Reheating via gravitational particle production in simple models of quintessence or ¿CDM Inflation

arXiv:1711.03788v1 [gr-qc] 10 Nov 2017 Reheating via gravitational particle production in simple models of quintessence or ΛCDM inflation Jaume Haroa,∗ , and Llibert Arest´e Sal´oa, † 13th November 2017 aDepartament de Matem`atica Aplicada, Universitat Polit`ecnica de Catalunya Diagonal 647, 08028 Barcelona, Spain Abstract We have tested some simple ΛCDM (the same test is also valid for quintessence) inflation models, imposing that they match with the recent observational data provided by the BICEP and Planck’s team and leading to a reheating temperature, which is obtained via gravitational particle production after inflation, supporting the nucleosynthesis success 1 Introduction There are several candidates to unify the early and late time acceleration of the universe, such as modified gravity [1] or via phantom scalar fields [2]. However, from our viewpoint the best one to unify inflation and the current cosmic acceleration of the universe is Quintessence or ΛCDM Inflation [3, 4, 5, 6, 7]. In this theory, the simplest way to construct the potential was performed for the first time in the seminal paper [8] matching an inflationary one (used to explain the early acceleration of the universe) with a quintessential one, which takes into account the current cosmic acceleration. For this kind of potentials, at early times, the inflationary acceleration is the one that dominates and it ceases to be dominant in an abrupt phase transition. We note that the phase transition needs to be abrupt in order to break the adiabatic regime, thus allowing a sufficient gravitational particle production . ∗E-mail: [email protected] †E-mail: llibert.[email protected].edu Moreover, an analytic expression of this amount of particles is only obtained when the derivative of the potential presents some discontinuities [9, 10, 11, 12] when the universe enters in the kination regime [13]. The key point is that the energy density of the background decreases faster than the one of the produced particles, meaning that the energy density of the created particles will eventually dominate, becoming the universe reheated and matching with the current hot Friedmann model. Finally, at very late times, the quintessential potential dominates and the universe starts to accelerate again. We point out that this kind of behavior could also be obtained by considering a universe with a small cosmological constant and by choosing a positive inflationary potential vanishing at some point, which is extended to zero for the other values of the field [14]. This comes from the fact that, when the potential is zero, the universe enters in a kination regime which lasts until the energy density of the created particles at the phase transition starts to dominate, thus matching with the current ΛCDM model, in which the square root of the cosmological constant must be of the same order as the current value of the Hubble parameter in order to take into account the cosmic coincidence. In order to assure the viability of these models, they are required to fit well with the recent observational data provided by the BICEP and Planck’s teams [15, 16], but also the reheating temperature has to be compatible with nucleosynthesis, meaning that it is constrained approximately between 1MeV and 109GeV. This is the main goal of the present work, namely to study the viability of some well-known inflationary potentials adapted to quintessence or to ΛCDM model. Since our interest is to test the inflationary spectral parameters (spectral index, ratio of tensor to scalar perturbations, number of e-folds and reheating temperature of the universe), the quintessence piece of the potential plays no role in our calculations and, therefore, the two most important points to take into account are the inflationary phase and a phase transition to kination. For convenience, we choose positive potentials