one obtained when the reheatingis due to the production of superheavy particles [26], one gets M ∼
18 GeV. Moreover, as we show in Figure 3 thevalues of the power spectrum and the ratio of tensor to scalar perturbations stand within 2 σ confidence level for somegiven Planck likelihoods but not if we consider all the ones available in the 2018 Planck results [59]. For that purpose,one would need to consider plateau potentials [60] or α -attractors [61, 62], such as an Exponential SUSY Inflationtype potential V α ( ϕ ) = λM pl (cid:18) − e αϕ/M pl + (cid:16) MM pl (cid:17) (cid:19) for ϕ ≤ λ M ϕ + M for ϕ ≥ , (4)or, a Higgs Inflation-type potential V α ( ϕ ) = λM pl (cid:18) − e αϕ/M pl + (cid:16) MM pl (cid:17) (cid:19) for ϕ ≤ λ M ϕ + M for ϕ ≥ . (5)For both potentials one can calculate the spectral index and the ratio of tensor to scalar perturbations, obtaining n s ∼ = 1 − N , r ∼ = 8 α N , (6)which implies that for α ∼ O (1) and for a number of e -folds greater than 60, which is typic in quintessential inflationdue to the kination phase, the ratio of tensor to scalar perturbations is less than 0 . σ CL for thePlanck TT, TE, EE + low E + lensing + BK14 + BAO likelihood.The dynamics of the first potential (and also the second one) is not difficult to understand. When ϕ (cid:28) − M pl ,the field slowly rolls and thus the universe inflates; after the inflation a phase transition from inflation to kination[27] occurs about ϕ ∼ = − M pl and the particles are produced. Since in a kination regime the energy density of thebackground decays as a − , this allows a relativistic plasma in thermal equilibrium, whose energy density evolves as a − , to eventually become dominant, and the universe is thus reheated. Finally, at the present time, the potentialenergy of the scalar field ϕ becomes dominant once again and the universe accelerates, depicting the current cosmicacceleration. Thus, as a result we have a unified framework where at the early time the universe experiences a rapidaccelerating phase and at late time another accelerating phase leading to the current dark energy era. Note also that FIG. 3: Marginalized joint confidence contours for ( n s , r ) at 68% and 95% confidence level. Considering the inflationary pieceof the potential as V = λφ β , in quintessential inflation, for the values of β = 2 , / , , /
3, we have drawn the curves from 65to 75 e-folds (see the black curves). And when one considers the standard inflation, for β = 2 ,
1, the curves have been drawnin red from 50 to 60 e -folds. As one can see, the quadratic potential ( V ∝ φ ), which is disregarded in standard inflation atgreater than 95% CL from a combination of Planck and BICEP2 limits on the tensor-to-scalar ratio [63], is favored for somelikelihoods in quintessential inflation. In the lower part of the image there are the curves for the values of α = 1 , r is nearly 0 and, if considering all Planck likelihoods, they stand within the 1 σ CL for alow number of e-folds (65 (cid:46) N (cid:46) σ CL for the other values of N . for the second model the second derivative of the potential is discontinuous at the beginning of kination. So, usingthe conservation equation, one can deduce that the third temporal derivative of the scalar field is discontinuous atthe begininning of kination, as well as the third temporal derivative of the Hubble parameter, as one can infere fromRaychaudhuri equation. The first potential is more abrupt and at the beginning of kination the second derivative ofthe Hubble parameter is discontinuous. So, dealing with the first one, the third derivative of the frequency ω k ( τ ) isdiscontinuous at the beginning of kination, namely τ kin .A key point is related to the initial conditions. It is well-known that at temperatures of the order of the Planck’smass quantum effects become very important and the classical picture of the universe is not possible of course.However, at temperatures below M pl , for example at GUT scales (i.e., when the temperature of the universe is ofthe order of T ∼ × − M pl ∼ GeV), the beginning of the Hot Big Bang (HBB) scenario is possible. Sincefor the flat FLRW universe the energy density of the universe, namely ρ , and the Hubble parameter H of the FLRWuniverse are related through the Friedmann equation ρ = 3 H M pl and the temperature of the universe is relatedto the energy density via ρ = ( π / g ∗ T (where g ∗ = 106 .
75 is the number degrees of freedom for the energydensity in the Standard Model), one can conclude that a classical picture of the universe might be possible when H ∼ = 5 × − M pl ∼ = 10 GeV. Then, if inflation starts at this scale, i.e. taking the value of the Hubble parameterat the beginning of inflation as H B = 5 × − M pl , we will assume as a natural initial condition that a superheavymassive quantum χ -field, whose decay products are the responsible of the reheating of the universe, is in the vacuumat the beginning of inflation. We will also choose the mass of the χ -field one order greater than this value of theHubble parameter ( m χ = 5 × − M pl ∼ = 10 GeV, which is a mass of the same order as those of the vectormesons responsible to transform quarks into leptons in simple theories with SU(5) symmetry [64]) because, as we willimmediately see, the polarization terms will be sub-dominant and do not affect the dynamics of the inflation field.So, we have m (cid:28) H B (cid:28) m χ (cid:28) M pl .To obtain the energy density of the produced particles by the χ -field (see formula (A9) of Appendix A) we have tocalculate the value of the β -Bogoliubov coefficient, whose expression has been derived in formula (A10). To performit, we have to integrate by parts two times, then before the beginning of kination one has β k ( τ ) = − ω (cid:48) k ( τ )4 iω k ( τ ) e − i (cid:82) τ ω k (¯ η ) d ¯ η + (cid:90) τ (cid:18) ω (cid:48) k ( η )4 iω k ( η ) (cid:19) (cid:48) e − i (cid:82) η ω k (¯ η ) d ¯ η dη = (cid:32) − ω (cid:48) k ( τ )4 iω k ( τ ) + 18 ω k ( τ ) (cid:18) ω (cid:48) k ( τ ) ω k ( τ ) (cid:19) (cid:48) + 116 iω k ( τ ) (cid:32) ω k ( τ ) (cid:18) ω (cid:48) k ( τ ) ω k ( τ ) (cid:19) (cid:48) (cid:33) (cid:48) + .... (cid:33) e − i (cid:82) τ ω k (¯ η ) d ¯ η . (7)However, after kination the β -Bogoliubov coefficient, in order to be continuous in time must be given by β k ( τ ) = (cid:32) − ω (cid:48) k ( τ )4 iω k ( τ ) + 18 ω k ( τ ) (cid:18) ω (cid:48) k ( τ ) ω k ( τ ) (cid:19) (cid:48) + 116 iω k ( τ ) (cid:32) ω k ( τ ) (cid:18) ω (cid:48) k ( τ ) ω k ( τ ) (cid:19) (cid:48) (cid:33) (cid:48) + .... (cid:33) e − i (cid:82) τ ω k (¯ η ) d ¯ η + C, (8)where the constant C has to be chosen in order that the β -Bogoliubov coefficient becomes continuous at τ kin becausethe equation (A8) [see Appendix A] is a first order differential equation. So, one has to impose continuity at thebeginning of kination in the same way that happens when one matches the modes. In this case, since they satisfy thesecond order K-G differential equation, the matching involves the continuity of the first derivative. Therefore, for thefirst potential one has C = (cid:32) iω k ( τ − kin ) (cid:32) ω k ( τ − kin ) (cid:18) ω (cid:48) k ( τ − kin ) ω k ( τ − kin ) (cid:19) (cid:48) (cid:33) (cid:48) − iω k ( τ + kin ) (cid:32) ω k ( τ + kin ) (cid:18) ω (cid:48) k ( τ + kin ) ω k ( τ + kin ) (cid:19) (cid:48) (cid:33) (cid:48) + .... (cid:33) e − i (cid:82) τkin ω k (¯ η ) d ¯ η = (cid:18) ω (cid:48)(cid:48)(cid:48) k ( τ − kin ) − ω (cid:48)(cid:48)(cid:48) k ( τ + kin )16 iω k ( τ kin ) + .... (cid:19) e − i (cid:82) τkin ω k (¯ η ) d ¯ η = (cid:32) m χ a kin ( a (cid:48)(cid:48)(cid:48) ( τ − kin ) − a (cid:48)(cid:48)(cid:48) ( τ + kin ))16 iω k ( τ kin ) + .... (cid:33) e − i (cid:82) τkin ω k (¯ η ) d ¯ η = (cid:32) m χ a kin ( ¨ H ( τ − kin ) − ¨ H ( τ + kin ))16 iω k ( τ kin ) + .... (cid:33) e − i (cid:82) τkin ω k (¯ η ) d ¯ η = (cid:32) m χ m a kin iω k ( τ kin ) + .... (cid:33) e − i (cid:82) τkin ω k (¯ η ) d ¯ η , (9)where a kin ≡ a ( τ kin ) and having used that¨ H ( τ + kin ) − ¨ H ( τ − kin ) = − ˙ ϕ kin M pl ( ¨ ϕ ( τ + kin ) − ¨ ϕ ( τ − kin )) = − ˙ ϕ kin M pl V ϕ ( − M − pl ) = m ˙ ϕ kin M pl = m , with ˙ ϕ ( τ kin ) ≡ ˙ ϕ kin , (10)with the assumption that there is no substantial drop of energy density between the end of inflation and the beginningof kination. Thus, at τ kin all the energy density is kinetic and given by √ m M pl because at the end of inflation,where all the energy density is potential, one has ϕ end = − (cid:112) √ M pl . The terms that do not contain C leadto a sub-leading geometric quantities in the energy density. Effectively, the term − ω (cid:48) k ( τ )4 iω k ( τ ) leads to the followingcontribution to the energy density m χ H π (cid:28) M pl H . The same happens with the term ω k ( τ ) (cid:0) ω (cid:48) k ( τ ) /ω k ( τ ) (cid:1) (cid:48) whichleads to a term of order H , which means that H M pl (cid:28) H . The product of the first and second term generates in theright-hand side of the modified semi-classical Friedmann equation a term of the order H m χ M pl , which is also sub-leadingcompared with H . Finally, the third term of (8) leads in the right-hand side of the semi-classical Friedmann equationto the sub-leading term H m χ M pl .Fortunately, this does not happen with C , whose leading term is m χ m a kin iω k ( τ kin ) , leading to the contribution (see [26]and the appendix of [65] for a detailed derivation of this result) (cid:104) ρ ( τ ) (cid:105) ∼ = (cid:40) τ < τ kin − (cid:16) mm χ (cid:17) m (cid:16) a kin a ( τ ) (cid:17) when τ ≥ τ kin , (11)which at the beginning of kination is sub-dominant with respect to the energy density of the background but eventuallyit will dominate because the one of the background, during kination, decreases as a − ( τ ). Remark 2.1
The authors of the diagonalization method assume that during the whole evolution of the universequanta named quasiparticles are created and annihilated due to the interaction with the quantum field with gravity [53]. Following this interpretation, the number density of created quasiparticles at time τ is given by (cid:104) N ( τ ) (cid:105) = π a ( τ ) (cid:82) ∞ k | β k ( τ ) | dk . However, one has to be very careful with this interpretation and specially keep in mindthat real particles are only created when the adiabatic regime breaks. Effectively, before the beginning of kinationthe main term of the β k -Bogoliubov coefficient is given by − ω (cid:48) k ( τ )4 iω k ( τ ) , whose contribution to the energy density is m χ H π , and to the number density of quasiparticles is m χ H π , and thus, at time τ before the beginning of kination (cid:104) ρ ( τ ) (cid:105) (cid:54) = m χ (cid:104) N ( τ ) (cid:105) . On the contrary, during kination the leading term of (cid:104) N ( τ ) (cid:105) is given by − (cid:16) mm χ (cid:17) m (cid:16) a kin a ( τ ) (cid:17) ,so we have (cid:104) ρ ( τ ) (cid:105) = m χ (cid:104) N ( τ ) (cid:105) and the decay follows a − ( τ ) , which justifies the interpretation of massive particleproduction. Finally, for the second potential a similar calculation leads to | β k ( τ ) | ∼ = m χ a ( τ kin )(... H ( τ − kin ) − ... H ( τ + kin )) ω k ( τ kin ) = m χ m a kin ω k ( τ kin ) , (12)and a simple calculation shows that, after the beginning of kination, the energy density is given by (cid:104) ρ ( τ ) (cid:105) ∼ = 8 × − (cid:18) mm χ (cid:19) m (cid:18) a kin a ( τ ) (cid:19) , (13)which is smaller than the one obtained from the first, more abrupt, potential.
