Power-law Inflation in the f(R) Gravity

We investigate a form of f(R)=R1+δ/Rcδ and study the viability of the model for inflation in the Jordan and the Einstein frames. We have extended this form to f(R)=R+R1+δ/Rcδ in an attempt to solve the problems of the former model. This model is further analyzed by using the power spectrum indices of inflation and the reheating temperature. During the inflationary evolution, the model predicts a value of the δ parameter very close to one (δ = 0.98), while the reheating temperature Tre∼1016 GeV at δ = 0.98 is consistent with the standard approach to inflation and observations. We calculate the slow roll parameters for the minimally coupled scalar field within the framework of our models. It is found that the values of the scalar spectral index and tensor-to-scalar ratio are very close to the recent observational data, including those released by Planck. Further, we find the scalar spectral index and the tensor-to-scalar ratio are exactly the same in the first model because the Jordan and the Einstein frames are conformally equivalent. We also attempt to provide a constraint through the non-Gaussianity parameter.


Introduction
In study of the very early universe, inflation was introduced to solve the horizon problem, flatness problem, monopole problem, entropy problem, etc. These are among the most pronounced problems of the hot Big Bang model in research in cosmology at present. Of course, even though there exist several competing solutions for these problems of the hot Big Bang model, still we do not have a completely viable inflationary model. In the literature, there are several models such as the Starobinsky model, the Chaotic inflationary model, the Plateau type inflationary model, etc., which attempt to solve these issues. Among these models, the Starobinsky model merits to be considered as the most significant one (Starobinsky 1987).
Several inflationary models set the scalar field to have constant energy density during inflation just as the cosmological constant or the vacuum energy density has (Linde 1974). Models with the cosmological constant varying through interaction with the background in an intermediate phase sandwiched between early inflation and the present accelerated phase have also been proposed (Verma 2010). Some authors assume that the universe was supercooled as a vacuum in the very early universe. Its source was considered to be the entropy (Linde 1979). Following this, Guth proposed an inflationary model in 1981 (Guth 1987). It was also based on supercooling in the false vacuum state, in which the universe enters a reheating phase by means of bubble collision (Kirzhnits & Linde 1976). However, this approach does not work well because it is affected by the reheating problem that needs to be solved.
A viable inflationary model should be able to reheat the universe. Reheating begins after the end of the inflationary phase, and this phase is very crucial for our universe because it increases the temperature of the very cold universe. The Grand Unification Theory (GUT) energy scale lies between 10 13 and 10 16 GeV and the electroweak Spontaneous Symmetry Breaking phase transition occurs at ∼300 GeV; therefore, the reheating temperature should be 10 16 GeV (Narlikar 2017;Kolb & Turner 2019). According to particle field theories and nuclear synthesis, the temperature of the universe should be greater than 100 GeV after the reheating process ends (Narlikar 2017). Until 1982, no viable model could solve the problems of the hot Big Bang model as well as the graceful exit problem of the old inflationary model.
In 1982, Linde proposed an inflationary model (Linde 1987) known as the "new" inflationary model. This model offers a solution to the hot Big Bang model's problems, including the graceful exit problem. The basic difference between the old and new inflationary models is that the universe becomes homogeneous in the new inflationary model, whereas it was inhomogeneous in the old inflationary theory. The recent observations of the Cosmic Microwave Background power spectrum show it is uniform on the order of 10 −5 (Penzias & Wilson 1965;Boggess et al. 1992), so the new inflationary model is more successful than the old inflationary model. Linde proposed a chaotic inflationary theory also in 1983 (Linde 1983).
There is a class of theories to explain the inflation based upon the scalar field theory by , Liddle & Lyth (2000), Carroll (2017), as well as modified gravity theories (Starobinsky 1987;Lyth & Riotto 1999;Capozziello 2002;Bezrukov & Shaposhnikov 2008;Gorbunov & Rubakov 2011). Starobinsky's inflationary model became a viable model, particularly after the release of the Planck 2018 data ). Starobinsky proposed the first modified gravity model for inflation in 1980 (Starobinsky 1987), and several other authors also published valuable work on inflation in the framework of modified gravity models (Brans & Dicke 1961;Appleby & Battye 2007;Hu & Sawicki 2007;Starobinsky 2007;Linder 2009). Scalar −tensor theory is a modified gravity model, and for the first time, Brans and Dicke introduced a scalar−tensor theory by replacing the inverse of the Newtonian gravitational constant G −1 by a scalar field f (Brans & Dicke 1961).