that vanish at some point and we extend them to zero in order to ensure a phase transition to kination. Hence, once we have these potentials, we calculate their spectral parameters, the number of e-folds (as we will show, in quintessential or ΛCDM inflation it has to be between 63 and 73, which is a greater range than the one given when the potential has a deep well) and its reheating temperature in two cases that will be analytically calculated: via gravitational production of massless particles nearly conformally coupled to gravity, which is the most studied case in the literature, and very heavy massive particles conformally coupled to gravity. The proceedings, which is essentially based on our recent paper [14], is organ- 2 ized as follows: Section 2 is devoted to the study of the reheating via gravitational particle production. Two cases are studied in detail: The gravitational production of heavy massive particles conformally coupled to gravity and the production of massless particles nearly conformally coupled to gravity. In Section 3 we calculate in two different ways the number of e-folds in quintessence inflation when there is a phase transition from inflation to kination, obtaining that it is constrainted to be between 63 and 73. In Section 4, we consider a universe with a small cosmological constant and we adapt the simplest inflationary potentials appearing in the Encyclopaedia Inflationaris [17] to quintessence, applying the results obtained in previous sections in order to study its viability. The units used throughout the paper are ~=c= 1 and, with these units, Mpl =1 √8πG is the reduced Planck’s mass. 2 Reheating in quintessence or ΛCDM inflation 2.1 Reheating via gravitational production of heavy massive particles conformally coupled to gravity In this section we will consider a pre-heating scenario that is not usually considered in quintessence: the creation of heavy massive particles conformally coupled to gravity that have no interaction with the inflaton field. In this situation, the frequency of the particles in the k-mode is ωk(τ) = qk2+m2 χa2(τ), where mχis the mass of the quantum field. During the adiabatic regimes, i.e., when H(τ)≪mχ=⇒ω′ k(τ)≪ω2 k(τ), one can use the WKB approximation [12] χW KB n,k (τ)≡s1 2Wn,k(τ)e−iRτWn,k(η)dη,(1) where nis the order of the approximation of the k-vacuum mode. When some high order derivative of the Hubble parameter is discontinuous, one has to match the kvacuum modes before and after this moment. To do this, one needs to use positive frequency modes after the breakdown of the adiabaticity, which is the cause of particle production. This is basically Parker’s viewpoint of gravitational particle production [18]. Then, since the classical picture can be used at scales lower than the Planck’s one, in order to preserve the condition H(τ)≪mχbefore the phase transition one has to choose mχ≥Mpl, but elementary particles with masses greater than the Planck’s one are micro black hole, whose behavior is unknown. Thus, to prevent the formation of these objects, we have to choose mχ∼Mpl [19]. 3 To clarify ideas and in order to obtain analytic expressions of the energy density of the produced particles, which allows us to calculate analytically the reheating temperature, we consider a phase transition from inflation to kination where the second derivative of the Hubble parameter is discontinuous. If we assume that the derivative of the potential is discontinuous at ϕE, due to the conservation equation the second derivative of the inflaton field is discontinuous at the transition time and, consequently, from Raychaudhuri equation ˙ H=−˙ϕ2 2M2 pl , one can deduce that the second derivative of the Hubble parameter is also discontinuous at the same time. In this case one only needs the first order WKB solution to approximate the k-vacuum modes before and after the phase transition χW KB 1,k (τ)≡s1 2W1,k(τ)e−iRτW1,k(η)dη,(2) where W1,k =ωk−1 4 ω′′ k ω2 k +3 8 (ω′ k)2 ω3 k .