3. THE REHEATING PROCESS
After the production of the heavy massive particles, they have to decay in lighter particles which after the ther-malization process form a relativistic plasma that depicts our hot universe. Two different situations may arise, asfollows:1. The decay is before the end of the kination regime, which happens at time τ r , when the energy density of theinflaton becomes equal to the one of the χ -field.2. The decay is after the end of the kination regime.Here we consider the decay of the χ -field into fermions ( χ → ψ ¯ ψ ), then the decay rate will be given by [64] Γ = h m χ π and the decay is finished at τ dec when Γ ∼ H ( τ dec ) ≡ H dec . Let us begin the discussion with the first potential. In this case, the energy density of the background, i.e. theone of the inflaton field, and the one of the relativistic plasma, when the decay is finished, that is when Γ ∼ H dec = H kin (cid:16) a kin a dec (cid:17) ∼ = √ √ √ m (cid:16) a kin a dec (cid:17) , will be ρ ϕ,dec = 3Γ M pl and (cid:104) ρ dec (cid:105) ∼ = 1 . × − (cid:18) mm χ (cid:19) Γ m m , (14)where we have used that there is no drop of energy density between the end of inflation and the beginning of kination,i.e., H ( τ kin ) ≡ H kin = √ m .Imposing that the end of the decay precedes the end of kination, that means, (cid:104) ρ dec (cid:105) ≤ ρ ϕ,dec , one gets h ≥ π × − (cid:18) mm χ (cid:19) (cid:18) mM pl (cid:19) , (15)which, for the value of the inflaton mass m ∼ = 5 × − M pl and the bare mass of the quantum field m χ ∼ = 5 × − M pl ,constrains the value of the coupling constant as h ≥ . × − . Moreover, since the decay is after the beginning ofthe kination, one has Γ ≤ H kin , obtaining h ≤ πH kin m χ , which for the values of H kin and m χ gives another restriction,namely h ≤ . × − . Thus, we have obtained that the parameter h is constrained as 5 . × − ≤ h ≤ . × − .Then the reheating temperature (i.e., the temperature of the universe when the relativistic plasma in thermalequilibrium starts to dominate, which happens when ρ ϕ,reh = (cid:104) ρ reh (cid:105) ⇐⇒ (cid:104) ρ dec (cid:105) ρ ϕ,dec = ( a dec /a reh ) ) will be T reh = (cid:18) π g ∗ (cid:19) / (cid:104) ρ reh (cid:105) = (cid:18) π g ∗ (cid:19) / (cid:104) ρ dec (cid:105) (cid:115) (cid:104) ρ dec (cid:105) ρ ϕ,dec ∼ = 2 × − g − / ∗ (cid:18) mm χ (cid:19) / (cid:16) m Γ (cid:17) / (cid:18) mM pl (cid:19) M pl , (16)where g ∗ is the number of degrees of freedom. Now, for the values of the masses involved in the process, the reheatingtemperature is of the order T reh ∼ = 3 . × − h − / g − / ∗ M pl ∼ = 8 h − / g − / ∗ GeV , (17)which, for the number of the degrees of freedom for the energy density in the Standard Model, i.e. g ∗ = 106 .
75, rangesbetween 4 GeV and 330 TeV.To end this subsection, we deal with the second potential, i.e., with equation (2), which has a smoother phasetransition compared to the first potential (1). As we have already showed, in this case the energy density of theproduced massive particles is given by (cid:104) ρ ( τ ) (cid:105) ∼ = 8 × − (cid:18) mm χ (cid:19) m (cid:18) a kin a ( τ ) (cid:19) , (18)and for the same decaying rate as in the previous cases the corresponding energy densities at the end of decay will be ρ ϕ,dec = 3Γ M pl , and (cid:104) ρ dec (cid:105) ∼ = 1 . × − (cid:18) mm χ (cid:19) Γ m . (19)Assuming, once again, that the end of the decay occurs before the radiation-domination epoch (i.e., (cid:104) ρ dec (cid:105) ≤ ρ ϕ,dec ),one obtains the relation h ≥ π × − (cid:18) mm χ (cid:19) (cid:18) mM pl (cid:19) , (20)which for the values m ∼ = 5 × − M pl and m χ ∼ = 5 × − M pl leads to the constraint h ≥ . × − . On the otherhand, together with the condition Γ ≤ H kin , it leads to, 5 . × − ≤ h ≤ . × − .Finally, if the thermalization of the relativistic plasma is instantaneous, the reheating temperature turns out to be T reh = (cid:18) π g ∗ (cid:19) / (cid:104) ρ dec (cid:105) / (cid:115) (cid:104) ρ dec (cid:105) ρ ϕ,dec ∼ = 6 . × − (cid:18) mm χ (cid:19) (cid:18) mM pl (cid:19) g − / ∗ h − / M pl ∼ = 5 . × − g − / ∗ h − / M pl ∼ = 12 g − / ∗ h − / MeV , (21)which for g ∗ = 106 .
75 ranges between 6 MeV and 5 TeV.