We transform the spacetime metric g μν from the Jordan frame to the Einstein frame by the conformal transformation. In the scalar−tensor theory, the scalar field arises as a new degree of freedom after the conformal transformation of the metric tensor g μν . In cosmology, there is a serious debate about the equivalence of the Jordan and the Einstein frames. Several approaches have been used to understand and settle this problem (Faraoni & Gunzig 1999;Bezrukov & Shaposhnikov 2009;White et al. 2012;Bahamonde et al. 2017;Nandi 2019aNandi , 2019b.
In the present paper, we consider the general function of the Ricci scalar, R, given by where R c is a constant and physically it has the dimension of mass squared, such as M 2 in the Starobinsky model f (R) = R + R 2 /6M 2 (De Felice & Tsujikawa 2010). We use it to normalize the action so that the forthcoming expressions following the action become dimensionally consistent. Also, δ is a dimensionless model parameter. This f (R) model has also been used to explain the dark matter problem in Yadav & Verma (2019). The above form of the f (R) model gives the exact solution of the cosmological perturbations, but it suffers from the graceful exit problem. Therefore, we consider a correction in the model given by Equation (1) as These forms of f (R) are inspired by the model f (R) = R 1+ δ (Bohmer et al. 2008), where the parameter δ is a function of the constant tangential velocity of the test particles (200−300 km s −1 ), and its value is on the order of (v 2 /c 2 ) ∼ 10 −6 . The viability of a similar f (R) model with a small δ has also been discussed for extended galactic rotational velocity profiles in the weak field approximation (Sharma et al. 2020(Sharma et al. , 2021. Therefore, a small correction in R may explain the issues that are otherwise solved by invoking dark matter. However, as we will notice further in this paper, we require a larger value of δ compared to 10 −6 to account for inflation at the early epoch in the universe. At δ = 0, we recover the Einstein−Hilbert action as , and if we take δ = 1 then Equations (1) and (2) transform into an R 2 model. Field equations in the f (R) gravity have de Sitter points in f (R) ∝ R 2 and the field oscillates at these points.
We use the f (R) model given by Equations (1) and (2) to address the issues of inflation and reheating in a modified gravity scenario. These models have been investigated by Costa & Nastase (2014), Chakravarty & Mohanty (2015), Motohashi (2015), and Rinaldi & Zerbini (2015) in the inflationary scenario.
In this paper, we attempt to obtain the value of the δ parameter from the spectral indices in the very early universe through some recent observations. We calculate the reheating temperature and non-Gaussianity parameter. Further, we also perform a comparative study between the models given by Equations (1) and (2) and the Starobinsky model.
The inflationary observational parameters such as the scalar spectral index n s , tensor-to-scalar ratio r, amplitude of the scalar power spectrum A s , etc. are the constraints on the model parameters and the reheating temperature . These are calculated from the Cosmic Microwave Background (CMB) power spectrum of the universe, mainly through Planck observations, Wilkinson Microwave Anisotropy Probe (WMAP) observations, etc.
The present paper is organized as follows. In Section 2, we focus our attention on inflation, reheating temperature, and power spectral indices in the scalar−tensor theory. We also calculate the value of the δ parameter in the Jordan and the Einstein frames in Sections 3 and 4, respectively, using the model of Equation (1). In Section 5, we study reheating and observational parameters in the model given by Equation (2). We compare the models with the Starobinsky model in Section 6 and discuss the results in the concluding Section 7.
Throughout, we use the Greek letters μ, ν, α, β = 0, 1, 2, 3 and Latin letters i, j, k = 1, 2, 3. The sign convention used for the metric is (−, +, +, + ) and the natural unit system is c = ÿ = k B = 1, where c, ÿ, and k B are the speed of light, reduced Planck constant, and Boltzmann's constant, respectively. A dot and prime indicate differentiation with respect to cosmic time and the Ricci scalar, respectively. We use

Inflation and Reheating Temperature
In the present work, we consider the scalar−tensor theory as an effective theory and study inflation within its framework. We further assume that the dynamical universe is homogeneous, isotropic, and spatially flat, which is defined by the Friedmann−Robertson−Walker (FRW) spacetime metric, given by the spacetime interval where a(t) is the scale factor, t is the cosmic time, and r, θ, f are the spherical coordinates. We know that the strong energy condition is ρ + 3P 0, such that the universe undergoes deceleration. The equation of state (EoS) w of all of the known normal matter components of the universe is −1/3. However, an interpretation of the observational data of supernovae type Ia (SN Ia) in 1998 suggests that the present universe is in an accelerating phase (Perlmutter et al. 1997;Riess et al. 1998).