(3) Before the transition time, namely τ= 0, vacuum is depicted by χW KB 1,k (τ), but after the phase transition this mode becomes a mix of positive and negative frequencies of the form αkχW KB 1,k (τ) + βk(χW KB 1,k )∗(τ). The βk-Bogoliubov coefficient can be obtained matching both expressions at τ= 0, namely βk=W[χW KB 1,k (t− E), χW KB 1,k (t+ E)] W[(χW KB 1,k )∗(t+ E), χW KB 1,k (t+ E)],(4) where W[f(t− E), g(t+ E)] = f(t− E)g′(t+ E)−f′(t− E)g(t+ E)is the Wronskian of the functions fand gat the transition time, and (+) (resp. (−)) means immediately after (resp. before) the phase transiton. The square modulus of the β-Bogoliubov coefficient will be given approximately by |βk|2∼ = m4a10 E¨ H+ E−¨ H− E2 256(k2+m2 χa2 E)5,(5) with ¨ H+ E−¨ H− E=−˙ϕE M2 pl ( ¨ϕ+ E−¨ϕ− E) = −˙ϕE M2 pl Vϕ(t− E),(6) 4 where we have used that in the kination phase the potential vanishes. This quantity, as we will see, is of the order H3 Ewhen dealing with quintessence models. Then the number density of the produced particles and their energy density, as has been rigorously proved in [20], will be nχ(t) = 1 2π2a3Z∞ 0 k2|βk|2dk ∼¯ λ3H6 E m3 χaE a(t)3 , ρχ(t) = 1 2π2a4Z∞ 0 ωk(t)k2|βk|2dk ∼mχnχ(t),(7) being ¯ λa dimensionless constant, which depends on the model. These massive particles will decay into lighter ones, which after some interactions become a fluid in thermal equilibrium. To calculate the moment when this occurs, we assume that the particles interact by exchange of gauge bosons and we use the thermalization rate Γ = nχ(0)σ(see [21] and also [8]), where the crosssection for emitting a gauge boson (whose typical energy is E∼ρ 1 4 χ(0)) from a scattering of two fermions is given by σ∼α3 E2, being αa coupling constant with typical values α∼10−2−10−1. Thus, Γ = α3nχ(0) mχ1 2=α3¯ λ3/2H3 E m2 χ .(8) Since equilibrium is reached when Γ∼H(teq) = HEaE aeq 3, we will have ρχ(teq)∼α3¯ λ9/2H8 E m4 χ , ρ(teq)∼3α6¯ λ3H6 EM2 pl m4 χ ,(9) and the universe will become reheated when both energy densities are of the same order, which will happen when aeq aR∼qρχ(teq ) ρ(teq ), and so, TR∼ρ 1 4 χ(tR)∼ρ 1 4 χ(teq)sρχ(teq) ρ(teq)∼5×10−1α−3/4¯ λ15/8H3 E mχMpl ∼5×10−1α−3/4¯ λ15/8H3 E M3 pl Mpl,(10) where we have used that mχ∼Mpl. 2.2 Reheating via gravitational production of massless particles nearly conformally coupled to gravity In this situation, the Klein-Gordon equation is given by ¯χ′′ k(τ) + k2+ξ−1 6a2(τ)R(τ)¯χk(τ) = 0,(11) 5 where ξis the coupling constant and Ris the scalar curvature. To define the vacuum modes before and after the phase transition, we use the in-out formalism (see [22], [23], [12] or [24] for a review), where the behavior of these modes at early and late times is respectively ¯χb,k(τ)≃e−ikτ √2k(when τ→ −∞),¯χa,k(τ)≃e−ikτ √2k(when τ→+∞).(12) Since we are considering particles nearly conformally coupled to gravity, we can consider the term (ξ−1/6)a2(τ)R(τ)as a perturbation, and we can approximate the “b” and “a” modes by the first order Picard’s iteration as ¯χb,k(τ)∼ =e−ikτ √2k−ξ−1/6 k√2kZτ −∞ a2(τ′)R(τ′) sin(k(τ−τ′))e−ikτ′dτ′,(13) ¯χa,k(τ)∼ =e−ikτ √2k+ξ−1/6 k√2kZ∞ τ a2(τ′)R(τ′) sin(k(τ−τ′))e−ikτ′dτ′,(14) which will represent, respectively, the vacuum before and after the phase transition. Then, after the phase transition, we can write the “in” mode as a linear combination of the “out” mode and its conjugate as follows ¯χb,k(τ) = αk¯χa,k(τ) + βk¯χ∗ a,k(τ).