Reheating after an Inflationary Universe from a Perfect Fluid and Its Comparison with Observational Data

The definition of slow-roll parameters , η, ξ in terms of potential V(ϕ) are as follows

$$\begin{eqnarray}\begin{array}{rcl}\epsilon & \equiv & \displaystyle \frac{1}{2{\kappa }^{2}}{\left(\displaystyle \frac{{V}^{{\prime} }(\phi )}{V(\phi )}\right)}^{2},\eta \equiv \displaystyle \frac{1}{{\kappa }^{2}}\displaystyle \frac{{V}^{{\prime\prime} }(\phi )}{V(\phi )},\\ {\xi }^{2} & \equiv & \displaystyle \frac{1}{{\kappa }^{4}}\displaystyle \frac{{V}^{{\prime} }(\phi ){V}^{\prime\prime\prime }(\phi )}{{\left(V(\phi )\right)}^{2}}.\end{array}\end{eqnarray} \tag{ 2 }$$

The prime denotes the derivative with respect to the argument of the functions such as ${V}^{{\prime} }(\phi )\equiv \tfrac{\partial V(\phi )}{\partial \phi }$, etc. The expressions of the scalar spectral index n_s of the curvature perturbations, the tensor-to-scalar ratio r of the density perturbations, and the running of the spectral index α_s , for a single scalar field model are obtained as follow

$$\begin{eqnarray}&&\begin{array}{l}{n}_{s}-1\sim -6\epsilon +2\eta ,r=16\epsilon ,\\ {\alpha }_{s}\equiv \displaystyle \frac{{{dn}}_{s}}{d\mathrm{ln}k}\sim 16\epsilon \eta -24{\epsilon }^{2}-2{\xi }^{2}.\end{array}\end{eqnarray} \tag{ 3 }$$

There is increasing interest in constructing a scalar field inflationary model to determine the number of e-folds N, the slow-roll parameter , or the tensor-to-scalar ratio r by the observables (Lyth & Riotto 1999; Nojiri & Odintsov 2006). We assume the flat Friedmann–Lemaitre–Robertson–Walker (FLRW) metric

$$\begin{eqnarray}&&{{ds}}^{2}=-{{dt}}^{2}+{a}^{2}(t)\displaystyle \sum _{i=1,2,3}{\left({{dx}}^{i}\right)}^{2},\end{eqnarray} \tag{ 4 }$$

where the Hubble parameter is defined as $H=\tfrac{\dot{a}}{a}$, and a(t) is the scale factor. The dot shows the time t derivative. The gravitational field equations in the chosen background are given by

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{3}{{\kappa }^{2}}{H}^{2} & = & \displaystyle \frac{1}{2}{\dot{\phi }}^{2}+V(\phi ),\\ & & -\displaystyle \frac{1}{{\kappa }^{2}}\left(3{H}^{2}+2\dot{H}\right)=\displaystyle \frac{1}{2}{\dot{\phi }}^{2}+V(\phi ).\end{array}\end{eqnarray} \tag{ 5 }$$

In Bamba et al. (2014c, 2014d) and Bamba & Odintsov (2016), the scalar field inflation model was reconstructed to an inflationary universe with perfect fluid component. By redefining a new scalar field φ, which is identical to the number of e-folds N so that ϕ = ϕ (φ), the field equations can be rewritten as follows

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{3}{{\kappa }^{2}}{\left(H(N)\right)}^{2} & = & \displaystyle \frac{1}{2}\omega (\varphi ){\left(H(N)\right)}^{2}+V(\phi (\varphi )),\\ & & -\displaystyle \frac{1}{{\kappa }^{2}}\left(3{\left(H(N)\right)}^{2}+2{H}^{{\prime} }(N)H(N)\right)\\ & = & \displaystyle \frac{1}{2}\omega (\varphi ){\left(H(N)\right)}^{2}-V(\phi (\varphi )).\hspace{0.5cm}\end{array}\end{eqnarray} \tag{ 6 }$$

In these reconstructed equations we consider $\omega (\varphi )\,\equiv {\left(\tfrac{d\phi }{d\varphi }\right)}^{2}$, and if the Hubble expansion rate H is a function of N, H(N), we can express ω (φ) and V(ϕ (φ)) in the following form

$$\begin{eqnarray}\begin{array}{rcl}\omega (\varphi ) & = & -\displaystyle \frac{2{H}^{{\prime} }(N)}{{\kappa }^{2}H(N)}\left|{}_{N=\varphi },\right.\\ V(\varphi ) & = & \displaystyle \frac{1}{{\kappa }^{2}}\left[3{\left(H(N)\right)}^{2}+H(N){H}^{{\prime} }(N)\right]{\left.\right|}_{N=\varphi }.\end{array}\end{eqnarray} \tag{ 7 }$$

