Quintessential inflation at low reheating temperatures

We have tested some simple quintessential 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. Moreover, for the models coming from supergravity one needs to demand low temperatures in order to avoid problems such as the gravitino overproduction or the gravitational production of moduli fields, which are obtained only when the reheating temperature is due to the production of massless particles with a coupling constant very close to its conformal value.


INTRODUCTION
Quintessential inflation [39] is a good candidate to unify the early and late time acceleration of our universe. These models are a combination of an inflationary potential, used to explain the early acceleration of the universe, and a quintessential one -that could be only a cosmological constant-, which takes into account the current cosmic acceleration. At early times the inflationary acceleration is the one that dominates and it ceases to be dominant in a phase transition where the universe enters in a kination regime [2]. At this moment, particles coupled with gravity are produced to reheat the universe and match with the current hot one. Finally, at very late times, the quintessential potential dominates and the universe starts to accelerate again.
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 [3,4], but also the reheating temperatures have to be compatible with the nucleosynthesis and baryogenesis bounds, preventing the increase of entropy due to the decays of gravitational relics such as moduli fields and gravitinos. It is well known that the success of nucleosynthesis constraints the reheating temperature to be between 1 MeV and 10 9 GeV [5][6][7][8][9][10][11], whereas in order to overcome the overproduction of very light gravitinos (particles which appear in supersymmetric gravitational theories), i.e., with masses in the range 10 −3 MeV m 3/2 10 −1 MeV, it is stated that the reheating temperature has to be less than 10 2 GeV [12]. It is more complicated to avoid the increase of entropy due to the production of moduli fields, which also appears in supergravity models, during the inflationary period; in that case, as it was showed in [16], the reheating temperature has to be less than 1 GeV. However, this low temperature seems to be problematic with baryogenesis, according to which it is commonly assumed that the reheating temperature is at least of the order of the electroweak scale (10 2 GeV). But, fortunately, there are mechanisms that explain the baryon asymmetry at very low temperatures [14]. For all these reasons, we will consider that those supergravity models with a reheating temperature in the MeV regime, i.e., between 1 MeV and 1 GeV, succeed in solving all these problems.
Having this in mind, the main goal of the present work is to study the viability of some well-known inflationary potentials, some of them coming from supersymmetric theories, adapted to quintessence. To do it in a simple way, first of all we consider a universe with a cosmological constant, we choose positive inflationary potentials that vanish at some value of the scalar field and then we extend them to zero for the other values of the field. Hence, we obtain a potential with a phase transition that models an inflationary universe at early times, which reheats the universe via gravitational particle production after inflation and it is finally dominated by the cosmological constant. Once we have these potentials, we calculate their spectral parameters, the number of efolds, which has to be in quintessential inflation between 63 and 73 as we show, and its reheating temperature in three cases that will be analytically calculated: via the production of heavy massive particles conformally coupled with gravity, massless nearly conformally coupled particles and particles far from the conformal coupling. This last case has also been studied in [15] in the context of braneworld inflation for exponential and power law potentials. Note that there is another way to reheat the universe, via the so-called "instant preheating" [16], which has been applied to quintessential inflation in [17,18] for exponential and power law potentials. This kind of reheating deserves future investigation for different potentials such as the ones studied in this paper, which we will deal with in a future work.
The paper is organized as follows: In Section II we build a non singular model based on a universe filled with a barotropic fluid with a non-linear equation of state where the universe starts and finishes in a de Sitter phase. Via the reconstruction method we find the potencial that mimics these dynamics when the universe is filled with a scalar field, we find the analytic solution that mimics the fluid model, and we see that it provides spectral quantities such as the spectral index, its running and the tensor to scalar ratio that match with the current observational data. However, this solution appears to be unstable at early times, in the sense that any small perturbation leads to a solution which is singular at early times, in the same way that happens with the Starobinsky model [19,20]. Moreover, with our criterion, the model only supports a reheating via the creation of massless particles nearly conformally coupled with gravity. In Section III, we consider a universe with a small cosmological constant and we adapt three simple inflationary potentials (Exponential SUSY Inflation potential (ESI), the Higgs Inflation Potential in Einstein Frame (HI), and the Power Law Inflation potential (PLI)) to quintessence, showing that the HI potential leads to an unacceptable number of efolds, the ESI potential, which comes from a supersymmetric potential and thus suffers the gravitino and moduli problems, only supports a reheating via the production of massless particles nearly conformally coupled with gravity, and the PLI, which is only viable in the quadratic case, supports the three kind of reheating studied in the work because, since it does not come from any supergravity potential, gravitinos or relic moduli fields do not appear in this theory. In the last Section we adapt other well known potentials (Witten-O'Raifeartaigh Inflation (WRI), Kähler Moduli Inflation I (KMII), Open String Tachionic Inflation (OSTI), Brane Inflation (BI) and Loop Inflation (LI)) coming from inflation, to quintessence and we study its viability. Finally, in the appendix we review from a critical viewpoint the overproduction of gravitational waves in quintessential inflation.
The units used throughout the paper are = c = 1 and, with these units, M pl = 1 √ 8πG is the reduced Planck's mass.

