Quintessential Inflation in a thawing realization

We study quintessential inflation with an inverse hyperbolic type potential $V(\phi) = {V_0}/{\cosh \left( {\phi^n}/{\lambda^n} \right)}$, where $V_0$, $\lambda$ and"n"are parameters of the theory. We obtain a bound on $\lambda$ for different values of the parameter n. The spectral index and the tensor-to-scalar-ratio fall in the $1 \sigma$ bound given by the Planck 2015 data for $n \geq 5$ for certain values of $\lambda$. However for $3 \leq n<5$ there exist values of $\lambda$ for which the spectral index and the tensor-to-scalar-ratio fall only within the $2 \sigma$ bound of the Planck data. Furthermore, we show that the scalar field with the given potential can also give rise to late time acceleration if we invoke the coupling to massive neutrino matter. We also consider the instant preheating mechanism with Yukawa interaction and put bounds on the coupling constants for our model using the nucleosynthesis constraint on relic gravity waves produced during inflation.

We study quintessential inflation with an inverse hyperbolic type potential V (φ) = V0/cosh (φ n /λ n ), where V0, λ and "n" are parameters of the theory. We obtain a bound on λ for different values of the parameter n. The spectral index and the tensor-to-scalar-ratio fall in the 1σ bound given by the Planck 2015 data for n ≥ 5 for certain values of λ. However for 3 ≤ n < 5 there exist values of λ for which the spectral index and the tensor-to-scalar-ratio fall only within the 2σ bound of the Planck data. Furthermore, we show that the scalar field with the given potential can also give rise to late time acceleration if we invoke the coupling to massive neutrino matter. We also consider the instant preheating mechanism with Yukawa interaction and put bounds on the coupling constants for our model using the nucleosynthesis constraint on relic gravity waves produced during inflation.

I. INTRODUCTION
The standard model of the Universe needs to be modified at early and late times in order to address the problems therein. It is interesting to note that both the early and late time shortcomings of the model can be successfully circumvented by introducing an early phase of accelerated expansion called inflation together with late time cosmic acceleration. The inflationary paradigm resolves the horizon problem, the flatness problem and the monopole problem and provides a mechanism for generation of primordial perturbations [1][2][3][4][5][6][7]. Late time cosmic acceleration is a recent discovery [8,9] backed up by observations of type Ia supernovae, CMB background and galaxy clustering. It requires the presence of an exotic matter of energy momentum tensor with a large negative pressure dubbed dark energy [10][11][12][13][14][15][16][17]. Cosmological constant and slowly rolling scalar fields provide a viable example of dark energy; the effect can also be mimicked by large scale modification of gravity. The presence of a late time phase of accelerated expansion is essential in order to resolve the age crisis [18] which arises in the hot big bang model. Clearly, accelerated expansion plays an important role in the history of our Universe− the big bang model is sandwiched between two phases of accelerated expansion To unify these two important concepts of inflation and late time acceleration, the notion of "Quintessential Inflation" was introduced where the inflaton field behaves like quintessence during late times [10,[19][20][21]. A major problem with quintessential inflation is the "cosmic coincidence" problem which occurs due the fact that the energy density of the scalar field and the matter energy density have comparable values today and for this to occur the conditions in the early universe have to be very precisely predetermined. This tight constraint on the initial conditions can be relaxed if we introduce the notion of the tracker field whose equation of motion has attractor type behavior [16,[22][23][24]. On the contrary, the thawing dynamics has a strong dependence on the initial values. However, as pointed out by Weinberg [25], the scalar field irrespective of its behaviour can not address the fine tuning problem associated with the cosmological constant. Indeed, replacing the cosmological constant by a scalar field translates the cosmological constant problem into the problem of naturalness of the scalar field.