As a solution for the field equation ϕ or φ, we have H = H(N) and φ = N and it is explicit that H^' < 0 (because ω (φ) > 0). Now we can write the slow-roll parameters , η, and ξ in terms of H

$$\begin{eqnarray}\begin{array}{rcl}\epsilon & = & \displaystyle \frac{1}{2{\kappa }^{2}}{\left(\displaystyle \frac{d\varphi }{d\phi }\right)}^{2}{\left[\displaystyle \frac{{V}^{{\prime} }(\varphi )}{V(\varphi )}\right]}_{\varphi =N}^{2}\\ & = & -\displaystyle \frac{H(N)}{4{H}^{{\prime} }(N)}{\left[\displaystyle \frac{6\tfrac{{H}^{{\prime} }(N)}{H(N)}+\tfrac{{H}^{{\prime\prime} }(N)}{{H}^{{\prime} }(N)}+{\left(\tfrac{{H}^{{\prime} }(N)}{H(N)}\right)}^{2}}{3+\tfrac{{H}^{{\prime} }(N)}{H(N)}}\right]}^{2},\end{array}\end{eqnarray} \tag{ 8 }$$

and

$$\begin{eqnarray}&&\begin{array}{l}\eta =\displaystyle \frac{1}{{\kappa }^{2}V(\varphi )}{\left[\displaystyle \frac{d\varphi }{d\phi }\displaystyle \frac{d}{d\varphi }\left(\displaystyle \frac{d\varphi }{d\phi }\right){V}^{{\prime} }(\varphi )+{\left(\displaystyle \frac{d\varphi }{d\phi }\right)}^{2}{V}^{{\prime\prime} }(\varphi )\right]}_{\varphi =N}\\ =-\displaystyle \frac{1}{2}{\left(3+\displaystyle \frac{{H}^{{\prime} }(N)}{H(N)}\right)}^{-1}\\ \times \,\left[9\displaystyle \frac{{H}^{{\prime} }(N)}{H(N)}+3\displaystyle \frac{{H}^{{\prime\prime} }(N)}{H(N)}+\displaystyle \frac{1}{2}{\left(\displaystyle \frac{{H}^{{\prime} }(N)}{H(N)}\right)}^{2}\right.\\ \left.-\displaystyle \frac{1}{2}{\left(\displaystyle \frac{{H}^{{\prime\prime} }(N)}{{H}^{{\prime} }(N)}\right)}^{2}+3\displaystyle \frac{{H}^{{\prime\prime} }(N)}{{H}^{{\prime} }(N)}+\displaystyle \frac{{H}^{\prime\prime\prime }(N)}{{H}^{{\prime} }(N)}\right],\end{array}\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}\begin{array}{rcl}{\xi }^{2} & = & \displaystyle \frac{{V}^{{\prime} }(\varphi )}{{\kappa }^{4}V{\left(\varphi \right)}^{2}\omega {\left(\varphi \right)}^{2}}\\ & & \times \left\{\left[-\displaystyle \frac{{\omega }^{{\prime\prime} }(\varphi )}{2\omega (\varphi )}+{\left(\displaystyle \frac{{\omega }^{{\prime} }(\varphi )}{\omega (\varphi )}\right)}^{2}\right]{V}^{{\prime} }(\varphi )\right.\\ & & \left.-\displaystyle \frac{3{\omega }^{{\prime} }(\varphi )}{2\omega (\varphi )}{V}^{{\prime\prime} }(\varphi )+{V}^{\prime\prime\prime }(\varphi )\right\}\left|{}_{\varphi =N}\right.\\ & = & \displaystyle \frac{6\tfrac{{H}^{{\prime} }(N)}{H(N)}+\tfrac{{H}^{{\prime\prime} }(N)}{{H}^{{\prime} }(N)}+{\left(\tfrac{{H}^{{\prime} }(N)}{H(N)}\right)}^{2}}{4{\left(3+\tfrac{{H}^{{\prime} }(N)}{H(N)}\right)}^{2}}\\ & & \times \left\{3\displaystyle \frac{H(N){H}^{\prime\prime\prime }(N)}{{H}^{{\prime} }{\left(N\right)}^{2}}+9\displaystyle \frac{{H}^{{\prime} }(N)}{H(N)}\right.\\ & & -2\displaystyle \frac{H(N){H}^{{\prime\prime} }(N){H}^{\prime\prime\prime }(N)}{{H}^{{\prime} }{\left(N\right)}^{3}}+4\displaystyle \frac{{H}^{{\prime\prime} }(N)}{H(N)}\\ & & +\displaystyle \frac{H(N){H}^{\prime\prime\prime }{\left(N\right)}^{3}}{{H}^{{\prime} }{\left(N\right)}^{4}}+5\displaystyle \frac{{H}^{\prime\prime\prime }(N)}{{H}^{{\prime} }(N)}-3\displaystyle \frac{H(N){H}^{{\prime\prime} }{\left(N\right)}^{2}}{{H}^{{\prime} }{\left(N\right)}^{3}}\\ & & -\left.{\left(\displaystyle \frac{{H}^{{\prime\prime} }(N)}{{H}^{{\prime} }(N)}\right)}^{2}+15\displaystyle \frac{{H}^{{\prime\prime} }(N)}{{H}^{{\prime} }(N)}+\displaystyle \frac{H(N){H}^{{\prime} ^{\prime\prime\prime} }(N)}{{H}^{{\prime} }{\left(N\right)}^{2}}\right\}.\end{array}\end{eqnarray} \tag{ 10 }$$