A NONSINGULAR MODEL
In this Section we are going to study inflation coming from fluids [21,22] and to develop the idea of a nonsingular universe proposed in [23], where the universe was filled with a barotropic fluid whose non-linear Equation of State (EoS), namely P = P (ρ) being P the pressure and ρ the energy density, satisfies P + ρ = 0 at two different scales, leading, at early and late times, to two de Sitter eras.
The simplest realization of this model, which could be generalized introducing viscosity [24], is to consider three parameters H i H E H f , where H i and H f are two fixed points of the dynamical system and H E is the value of the Hubble parameter when the transition from inflation to kination is produced. To simplify, we assume that H i = M pl , i.e., the universe starts, in a de Sitter phase, at Planck scales, and in order to reproduce the current cosmic acceleration we have to take H f ∼ H 0 . The simplest way to obtain the model is to consider the following differential equationḢ where we have chosen k = to ensure the continuity ofḢ. Regarding the value of k, it will be determined in next section so that our model matches with the current observational data. The dynamical system can be analytically solved leading to the following Hubble parameter and the corresponding scale factor is Moreover, the effective EoS parameter, namely w ef f , which is defined as w ef f ≡ P ρ = −1 − 2Ḣ 3H 2 , for our model is given by where (end) stands for the end of inflation, i.e., end = 1, that is, Thus, by using the calculated values of k and H * , we obtain a number of e-folds satisfying 52 < N < 77. Now, we are going to compare this value with the one that we will obtain from the following equation [29] in an analogous way as in [30]: where R and M symbolize the beginning of radiation era and the beginning of the matter domination era and we have used relations (a E /a R ) 6 = ρ R /ρ E and (a R /a M ) 4 = ρ M /ρ R . We use that H 0 ≈ 2 × 10 −4 Mpc −1 and, as usual, we choose a 0 = 1 taking as a physical value of the pivot scale k phys = 0.02 Mpc −1 (value used by Planck2015 [4]). Then one has that, in co-moving coordinates, the pivot scale will be k * ≡ a 0 k phys = 0.02 Mpc −1 . Moreover, we know that the process after reheating is adiabatic, i.e. T 0 = a M a0 T M , as well as the relations ρ M ≈ π 2 15 g M T 4 M and ρ R ≈ π 2 30 g R T 4 R (where {g i } i=R,M are the relativistic degrees of freedom [31]). Hence, We use that H 0 ∼ 6 × 10 −61 M pl and, from Equation (8), we infere that H * ∼ 4 × 10 4 √ * M pl . We know as well that T 0 ∼ 2 × 10 −13 GeV and g M = 3.36 [31]. Also, g R = 107, 90 and 11 for T R ≥ 135 GeV, 175 GeV ≥ T R ≥ 200 MeV and 200 MeV ≥ T R ≥ 1 MeV, respectively [31]. On the other hand, Thus, we obtain that Therefore, with the values in our model and with the range 1 MeV ≤ T R ≤ 10 9 GeV required in order to have a successful nucleosynthesis [32], we find that 63 N 73. With equation (9), this range is verified in our values for 0.969 < n s < 0.973.
In conclusion, with the intersection of the bounds obtained from the constraints r < 0.12 and 63 N 73, in our model we have that 0.970 < n s < 0.973 with the corresponding values of k ∼ 8 × 10 −12 and H * ∼ 3 × 10 −5 M pl (corresponding to H E ∼ 2 × 10 −6 M pl ), thereby having a number of e-folds of 66 ≤ N ≤ 73. Hence, our model satisfies all the values obtained from recent observations. Moreover, one can see in Figure 1 that our model also provides theoretical values that enter in the marginalized 2σ C.L. contour in the plane (n s , r).

