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Quintessential Inflation for Exponential Type Potentials: Scaling and Tracker Behavior

We will show that for exponential type potentials, which are used to depict quintessential inflation, the solutions whose initial conditions take place during the slow roll phase in order to describe correctly the inflationary period do not belong to the basin of attraction of the scaling solution -a solution of the scalar field equation whose energy density scale as the one of the fluid component of the universe during radiation or the matter domination period-, meaning that a late time mechanism to exit this behavior and depict correctly the current cosmic acceleration is not needed.


INTRODUCTION
Quintessence [1-7] is a theory used to reproduce the current cosmic acceleration without the need of a cosmological constant. In quintessence it has been shown that, for exponential potentials V (ϕ) ∼ e −γϕ/M pl with γ > 2, there exists a solution whose energy density scales as the one of radiation [8]. And, recently, it has been proved that, for more general exponential potentials V (ϕ) ∼ e −γϕ n /M n pl , there also exists an approximate scaling solution [9,10]. Such solution, termed as scaling solution (see [11] for a detailed classification of the potentials that lead to scaling solutions), is important in order to deal with the coincidence problem because, due to the attractor behavior of the scaling solution, if the scalar field is at the beginning of radiation in the basin of attraction of this scaling solution, it evolves as a radiation fluid. Therefore, since in standard quintessence we have two fields, the inflaton (which vanishes after releasing its energy when it oscillates in the deep well of the potential) and the quintessence scalar field, one can assume initial conditions for this field which lead it to enter in the basin of attraction of the scaling solution.
However, so that the universe enters in the late time accelerated phase, the quintessence field has to leave the scaling behavior, which could be done in several ways. Taking into account that for 0 < γ < √ 2 during the matter domination era there exists a tracker solution [2,8,12,13] leading to an accelerating late time universe, one could add to the potential the term e −γϕ/M pl , with 0 < γ < √ 2. In this situation, one can show that the first term of the potential dominates during the radiation dominated era and the second term dominates during the matter dominated one [4]. Alternatively, one could introduce a non-minimal coupling between the quintessence field and massive neutrinos, whose effect is to modify the potential in the matter domination era [9,10], but in that case, as we will see, the current cosmic acceleration is due to an effective cosmological constant.
On the contrary, in quintessential inflation [14][15][16][17][18][19][20][21][22][23][24] there is only one scalar field -the inflaton-driving the evolution of the universe by depicting both the early-and late-acceleration of the universe. Due to the attractor behavior of inflation, the initial condition of the scalar field has to be taken to belong to the basin of attraction of the slow roll solution.
Then, using a quintessential inflation model based on the exponential type potentials proposed in [10], where the authors showed that there exists an approximately scaling solution, we will show that at the beginning of the radiation era the scalar field is not in the basin of attraction of the scaling solution. In fact, the value of the effective Equation of State (EoS) parameter for the inflaton field is 1 during the radiation epoch, that is, it does not scale as the relativistic plasma whose energy density dominates during this period. As a consequence, in quintessential inflation a mechanism to exit the scaling behavior is not needed.
The only thing needed to reproduce the evolution of the universe is an inflationary potential leading to a spectral index (n s ) and a ratio of tensor-to-scalar perturbations (r) entering into the two dimensional marginalized joint confidence contour at 2σ confidence-level (CL) provided by Planck data [25,26] combined with a quintessence potential will be dominant at late times in order to depict correctly the current cosmic acceleration.
The paper is organized as follows. In Section 2, we study the exponential type potentials introduced in [9], we calculate the reheating temperature of the universe using the mechanism of instant preheating [27,28] because the potential is very smooth and gravitational particle production of neither light nor superheavy particles is not effective for this kind of potentials [29][30][31][32][33]. With this reheating temperature we compute the evolution of the inflaton field during the kination regime [34] to obtain its initial conditions at the beginning of the radiation epoch. Finally, with this initial data we integrate numerically the dynamical system to show that the dynamics of the inflaton field is completely different to the one of the scaling solution. Section 3 is devoted to the study of a viable model of quintessential inflation whose potential is the combination of an exponential type potential -which stands for inflation-with a pure exponential potential which will reproduce the late time acceleration of the universe. To obtain numerically the value of the parameter on which the model depends we use the current observation data such as the red-shift at the beginning of the matter-radiation equality, the current values of the Hubble rate and the ratio of the matter energy density to the critical one. Finally, in Section 4 we present the conclusions of our work.
The units used throughout the paper are = c = 1 and we denote the reduced Planck's mass by M pl ≡ 1 √ 8πG ∼ = 2.44 × 10 18 GeV.