We can solve the slow-roll parameters with respect to H(N), but it is not so straightforward in (8)–(10). It is possible to recover the corresponding scalar field theory by using (7). In this reconstructed model, we explore the case that has the following condition

$$\begin{eqnarray*}&&H(N)\gg {H}^{{\prime} }(N)\gg {H}^{{\prime\prime} }(N)\gg {H}^{\prime\prime\prime }(N).\end{eqnarray*}$$

In this case, we achieve

$$\begin{eqnarray}\begin{array}{rcl}\epsilon (N) & \sim & -\displaystyle \frac{{H}^{{\prime} }(N)}{H(N)}\,\,{or}\\ H(N) & \sim & {H}_{0}\,\exp \left(-{\int }^{N}d\hat{N}\epsilon (\hat{N})\right),\end{array}\end{eqnarray} \tag{ 11 }$$

where H₀ is a constant. The other slow-roll parameters have also become as follows

$$\begin{eqnarray}\begin{array}{rcl}\eta (N) & \sim & -\displaystyle \frac{3}{2}\displaystyle \frac{{H}^{{\prime} }(N)}{H(N)}\sim \displaystyle \frac{3}{2}\epsilon (N),\\ {\xi }^{2}(N) & \sim & \displaystyle \frac{3}{2}{\left(\displaystyle \frac{{H}^{{\prime} }(N)}{H(N)}\right)}^{2}\sim \displaystyle \frac{3}{2}{\epsilon }^{2}.\end{array}\end{eqnarray} \tag{ 12 }$$

Then, by using Equation (8), we can obtain

$$\begin{eqnarray}\begin{array}{rcl}\omega (\varphi ) & \sim & \displaystyle \frac{2}{{\kappa }^{2}}\,\epsilon (N)\left|{}_{N=\varphi },\right.\\ V(\varphi ) & \sim & \displaystyle \frac{3{H}_{0}^{2}}{{\kappa }^{2}}\exp \left(-2{\int }^{N}d\hat{N}\epsilon (\hat{N})\right)\left|{}_{N=\varphi }\right..\end{array}\end{eqnarray} \tag{ 13 }$$

After combining Equations (3), (11) and (12), we can find the observable parameters in terms of the slow-roll parameter as n_s − 1 ∼ − 3, and α_s ∼ − 3². So, an appropriate definition of H(N) in an inflationary model can link the reconstructed model and the scalar field theory.

To obtain the slow-roll parameters according to perfect fluid, we consider the gravitational field equations of fluid in the FLRW spacetime

$$\begin{eqnarray}&&\displaystyle \frac{3}{{\kappa }^{2}}{\left(H(N)\right)}^{2}=\rho ,\end{eqnarray} \tag{ 14 }$$

where ρ is the energy density of the perfect fluid.

$$\begin{eqnarray}&&-\displaystyle \frac{2}{{\kappa }^{2}}H(N){H}^{{\prime} }(N)=\rho +P,\end{eqnarray} \tag{ 15 }$$

P is the pressure of the perfect fluid. We assume an arbitrary function f(ρ) (in terms of the energy density of the perfect fluid), and a rather general EOS parameter as follows

$$\begin{eqnarray}&&P(N)=-\rho (N)+f(\rho ).\end{eqnarray} \tag{ 16 }$$

With this assumption, Equation (15) takes the following form

$$\begin{eqnarray}&&-\displaystyle \frac{2}{{\kappa }^{2}}H(N){H}^{{\prime} }(N)=f(\rho ).\end{eqnarray} \tag{ 17 }$$

By using the conservation law, we obtain ρ' = − 3f(ρ). If we substitute it in Equation (17), a new relation is provided

$$\begin{eqnarray}&&\displaystyle \frac{2}{{\kappa }^{2}}\left[{\left({H}^{{\prime} }(N)\right)}^{2}+H(N){H}^{{\prime\prime} }(N)\right]=3{f}^{{\prime} }(\rho )f(\rho ).\end{eqnarray} \tag{ 18 }$$