The scalar field
In this section we are going to mimic the perfect fluid that fills the FLRW universe with a scalar field. If we represent the energy density and the pressure by the notations ρ ϕ , p ϕ , respectively, then they assume the following simplest forms: Now, we aim to analyze the stability of the solution to our model. Using the potential found in Eq. (17), we will study the dynamics of the equation:φ where H(ϕ,φ) = 1 The numerical results are presented in Figure 2. For clarity in the understanding of the dynamical system, we have used a value of k which differs in various orders of magnitude from the one in our model. Though, the behaviour is exactly the same as with the real values. Therefore we observe that the analytical orbit is the only one with 2 de Sitter points. All the other orbits start with an infinite energy at t → −∞, they experience a phase transition at ϕ = ϕ E and then asymptotically approach the analytical orbit for t → ∞, reaching at t → ∞ the de Sitter point H = H f . An important remark is in order: From the phase space portrait we can see that there is only one non-stable non-singular solution, i.e., if one takes initial conditions near the analytical solution one obtains a past singular orbit. This is exactly the same that happens in the Starobinsky model [19] (see Figure 3 of [20] for details).

Reheating constraints
Before studying the reheating constraints in our model, we are going to carry out some useful approximations that will help us in our further calculations. Since k ∼ = On the other hand, it is also fulfilled that 0 ∼ = αM 2 pl − 2H 2 * and, thus, we obtain all the following approximative expressions and, hence, r = 16 * ∼ = 16k Moreover, the number of e-folds is but using equation (13) we also obtain Equaling both quantities one obtains the equation where we have introduced the notation Y = 6 1−ns .
Hence, from this equation we obtain the following relation between the cosmological parameters (n s ,r) and T R , showing that the suitable constraints will be verified for 1 MeV ≤ T R ≤ 10 5 GeV, as we can see in Figure 3. Now, let us consider the gravitational production of χ-particles. Firstly, we treat the case when the produced particles are very massive and conformally coupled with gravity. Since a classical picture of the universe is only possible at energy densities less than the Planck's one, it seems natural to choose that at this scale the quantum field χ is in the vacuum. In fact, we will choose as a vacuum state the adiabatic one, defined by the modes [33] where Ω k satisfies the equation .
Then, to obtain an approximate expression of Ω k , one can use the WKB approximation, which holds when m ≥ H. For this reason, since at the Planck epoch H ∼ M pl , in order to obtain the WKB solution one has to assume that the mass of the field, namely m, satisfies m ≥ M pl . However, since for m > √ 4πM pl the produced particles are micro Black Holes [34], whose thermodynamical description is unknown [35], one has to choose m ∼ M pl so as to avoid this Black Holes production.
The reheating temperature caused by the decay of these particles into lighter ones, with a thermalization rate -the same used in [5,36] [25], is of the order (see for details [30,37]) and, inserting this expression in (22), one obtains which leads to a spectral index n s ∼ = 0.9716, a tensor/scalar ratio r ∼ = 0.1136 and a reheating temperature T R ∼ 11 GeV. Since this model does not come from supergravity, we do not need to take into account the gravitino and moduli fields problems. Hence, these results lead to a viable model.
On the other hand, when considering massless particles nearly conformally coupled with gravity, the reheating temperature becomes [36,38] with ξ the coupling constant, N = 1 where R is the scalar curvature. It has been numerically computed that N ∼ 1. So as to verify the bounds 1 MeV ≤ T R ≤ 10 5 GeV (coming from the restriction r ≤ 0.12 in our model), the coupling constant ξ must satisfy If we consider now massless particles far from the conformal coupling with gravity, using the results of [39,40] which we will discuss in the appendix, the reheating temperature was calculated in [37], leading to In this case combining equations (22) and (29), one obtains whose solution is n s = 0.970, corresponding to r = 0.119, coinciding thus with the limit where our model starts being valid. Hence, our model supports the presence of these particles.