At the onset, it sounds quite plausible to unify the early and late time evolution using a single scalar field that does not disturb the thermal history of Universe. For instance, a field, whose potential is initially shallow at early times followed by a steep behaviour and then finally shallow again at late times, should full fill the said requirement. Unfortunately, the generic potentials do not change their shape that frequently− they are either shallow at early times followed by a steep behaviour thereafter or vice-versa. In the first category, one can realize inflation followed by a scaling solution; a suitable mechanism is then required to exit from the scaling regime in order to obtain late time acceleration. In the second category, one requires extra damping at early times to facilitate inflation. Such an effect, in particular, could be induced by high energy corrections on the Randall-Sundrum brane. Unfortunately, this scenario is ruled out by observation− the tensor to scalar ratio of perturbations is too high in this case. However, the first category of models could give rise to a viable scenario of quintessential inflation.
Let us note that in the first category of models, one can exit from the scaling regime to obtain late time acceleration by coupling the field non minimally to matter. The coupling builds up dynamically in the matter era giving rise to a minimum in the potential where the field could settle down leading to a de Sitter like solution. However, coupling to cold dark matter might destroy the matter phase as the minimum in this case would be induced as soon as the matter phase is established whereas there is no such issue associated with the non-minimal coupling of the field to massive neutrino matter. On the other hand, neutrinos become non-relativistic at late times and as a result the coupling of massive neutrinos to the scalar fields becomes more prominent. The latter ensures the appearance of a minimum in the field potential at late times [23,26,27] which is desirable for the commencement of late time cosmic acceleration. In this paper we follow in the footsteps of the above authors and discuss a model with inverse cosine-hyperbolic potential that might successfully give rise to quintessential inflation.
The inverse cosine-hyperbolic potential is a tachyonic type potential and is inspired from Sring theory [28,29]. Some of the works related to tachyonic potential can be found in Refs. [30,31]. The inverse cosine hyperbolic type of potential is mentioned in the Refs. [28,29]. However, it is not studied in detail by considering the cosmological data of Planck 2015. Further, the late time evolution of the Universe is not studied yet. In this paper we generalize the inverse cosine hyperbolic potential to describe quintessential inflation.

II. INFLATION AND OBSERVATIONAL CONSTRAINTS
As mentioned in the introduction, we are looking for a scenario that would facilitate slow roll at early times followed by scaling behavior in the post inflationary regime such that the exit to late time acceleration is caused by non-minimally coupled massive neutrinos. To this effect, we shall consider the following action, with, (2) where λ = αM pl ; β = 1 α and α, β are dimensionless parameters and n is a positive integer.
Eq. (1) includes the actions for standard matter (S m ), massive neutrino (S ν ), and radiation (S R ). We have assumed that the scalar field is directly coupled to massive neutrino matter whereas the dark matter is minimally coupled. As for inflation, a remark about the matter actions is in order. Since matter is generated during reheating, the said actions should be dropped while disusing inflation.
In what follows, we specialize to a flat FRW background to obtain the evolution equations for the action (1), andφ + 3Hφ + dV dφ = 0.
The slow roll parameters for a potential V (φ) are defined as usual [23,32], and The end of inflation is marked by, where "end" represents the value at the end of inflation. Let us consider a period which begins when the modes cross the horizon and ends with the end of inflation. Then the number of e-foldings during this period is given by [23,32], The tensor to scalar ratio r is given by, and the scalar spectral index n s , which is defined as, where P R is the spectrum of curvature perturbations, is reduced to the form, We now study a model based on a potential given by Eq.
(2) for general n and derive expressions for the slow roll parameters and the spectral index in terms of n. Introducing a dimensionless scalar field χ = φ M pl and using Eq. (7), Eq. (8) and Eq. (2), we obtain, and From Eq. (14), Eq. (15) and Eq. (17) we also obtain the spectral index Eq. (9) gives us From Eq. (11) we obtain, We now set the number of e-foldings to be 60. Comparing the theoretical value of the power spectrum, where M pl = (8πG N ) − 1 2 = 2.4 * 10 18 GeV, with its observational value P R ≈ 2.19 * 10 −9 and combining it with Eq. (2) and Eq. (15), one derives an estimate for V 0 , where, χ i is the value of χ at the beginning of inflation.