(15) Imposing the continuity of ¯χand its first derivative at the transition time we obtain, up to order (ξ−1/6)2, that the value of these coefficients is [9] αk∼ =1−i(ξ−1 6) 2kZ∞ −∞ a2(τ)R(τ)dτ, βk∼ =i(ξ−1 6) 2kZ∞ −∞ e−2ikτ a2(τ)R(τ)dτ. (16) Finally, in order to define the energy density, the in-out formalism is also used, that is, when the universe is asymptotically static, the energy density of the produced particles at late times is given by 1 2π2a4 ∞R∞ 0k3|βk|2dk, where a∞is the value of the scale factor at late times. Then, despite not coinciding with the definition of energy density because the 1-loop effects are disregarded (see for instance [25] where the energy density for a transition from de Sitter to radiation is calculated), it is assumed that the energy density of the produced particles due to the phase transition is given by [24] ρχ=1 2π2a4Z∞ 0 k3|βk|2dk. (17) The integral of the β-Bogoliubov coefficient (16) is convergent because at early and late times the term a2(τ)R(τ)converges fast enough to zero. Moreover, if at 6 the transition time tEthe first derivative of the Hubble parameter is continuous, one has βk∼ O(k−3), which means that the energy density of the produced particles is not ultra-violet divergent. Then, the energy density of the produced particles approximately becomes ρχ(t)∼ =ξ−1 62 NH4 EaE a(t)4 ,(18) where Nis a dimensionless numerical factor and HEand aEare respectively the value of the Hubble parameter and the scale factor at the phase transition time. Remark: The number Nis clearly model dependent. In the case proposed by Ford in [9], the author considers a toy model where there is a transition from de Sitter to matter domination modelled by a2(τ)R(τ)≡12 τ2+τ2 0 and where Ncan be analytically calculated giving as a result 9 8. In [10], a toy model based on an abrupt transtion from de Sitter to radiation is considered, obtaining a result of the same order as Ford. However, note that in both cases reheating is impossible because the energy density of the produced particles decreases faster or equal than those of the background. We have calculated numerically this factor for some simple models that have a transition from an inflationary regime to kination and in all cases Nis of the order 1. Finally, assuming that N ∼ 1, the reheating temperature, in this case, is given by TR∼ξ−1 63/2HE Mpl 2 Mpl.(19) 3 Detailed calculation of the number of e-folds The number of e-folds can be calculated in two different ways: 1. By definition this quantity is equal to N=Rtend t∗Hdt, where (∗)denotes when the pivot scale leaves the Hubble radius and (end)stands for the end of inflation. During the slow roll this quantity is given by N=Ztend t∗ Hdt ∼ =1 M2 pl Zϕend ϕ∗ V Vϕ dϕ. (20) 2. By using the whole history of the universe; in our case the transition from inflation to kination, passing though radiation and matter domination up to 7 the present. We start with the following equation [26] k∗ a0H0 =e−NH∗ H0 aend aE aE aR aR aM aM a0 =e−NH∗ H0 aend aE ρ−1/12 Rρ1/4 M ρ1/6 E aM a0 ,(21) where R,Mand 0symbolize the beginning of radiation era, the beginning of the matter domination era and the present time, and having used relations (aE/aR)6=ρR/ρE(kination era) and (aR/aM)4=ρM/ρR(radiation domination). We use, as well, that H0≈2×10−4Mpc−1and take as a physical value of the pivot scale kphys ≡k∗ a0= 0.02 Mpc−1(value used by Planck2015 [16]). Moreover, we know that the process after reheating is adiabatic, i.e. T0=aM a0TM, as well as the relations ρM≈π2 15 gMT4 M and ρR≈π2 30 gRT4 R(where {gi}i=R,M are the relativistic degrees of freedom [27]). Hence, N=−4.61 + ln H∗ H0+ ln aend aE+1 4ln 2gM gR+1 6ln ρR ρE+ ln T0 TR.(22) We use that H0∼6×10−61Mpl and, from the value of the power spectrum [28, 29] P≈H2 ∗ 8π2ǫ∗M2 pl ∼2×10−9, we infere that H∗∼4×104√ǫ∗Mpl, where ǫ∗is the main slow roll parameter evaluated when the pivot scale leaves the Hubble radius. We know as well that T0∼2×10−13 GeV and gM= 3.36 [27]. Also, gR= 107,90 and 11 for TR≥135 GeV, 175 GeV ≥ TR≥200 MeV and 200 MeV ≥TR≥1MeV, respectively [27]. On the other hand, assuming that the transition phase occurs immediately after the end of inflation and that there is not a substantial drop of energy, one obtains N≈54.5 + 1 2ln ǫ∗−1 3ln g1/4 RTRHend M2 pl !.(23) Therefore, with the values in our model and with the range 1MeV ≤TR≤ 109GeV required in order to have a successful nucleosynthesis [30], and taking, as usual, Hend ∼10−6Mpl and ǫ∗∼10−2, we find that 63 ≤N≤ 73. Moreover, the equations (20) and (23), as we will see, are functions of the spectral index ns. Then, equaling both equations, one will obtain some constraints for the spectral values in each model. 