SIMPLE QUINTESSENTIAL INFLATION MODELS
In this section, we are going to proceed in an inverse way from the one we have just done. We are going to take some simple and well-known potentials, some of them with a similar form from the one already studied, and we are going to verify whether we can adjust the parameters so as to fulfill all the corresponding constraints for the cosmological parameters. To simplify, we will consider some well-known positive inflationary potentials that vanish at some value of the scalar field, namely ϕ E , and we will extend it to zero for the other values of the field. Moreover, we introduce a cosmological constant Λ ∼ 3H 2 0 (being H 0 the current Hubble cosmological constant) to ensure the current cosmic acceleration.

Exponential SUSY Inflation (ESI)
The first potential we are going to study is an Exponential SUSY Inflation (ESI) style potential [41,42], being λ a dimensionless positive parameter. By using the following approximate expressions of the slow-roll parameters as a function of the potential, we obtain * = M pl . Hence, we can compute the associated inflationary parameters, analogously as we did in Section 3: where we have used for the calculation of α s thatḢ = −φ 2 2M 2 pl . It is also straightforward to calculate the power spectrum: where we have used that V (ϕ * ) ≈ 3M 2 pl H 2 * . We will guarantee that the value P ∼ 2 × 10 −9 is verified in our model by taking λ = . Finally, regarding the number of e-folds, So as to fit our model with the recent observational values, we will initially take, as before, the spectral index obtained by Planck2015 data, n s = 0.968 ± 0.006 and we will find the corresponding value ϕ * from the first equation in (33), which is −4.37 ≤ ϕ * M pl ≤ −3.99. Therefore, the other cosmological parameters are 0.0013 ≤ r ≤ 0.0028 and −7 × 10 −4 ≤ α s ≤ −3 × 10 −4 . And, finally, the corresponding number of e-folds is 49 N 73.
Hence, all the values of r and α s fit to our restrictions and, regarding the number of e-folds, in order to have 63 N 73 we need to take the range of 0.970 ≤ n s ≤ 0.974 for the spectral index, obtaining thus 0.0013 < r < 0.0017 and −5 × 10 −4 < α s < −3 × 10 −4 . Moreover, the theoretical values provided by our model enters in the marginalized 1σ C.L. contour in the plane (n s , r) as one can see in Figure 4. Since the approximative expression of the cosmological parameters is only valid during the inflation process, we cannot directly calculate the Hubble constant at the transition, namely H E . As we can verify in the former model where we had the analytical form of the dynamical system, it is of the same order as at the end of the inflation (H end ).
Since end = 1 and ϕ end = ln So as to study the reheating constraints for this model, we are going to proceed as in the former case, i.e, with the approximative expressions r . Now we can not reproduce the same calculation as we did for the former case so as to obtain Equation (13), because we do not have an analytic expression to perform the calculations given in (12), but since this value is negligible we can continue using it. Therefore, by combining equation (13) and the just obtained approximate value of N , we have that again using that Y = 6 1−ns . Hence, from solving this equation, one can see in Figure 5 that the constraints on r and n s are fulfilled, as we had already pointed out. Now, with regards to the case when the produced particles are very massive and conformally coupled with gravity (again choosing m ∼ M pl for the same argument as in previous section), we have to proceed as follows: The square modulus of the β-Bogoliubov coefficient is given by the expression [30] where a E is the value of the scale factor at the transition phase, andḦ ± E are the value of the second derivative of the Hubble parameter after and before the phase transition. Since we do not have an analytic expression of the Hubble parameter, to obtain its second derivative we use the equationḦ = −φφ M 2 pl . From the conservation equation we deduce On the other hand, at the transition time all the energy density is kinetic, which pl ∼ = 32H 6 end . A simple integration leads to an energy density for the produced particles of ρ χ ∼ = 5H 6 end 16 3 πm 2 , and following the calculations made in [30] one gets the following reheating temperature which leads, by combination with (36), to obtaining that n s ∼ = 0.9720, r ∼ = 1.56 × 10 −3 and T R ∼ 5 GeV. This result again satisfies the gravitino-overproduction problem but not the moduli fields one.
In order to study massless particles nearly conformally coupled with gravity, we will use the same formula as used in (27) and, since we do not have the analytical expression of the Hubble constant throughout all the time, we will assume that N ∼ 1 as in the other case. Hence, the reheating temperature is obtaining, thus, that for the bounds of T R coming from the nucleosynthesis it should be verified that 10 −7 ξ − 1 6 16. This only gives us a restriction for the lower bound, since given that we have considered the particles to be nearly conformally coupled with gravity ξ − 1 6 10 −1 , which adds the constraint that T R ≤ 7 × 10 5 GeV. By considering a reheating temperature in the MeV scale, one would obtain that 10 −7 ξ − 1 6 10 −5 . Finally, regarding massless particles far from the conformal coupling with gravity, the reheating temperature becomes which leads us, using also Equation (36), that whose solution is n s = 0.970, corresponding to r = 0.0018, which now clearly falls within the bounds imposed in our model so as to satisfy the observational data, but again fails to fulfill the bound T R ≤ 1 GeV suggested in the introduction.
Finally we are going to briefly study the dynamics of the equationφ + 3H(ϕ,φ)φ + V ϕ = 0. In Figure 6 we have represented in blue some orbits which are solution of these dynamical system, observing that, as in the former case, they start with infinite energy at t → −∞, they experience a phase transition at ϕ = 0 and then evolve towards a de We observe that there is a long period in inflation where orbits coincide with the slow-roll one. Furthermore, as happened with the analytical solution in the former case we studied, the slow roll solution is a limit orbit that starts with a de Sitter phase at t → −∞, since all the other orbits start with infinite energy.