According to the Planck 2015 results [33], n s = 0.968± 0.006 at the 1σ confidence level. For n = 1, Eqns. (19) and (18) and respectively, where the expression for n s is derived analytically by inverting the equation, to obtain sinh (βχ) and hence tanh (βχ). Since tanh (βχ end ) can take a maximum value of one, therefore, for inflation to end and hence for ǫ to be unity, The above parametric plot shows the variation of the spectral index n s (β) with respect to the tensor-to-scalar ratio r(β) for different values of n with β lying in the range 0.2 -7.19 where n takes values from 2 to 6. For n is equal to 2, 3, 4, 5, 6; we obtain χ end is equal to 17.68, 7.68, 6.09, 5.55, 5.29 and χ i is equal to 1.80, 2.18, 2.55, 2.83, 3.04 respectively for β = 0.2. For n is equal to 2, 3, 4, 5, 6; we obtain χ end is equal to 0.06, 0.08, 0.09, 0.097, 0.1 and χ i is equal to 0.001, 0.01, 0.02, 0.03, 0.04 respectively for β = 7.19. Both χ i and χ end decrease monotonically with increase in β for all n. The plot also shows the 1 − σ (grey shaded area) and 2 − σ ( yellow shaded area) intervals as given by Planck 2015. The plot shows that n = 3 (black dotted-dashed) and n = 4 (green-dashed) lie in the 2 − σ interval, n = 5 (red-dotted) and n = 6 (blue-solid) lie in the 1 − σ interval and n = 2 (black-thin) does not lie even in the 2 − σ interval.  2)) the theoretical value of the spectral index n s is in agreement with the Planck 2015 data up-to the 1σ confidence level. For β < 0.9, we have φ f > φ i > M pl whereas β 0.9, φ i < φ f < M pl for n = 5 and 6. For higher value of β, φ becomes much smaller than M pl and also the spectral index n s attains a constant value of 0.9625 and 0.9633 respectively for n = 5 and 6. The tensor-to-scalar ratio r satisfies the Planck 2015 data [33], i.e., r < 0.1 for all the values of n discussed here.
We should emphasize that the permitted range of parameters in the model leads to small numerical values of r, thereby, the Lyth bound [34] is easily satisfied in the sub-Planckian region. Indeed, using Eq. (11), For r ∼ 10 −6 , N = 60, we find that δφ .02M pl and Lyth bound is satisfied as stated.

III. PREHEATING BASED UPON INSTANT PARTICLE PRODUCTION
In the paradigm of quintessential inflation, the underlying potential is typically of a run away type. Thereby the standard mechanism of (p)reheating [35][36][37][38] can not be implemented in this case. One of the possible alternatives is provided by gravitational particle production. However, this process is inefficient and as a result the field spends a long time in the kinetic regime (ρ φ ∼ 1/a 6 ). The energy density of gravity waves which were produced at the end of inflation becomes sufficiently larger than ρ φ during the kinetic regime and this leads to a violation of the nucleosynthesis constraint. The nucleosynthesis constraint restricts the duration of the kinetic regime and we can also put a lower bound on the reheating temperature using this constraint. Therefore, we require an efficient reheating mechanism to address the issue. The instant preheating mechanism turns out to be suitable.
In what follows, we shall consider a scenario where the inflaton field φ interacts with another scalar field χ and we assume that the scalar field χ interacts with the matter field ψ such that [39], Let us choose the following interaction terms in the Lagrangian, in order to implement the aforesaid. Here g and h are positive coupling constants [34,[40][41][42][43]. The form of the Lagrangian is chosen such that χ field has no bare mass and its effective mass is given by m χ = g |φ| [44]. Particle production commences after inflation provided that m χ changes non-adiabatically,ṁ χ m 2 χ . The mass of produced χ particles grows as the field evolves to larger values after the end of inflation and χ decays into various species of particles. Assuming that all the energy is instantaneously converted into radiation and thermalized, one can obtain the following estimate(see Appendix A for details), As mentioned before, the energy density of gravity waves becomes significantly larger than ρ φ during the kinetic regime and this might later lead to a violation of the nucleosynthesis constraint during the radiative regime. The said constraint, at the end of inflation. This gives us a lower bound on (ρ r ) end , (ρ r ) end 2.48 × 10 49 GeV 4 and hence a bound on the reheating temperature, (T r ) end 2.2 × 10 12 GeV , is obtained. Using Eq. (29) together with Eq. (31), we find out the lower limit on g. The limit on h can be set by avoiding the back reaction of χ particles on the post inflationary dynamics. We find a comfortable range in the parameter space (g, h) which can give rise to efficient preheating after inflation (see Appendix A for details).