Prime operating means the derivative with respect to the energy density of the perfect fluid $\left({f}^{{\prime} }(\rho )\equiv \tfrac{{df}(\rho )}{d\rho }\right)$, whereas $H^{\prime} (N)\equiv \tfrac{{dH}(N)}{{dN}}$ and $\rho ^{\prime} (N)\equiv \tfrac{d\rho (N)}{{dN}}$. Then we can rewrite the slow-roll parameters in terms of ρ (N) and f(ρ (N)).

$$\begin{eqnarray}\begin{array}{rcl}\epsilon & = & \displaystyle \frac{3}{2}\rho (N)f(\rho ){\left[\displaystyle \frac{{f}^{{\prime} }(\rho )-2}{2\rho (N)-f(\rho )}\right]}^{2},\\ \eta & = & \displaystyle \frac{3\rho }{2\rho (N)-f(\rho )}\\ & & \times \left[\displaystyle \frac{2f(\rho )}{\rho (N)}+\displaystyle \frac{f(\rho ){f}^{{\prime} }(\rho )}{\rho (N)}-\displaystyle \frac{1}{2}{\left({f}^{{\prime} }(\rho )\right)}^{2}\right.\\ & & \left.+\displaystyle \frac{1}{3}{f}^{{\prime\prime} }(\rho )-\displaystyle \frac{1}{6}\displaystyle \frac{{f}^{{\prime} }(\rho )}{\rho (N)}\right].\end{array}\end{eqnarray} \tag{ 19 }$$

The observables for the inflationary model of perfect fluid, n_s, r, and α_s expressed in (3). These relations in terms of ρ and f(ρ), become new forms (Bamba et al. 2014d). If ρ and f(ρ) are affected by a very slow condition in the inflationary era as $\tfrac{f(\rho )}{\rho (N)}\ll 1$, the relations in (3) are reduced to a simple form as

$$\begin{eqnarray}&&{n}_{s}\sim 1-6\displaystyle \frac{f(\rho )}{\rho (N)},r\approx 24\displaystyle \frac{f(\rho )}{\rho (N)},{\alpha }_{s}\approx -9{\left(\displaystyle \frac{f(\rho )}{\rho (N)}\right)}^{2}.\end{eqnarray} \tag{ 20 }$$

First, we examine the linear form for H²

$$\begin{eqnarray}&&{H}^{2}(N)={G}_{0}N+{G}_{1},\end{eqnarray} \tag{ 21 }$$

where G₀(<0) and G₁(>0) are constants. The scale factor for the slow-roll exponential inflation is given by $a=\overline{a}\exp ({H}_{\inf }t)$ ($\overline{a}$ is a constant) in which H_inf is the Hubble parameter at the inflationary stage. In this stage, the Hubble parameter is approximately constant, so that its time dependence is weak. This weak time dependence of H is considered in Equation (21), in which the number of e-folds N, is assumed to play the role of time. During inflation, if we have $\tfrac{{G}_{1}}{{G}_{0}}\ll N$, then the time dependence of the Hubble parameter is negligible. By using the gravitational fields equations for the perfect fluid in Equations (14) and (15) with Equation (21)) and H > 0, we obtain

$$\begin{eqnarray}\begin{array}{rcl}\rho (N) & = & \displaystyle \frac{3}{{\kappa }^{2}}\left({G}_{0}N+{G}_{1}\right),\\ P(N) & = & -\displaystyle \frac{1}{{\kappa }^{2}}\left[(3N+1){G}_{0}+3{G}_{1}\right].\end{array}\end{eqnarray} \tag{ 22 }$$

After omitting N from these equations, and from Equation (21)) we obtain $f(\rho )=-\tfrac{{G}_{0}}{{\kappa }^{2}}$. Therefore, by combining Equations (16) and (22), we acquire the EOS parameter as

$$\begin{eqnarray}\begin{array}{rcl}\omega (N) & \equiv & \displaystyle \frac{P(N)}{\rho (N)}=-1+\displaystyle \frac{f(\rho )}{\rho (N)}\\ & = & -\displaystyle \frac{(3N+1){G}_{0}+3{G}_{1}}{3({G}_{0}N+{G}_{1})}.\end{array}\end{eqnarray} \tag{ 23 }$$

The slow-roll parameters, along with the slow-roll conditions, help us to determine the logical values of the constants parameters of the linear perfect fluid inflationary model. The slow-roll parameters and η in terms of the energy density and the e-folds number in this model defined as

$$\begin{eqnarray}&&\epsilon =-\displaystyle \frac{18{G}_{0}\left({G}_{0}N+{G}_{1}\right)}{{\left[6({G}_{0}N+{G}_{1})+{G}_{0}\right]}^{2}}=-\displaystyle \frac{3\rho {G}_{0}}{{\left(2\rho +{G}_{0}\right)}^{2}}.\end{eqnarray} \tag{ 24 }$$

and

$$\begin{eqnarray}&&\eta =\displaystyle \frac{-3{G}_{0}}{6({G}_{0}N+{G}_{1})+{G}_{0}}=-\displaystyle \frac{6{G}_{0}}{2\rho +{G}_{0}}.\end{eqnarray} \tag{ 25 }$$