Now we assume that the decay of the χ -field is after the end of kination. Then, one has to impose Γ ≤ H ( τ r ) ≡ H r ,where we have denoted by τ r the time at which kination ends. Taking this into account, one has H r = 2 ρ ϕ,r M pl and ρ ϕ,r = ρ ϕ,kin (cid:18) a kin a r (cid:19) = 3 H kin M pl Θ , (22)in which, taking into account that during kination the energy density of the inflaton field decays as a − and the one ofthe produced particles as a − , we have introduced the so-called heating efficiency , defined as Θ ≡ ( a kin /a r ) = (cid:104) ρ kin (cid:105) ρ ϕ,kin .Consequently, from equation (22), one can easily have H r = √ H kin Θ and, sinceΘ = (cid:26) . × − , for potential 1 , . × − , for potential 2 , (23)one obtains that the parameter h has to be very small satisfying h ≤ . × − √ Θ, which means that for the firstpotential h ≤ × − while for the second potential, h ≤ . × − . Assuming once again the instantaneousthermalization, the reheating temperature (i.e., the temperature of the universe when the thermalized plasma startsto dominate) becomes T reh = (cid:18) π g ∗ (cid:19) / (cid:104) ρ dec (cid:105) / = (cid:18) π g ∗ (cid:19) / (cid:112) Γ M pl , (24)where we have used that after τ r the energy density of the produced particles dominates the one of the inflaton field.Then, we will have T reh ∼ = 7 × − hg − / ∗ M pl . (25)Consequently, assuming that the BBN epoch occurs at the 1 MeV regime and taking g ∗ = 106 .
75, one can find thatthe value of h resides in the interval 10 − (cid:46) h (cid:46) − .
4. PRODUCTION OF GRAVITATIONAL WAVES
In this Section we study the production of gravitational waves (GWs), which is the same as the gravitational particleproduction of massless particles minimally coupled to gravity, due to a sudden phase transition from a de Sitter phaseto an exact kination regime, i.e., when the EoS parameter is exactly 1.The model is given by the following dynamics. The conformal Hubble parameter for this model evolves as H ( τ ) = − τ for τ < τ kin < τ − τ kin ) for τ ≥ τ kin , (26)and the scale factor evolves with a ( τ ) = − H kin τ for τ < τ kin < a kin (cid:113) τ − τ kin − τ kin for τ ≥ τ kin , (27)where H kin is the value of the Hubble parameter during the de Sitter phase and a kin = − H kin τ kin . The k -mode isgiven by χ k ( τ ) = (cid:40) √ k e − ikτ (cid:0) − ikτ (cid:1) for τ < τ kin < α k (cid:113) π ( τ − τ kin )4 H (2)0 (cid:0) k ( τ − τ kin ) (cid:1) + β k (cid:113) π ( τ − τ kin )4 H (1)0 (cid:0) k ( τ − τ kin ) (cid:1) for τ ≥ τ kin , (28)where H (1)0 and H (2)0 are the Hankel’s functions. These modes satisfy the equation χ (cid:48)(cid:48) k + Ω k ( τ ) χ k = 0 , (29)where we have introduced the notation Ω k ( τ ) ≡ k − a (cid:48)(cid:48) a .From a simple calculation one could find that a (cid:48)(cid:48) a ∝ a H , which shows that the modes well inside the Hubble radius( k (cid:29) aH = H ∝ τ ) do not feel gravity and, thus, no particles are produced during the phase transition. So, onlythe ones well outside of the Hubble radius have to be used to compute the energy density of the produced particles,which is actually given by [49] (cid:104) ρ GW ( τ ) (cid:105) = 14 π a ( τ ) (cid:90) H kin (cid:8) ( | χ (cid:48) k | + k | χ k | − k ) − (cid:2) H ( | χ k | ) (cid:48) − H | χ k | (cid:3)(cid:9) k dk = 14 π a ( τ ) (cid:90) H kin (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) χ k ( τ ) a ( τ ) (cid:19) (cid:48) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + k (cid:12)(cid:12)(cid:12)(cid:12) χ k ( τ ) a ( τ ) (cid:12)(cid:12)(cid:12)(cid:12) − k k dk, (30)where as in the massive case, the zero-point oscillations of the vacuum have been substracted.The calculation has to be done in three steps:01. For modes that are outside the Hubble radius at the beginning of kination and re-enter it during kination, i.e.,satisfying H r < k < H kin (where we have denoted by H r the value of the conformal Hubble parameter at theend of kination), when τ (cid:38) τ r one has (cid:46) τ τ r ∼ = τ H r < kτ , so the modes practically do not feel gravity and,thus, we can make the approximation χ k ( τ ) = α k e − ikτ √ k + β k e ikτ √ k . (31)2. For modes that are outside of the Hubble radius at the end of kination ( k < H r ), we can use the small argumentapproximation of Hankel’s functions and obtain χ k ( τ ) = α k (cid:115) π ( τ − τ kin )4 (cid:18) − iπ (cid:18) γ + ln (cid:18) k ( τ − τ kin )2 (cid:19)(cid:19)(cid:19) + β k (cid:115) π ( τ − τ kin )4 (cid:18) iπ (cid:18) γ + ln (cid:18) k ( τ − τ kin )2 (cid:19)(cid:19)(cid:19) . (32)3. As we have already explained the relevant modes satisfy k < H kin ⇐⇒ k | τ kin | <