IV. LATE TIME ACCELERATION WITH NON-MINIMAL COUPLING TO MASSIVE NEUTRINO MATTER
In the absence of non-minimal coupling, the scalar field with a potential given by Eq. (2) can not give rise to late time acceleration as the potential is steep. To obtain late time acceleration one needs to introduce a feature in the potential which can be achieved by coupling the scalar field non minimally to massive neutrino matter. The coupling of the scalar field with neutrinos is not prominent in the radiation era while it plays a crucial role during the late stages of evolution where it induces a minimum in the potential giving rise to a de Sitter solution. The result is a late time attractor. As mentioned before, it is desirable to leave the matter era intact and assume that the field couples to massive neutrino matter only.
The neutrinos here are relativistic p ν = ρν 3 like radiation during early times but turn non relativistic (p ν = 0) at late times. This behaviour can be mimicked by the following ansatz for parametrization of ω ν (z), The equation of state ω ν (z) is thus chosen keeping in mind the phase transition from the relativistic state to the non relativistic one. z eq and z dur give the time and duration of the transition. The equation of continuity for massive neutrinos, in presence of coupling, takes the form, [23,34].
Varying the action (1) with respect to φ and following the Ref. [34], we obtain, We note that in the radiation era when the neutrino is relativistic, in Eqs (33) and (34), the coupling term of the scalar field and neutrinos vanishes since ρ ν − 3p ν = 0. Thus the coupling becomes effective only when the neutrinos become non-relativistic. We thus have a system of equations, ρ i + 3H (ρ i + p i ) = 0; i = matter and rad., which we can solve to obtain the evolution of the various energy densities. To solve this system of differential equations, we introduce the dimensionless variables x, y, ω ν , Ω m , Ω r and λ where x =φ √ , Ω m = ρm , Ω r = ρr The evolution equations can then be cast in a convenient form, Here, prime ′ is derivative w.r. to N and we have used the constraint, Ω ν = 1 − Ω r − Ω m − x 2 − y 2 . We now solve this dynamical system numerically for β = 1 and plot these variables taking the initial conditions at the end of inflation. The initial values of x, y, z, ω ν Ω m and Ω r are approximately 10 −51 , 10 −51 , 0.88, 0.3, 10 −23 and 1 respectively. z dur and γ are 6.9 and 100 respectively. The initial value of Ω ν is 10 −7 . From the plots of Fig. (3), it is clear that we can obtain late time acceleration with neutrino coupling using these initial conditions, since we can obtain the current values of Ω φ , Ω m and ω φ which are approximately 0.7, 0.3 and −1 respectively. Similar graphs were plotted for β = 2, 3 etc. and the same result was obtained. A comment regarding the thawing nature of evolution is in order. The potential (2) belongs to the class of tracker potentials. Indeed, One can easily check that our potential behaves as V ∼ Exp(−β n φ n /M n p ) [45] asymptotically for large values of φ. However, on the contrary, we see a thawing behaviour. It is clear from subfigure (3d) that after inflation ends, the potential drops sharply resulting in a deep overshoot of ρ φ with respect to the background energy density. Clearly, it takes long time for the field to recover from freezing which happens when the background energy density become comparable to ρ φ . When it happens, the field has already evolved to the slow roll regime near the minimum of the potential and consequently, tracker regime is missed out.
In this case nucleosynthesis poses no constraint on the slope of the potential unlike the case of standard exponential potential.