We expect the perfect fluid inflation obtained by a reconstructed formalism from the scalar field inflationary model gives rise to realizing inflation compatible with Planck2018 data. In this regard, we must first be able to determine the limits of the fixed parameters of Equation (21), G₀ and G₁. By using the slow-roll conditions,  ≪ 1 and ∣η∣ ≪ 1, the amount of energy density ρ, the number of e-folds N and G parameters, at the end of the inflationary era is explicitly indicated in Figure 1, which are summarized in Table 1. In this case, the energy density at the end of the inflationary era is approximately equal to 1.44 GeV, which is necessary for our calculations in the reheating epoch in the next section. To check the acceptability of our perfect fluid inflationary model, we compare the model's observables with Planck2018 observational data. We have plotted r versus n_s for the linear perfect fluid inflationary model in the background of Planck2018 data and performed some numerical analysis on the tensor-to-scalar ratio and the scalar spectral index as acquired in Figure 2. The blue rods in this figure, show the available ranges of (r − n_s) for various values of G parameters in the allowed interval $\left(-145\lesssim \tfrac{{G}_{1}}{{G}_{0}}\lesssim -100\right)$. These results in the linear perfect fluid model seem to be near the results of the scalar field models with (V ∝ ϕⁿ) potential for ($n=1,2,\tfrac{4}{3}$), between (50–70) e-folds, which are illustrated in Akrami et al. (2020b). Likewise, the rods in the interval $\left(-100\lesssim \tfrac{{G}_{1}}{{G}_{0}}\lesssim -80\right)$, show some behavior similar to the power-law inflation. G parameters in this model can be related to the conventional scalar field models, through the expressions in (11) and (13). Additionally, we compare the running of scalar spectral index α_s in this linear model with Planck2018 data. In this plot, we adopted $\tfrac{{G}_{1}}{{G}_{0}}\simeq -120$, from the allowed range for the linear model in Table 1. We can see that the linear perfect fluid inflationary model is consistent with Planck2018 observation data, at least in some subspaces of the model parameter space.

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Table 1.  The Acceptable Ranges of the Constant Parameters G₀ and G₁, As Well As the Admissible Intervals of the Energy Density ρ, and the Number of e-folds N, in the Linear Perfect Fluid Model

		G0	ρ (GeV)	$\tfrac{{G}_{1}}{{G}_{0}}$	N
Linear form of perfect fluid		−0.6 ≲ G0 ≲ 0	1.44 ≲ ρ ≲ 1019	$-145\lesssim \tfrac{{G}_{1}}{{G}_{0}}\lesssim -80.7$	65 ≲ N ≲ 80

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In this case, we study the following exponential form for H²

$$\begin{eqnarray}&&{H}^{2}(N)={G}_{2}{e}^{\beta N}+{G}_{3},\end{eqnarray} \tag{ 26 }$$

where G₂ (<0), G₃(>0), and β (>0) are constants. There is also a motivation that convinces us to choose the exponential form of the perfect fluid inflationary model. The scale factor in the power-law inflation is expressed as $a=\overline{a}{t}^{\hat{p}}$, where p is a constant. The square of the Hubble parameter during inflation becomes ${H}^{2}={\left(\tfrac{\hat{p}}{t}\right)}^{2}={\hat{p}}^{2}\exp \left(-\tfrac{2N}{\hat{p}}\right)$, which is equivalent to the form in Equation (26) with ${G}_{2}={\hat{p}}^{2}$, $\beta =-\tfrac{2}{\hat{p}}$, and G₃ = 0. Therefore, this exponential form can describe the power-law inflation for the perfect fluid. By using the gravitational fields equations for the perfect fluid, Equations (14) and (15), with Equation (26) and H > 0, we obtain

$$\begin{eqnarray}\begin{array}{rcl}\rho (N) & = & \displaystyle \frac{3}{{\kappa }^{2}}\left({G}_{2}{e}^{\beta N}+{G}_{3}\right),\\ P(N) & = & -\displaystyle \frac{1}{{\kappa }^{2}}\left[(3+\beta ){G}_{2}{e}^{\beta N}+3{G}_{3}\right].\end{array}\end{eqnarray} \tag{ 27 }$$