1. Thus, in order to calculatethe Bogoliubov coefficents, which are obtained matching the modes at its first derivative at τ kin , one can usethe small argument approximation of Hankel’s functions and obtain that α k = ie − ikτ kin √ π (cid:34) (cid:18) H kin k (cid:19) / + 12 (cid:18) H kin k (cid:19) − / (cid:18) γ + ln (cid:18) k H kin (cid:19)(cid:19) − i (cid:32)(cid:18) H kin k (cid:19) / + π (cid:18) H kin k (cid:19) − / (cid:33) (cid:35) (33) β k = ie − ikτ kin √ π (cid:34) (cid:18) H kin k (cid:19) / + 12 (cid:18) H kin k (cid:19) − / (cid:18) γ + ln (cid:18) k H kin (cid:19)(cid:19) − i (cid:32)(cid:18) H kin k (cid:19) / − π (cid:18) H kin k (cid:19) − / (cid:33) (cid:35) . (34)Note that the Bogoliubov coefficients satisfy the well known relation | α k | − | β k | = 1 and the leading term of β k is i √ π (cid:0) H kin k (cid:1) / .For modes satisfying H r < k < H kin , the contribution to the energy density when τ (cid:38) τ r is12 π a ( τ ) (cid:90) H kin H r k | β k | dk − π a ( τ ) (cid:90) H kin H r (cid:2) H ( | χ k | ) (cid:48) − H | χ k | (cid:3) k dk. (35)The first term leads to π H kin (cid:16) a kin a ( τ ) (cid:17) and the second one is bounded by π a ( τ ) (cid:82) H kin H r [ k H + H ]( | α k | + | β k | ) kdk. Then, taking the leading terms of the Bogoliubov coefficients, one gets14 π a ( τ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) H kin H r (cid:2) H ( | χ k | ) (cid:48) − H | χ k | (cid:3) k dk (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ π H kin ln (cid:18) a r a kin (cid:19) (cid:18) a kin a ( τ ) (cid:19) + 12 π H kin a kin a r a ( τ ) (36)and, taking into account the bounds (cid:16) a kin a r (cid:17) (cid:28) (cid:16) a r a kin (cid:17) (cid:16) a kin a r (cid:17) (cid:28)
1, one can see that the first term of (35)is the leading one and its contribution to the energy density of GWs is π H kin (cid:16) a kin a ( τ ) (cid:17) .For modes satisfying k < H r , using the small argument approximation and the formulas α k χ k ( τ ) + β k χ ∗ k ( τ ) a ( τ ) = e − ikτ kin a kin √ H kin (cid:34)(cid:18) H kin k (cid:19) / + i (cid:40)(cid:18) H kin k (cid:19) / + 12 (cid:18) H kin k (cid:19) − / ln (cid:18) HH kin (cid:19)(cid:41)(cid:35) (37)1and (cid:18) α k χ k ( τ ) + β k χ ∗ k ( τ ) a ( τ ) (cid:19) (cid:48) = − ie − ikτ kin a kin √ H kin τ − τ kin (cid:18) H kin k (cid:19) − / = − ie − ikτ kin (cid:16) a kin a (cid:17) (cid:114) H kin a kin (cid:18) H kin k (cid:19) − / , (38)one obtains the following contribution to the energy density, H kin π (cid:18) a kin a ( τ ) a r + 2 a kin a ( τ ) a r + (cid:18) (cid:18) a kin a ( τ ) (cid:19)(cid:19) a kin a ( τ ) a r + 13 H kin ln (cid:18) a kin a ( τ ) (cid:19) a kin a ( τ ) a r (cid:19) , (39)which is sub-leading compared to π H kin (cid:16) a kin a ( τ ) (cid:17) and, thus, one can conclude that the energy density of GWs when τ (cid:38) τ r turns out to be (cid:104) ρ GW ( τ ) (cid:105) ∼ = H kin π (cid:18) a kin a ( τ ) (cid:19) ∼ = 10 − H kin (cid:18) a kin a ( τ ) (cid:19) . (40)We close this section with a short remark on the β -Bogoliubov coefficient. We noted that the β -Bogoliubovcoefficient calculated by us mildly differs from [68]. In particular, eqn. (C.2) of Appendix C of [68] has a very mildmismatch with us. However, such difference does not affect the main results and conclusions of [68] apart from afactor in the BBN bound. However, inspite of that for interested readers we present our calculations in Appendix C. The success of the BBN demands that the ratio of the energy density of GWs to the one of the produced particlesat the reheating time satisfies [41] (cid:104) ρ GW,reh (cid:105)(cid:104) ρ reh (cid:105) ≤ − . (41)This bound could never be accomplished when reheating is due to the gravitational production of massless particlesbecause the energy density of those particles decreases as the one of GWs [13], i.e., as we have already seen in theprevious section, close to the end of kination the energy density decreases as 10 − H kin ( a kin /a ( τ )) .In the same way, dealing with heavy massive particles, first of all we see that the constraint (41) is never overpassedwhen the decay of the massive particles is previous to the end of kination. Effectively, if the decay occurs after theend of kination one can calculate (cid:104) ρ GW ( τ ) (cid:105)(cid:104) ρ ( τ ) (cid:105) at the end of kination. Precisely, using equation (22) and the fact thatΘ = ( a kin /a r ) , one finds (cid:104) ρ GW,r (cid:105)(cid:104) ρ r (cid:105) = 13 10 − (cid:18) H kin M pl (cid:19) Θ − / ∼ = (cid:26) . × − , for potential 1 , . , for potential 2 . (42)This result shows that, if the decay occurs before the end of kination, the constraint (41) is never achieved becauseafter the decay the energy density of the produced particles decreases as the one of the GWs, so in that case (cid:104) ρ GW,reh (cid:105)(cid:104) ρ reh (cid:105) is greater than 3 . × − for the first potential and it is also greated than 2 . (cid:104) ρ dec (cid:105) =3Γ M pl and H dec = H r (cid:18) a r a dec (cid:19) / = ⇒ (cid:18) a r a dec (cid:19) / = Γ √ H kin Θ , (43)we will have (cid:104) ρ GW,dec (cid:105) = (cid:104) ρ GW,r (cid:105) (cid:18) a r a dec (cid:19) = (cid:104) ρ GW,r (cid:105) (cid:18) Γ √ H kin Θ (cid:19) / = 10 − H kin Θ − / (cid:18) Γ √ H kin (cid:19) / , (44)2and thus, (cid:104) ρ GW,reh (cid:105)(cid:104) ρ reh (cid:105) ∼ = 10 − (cid:18) h Θ (cid:19) / m / χ H / kin M pl ∼ = (cid:26) × h / , for potential 12 . × h / , for potential 2 , (45)from which one can see that the constraint (41) is satisfied for h ≤ . × − (for the first potential) and for h ≤ × − (for the second potential). Therefore, for g ∗ = 106 .