8 4 Application to some simple ΛCDM inflation potentials In this section we consider the simplest inflationary potentials that appear in [17] and, as we have explained in the introduction, we adapt them in order to have, after inflation, a kination phase followed by the standard ΛCDM regime. The way to obtain this kind of potentials is to choose, in the extensive list that appears in [17], the simplest and most well-known positive inflationary potentials that vanish at some value of the scalar field and extend them to zero for the other values of the field. Moreover, in order to ensure the late time cosmic acceleration and coincidence, we introduce a cosmological constant Λ∼H2 0, being H0the current value of the Hubble parameter. 4.1 Exponential SUSY Inflation (ESI) The first potential we are going to study is an Exponential SUSY Inflation (ESI) style potential [31, 32], V(ϕ) = (λM4 pl(1 −e ϕ Mpl )ϕ < 0 0ϕ≥0,(24) being λa dimensionless positive parameter. By using the following approximate expressions of the slow-roll parameters as a function of the potential, ǫ≈M2 pl 2Vϕ V2 η≈M2 pl Vϕϕ V,(25) we can compute the spectral index (ns) and the ratio of tensor to scalar perturbations (r): ns−1 = −s∗3 2s∗+ 2∼ =−2s∗r= 16ǫ∗,(26) where we have introduced the notation s∗=e ϕ∗ Mpl 1−e ϕ∗ Mpl ∼ =e ϕ∗ Mpl . It is also straightforward to calculate the power spectrum: P≈H2 ∗ 8π2ǫ∗M2 pl ≈λ 3π2(1 −ns)2,(27) where the constraint P∼2×10−9is verified by choosing λ= 6×10−9π2(1−ns)2 . Finally, regarding the number of e-folds, N=e−ϕ∗ Mpl −1 + √2 √2+ϕ∗ Mpl −ln √2 1 + √2!≈2 1−ns + ln 1−ns 2.(28) 9 4.8 Loop Inflation (LI) In this case the potential behaves as [39] V(ϕ) = (0, ϕ ≤ϕE≡Mple−1 α λM4 pl 1 + αln ϕ Mpl , ϕ ≥ϕE≡Mple−1 α,(52) where λand αare positive dimensionless constants. We consider two different asymptotic cases: 1. 0< α ≪1: In this case one has ns−1∼ =2α x2 ∗, where we have introduced the parameter x≡ϕ Mpl . For the number of e-folds one has N∼ =x2 ∗ 2α∼ =1 1−ns ,(53) which leads, as in the case of HI, to a not high enough number of e-folds. 2. α≫1: The spectral index and the tensor/scalar ratio will be as a function of x∗ 1−ns=1 x2 ∗ln2x∗ (3 + 2 ln x∗), r =8 x2 ∗ln2x∗ .(54) Then, at 2σC.L., for the allowed values of the spectral index, we can see, after some numerics, that x∗ranges in the domain 6.94 ≤x∗≤7.98. On the other hand, the number of e-folds is N=x2 ∗ 2ln x∗−1 2−x2 end 2ln xend −1 2.(55) Using the range of values for x∗one finds that 34 ≤N≤50, which comes out of the viable range. 16 5 Discussion We have adapted some inflationary potentials to ΛCDM inflation, extending them to zero after they vanish and adding a small cosmological constant. Once we have done it, we have tested the models imposing that: 1. They fit well with the current observational data provided by BICEP and Planck teams. 2. The number of e-folds must range between 63 and 73. This number is larger than the usual one used for potentials with a deep well, due to the kination phase after inflation. 3. The reheating temperature due to the gravitational particle production during the phase transition from inflation to kination has to be compatible with the nucleosynthesis success, i.e., it has to range between 1MeV and 109GeV. Our study shows that the potentials WRI and KMII lead to a too high number of e-folds, while for HI and LI potentials this number is too small. Other potentials such as ESI, PLI (only when the potential is quadratic), OSTI and BI satisfy the prescriptions 1and 2. Moreover, dealing with the reheating temperature, we have showed the viability of the reheating via the gravitational particle production of heavy massive particles conformally coupled to gravity and also via the production of massless particles nearly conformally coupled. Acknowledgments. 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