From the above discussion, we can conclude that normal matter does not produce an accelerated expansion of the universe. However, the acceleration can still be obtained by adding some exotic matter or by making a "correction" in the curvature part of the Einstein−Hilbert action. There are different ways to solve this problem in emergent gravity models, steady-state theory, string theory, etc. The aim of this "correction" is to attain the accelerated expansion of the universe with the EoS of matter w < −(1/3). Such an equation of state can be obtained by the violation of the strong energy condition, i.e., by ensuring ρ + 3P < 0. In this paper, we discuss inflation in a modified gravity model. Such a universe in the present accelerated phase can be realized by the Λ cosmological constant. Of course, the Λ model alone cannot be a cosmologically viable model for accelerated expansion of the early universe because exponential expansion will never end (Amendola & Tsujikawa 2010). Therefore, we do not need an exactly constant, but rather an approximately constant, energy density, H ; constant, and a quasi de Sitter spacetime during intervals of very early epochs.
The observational values of the inflationary parameters from the recent Planck 2018 data analysis (Amin et al. 2016;Akrami et al. 2020 In this paper, we compare the calculated values of the power spectrum indices with the observations and check the consistency of this proximity.

Reheating Temperature T re
In a viable inflationary model reheating must begin at the end of inflationary phase. This phase is quite crucial for our universe because it increases the temperature of the super cold universe, which is necessary for further evolution. Therefore, we calculate the reheating temperature.
We assume that the entropy (S = sV, where S is entropy, s is entropy density, and V is volume of the patch) does not change during adiabatic expansion after the end of the reheating until now: = s a s a o o re re 3 3 , where s re , s o , a re , and a o denote entropy density at the end of reheating, the present entropy density, the scale factor at the end of reheating, and the present scale factor, respectively. We can thus connect the reheating and present epochs via entropy. After reheating, the universe was dominated by relativistic species, and the total entropy was almost carried by these relativistic species until the present epoch. The entropy density of relativistic species can be given as (Gorbunov & Rubakov 2011) , 7 s re 2 ,re re 3 where g s,re is total relativistic degree of freedom at the end of reheating and T re is the reheating temperature. Today, the entropy density of relativistic species is where g s,o = (43/11); (Gorbunov & Rubakov 2011) refers to the total number of relativistic degrees of freedom after neutrino decoupling and T o is the CMB temperature today. Now, using Equations (7) and (8) As usual, the energy density of the universe varies as where w i is the EoS of the fluid and the subscript i denotes the corresponding species of matter or radiation. Therefore, we have a relation between ρ en and r re as where ρ en is the energy density at the end of inflation, r re is the energy density at the end of reheating, and w re is the EoS parameter during reheating.
We have a relation for the reheating number of e-foldings N re in terms of ρ en and r re by using Equation (13) ln . 14 re re en re Assuming that the total energy density of the inflation field gets completely converted into the energy density of relativistic species during reheating, the energy density of relativistic species at the end of reheating is given by where g re is the relativistic degree of freedom. Now, putting the above value of r re into Equation (14), And with the value ofe N re from Equation (16) in Equation (12), the reheating temperature is given by Further, assuming = w 0 re during the reheating phase, T re becomes Thus, the reheating temperature depends on H k , N k , and ρ en . This leads us to calculate the reheating temperature by using these values in the Jordan and the Einstein frames.