By eliminating N from these equations, we can obtain P(N) in terms of ρ (N)

$$\begin{eqnarray}&&P(N)=-\left(1+\displaystyle \frac{\beta }{3}\right)\rho (N)+\displaystyle \frac{{G}_{3}\beta }{{\kappa }^{2}}.\end{eqnarray} \tag{ 28 }$$

By using Equation (27) and comparing Equation (28) with Equation (16), we obtain

$$\begin{eqnarray}&&f(\rho )=-\displaystyle \frac{\beta }{3}\rho (N)+\displaystyle \frac{{G}_{3}\beta }{{\kappa }^{2}}.\end{eqnarray} \tag{ 29 }$$

So the EOS parameter, in this case, is as follows

$$\begin{eqnarray}&&\omega (N)=-\displaystyle \frac{(3+\beta ){G}_{2}{e}^{\beta N}+3{G}_{3}}{3({G}_{2}{e}^{\beta N}+{G}_{3})}.\end{eqnarray} \tag{ 30 }$$

The slow-roll parameter and η, in terms of the energy density and the e-folds number in the exponential form of the perfect fluid inflationary model, defined as

$$\begin{eqnarray}\begin{array}{rcl}\epsilon & = & -\displaystyle \frac{\beta {\left(\beta +6\right)}^{2}{G}_{2}{e}^{\beta N}({G}_{2}{e}^{\beta N}+{G}_{3})}{2{\left[{G}_{2}(\beta +4){e}^{\beta N}+4{G}_{3}\right]}^{2}}\\ & = & -\displaystyle \frac{1}{2}\rho \beta \left(\rho -\displaystyle \frac{3{G}_{3}}{{\kappa }^{2}}\right){\left[\rho -\displaystyle \frac{3\beta {G}_{3}}{{\kappa }^{2}(6+\beta )}\right]}^{-2},\end{array}\end{eqnarray} \tag{ 31 }$$

and

$$\begin{eqnarray}\begin{array}{rcl}\eta & = & \displaystyle \frac{-{G}_{2}}{{G}_{2}\left(\beta +3\right){e}^{\beta N}+{G}_{3}}\left\{\beta {e}^{\beta N}\left(3+\displaystyle \frac{5}{4}\beta \right)\right.\\ & & \left.+\displaystyle \frac{{G}_{2}{\beta }^{2}{e}^{2\beta N}}{{G}_{2}{e}^{\beta N}+{G}_{3}}+\displaystyle \frac{\beta }{{G}_{2}}\left(4-\displaystyle \frac{\beta }{2}\right)({G}_{2}{e}^{\beta N}+{G}_{3})\right\}.\end{array}\end{eqnarray} \tag{ 32 }$$

As in the previous case, we perform an investigation to determine acceptable intervals for the constant parameters G₂ and G₃ in this exponential form. Of course, it must be determined what space of other parameters, ρ and N, can produce standard inflation in this perfect fluid model. These goals are achieved by applying slow-roll conditions that are evaluated in Figures 3 and 4 for β = 0.001 and β = 0.01. The results of these evaluations are summarized in Table 2. In this case, the energy density at the end of the inflationary era for β = 0.001 and β = 0.01 is approximately equal to 2.85 and 0.45 GeV, respectively, which is necessary for our calculations in the reheating epoch in the next section. For values more or less than the values we have considered for the β parameter, we could not have acceptable results for the model. To test the viability of our exponentially perfect fluid inflationary model, we compare the model's observables with Planck2018 observational data. We have plotted r versus n_s in the background of Planck2018 data and performed some numerical analysis on the tensor-to-scalar ratio and the scalar spectral index as acquired in Figure 5. There is a set of plots in acceptable ranges of G parameters that are illustrated by green rods in Figure 5(a). When we adopt $-3\lesssim \tfrac{{G}_{3}}{{G}_{2}}\lesssim 0$, the rods are placed in limits similar to the scalar field models with V(ϕ) ∝ ϕⁿ, for ($n=1,\tfrac{4}{3}$) between 50 and 70 e-folds. The same scenario happens for the exponential perfect fluid model with β = 0.01. The plots in Figure 5(b) reveal a wide range of r − n_s for $\tfrac{{G}_{3}}{{G}_{2}}\simeq -5$ which is compatible with Planck2018 data. There is also a comparison between the running of the scalar spectral index of the exponential perfect fluid model and the Planck2018 data. We have adopted values for G₂ and G₃ from the interval in Table 2. We see that these exponential perfect fluid inflationary models are consistent with Planck2018 data at least in some subspaces of the model parameter space.