75 and using the equation (25), one can see that themaximum reheating temperature in the case of the first potential turns out to b, T reh ∼ = 57 TeV, while for the secondpotential T reh ∼ = 3 TeV.A final remark is in order: After the discovery of the Higgs boson, it is well-know that there exists at least oneother scalar field, which during inflation it appears to be a spectator field with no dynamical role [70]. The StandardModel Higgs doublet could be parametrized with a single scalar degree of freedom, namely φ , whose potential forlarge amplitudes is just given by a quadratic potential [71] V ( φ ) = λ φ , (46)where λ is the self-coupling constant.It has been showed in section 2.1 of [70] (see also section II A of [71]) that at the end of inflation the energy densityof the Higgs field, namely ρ φ , is ρ φ ∼ − H ∗ , where H ∗ is the Hubble scale at the end of inflation [72], that is, H ∗ = H end ∼ = H kin , because there is not substantial drop of energy between the end of inflation and the beginning ofkination. So, at the beginning of kination the energy density of the Higgs scalar is approximately one order less thanthe energy density of the GWs (see formula (40)).On the other hand, assuming that the Higgs field starts to oscillate immediately after the end of inflation, then sincethe potential is quartic, using the Virial Theorem we can deduce that during the oscillations its effective Equation ofState parameter is given by w eff = 1 / ρ φ ( τ ) ≤ (cid:104) ρ GW ( τ ) (cid:105) after the beginning of kination. This means thatat the reheating time ρ φ,reh (cid:104) ρ reh (cid:105) ≤ (cid:26) × h / , for potential 12 . × h / , for potential 2 , (47)and for a very low reheating temperature, for example 1 MeV, which corresponds to h ∼ − (see below formula(25)), one gets ρ φ,reh (cid:104) ρ reh (cid:105) ≤ (cid:26) . × − , for potential 11 . × − , for potential 2 , (48)that is, at the reheating time the energy density from the Higgs condensate decay is completely negligible comparedwith the energy density of the decay products of the superheavy χ -field.
5. CONCLUSIONS
The description of both early inflationary phase and late quintessence phase in a single framework was named asquintessential inflationary models by Peebles and Vilenkin. This class of unified cosmic models has gained a robustattention to the cosmological community since its appearance. Later on, the developments of the observational datahave clarified many issues, including the shortcomings of those models, and eventually the quintessential inflationarymodels have been revised either by replacing the inflationary piece of the models or by introducing a different reheatingmechanism via gravitatational particle production. The present work has aimed to discuss the understanding of thegravitational particle production in such models.Thus, assuming two quintessential inflationary models, we study the creation of superheavy massive particlesconformally coupled to gravity at the beginning of kination regime, where the adiabatic regime is broken. First of allwe have shown how to perform the calculation of the energy density of the produced particles using the well-knowndiagonalization method, proving that before the beginning of kination the one-loop energy density of the vacuum onlycontains sub-dominant geometric polarization terms, i.e., terms that do not affect the classical Friedmann equation.3Only after the beginning of kination, where the adiabatic regime is broken, particles are created and its energy densityis calculated.We also show that the same energy density of the produced particles could be obtained by approximating thevacuum modes using the WKB approximation and performing the matching of the modes at its first derivative atthe beginning of kination. Since these superheavy particles have to decay in lighter ones to form a relativistic plasmawhich eventually becomes dominant and matches with the hot big bang universe, two different situations arise, namely,when the decay occurs before the end of kination regime and when the decay occurs after the end of the kinationregime. Thus, for both situations we have calculated the reheating temperature of the universe, i.e., the temperatureof the universe when the energy density of the inflaton field is of the same order as the relativistic plasma as a functionof the decay rate.Finally, we have also reviewed with all the details the calculation of the energy density of the produced GWs dueto the phase transition from inflation to kination, obtaining a β -Bogoliubov coefficient differing by a logarithmic term[68]. Such a difference plays no effective role because, apart from a numerical factor in the BBN bound, nothingactually changes. Moreover, we have also shown that, in order that this overproduction of GWs does not affectthe BBN success, the decay of the heavy massive particles must be after the end of kination, obtaining reheatingtemperatures in the TeV regime. Acknowledgments
This investigation has been supported by MINECO (Spain) grants MTM2014-52402-C3-1-P and MTM2017-84214-C2-1-P, and in part by the Catalan Government 2017-SGR-247. SP acknowledges the research grant under FacultyResearch and Professional Development Fund (FRPDF) Scheme of Presidency University, Kolkata, India. The authorsthank Prof. M. Giovannini and Prof. J. D. Barrow for useful correspondence.