Inflationary Dynamics in the Scalar-Tensor Theory
In generalized scalar−tensor theories, an action without a matter field can be written as (Hwang & Noh 1996) where f (R, f) is a function of the Ricci scalar and scalar field, and ω is a parameter that is ≠ 1 for a noncanonical scalar field, and = 1 for a canonical scalar field. Varying the action in Equation (19) with respect to the scalar field f and metric tensor with ω = 1, we obtain the equations of motion as (Hwang & Noh 1996) ( ) ( ) where In f (R, f) theories, the slow roll parameters are defined as (Hwang & Noh 1996;Nojiri et al. 2017) We can write slow roll parameters in the generalized form as ε i , where i = 1, 2, 3, and 4, and if the slow roll parameter ε 1 is constant, we can take  e = 0 i (Nojiri et al. 2017). Cosmological perturbations in f (R) theory have been studied in Hwang & Noh (1996). Perturbations generated during inflation are strong evidence of the inflation. These perturbations freeze after leaving the Hubble radius. They are imprinted on the CMB and used as a fingerprint of the inflation. These fingerprints are known as power spectral indices (e.g., the tensor-to-scalar ratio, scalar spectral index, tensor spectral index, etc.). The power spectrum of the CMB is almost scale invariant (n s ∼ 1) during inflation.
The scalar spectral index n s and tensor spectral index n t during inflation in modified gravity theories are given, respectively, by (De Felice & Tsujikawa 2010) where ν s and ν t are, respectively, given as Now, the tensor-to-scalar ratio r is (De Felice & Tsujikawa 2010) H F F, remains invariant,=  , under the conformal transformation and so does the tensor perturbation. Therefore, both the tensor-toscalar ratio and the scalar spectral index in the Jordan frame are identical with those in the Einstein frame (De Felice & Tsujikawa 2010).

Inflationary Dynamics in the Jordan Frame Using the
Model Action in f (R) theories of gravity without a matter field during inflation is given as where f (R) is the function of the Ricci scalar R, and we obtain the field equations after varying the action (32) with respect to the metric tensor where F(R) = ∂f/∂R, ∇ μ is the covariant derivative, and , ≡ ∇ μ ∇ μ is the covariant D'Alembertian operator. The trace of Equation (33) is and the equations of motion in f (R) gravity are given as Energy density and pressure in the universe in f (R) theory are found to be  (2005), , in Equation (39) and obtain It is seen that the first slow roll parameter is constant with the vanishing derivative  e 1 and such an accelerated expansion stays forever. But we need to end the accelerated expansion, and universe should go into a deceleration phase. It is a serious problem of the model given by Equation (1). Now, we use Equation (40)  for ε 1 > 0. But if we consider δ = 1 in Equation (40), then the expansion becomes exactly a de Sitter type; if δ ≠ 1, then we obtain the power-law accelerated expansion from Equation (42) at the very early epoch.
The parameter δ has the range ( ) d > -+ 1 3 2 or (0.366 < δ < 1) from the condition of the acceleration ε 1 < 1. However, we do not consider δ > 1 to avoid the state of superinflation (De Felice & Tsujikawa 2010). In Equation (23), the second and fourth slow roll parameters become ε 2 = 0 and  e = F HF 4 , respectively, due to the fact that  f is absent in the Jordan frame, and Equation (24)  We can write Equation (36) in terms of slow roll parameters as We find the scalar spectral index n s and tensor-to-scalar ratio r in f (R) gravity theories in the Jordan frame after using Equation ( Next, we have calculated the slow roll parameters (Equation (43)) and power spectrum index by using Equation (1). These slow roll parameters are given by We can write ε 3 and ε 4 parameters in terms of ε 1 as We obtain the scalar spectral index by putting the values of ε 1 , ε 3 , and ε 4 in Equation (46) from Equation (48): After putting the value of ε 3 from Equation (48) into Equation (47), we obtain the tensor-to-scalar ratio in the Jordan frame, We have also found the relation between n s and r by using Equations (50) and (51): This relation shows the dependence on the δ parameter. These are the exact results of the power spectrum indices. We have used the relations (50) and (51) to obtain plots of δ versus n s and r. It can be seen that both n s and r depend only on the model parameter δ. We have found that both are constant, meaning that they do not change from the epoch of horizon exit until the present epoch and will stay the same forever. Figure 1 is a plot of δ versus n s , showing that the allowed values of the parameter δ are fixed by the observational upper and lower limit of n s . The range of δ is 0.8061 δ 0.8246. The value of n s increases from 0.9607 to 0.9691 between the points A and C and decreases between the points D and B.