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Table 2.  The Acceptable Ranges of the Constant Parameters G₂ and G₃, As Well As the Admissible Intervals of the Energy Density ρ, and the Number of e-folds N, in Exponential Form of the Perfect Fluid Model

		G3	ρ (GeV)	$\tfrac{{G}_{3}}{{G}_{2}}$	N
Exponential form of the perfect fluid with β = 0.001		0 ≲ G3 ≲ 1200	2.85 ≲ ρ ≲ 1019	$-6\lesssim \tfrac{{G}_{3}}{{G}_{2}}\lesssim 0$	50 ≲ N ≲ 58
Exponential form of the perfect fluid with β = 0.01		0 ≲ G3 ≲ 19	0.45 ≲ ρ ≲ 1019	$-10.8\lesssim \tfrac{{G}_{3}}{{G}_{2}}\lesssim 0$	44 ≲ N ≲ 55

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The reheating stage related to a single scalar field has been studied in several papers (Dai et al. 2014; Cai et al. 2015; Munoz & Kamionkowski 2015; Ueno & Yamamoto 2016), and we are interested in extending these processes to the perfect fluid inflationary models. In this subsection, we examine the reheating process in a linear perfect fluid inflationary model. We investigate the additional constraints on inflationary parameters of the perfect fluid model by following plots that come from the reheating era. The number of e-folds in the linear perfect fluid model are defined as

$$\begin{eqnarray}&&N=\displaystyle \frac{{\kappa }^{2}\rho -3{G}_{1}}{3{G}_{0}}.\end{eqnarray} \tag{ 50 }$$

Now, by using Equations (24), (25), and (3), we find

$$\begin{eqnarray}&&{n}_{s}=1+\displaystyle \frac{108{G}_{0}\left({G}_{0}N+{G}_{1}\right)}{{\left[6({{NG}}_{0}+{G}_{1})+{G}_{0}\right]}^{2}}-\displaystyle \frac{6{G}_{0}}{6({{NG}}_{0}+{G}_{1})+{G}_{0}}\end{eqnarray} \tag{ 51 }$$

and

$$\begin{eqnarray}&&r=-\displaystyle \frac{288{G}_{0}\left({{NG}}_{0}+{G}_{1}\right)}{{\left[6({{NG}}_{0}+{G}_{1})+{G}_{0}\right]}^{2}}.\end{eqnarray} \tag{ 52 }$$

After obtaining these quantities, we can show the parameter space of the linear perfect fluid model in the reheating stage. By these equations, we can write the number of e-folds during reheating N_re, and the final reheating temperature T_re, in terms of the scalar spectral index n_s. The results are shown in Figure 6. In this model, we adopt the value $\tfrac{{G}_{1}}{{G}_{0}}\simeq -135$ from the ranges that are determined in Table 1. The behavior of the reheating processes of this model for $-145\lesssim \tfrac{{G}_{1}}{{G}_{0}}\lesssim -135$ coincides with the reheating results of the scalar field model with (V(ϕ) ∝ ϕ²) (Dai et al. 2014). The left panel of Figure 6 indicates the reheating e-folds number N_re, and the middle panel shows reheating final temperature T_re, versus the scalar spectral index n_s, for various values of the EOS parameter ${\omega }_{{re}}$. In this model, more wide ranges of the reheating e-folds number belong to ${\omega }_{{re}}=\tfrac{1}{6}$ and ${\omega }_{{re}}=\tfrac{2}{3}$, which is compatible with the Planck2018 data, i.e., these ranges indicate longer lifetimes than the others. The canonical reheating scenario, for ${\omega }_{{re}}=0$, in the linear perfect fluid model has a shorter lifetime. The convergence point of all the lines in temperature plots, Figure 6(b), suggests the limit of ${N}_{{re}}\to 0$, which presents the instantaneous reheating in each case. This temperature corresponds to the maximum temperature at the end of reheating. In this model, there is also more wide ranges of temperature for ${\omega }_{{re}}=\tfrac{1}{6}$ and ${\omega }_{{re}}=\tfrac{2}{3}$. In Figure 6(c), the e-folds number during reheating N_re versus the effective EOS parameter ${\omega }_{{re}}$, shows the parameter space of reheating phase specified by the observational constraints, n_s and r. Instantaneous reheating, equivalent to ${\omega }_{{re}}=\tfrac{1}{3}$ and ${N}_{{re}}=0$, is the point in which all of the curves converge together. In this figure, the colored region specified by requiring 0.9605 < n_s < 0.9693 and r < 0.11, illustrates the allowed region for ${\omega }_{{re}}$ in the reheating process. The acceptable ranges of the number of e-folds and final temperature of the reheating stage for the linear perfect fluid model compared with Planck2018 observational data are summarized in Table 3.