Appendix A: The diagonalization method
The diagonalization method was developed during the seventies of last century by the Russian scientists Grib,Frolov, Mamayev, Mostepanenko [29–31] and also by Zeldovich and Starobinsky [54]. Principally, for a quantumscalar field of superheavy particles conformally coupled to gravity, namely χ , the Klein-Gordon (K-G) equation in theflat Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) spacetime follows χ (cid:48)(cid:48) + 2 H χ (cid:48) − ∇ χ + (cid:18) m χ a + a (cid:48)(cid:48) a (cid:19) χ = 0 , (A1)where the prime attached to any quantity denotes the derivative with respect the conformal time τ ; H ≡ a (cid:48) /a , is theconformal Hubble parameter and m χ is the mass of the scalar field. Now, writing the quantum field in Fourier space, χ ( x , τ ) = 1(2 π ) / a (cid:90) d k (cid:16) ˆ a k χ k ( τ ) e − i k . x + ˆ a † k χ ∗ k ( τ ) e i k . x (cid:17) , (A2)where d k = dk dk dk , k = ( k , k , k ), x = ( x , x , x ), k = (cid:112) k + k + k and ˆ a k is the annihilation operatorcorresponding to the vacuum state at a given initial time τ i , which is defined by the condition χ k ( τ i ) = 1 (cid:112) ω k ( τ i ) e − i (cid:82) τi ω k (¯ η ) d ¯ η , χ (cid:48) k ( τ i ) = − iω k ( τ i ) χ k ( τ i ) , (A3)with ω k ( τ ) = (cid:113) k + m χ a ( τ ), the Klein-Gordon equation (A1) becomes χ (cid:48)(cid:48) k ( τ ) + ω k ( τ ) χ k ( τ ) = 0 , (A4)which is the equation of a harmonic oscillator with time dependent frequency ω k ( τ ). Additionally, the energy densityof the vacuum is given by [49] (cid:104) ρ ( τ ) (cid:105) ≡ (cid:104) | ˆ ρ ( τ ) | (cid:105) = 14 π a ( τ ) (cid:90) ∞ k dk (cid:0) | χ (cid:48) k ( τ ) | + ω k ( τ ) | χ k ( τ ) | − ω k ( τ ) (cid:1) , (A5)where in order to obtain a finite energy density [53] we have subtracted the energy density of the zero-point oscillationsof the vacuum π ) a ( τ ) (cid:82) d k ω k ( τ ).4 Remark A.1
For a quantum field not conformally coupled to gravity, it is not enough to subtract the energy densityof the zero-point oscillations of the vacuum to get a finite energy density. In that case one needs a more complicatedregularization process such as the subtraction of adiabatic terms up to the four order [49], the point splitting method[55, 56] or the n − wave procedure [53]. We follow the method developed in [54] (see also Section 9 . χ k ( τ ) = α k ( τ ) e − i (cid:82) τ ω k (¯ η ) d ¯ η (cid:112) ω k ( τ ) + β k ( τ ) e i (cid:82) τ ω k (¯ η ) d ¯ η (cid:112) ω k ( τ ) , (A6)where α k ( τ ) and β k ( τ ) are the time-dependent Bogoliubov coefficients. Now, imposing that the modes satisfy thecondition χ (cid:48) k ( τ ) = − iω k ( τ ) (cid:32) α k ( τ ) e − i (cid:82) τ ω k (¯ η ) d ¯ η (cid:112) ω k ( τ ) − β k ( τ ) e i (cid:82) τ ω k (¯ η ) d ¯ η (cid:112) ω k ( τ ) (cid:33) , (A7)one can show that the Bogoliubov coefficients must satisfy the system (cid:40) α (cid:48) k ( τ ) = ω (cid:48) k ( τ )2 ω k ( τ ) e i (cid:82) τ ω k (¯ η ) d ¯ η β k ( τ ) β (cid:48) k ( τ ) = ω (cid:48) k ( τ )2 ω k ( τ ) e − i (cid:82) τ ω k (¯ η ) d ¯ η α k ( τ ) , (A8)and thus the expression (A6) is the solution of the equation (A4). Remark A.2
Since the Wronskian is conserved and W [ χ k ( τ i ) , χ ∗ k ( τ i )] ≡ χ k ( τ i )( χ ∗ k ) (cid:48) ( τ i ) − χ (cid:48) k ( τ i ) χ ∗ k ( τ i ) = i , one cansee that the Bogoliubov coefficients satisfy the equation | α k ( τ ) | − | β k ( τ ) | = 1 . Finally, inserting (A6) into the expression for vacuum energy density (A5), one finds that (cid:104) ρ ( τ ) (cid:105) = 12 π a ( τ ) (cid:90) ∞ k ω k ( τ ) | β k ( τ ) | dk. (A9)Coming back to the equation (A8), in the first approximation taking α k ( τ ) = 1, we get β k ( τ ) = (cid:90) τ ω (cid:48) k ( η )2 ω k ( η ) e − i (cid:82) η ω k (¯ η ) d ¯ η dη. (A10)Finally, it is important to stress that the classical Friedmann equation is modified by the following semi-classicalequation H = M pl ( ρ + (cid:104) ρ (cid:105) ). Appendix B: The use of the WKB approximation to calculate particle production
The Wentzel-Kramers-Brilloui (WKB) approximation applied to cosmology (see for instance [66, 67], and referencestherein) shows that the vacuum mode during the adiabatic regime can be approximated by χ W KBn,k ( τ ) ≡ (cid:115) W n,k ( τ ) e − i (cid:82) τ W n,k ( η ) dη , (B1)where n is the order of the approximation and W n,k ( τ ) is calculated as follows (see for more details [67]). First of all,instead of equation (A4) we consider the following equation (cid:15) ¯ χ (cid:48)(cid:48) k + ω k ( τ ) χ k = 0 , (B2)where ¯ (cid:15) is a dimensionless parameter that one may set ¯ (cid:15) = 1 at the end of calculations. Looking for a solution of (B2)of the form χ W KBn,k ( τ ; ¯ (cid:15) ) = 1 (cid:112) W n,k ( τ ; ¯ (cid:15) ) e − i ¯ (cid:15) (cid:82) τ W n,k ( η ;¯ (cid:15) ) dη , (B3)5where W ,k ( τ ; ¯ (cid:15) ) ≡ ω k ( τ ), inserting (B3) into (B2) and collecting the terms of order ¯ (cid:15) n , one arrives at the iterativeformula W n,k ( τ ; ¯ (cid:15) ) = terms up to order ¯ (cid:15) n of (cid:118)(cid:117)(cid:117)(cid:116) ω k ( τ ) − ¯ (cid:15) (cid:34) W (cid:48)(cid:48) n − ,k ( τ ; ¯ (cid:15) ) W n − ,k ( τ ; ¯ (cid:15) ) −