A QUINTESSENTIAL INFLATION MODEL
In this work we will consider the same Exponential Inflation-type potentials studied, for the first time, in [9], where λ is a dimensionless parameter and n is an integer.
For this model the power spectrum of scalar perturbations, its spectral index and the ratio of tensor to scalar perturbations are given by (see for details of the calculations [10]) and An important relation is obtained combining the equations (3) and (4), which leads to the formula for the power spectrum and, thus, which for the viable values of n s = 0.9649 (the central value of the spectral index) and It is important to realize that a way to find theoretically the possible values of the parameter λ is to combine the equations (4) and (5) to get And, using the theoretical values r ≤ 0.1 and n s = 0.9649±0.0042 (see for instance [25,26]), one can find the candidates of λ at 2σ C.L. These values have to be checked for the joint contour in the plane (n s , r) at 2σ C.L., when the number of efolds is approximately between 60 and 75, which is what happens in quintessential inflation due to the kination phase [20,35] -the energy density of the scalar field is only kinetic [34,36]-after the inflationary period.
For this kind of potentials, in order to compute the number of efolds we need to calculate the main slow-roll parameter whose value at the end of inflation is EN D = 1, meaning that at the end of this epoch the field reaches the value ϕ EN D = 2 Then, the number of efolds is given by and, thus, combining the equations (3), (4) and (11) one obtains the spectral index and the tensor/scalar ratio as a function of the number of efolds and the parameter λ.
On the other hand, since inflation ends at ϕ EN D = and the corresponding value of the Hubble rate is given by which will constraint very much the values of the parameter λ because in all viable inflationary models at the end of inflation the value of the Hubble rate is of the order of 10 −6 M pl [37]. In fact, when (13) is of the order 10 −6 M pl one gets Then, to perform numerical calculations, throughout the paper we will use the values of n = 10 and r = 0.02 and, thus, for λ = 4.1 × 10 −16 we have obtained approximately 67.4 efolds, which is a viable value in quintessential inflation. With these values, if the particles are created via instant preheating [27,28]-which seems the best mechanism due to the smoothness of the potential-we have to obtain the Enhanced Symmetry Point (EPS), which is the value of the field at which its temporal derivative is maximum. In our case, taking initial conditions during the slow roll period (recall the slow roll solution is an attractor, so the evolution of the inflation field is the same for all initial conditions in the basin of attraction of the slow roll solution) we have obtained by integrating numerically the dynamical system that approximatelyφ max ∼ = 2.9 × 10 −6 M 2 pl at ϕ max ∼ = 37M pl .
After this, we have to find out the moment when kination starts, which could be chosen at the moment when the effective EoS parameter is very close to 1. Assuming for instance that kination starts at w ef f ∼ = 0.99, we have numerically obtained ϕ kin ∼ = 47M pl anḋ ϕ kin ∼ = 5.6 × 10 −9 M 2 pl , and hence H kin ∼ = 2.3 × 10 −9 M pl .
On the other hand, since in instant preheating the effective mass of the quantum field χ field coupled with the inflaton ϕ is given by g(ϕ − ϕ max ), where g is the dimensionless coupling constant, at the beginning of kination the energy density of the created superheavy particles is given by where the density of produced particles is [28] n χ,max = and a max a kin has been calculated numerically.
Then, at the beginning of the kination regime we have and, denoting by Γ the decay rate -the superheavy particles must decay into lighter ones in order to obtain a relativistic plasma needed to match with the hot Friedmann universe-, and taking into account that the inflaton field ϕ is nearly frozen during kination, we get ρ ϕ,dec = 3Γ 2 M 2 pl and ρ χ,dec ∼ = 1.04 × 10 −13 g 5/2 M 3 pl Γ.
In addition, in order to avoid a second inflationary phase we have to impose that the decay was before the end of the kination [28] (see also [38] for a detailed explanation), i.e., we have to assume ρ χ,dec ≤ ρ ϕ,dec , which leads to the following constraint, Then, following for example Section II of [38], the reheating temperature is given by where g rh = 106.75 are the degrees of freedom for the Standard Model.
On the other hand, as we have already explained, the decay must be before the end of the kination phase, meaning that Γ ≤ H kin ∼ = 2.3 × 10 −9 M pl , which leads to the lower bound T rh ≥ 6.27 × 10 −9 g 15/8 M pl = 1.53 × 10 10 g 15/8 GeV.
Moreover, to preserve the BBN success the reheating temperature has to be approximately constrained between 1 MeV and 10 9 GeV [39], so we get the bound 3.08 × 10 −12 ≤ g ≤ 0.23.
Finally, to fix the reheating temperature, we choose the following compatible values of the parameters, g = 10 −2 and Γ = 10 −12 M pl , obtaining a reheating temperature of