Higgs Inflation (HI)
Another potential that could work would be the following Higgs Inflation (HI) style potential in the Einstein Frame [43].
being λ a dimensionless positive parameter. In this case, the slow-roll parameters are * = 2s 2 * and η * = −s * (1 − s * ), being s * the same as above. In this case, the running, power spectrum and number of e-folds are However, in this model now we have N ∼ 1 s * ∼ = 1 1−ns , which is bounded by 50 for the allowed values of the spectral index. Thus, the potential is not a viable quintessential inflation model because it leads to an insufficient number of e-folds.

Power Law Inflation (PLI)
Now, we are going to study a Power Law Inflation (PLI) [44] potential, adapted to quintessence With the same procedure as in the former cases, we obtain * = 2n 2 M pl ϕ * 2 and η = 2n(2n − 1) and N = n+1 1−ns − n 2 . Hence, we can easily verify that, so as to obtain a number of e-folds 63 ≤ N ≤ 73 with a spectral index n s = 0.968 ± 0.006, we need that 0.65 < n < 1.35. Therefore, taking n = 1 is a good choice in order to match our model with the observational results. Thus, if we consider the range of spectral index 0.9685 ≤ n s ≤ 0.973, we obtain the desired number of e-folds. Regarding the ratio of tensor to scalar perturbations r = 16 * = 4(1 − n s ), the constraint r ≤ 0.12 is verified for n s ≥ 0.97. And so does the running, which becomes α s = − 8 , resulting that −3.7 × 10 −3 ≤ α s ≤ −3.1 × 10 −3 for 0.97 ≤ n s ≤ 0.973. Finally, the power spectrum has the following expression and, thus, since P ∼ 2 × 10 −9 , we can determine λ, namely λ ∼ 10 −11 .
Remark: We can see that the power law potential for n = 2 is equivalent during the inflation, i.e. for ϕ −|ϕ E | to the Double Well Inflation (DWI) [45] potential, namely Therefore, we have seen that the DWI is not a valid model that matches with the corresponding observational results. Now, we are going to proceed analogously as for the potential previously studied. Therefore, we approximate the Hubble constant at the transition point by H E ∼ H end = M pl √ λ ≈ 10 −4 (1 − n s )M pl . Now, combining Equation (13) and the number of e-folds for this particular potencial, we obtain that where, as before, Y = 6 1−ns . Thus, as we can see in Figure 7, the bounds coming from the observational values used throughout all this paper are satisfied for 1 MeV ≤ T R ≤ 10 4 GeV (corresponding to 0.970 ≤ n s ≤ 0.973). If we start considering the production of massless particles nearly conformally coupled to gravity, then the reheating temperature becomes Hence, for 1 MeV ≤ T R ≤ 10 4 GeV, we obtain that 10 −7 ξ − 1 6 5 × 10 −3 . On the other hand, in the case of heavy massive particles (m ∼ M pl ), since the first derivative of the potential is continuous at the transition phase, one has to use, for the β-Bogoliubov coefficient the expression given in [38], To obtain the third derivative of the Hubble parameter first of all we use the formula ...
where we have used that at the transition time all energy density is kinetic. Then we have ( where we have used that H end = √ λM pl . Therefore, the energy density of produced particles is equal to ρ χ ∼ = 10 −3 π H 8 end m 4 , thus, following step by step the calculations made in [30] and, using the thermalization rate introduced in Section 2.3, one gets the following reheating temperature Remark 3.1 Note that the formula of the reheating temperature is a little bit different from the one obtained in [38], since there another thermalization rate has been used. By combining it with equation (48), namely one obtains that n s ∼ = 0.9721, r ∼ = 0.1117 and T R ∼ 11 MeV, which means that this potential supports the production of heavy massive particles. If we consider massless particles far from the conformal coupling, then the reheating temperature becomes Therefore, it falls out of the bounds that we have previously found (1 MeV ≤ T R ≤ 10 4 GeV) and, thus, our model does not support the production of massless particles far from the conformal coupling.
Regarding the dynamics of the equationφ + 3H(ϕ,φ)φ + V ϕ = 0, we have also proceeded analogously as with the other potential. In Figure 8 we have represented some blue orbits solution of the dynamical system which start with infinite energy at t → −∞ and then evolve to the de Sitter phase for t → ∞ after having gone through a phase transition at ϕ = 0. In this case, the slow roll solution (in black) isφ = − Vϕ 3H = M 2 pl 4λ 3 . We note that in this case the slow roll solution does not start with a de Sitter phase. We can compare this power law potential with the one studied in [37], namely Firstly we note that for ϕ ϕ E , i.e during all the inflation, the behaviour will be analogous. Regarding the reheating temperature, the upper bound T R ≤ 10 4 GeV coincides. It only differs the fact that for heavy massive particles nearly conformally coupled with gravity the reheating temperature is in the GeV regime, instead of the MeV one, for the model coming from the potential in Equation (54).