From Figure 1, we can see that initially n s increases with increasing δ and reaches the maximum at δ = 1, implying that there is no tilt in the CMB power spectrum. Now, using the observational value of the scalar spectral index n s = 0.9691 at Figure 1. Plot of scalar spectral index n s vs. δ. The straight dashed line shows an observational upper limit of n s = 0.9691 and the straight dotted line is a lower limit of n s = 0.9607. In this paper, we do not consider the value of delta greater than one; therefore, the value of δ must lie between δ A δ δ C . As δ rises above 0.80745, the value of n s crosses its upper limit. δ = 0.98, we have the tensor-to-scalar ratio r ; 0.12 from Equation (52). This is consistent with Akrami et al. (2020), but is disfavored with respect to the combined BICEP2/Keck Array BK15 (Ade et al. 2014) results. Figure 2 is a plot of the tensor-to-scalar ratio r versus δ. We have a range of the δ parameter 0.9006 δ 1.1241 obtained from the observational value of r < 0.06, whereas for r = 0.1 it is 0.8795 δ 1.1587. If the value of delta increases, the value of r falls to zero before rising again. At r = 0 with δ = 1, primordial gravitational waves cannot be produced. As r increases further from 0 to 0.1 with δ from 1 to 1.1587, the gravitational wave component reaches its uppermost observed value r = 0.1, even though this δ would imply superinflation. Therefore, we can set the limit of δ as 0.9006 δ < 1 for inflation, but if r = 0.1 then the range of δ becomes 0.8795 δ 1.1587.
However, this is inconsistent with the range of δ allowed in Figure 1. It can be seen that r < 0.1 allows for a better consistency of r and n s with respect to their variation with δ.
Thus, it is interesting to check it further, and so we plot the tensor-to-scalar ratio r and n s in Figure 3 with two different values of the model parameter δ. Assuming δ ; 0.98 gives a constraint on the values of n s and r. Both curves intersect at the same points A(n s , r) = (0.9837, 0.065) and B(n s , r) = (0.974, 0.1) at δ = 0.98. However, while it is satisfactory for r, the value of n s is inconsistent with the observed range of n s discussed in Figure 1. Again, consistency can be made stronger by allowing r < 0.1, as shown in our model from Figure 3. (32) gives the action as

Inflationary Dynamics in the Einstein Frame Using the
where χ is an auxiliary field.
After defining j ≡ f (χ), χ Equation (53) can be written as where U(j) is the potential of the field (j) given by We can rewrite Equation (32) as Invoking a conformal transformation of the metric tensor = W , we obtain the action in the Einstein frame ( E ) as . We can rewrite Equation (58) by redefining the scalar field kf = F 3 2 ln and Ω 2 = F in the Einstein frame as˜˜( Equation (59) shows that the scalar field f is minimally coupled with the curvature. Since action (59) is the same for the canonical single scalar field, the dynamical equations therefore are equivalent in the Einstein frame for the slow roll inflation. We have evaluated the potential of the scalar field V(f) from Equation (60) by using the values of R, f (R), and F in terms of Interestingly, potentials of this type are able to produce slow roll inflation in the very early universe.
In Figure 4, slow roll inflation ends as f → −∞. Such a potential has a minimum at −∞, and therefore the scalar field (inflation field) does not oscillate about minimum. It goes on rolling toward infinity and perhaps decays into other particles via instant preheating, as described in Felder et al. (1999) and Dimopoulos et al. (2018). That said, in the standard reheating mechanism, the potential has a global minimum at f min and the inflation field oscillates about it and decays. It may decay by a direct coupling to the matter field and another scalar field. Alternatively, the reheating process may be followed by gravitational particle production (Copeland et al. 2000;Amendola & Tsujikawa 2010). However, in the present paper, we do not intend to discuss the process of particle production during reheating. Inflation never ends in the model given by Equation (1) because the first slow roll parameter is constant.
Slow roll parameters in scalar−tensor theory are defined by Equation (23)  where η is Further, η can be expressed in terms of the f (R) as (Miranda et al. 2019) The scalar spectral indexñ s in scalar−tensor theory is given by Equation (25); (Hwang & Noh 1996). Thus, using the values ofẽ 1 ,ẽ 2 ,ẽ 3 , andẽ 4 from Equation (62) In this Section, we calculateẽ 1 and η by using Equations (1), (64), and (68), respectively, as Clearly, these expressions show the dependence on the single parameter δ only, and, therefore, these are sensitive to the f (R) model parameter. Now, again we find thatẽ 1 is constant, as in the Jordan frame. These Equations (72) and (73) lead us to obtain the expressions for the scalar spectral indexñ s and the tensor-toscalar ratior using Equations (70) and (71), respectively, as Using Equations (74) and (75), we get the relation between the spectral index and the tensor-to-scalar ratio as Here, we discover that Equations (50), (51), and (52) are exactly same as Equations (74)  We notice from Table 1 that δ can be ∼0.82 and we can say that the value of the tensor-to-scalar ratio has a small deviation from the Planck 2018 result when r is < 0.1. However, if we consider the value of δ from Table 2, then the scalar spectral index becomes higher than the observational value of 0.9649.