Dynamical evolution of the scalar field
Next, we want to calculate the value of the scalar field and its derivative at the reheating time. Analytical calculations can be done disregarding the potential during kination because during this epoch the potential energy of the field is negligible. Then, since during kination one has a ∝ t 1/3 =⇒ H = 1 3t , using the Friedmann equation the dynamics in this regime will beφ Thus, at the reheating time, i.e., at the beginning of the radiation phase, one has And, using that at the reheating time, i.e., when the energy density of the scalar field and the one of the relativistic plasma coincide, the Hubble rate is given by H 2 rh = 2ρ rh 3M 2 pl , one gets where we have used that the energy density and the temperature are related via the formula ρ rh = π 2 30 g rh T 4 rh , where the number of degrees of freedom for the Standard Model is g rh = 106.75 [40].
As we have already commented, we will take as the reheating temperature T rh ∼ = 2 × 10 −7 GeV. Then, at the beginning of the radiation era we will have To calculate the value of the field and its derivative at the matter-radiation equality, namely ϕ eq andφ eq , we continue assuming that the potential is negligible (this situation has to be verified numerically integrating the full dynamical system, which we will show in the next subsection), i.e., we are assuming that the effective EoS parameter of the field pl , and at matter-radiation equality we will have ρ eq = 2ρ m,0 (1 + z eq ) 3 = 2.48 × 10 −110 M 4 pl = 8.8 × 10 −1 eV 4 . Now, using the relation at the matter-radiation equality ρ eq = π 2 15 g eq T 4 eq with g eq = 3.36 (see [40]), we get T eq = 3.25×10 −28 M pl = 7.81×10 −10 GeV. Thus, solving the dynamical equationφ+ 3 2tφ = 0, one obtains ϕ eq =φ rh t rh t eq t rh t eq = 4 3 M pl H eq H eq H rh = 4π 9 g eq 5 g eq g rh where once again we have used that T rh ∼ = 2 × 10 7 GeV.