OTHER QUINTESSENTIAL INFLATION POTENTIALS
In this last section we are going to study some other potentials that appear in [43] studying whether they can be implemented in the quintessential inflation.

Open String Tachionic Inflation (OSTI)
We consider the following adapted form of the OSTI potential [46] where |ϕ 0 | M pl . We note that this potential is only non-negative for ϕ > −|ϕ 0 |. Therefore, we will always move through this domain. Now, let's compute the cosmological parameters. * = 2 Since cancels at ϕ = −e −1/2 |ϕ 0 |, the inflation period will happen between this value and ϕ end (corresponding to = 1). Using that |ϕ 0 | M pl , we obtain that ϕ end ≈ √ 2M pl , ϕ * ≈ . Thus, a number of e-folds comprised between 63 and 73 corresponds to a spectral index 0.9685 < n s < 0.9728 and a ratio of tensor to scalar perturbations 0.109 < r < 0.126. Therefore, the constraint r ≤ 0.12 restricts this range to 66 N 73, corresponding to 0.97 n s 0.973, 0.11 < r < 0.12 and 1−ns is imposed to be P ≈ 2 × 10 −9 by choosing the suitable value of λ.

Now, we approximate the Hubble constant at the transition point by H
The combination of equation (13) and the number of e-folds for this potential leads to obtaining, as shown in Figure 9, that the valid bounds of the reheating temperature for this potential are 1 MeV ≤ T R ≤ 10 5 GeV. As in the case of the power-law potential with n = 1, this potential has a continuous derivative at the transition phase. Thus, when considering the production of massless particles nearly conformally coupled to gravity, the reheating temperature is Thus, for 1 MeV ≤ T R ≤ 10 5 GeV, we obtain that 10 −8 ξ − 1 6 3 × 10 −3 . However, dealing with heavy massive particles (m ∼ M pl ), one can see that the second derivative of the potential, and thus the third derivative of the Hubble parameter, diverges at the transition phase, which means that we cannot use the WKB solution to approximate the modes. Therefore, we are not able to compute, in this case, the reheating temperature. Regarding the case of massless particles far from the conformal coupling, the reheating temperature becomes which shows that this model neither supports the production of these particles.
Finally, as done with the other potentials, we study the dynamics of the equationφ + 3H(ϕ,φ)φ + V ϕ = 0. In Figure 10 we see again that the blue orbits (solution of the dynamics equation) start with infinite energy at t → −∞ and then evolve to the de Sitter phase after a phase transition at ϕ = 0. As happened with the PLI potential, the slow roll inflation, namelyφ = 2M 2

Witten-O'Raifeartaigh Inflation (WRI)
In this case, the version of the WRI [47,48] potential is where |ϕ E | M pl . The cosmological parameters are * = 2 and idem for end . Hence, we get that With regards to the number of e-folds, it can be exactly integrated as , which results in a too high number of e-folds, namely N 100.