Inflation and Reheating Temperature in
model is inflicted with a serious problem, the graceful exit from inflation. Viable inflationary models should have a reheating phase and, consequently, a radiation-dominated, matter-dominated, and late-time accelerating universe. Therefore, we write Equation (1) in another way: Such a model has been investigated by Motohashi (2015), Costa & Nastase (2014), and Chakravarty & Mohanty (2015), but we will further investigate these types of f (R) models in the light of recent (Ade et al. 2018;Akrami et al. 2020) observational data and calculate the reheating temperature. We have also obtained the non-Gaussian parameter f equi NL in the model given by Equation (2). We prefer to study this model in the Einstein frame because it is much simpler than in the Jordan frame. We obtain the potential after putting the value of the first derivative of Equation (2), , and Equation (2) into Equation (57), which is similar to that in Motohashi (2015) and is ( ) . 78 Taking the derivative of the potential in Equation ( We get the minima of the potential by using V, f (f) = 0: The above equation has three solutions: the potential has minima at kf = ¥ min , kf = 0 min , and kf = We get V, ff (f) at the minimum value of f after putting the value of ( We obtain f end by solving theẽ = 1 1 : f end ; 0.8744M p at δ = 0.98, as clearly seen from Figure 5. Now, using the equation from Liddle & Lyth (2000) and  we find the number of e-foldings:   and we obtain f i ; 5.36M p from Equation (87) for δ = 0.98 and = N 60. We can express Equations (84) and (85)   In the Einstein frame, the Hubble parameter in terms of s andr is given as˜˜˜( at the horizon exit. We have used the expression (18) to calculateT re by putting in˜= N 60 k ,H k from Equation (97), andr en from Equation (96).˜( We findT re is 1.01097 × 10 16 GeV after putting in the value of  =Ŕ M 1 2.44 10 c p 2 18 GeV and δ = 0.98 (Mishra et al. 2021). The value of R c can be obtained through the normalization of the primordial curvature perturbations (CMB temperature anisotropies) during inflation. Figure 6 shows that T re drops to 10 16 GeV and turns around at δ ; 1.2. It further begins to increase from 10 16 GeV with δ. That implies that the reheating temperature does not drop lower than 10 16 GeV for any value of the δ parameter in the model given by Equation (2). In Table 3, we find that δ = 0.98 is suitable for Equation (2), butT re is approximately the same for 0.81 < δ < 0.98. We can also take δ = 0.98 for the model given by Equation (1) Figure 7 shows the variation of f NL equi with model parameter δ. There are small differences between the three curves represented by the solid line for˜= N 60, the dashed line for = N 50, and the dotted line for˜= N 40. Each curve has different turning points at which f NL equi again starts to increase with δ. The non-Gaussianity parameter is very small, but we hope that it may be observed in upcoming projects. Thus, it can be a good consistency parameter with which to test the viability of inflationary models.
If we consider the matter distribution at the present epoch, then the effective potential of the scalar field can be expressed as . 100 where f 1 and f 2 are turning points at which We plot the curve between EoS w and the energy density of the scalar field ρ f in the Einstein frame for δ ; 0.98 and ρ m ∼ 10 −48 (GeV) 4 at cosmological scales. We have found that the value of w stays close to −1 at the energy density of DE ρ DE = ρ f ∼ 10 −48 (GeV) 4 (the present value of the relative density parameters are Ω f ; 0.68 and Ω m ; 0.05 ). Therefore δ ; 0.98 is the preferred value with respect to smaller values, to obtain the accelerated expansion at the present epoch, and it is reasonable to argue that it is this value that must be compared to similar dynamical effects at the inflationary epoch in the very early universe.