Numerical simulation during radiation
To show numerically that during radiation the scalar field is not in the basin of attraction of the scaling solution, first of all we calculate the value of the red-shift at the beginning of the radiation epoch a eq a rh = (1 + z eq ) ρ r,rh ρ r,eq where we have used that ρ r,eq = ρ r,rh a rh aeq 4 . Then, for the reheating temperature T rh = 2 × 10 7 GeV, we get z rh = −1 + 1.90 × 10 20 .
Moreover, at the beginning of radiation the energy density of the matter will be ρ m,rh = ρ m,eq a eq a rh 3 = ρ m,eq ρ r,rh ρ r,eq where we have used that ρ m,eq = ρ r,eq = ρ eq /2.
In this way, the dynamical equations after the beginning of the radiation can be easily obtained using as a time variable N ≡ − ln(1 + z) = ln a a 0 . Recasting the energy density of radiation and matter respectively as a functions of N , we get ρ r (a) = ρ r,rh a rh a 4 =⇒ ρ r (N ) = ρ r,rh e 4(N rh −N ) and where N rh denotes the value of the time N at the beginning of radiation and, as we have already obtained, ρ m,rh ∼ = 7.7 × 10 13 GeV 4 and ρ r,rh ∼ = 4.4 × 10 30 GeV 4 .
To obtain the dynamical system for this scalar field model, we will introduce the dimensionless variables where K is a parameter that we will choose accurately in order to facilitate the numerical calculations. So, taking into account the conservation equationφ + 3Hφ + V ϕ = 0, one arrives to the following dynamical system, where the prime is the derivative with respect to N , . It is not difficult to see that one can writē where we have defined the dimensionless energy densities asρ r = ρr Next, to integrate from the beginning of radiation up to the matter-radiation equality, i.e., from N rh ∼ = −50.57 to N eq ∼ = −8.121, we will choose KM pl = 10 −17 GeV 2 , yieldinḡ andV Finally, with the initial conditions for the field being x rh = 71.3 and y rh = 2.97 × 10 32 (these initial conditions are obtained in the equations (30) and (31)) and integrating numerically the dynamical system, we conclude that the value of w ϕ remains 1 during radiation, namely between N rh and N eq , thus proving that the inflaton field does not belong to the basin of attraction of the scaling solution.

A VIABLE MODEL
To depict the late time acceleration, we have to modify the original potential because it could not explain the current observational data. For this reason, we will consider the following modification by introducing a new exponential term, with M a mass which must be calculated numerically and 0 < γ < √ 2 in order to obtain that at late times the solution is in the basin of attraction of the tracker solution [12,13], which evolves as a fluid with effective EoS parameter w ef f = γ 2 3 − 1 (see [42] for a detailed deduction of the tracker solution).
with β > 0, which has a minimum where the inflation end its evolution, thus acting as an effective cosmological constant which stands for the current cosmic acceleration, our potential does not have a minimum and the scalar field continues rolling down the potential following the spirit of the quintessential inflation.
Now we have to solve the dynamical system (37) with initial conditions at the beginning of the matter-radiation era. We will take K = H 0 the current value of the Hubble rate, so the initial conditions at N eq = −8.121 will be x eq = 72.9 and y eq = 2.15 × 10 −11 . Finally, taking into account that nowH(N ) = H(N )/H 0 , to obtain the value of the parameter M one has to impose the value ofH to be 1 at N = 0.

Numerical calculations
For γ = 1, integrating the dynamical system and imposing thatH(0) = 1, we have obtained M = 1.94 × 10 −1 MeV. Thus, looking at formula (8), we realize that during the slow roll phase (ϕ ∼ M pl ) the second term of the potential (41) is sub-leading, that is, the responsible for inflation is the first term. On the contrary, for large values of the inflaton field ϕ (ϕ ∼ 70M pl ), the second term of the potential is dominant, meaning that it is the responsible for quintessence.
On the other hand, in Figure 1 we show the evolution of the respective energy densities and Ω's. One can see that the energy density of the scalar field is dominant and in Figure  FIG. 1: Left graph: The reduced densitiesρ m (blue curve),ρ r (red curve), andρ ϕ (green curve) in log scale, from reheating to matter-radiation equality. Right graph: The density parameters Ω m , Ω r , Ω ϕ (using the same color code), from reheating to matter-radiation equality. 2 one can deduce that the universe is accelerating because the effective EoS parameter is at the present time and future less than −1/3.

CONCLUDING REMARKS
We have shown that in quintessential inflation with exponential type potentials V ∼ e −λϕ n /M n pl the solutions obtained from initial conditions during the slow roll regime do not enter in the basin of attraction of the scaling solution.
In addition, we have seen that these potentials only depict the inflationary period. So, to obtain the current cosmic acceleration and describing all the evolution of the universe, we need to combine them with a quintessential potential. In our work, we have chosen as a quintessential potential an exponential potential of the type e −γϕ/M pl with 0 < γ < √ 2 in order that at late times the solution is in the basin of attraction of the tracker solution, thus depicting a late time accelerated universe.