Kähler Moduli Inflation I (KMII)
The expression of the Kähler Moduli Inflation I (KMII) potential [43] is where α is a positive dimensionless constant such that α ≥ e and ϕ E is the value of ϕ where ϕ M pl e −ϕ/M pl = 1/α such that ϕ E ≥ M pl . We note that, in contrast to the former potentials considered, in this one we are going to assume thatφ < 0. The cosmological parameters are = 1 2 where x = ϕ/M pl . By considering that x * 1, we obtain that n s − 1 ≈ 2αx * e −x * . Regarding the number of e-folds, they can be exactly integrated, namely where x end is the value of x where = 1 and E i is the exponential integral function, which verifies that for x * 1, x * , being this the dominant term in the previous equation. Hence, by using that x * e −x * ≈ ns−1 2α , one obtains that which does not fulfill our bounds for the number of e-folds and the spectral index. Therefore, we have proved that this potential is not a viable quintessential inflation model.

Brane Inflation (BI)
The Brane Inflation (BI) potential behaves as [49] V where µ and p are positive dimensionless parameters and ϕ E ≡ −µM pl We will consider for simplicity only p ∈ N. Thus, the cosmological parameters are where x = −ϕ µM pl . We are going to differ two cases: a) µ 1: In this case one will have that x * 1, and thus * ∼ = On the other hand, one has Taking into account that x end ∼ = p 2 (1−ns)(p+2) which enters in our range for values of p greater than 17. For the tensor/scalar ratio one has r ∼ = Hence, one can conclude that for all the values of 2σ CL of the spectral index it is verified that r < 0.12. We find as well that α s ≈ − (1−ns) 2

2
, which also fulfills the constraints for the running. And finally, by adjusting λ so that P ∼ 2 × 10 −9 , we can build a successful quintessential model.
Effectively, regarding the reheating constraints we obtain that for all the restricted values of the parameter p, H E ∼ 10 −4 √ 1 − n s M pl . So, as usual, one obtains that the reheating temperature bounds from nucleosynthesis give the constraint 0.968 ≤ n s ≤ 0.972. For massive particles we have that T R ∼ 10 3 GeV. In the case of massless particles nearly conformally coupled with gravity, we obtain that ξ − 1 6 10 −8 and the fact that ξ − 1 6 1 should be satisfied constraints our reheating temperature to be less than 10 7 GeV. Finally, considering massless particles far from conformal coupling, one finds that T R ∼ 10 7 GeV. b) µ 1: In this case by taking for example µ ≥ 100p, since * or η * have to be of the order of 10 −2 , it is verified that x * ∼ = 1 and, thus, So, given that * η * , 1−n s ∼ = 6 * ∼ = 1 µ 2 3p 2 (1−x p * ) 2 . This means that x * ∼ = 1−  In this case the potential behaves as [50] where λ and α are positive dimensionless constants. The slow roll parameters are where we have introduced the parameter x ≡ ϕ M pl . We consider to different asymptotic cases: In this case one has * ∼ = and thus, n s − 1 ∼ = −2η * ∼ = 2α . For the number of e-folds one has which leads, as in the case of HI, to a not high enough number of e-folds.
2. α 1: Now the slow roll parameters become * = and the spectral index and the tensor/scalar ratio will be as a function of x * 1 − n s = 1 Then, at 2σ C.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 Using the range of values for x * one finds that 34 N 50, which comes out of the nucleosynthesis bounds.