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Table 3.  The Acceptable Ranges of the Number of e-folds and Final Temperature of the Reheating Stage for the Perfect Fluid Models (Linear and Exponential form), Compared with Planck2018 Observational Data

Perfect Fluid	Linear Form		Exponential Form with β = 0.01		Exponential Form with β = 0.001
	Nre	Tre (GeV)	Nre	Tre (GeV)	Nre	Tre (GeV)
$\omega =-\tfrac{1}{3}$	N < 4.12	T > 106.87	N < 29.21	T < 1014.93	N < 24.05	T < 1014.93
ω = 0	N < 7.56	T > 10 3.35	N < 58.49	T < 1014.93	N < 49.14	T < 1014.93
$\omega =\tfrac{1}{6}$	N < 89.69	T > 10−4	N < 5.15	106.23 < T < 1014.93	N < 21.65	10−3.42 < T < 1014.93
$\omega =\tfrac{2}{3}$	N < 45.02	T > 10−4	N < 2.74	T > 1014.93	N < 8.93	105.21 < T < 1014.93
ω = 1	N < 21.99	T > 103.45	N < 1.71	T > 1014.93	N < 4.81	1012.05 < T < 1014.93

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In this subsection, we investigate the reheating process in an exponential form of the perfect fluid inflationary model. We present the additional constraints on inflationary parameters of this perfect fluid model by following plots that come from the reheating era. The number of e-folds in the exponential perfect fluid model defined as

$$\begin{eqnarray}&&N=\displaystyle \frac{1}{\beta }\mathrm{ln}\left[\displaystyle \frac{{\kappa }^{2}\rho -3{G}_{3}}{3{G}_{2}}\right].\end{eqnarray} \tag{ 53 }$$

Now, by using the Equations (31), (32), and (3), we find

$$\begin{eqnarray}&&\begin{array}{l}{n}_{s}=1+\displaystyle \frac{3{G}_{2}\beta {\left(\beta +6\right)}^{2}{e}^{\beta N}({G}_{2}{e}^{\beta N}+{G}_{3})}{2{\left[{G}_{2}(\beta +4){e}^{\beta N}+4{G}_{3}\right]}^{2}}\\ +\,\displaystyle \frac{-2{G}_{2}}{{G}_{2}(\beta +3){e}^{\beta N}+{G}_{3}}\left\{\beta {e}^{\beta N}\left(3+\displaystyle \frac{5}{4}\beta \right)\right.\\ +\,\displaystyle \frac{{G}_{2}{\beta }^{2}{e}^{2\beta N}}{{G}_{2}{e}^{\beta N}+{G}_{3}}+\beta {G}_{2}\left(4-\displaystyle \frac{\beta }{2}\right)\left({G}_{2}{e}^{\beta N}+{G}_{3}\right)\left.,\right\}.\end{array}\end{eqnarray} \tag{ 54 }$$

and

$$\begin{eqnarray}&&r=-\displaystyle \frac{8\beta {G}_{2}{\left(\beta +6\right)}^{2}{e}^{\beta N}({G}_{2}{e}^{\beta N}+{G}_{3})}{{\left[{G}_{2}(\beta +4){e}^{\beta N}+4{G}_{3}\right]}^{2}}.\end{eqnarray} \tag{ 55 }$$

We attempt to illustrate the parameter space of the exponential form of the perfect fluid inflationary model for β = 0.001 and β = 0.01, in the reheating stage. The results are shown in Figures 7 and 8, by the plots of N_re and T_re, versus the scalar spectral index n_s. In these models, we use the logical values of G₂ and G₃, which are determined in Table 2. In these two cases, we adopt $\tfrac{{G}_{3}}{{G}_{2}}\simeq -6$ and $\tfrac{{G}_{3}}{{G}_{2}}\simeq -7$ for the values of (β = 0.001) and (β = 0.01), respectively. These results are close to the conclusions of the reheating processes in the scalar field inflationary model with V(ϕ) ∝ ϕ², in Dai et al. (2014). For β = 0.01, it is clear in Figure 7(a), that most of the lifetime of this model belongs to the ${\omega }_{{re}}=-\tfrac{1}{3}$, and the canonical reheating, i.e, ${\omega }_{{re}}=0$. The same conditions apply to the exponential model with β = 0.001, which is illustrated in Figure 8(a). The maximum temperature at the end of reheating (or the instantaneous reheating), in the exponential perfect fluid inflationary model, in both cases β = 0.001 and β = 0.01, correspond to the viable ranges of the Planck data in Figures 7(b) and 8(b). The wider range of acceptable final temperature, in both cases, is relevant to ${\omega }_{{re}}=-\tfrac{1}{3}$ and ${\omega }_{{re}}=0$. The parameter space of the reheating phase, which is specified by the Planck data in Figures 7(c) and 8(c), shows wider ranges with respect to the linear perfect fluid model in the reheating epoch. The acceptable ranges of the number of e-folds and final temperature of the reheating stage for the exponential perfect fluid model with β = 0.001 and β = 0.01, compared with Planck2018 observational data are summarized in Table 3.

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