V. CONCLUSIONS
In this paper, we have considered a single scalar field model with a potential V 0 / cosh (φ n /λ n ) and demonstrated that the model can give rise to quintessential inflation. First, we have demonstrated the viability of the theory for inflation. In particular, we have shown that for lower values of n, the model is ruled out by observation. For instance if n = 2, the theoretical value of n s fails to lie in the observed range. However, situation improves for higher values of n; potential flattens significantly in this case. In particular, for n = 3 and 4, the considered potential provides a value of n s such that it can lie in the range of 2 − σ confidence level of Planck 2015 data. The value of β is greater than 0.17 and 0.085 for n = 3 and 4 respectively. The most profound results are obtained for n = 5 and 6. The theoretical value of n s satisfies the Planck 2015 data at the 1σ confidence level and its value becomes ∼ 0.9625 and ∼ 0.963 for β ≥ 0.16 and β ≥ 0.09 for n = 5 and 6 respectively. It should be noted that our model mimics the hill-top potential [32] for β ∼ 1 as φ << M p . It is therefore not surprising that the tensor-to-scalar ratio r is negligibly small in our case similar to that of the Starobinsky-model [46]. Nevertheless, the very small value of the tensor-to-scalar ratio is a salient feature of our model compared to other models discussed in the literature. If we take β to be much less than unity, we find that the value of r is significantly greater than 10 −6 . Further, we have plotted the evolution of normalized values of different components of the energy densities Ω i , ω ν , ω φ and the energy density of the scalar field ρ φ . The plots explain that the universe undergoes late time acceleration, i.e., for today, Ω φ ∼ 0.7, Ω m ∼ 0.3 and ω φ = −1. Though the underlying potential belongs to the tracker class, the thawing behaviour manifests itself in the post-inflationary region. This behaviour depends on the specific initial conditions taken at the end of inflation. The latter then leads to a deep overshoot of the field and consequently misses the tracker region of the potential. Thus, in the model under consideration (in a thawing realization), the scalar field which drives inflation in the early Universe is also responsible for late time acceleration once we couple the scalar field to massive neutrino matter. Finally, we considered the instant preheating mechanism for particle production where we assumed that the scalar field couples to the matter field via Yukawa interaction. We have put bounds on the coupling constants g and h defined in the Yukawa interaction which are found to be g ≥ 3.8 × 10 −5 and h ≥ 2.5 × 10 −3 / √ g such that (T r ) end ≥ 2.2 × 10 12 GeV . These values satisfy the nucleosynthesis constraint. We have also studied the spectrum of relic gravity waves in our model (see appendix B); the generic feature of the scenario includes the blue spectrum of gravity waves on scales smaller than comoving horizon scales. Fig. (4) shows the energy spectrum for n = 6 and β = 1 as a function of the wavelength λ together with the sensitivity curves of advLIGO and LISA.
Let us now confirm that the above condition is met in the model under consideration. To this effect, we estimatė φ end using the expression of slow roll parameter, ǫ = 4πGφ The condition for particle production then becomes, Using Eq. (49), we have following expression, The production time can be estimated [34,37,38,40,43] as which allows us then to estimate wave number using uncertainty relation, The occupation number for χ particles is given by the following expression, We can then find the number density and energy density of created particles, Now plugging |φ p | ≃ g − 1 2 φ end 1 2 in the above equation, one finds, Substituting φ end ≈ V 1 2 end in the above equation, ρ χ reduces to Using Eq. (48) and ρ φ =φ Combining Eq. (55) and Eq. (57) yields, Assuming that the χ field gets converted into radiation and thermalization takes place instantaneously, we have, We thus obtain, The nucleosynthesis constraint [34] dictates that where Combining Considering the form of our potential at the beginning and end of inflation, the above inequality reduces to, (ρ r ) end 400 3π Evaluating the expression on the right hand side for β = 1 and n = 6 as above, the above inequality is calculated as, ρ φ ρ r end 2.53 × 10 11 .