CONCLUSIONS
We have adapted some inflationary potentials to quintessence 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 gravitation 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 1 MeV and 10 9 GeV, although if one wants to remove the gravitino and moduli problems, which appear in supergravity, this temperature has to be less than 1 GeV.
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 1 and 2. Dealing with the reheating temperature, all the four models, namely ESI, PLI, OSTI and BI, lead to a temperature compatible with the nucleosynthesis bounds. However, since the ESI potential comes from a supersymmetric gravitational one, the problems related with the gravitino and moduli overproduction will appear, which can only be removed when the reheating temperature is in the MeV regime. Then, for this potential, when reheating is due to the production of massless nearly conformally coupled particles (the coupling constant is very close to 1/6), these problems are removed because the reheating temperature is very low (in the MeV regime). On the contrary, reheating due to the production of very heavy massive particles conformally coupled with gravity or massless particles far from the conformal coupling leads to a too high reheating temperature, which does not avoid the gravitino and moduli problems for the ESI potential.
Fortunately, the other three potentials, namely PLI, OSTI and BI, do not come from supersymmetric potentials and, thus, do not suffer these problems because gravitino and moduli fields only appear in supergravity, meaning that the reheating via very massive particles conformally coupled with gravity or via massless particles far from the conformal coupling is also viable for these three potentials.
Gravitons satisfy the same equation as scalar fields minimally coupled with gravity, so its energy density at the end of inflation is twice that of a single scalar field due to the two graviton polarization states. Experimental observations show that, at the end of inflation, the ratio of the energy density of the relic gravitational waves to the one of the produced particles is less than 10 −2 [17]. In [36], using the results of [39] and [40], the authors take as the energy density of the gravitons ρ g = 2 × 10 −2 H 4 E , where H E is the value of the Hubble parameter at the transition time, so assuming that there are N s different light scalar fields with a coupling constant near zero, one deduces that ρg ρχ = 2 Ns , where ρ χ denotes the energy density of the produced particles. Then, according to the observational bound, this means that the number of different scalar fields must be greater than 100, which only happens in some supersymmetric theories [36].
Note that in this situation the reheating temperature is of the order 10 3 N 3/4 s GeV ∼ 3 × 10 4 GeV, allowing the overproduction of gravitinos and moduli fields. The problem is even worse when one considers reheating via the production of heavy massive particles because this mechanism is less efficient than the one due to the creation of massless particles. The situation could be addressed when one considers models whose potential does not come from supergravity, which avoids the gravitino and moduli problems, and reheating is produced via oscillations of an auxiliary field in hibrid quintessential models [51] or via instant preheating [15,17,18] because these kinds of reheating are more efficient than the one due to gravitational particle production.
However, a more detailed investigation shows that the results used in [36] about the production of gravitons may not be correct and the conclusions could be wrong. Here, we summarize the problems that appear in this calculation 1.-Gravitational particle production of massless particles far from the conformal coupling is only considered in toy models, where there is a transition from the exact de Sitter phase to a kination or radiation one. This happens because the vacuum modes are represented by Hankel functions and one can perform analytic calculation, but, as it is well known, the inflationary period is not exactly de Sitter.
2.-In the calculations, only the energy density of the particles is considered and vacuum polarization effects (see [52] for a detailed calculation) are disregarded, but as has been showed in [16] they could be very important in some inflationary scenarios leading to undesirable effects. , and thus, where we have used that for negative small values of the coupling constant it is satisfied that ν = 3 2 + 4|ξ|. Since N has to be greater than N (the number of e-folds from the leaving of the pivot scale to the end of inflation), one can safely take N ≥ 100, then choosing for example N = 100, and ξ = −0.012 the bound is reached. So, a unique massless quantum field with a small negative coupling constant satisfies the bound ρg ρχ ≤ 10 −2 , meaning that gravitational supersymmetric potentials are not needed and the gravitino and moduli problems do not appear, allowing a greater reheating temperature. In this case, the reheating temperature, taking as usual H E ∼ 10 −6 M pl , is given by T R ∼ 2×10 4 e 8|ξ|N −1 |ξ|

3/4
GeV, leading for the values N = 100 and ξ = −0.012 to the reheating temperature T R ∼ 7 × 10 8 GeV, which enters in the viable range.
Summing up, the problem of the production of massless particles far from the conformal coupling and in particular the overproduction of gravitational waves in quintessential inflation is a problem that deserves future investigation for many reasons, but basically because nowadays we do not have any analytic method which allows us to calculate, in realistic models, the backreaction of a massless quantum field not conformally coupled with gravity, and only results that are valid for toy models which may not apply to viable models are used to perform analytic calculations.