34 ( W (cid:48) n − ,k ( τ ; ¯ (cid:15) )) W n − ,k ( τ ; ¯ (cid:15) ) (cid:35) . (B4)For the first potential (1) one only needs the first order WKB solution to approximate the k -vacuum modes beforeand after the beginning of kination, given by χ W KB ,k ( τ ) ≡ (cid:115) W ,k ( τ ) e − i (cid:82) τ W ,k ( η ) dη , (B5)where W ,k has the expression [67] W ,k = ω k − ω (cid:48)(cid:48) k ω k + 38 ( ω (cid:48) k ) ω k , (B6)because W ,k contains the first derivative of the Hubble parameter and, since the matching involves the derivative ofthe mode and the second derivative of the Hubble parameter is discontinuous at τ kin , the β -Bogoliubov coefficientdoes not vanish. Effectively, before the beginning of kination the vacuum mode is depicted by χ W KB ,k ( τ ), but after τ kin this mode becomes a mix of positive and negative frequencies of the form α k χ W KB ,k ( τ ) + β k ( χ W KB ,k ) ∗ ( τ ), whichis the manifestation of the particle production. The β k -Bogoliubov coefficient is obtained matching both expressionsat τ kin , leading to β k ( τ ) = W [ χ W KB ,k ( τ − kin ) , χ W KB ,k ( τ + kin )] W [( χ W KB ,k ) ∗ ( τ + kin ) , χ W KB ,k ( τ + kin )] = i W [ χ W KB ,k ( τ − kin ) , χ W KB ,k ( τ + kin )] , (B7)where W [ f, g ] = f g (cid:48) − f (cid:48) g denotes the Wronskian of the functions f and g , and we have introduced the notation f ( τ + kin ) = lim τ → τ kin ; τ>τ kin f ( τ ) and f ( τ − kin ) = lim τ → τ kin ; τ<τ kin f ( τ ).The square modulus of the β -Bogoliubov coefficient will be given approximately by [22] | β k ( τ ) | ∼ = m χ a kin (cid:16) ¨ H ( τ + kin ) − ¨ H ( τ − kin ) (cid:17) ω k ( τ kin ) , (B8)which coincides with the square modulus of the leading term of the integration constant C obtained in equation (9),as happens with the second potential. This shows the equivalence between the methods to obtain the energy densityof the produced particles. Appendix C: An additional remark on the β -Bogoliubov coefficient In Ref. [68], the author obtains that the leading value of the β -Bogoliubov coefficient is β k ∼ √ π (cid:0) H kin k (cid:1) / ln (cid:16) k H kin (cid:17) . However, it seems to us that there might be a very mild change in the β -Bogoliubovcoefficient which of course does not affect the main results and the conclusion of the paper apart from a factor inthe BBN bound. Hence, there is absolutely no worry at all. We find that the term containing (cid:0) H kin k (cid:1) / ln (cid:16) k H kin (cid:17) vanishes and the leading term becomes i √ π (cid:0) H kin k (cid:1) / . Effectively, using the long wave-length approximation one has χ k ( τ − kin ) = − i √ kkτ kin ; χ (cid:48) k ( τ − kin ) = i (cid:114) k k τ kin ; χ k ( τ + kin ) = − i (cid:114) − τ kin π ln (cid:18) − kτ kin (cid:19) ; χ (cid:48) k ( τ + kin ) = − i √− πτ kin ln (cid:18) − kτ kin (cid:19) − i (cid:114) − πτ kin . (C1)Then, since β k = i W [ χ k ( τ − ); χ k ( τ + )], a simple calculation proves our statement, i.e., β k ∼ = i √ π − kτ kin ) / = i √ π (cid:0) H kin k (cid:1) / . If one recalculates the computations done in [68] in order to obtain the β -Bogoliubov coefficient( A − ( k ) in its notation) one obtains the following expression:6 β k = − π √ e − i π ( ν +1) (cid:26) H (2)0 (cid:16) x (cid:17) (cid:20) H (2) ν ( − x ) + x (cid:16) H (2) ν +1 ( − x ) − H (2) ν − ( − x ) (cid:17)(cid:21) − x H (2)1 (cid:16) x (cid:17) H (2) ν ( − x ) (cid:27) , (C2)where x = − kτ kin and ν = 3 /
2. We note that, when | x | (cid:28) H (2) ν − ( − x ) is subdominant relative to H (2) ν +1 ( − x ).So, by ignoring this term, our expression almost coincides with the one in equation (C.2) in [68] with only differenceof a minus sign in front of H (2) ν ( − x ). This minus sign appears to be important as we show next. By using therecurrence relation αx Z α ( x ) = Z α − ( x ) + Z α +1 ( x ), being Z α any combination of Bessel functions of order α , wefind that H (2) ν ( − x ) = − x ν (cid:16) H (2) ν − ( − x ) + H (2) ν +1 ( − x ) (cid:17) . Therefore the dominant terms multiplying H (2)0 (cid:0) x (cid:1) getcancelled each other and, hence, the only remaining dominant term turns out to be β k ∼ π √ e − i π ( ν +1) x H (2)1 (cid:16) x (cid:17) H (2) ν ( − x ) ∼ i √ π e − iπ kτ kin ) / = i √ π e − iπ (cid:18) H kin k (cid:19) / , (C3)and thus | β k | ∼ π (cid:0) H kin k (cid:1) .From our viewpoint this mild mismatch in [68] may come from the fact that during the de Sitter phase theconformal time is negative. However, the author uses the vacuum mode e − iπν/ e − iπ/ (cid:112) πτ H (2) ν ( kτ ) (see for-mula (3 .
4) of [68]), which contains square roots of negative numbers that complicate the calculations instead of e iπν/ e iπ/ (cid:113) − πτ H (1) ν ( − kτ ), which has a positive argument that facilitates the calculations, obtaining β k = i π √ e i πν (cid:26) H (2)0 (cid:16) x (cid:17) (cid:20) H (1) ν ( x ) − x (cid:16) H (1) ν +1 ( x ) − H (1) ν − ( x ) (cid:17)(cid:21) − x H (2)1 (cid:16) x (cid:17) H (1) ν ( x ) (cid:27) ∼ − i π √ e i πν x H (2)1 (cid:16) x (cid:17) H (1) ν ( x ) ∼ √ π e iπ (cid:18) H kin k (cid:19) / . (C4)A consequence of such mild mismatch is that the energy density per logarithmic interval of longitudinal momentum,for H r < k < H kin , is now given by ρ ( k, τ ) = dρ GW ( k, τ ) d ln k = kdρ GW ( k, τ ) dk = k π a ( τ ) | β k | ∼ = 12 π H kin (cid:18) k H kin (cid:19) (cid:18) a kin a ( τ ) (cid:19) , (C5)which differs from a logarithmic term of the result obtained in formula (3 .
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