Comparison between the Models
Now we take forward our analysis of the model expressed by Equation (2) through a comparative study with the Starobinsky model . 103 2 2 If we put δ = 1 in Equation (2), then it reduces to Equation (103) where R c = 6M 2 . The scalar field potential (Equation (78)) also reduces to the potential of the Starobinsky model, and the expression becomes as in Equation (79). In our model, the potential has a minimum at f = 0 min as does the model in Equation (103) (2). Thus, we find that the model given by Equation (2) is closer to the Planck2018/BICEP2/Keck Array results when δ = 0.98. We found that the model which is given by (1) is consistent with ) (n s , r) = (0.97, 0.12) without taking BICEP2/Keck Array BK15. This indicates that the value of the model parameter δ = 0.98 is favorable for producing the effects of acceleration, and justifies the comparison between the inflation at the early epoch and the ongoing acceleration at cosmic scales.

Conclusion
We have ended this paper showing that our model in Equation (2) is closer to the Planck 2018/BICEP2/Keck Array results when δ = 0.98, and we have compared the models of Equations (2) and (103). We also examined the viability of the models expressed by Equations (1) and (2). We also have concluded that the model parameter δ is approximately equal to 1 from Tables 1, 2, and 3 and obtained the more favorable value at δ = 0.98.
In the foregoing discussion, we have seen the conformal equivalence between the Jordan and the Einstein frames through the relations of n s and r in the very early universe in the ( ) = does not have a global minimum, and therefore the scalar field keeps rolling down forever and we cannot explain reheating by invoking the standard mechanism. Indeed, this model counters the exit problem. We hope to study in our future work the mechanism to end inflation, or alternatively, a reheating mechanism without an end to inflation.
If we put δ = 1 in Equation (1), it turns into an αR 2 model. We obtained the exact scale invariant power spectrum, and it does not produce primordial gravitational wave at δ = 1. We calculated the model parameter δ during inflation and attempted to constraint the upper bound on the reheating temperature. We obtained the value of the model parameter δ = 0.98 from Tables 1, 2, and 3, which is more favorable.
We have studied the framework of ( ) models. These models provide values of the power spectrum indices that are very close to the observations for the same δ.
Clearly, inflationary slow roll parameters, scalar spectral indices, and the tensor-to-scalar ratio are sensitive to the δ parameter in the model given by Equation (1). In addition, the model given by Equation (2) depends on the number of e-foldings and δ. If δ = 1 in Equation (1), then n s =1, implying that tilt of the power spectrum is zero. We have calculated n s =0.99972 at δ = 0.98, but the observational constraints on the scalar spectral index from Planck 2018/BICEP2/Keck Array give n s = 0.9647 ± 0.0042. Thus, there is small difference between the observed and the calculated value of n s from Equation (1), which can be attributed to several factors, including statistical or systematic errors. We also see that the tensorto-scalar ratio r becomes zero if δ = 1 is put into Equation (1), which means that the amplitude of the tensor power spectrum is zero. Therefore, primordial gravitational waves are not produced during inflation if δ = 1. If we take δ = 0.98, thenñ s andr become ∼0.96398 and ∼0.011, respectively, in the model given by Equation (2). These results are closer than the Starobinsky model to recent Planck 2018/BICEP2/Keck Array observations.
Calculations show that the reheating temperature at δ = 0.98 is˜T 10 re 16 GeV in model (2). This value of the reheating temperature is required for grand unification symmetry breaking. However, it is interesting to see that the reheating temperature never drops belowT 10 re 16 GeV at = R M 1 c p 2 . It has been suggested by some authors (Rinaldi & Zerbini 2015) that the one-loop quantum correction in the f (R) ∝ R 2 model and a small deviation from it are required. Our finding of δ ; 0.98 is consistent with previous work (Motohashi 2015;Rinaldi & Zerbini 2015), showing that the f (R) model is not exactly of an R 2 type. However, the basic difference of our finding is that we are using the recent Planck 2018 data and the value of δ ; 0.98. We have also obtained the non-Gaussianity parameter | |  f 1 NL equi in view of the fact that the recent (Akrami et al. 2020) results do not completely exclude the appearance of non-Gaussianity in the CMB power spectrum.
We obtained the value of δ ; 0.98, which is large in comparison to the value of δ at the galactic scale. In fact, the effects usually attributed to dark matter were determined on the basis of galactic rotation curves, which were consistent with the very small value of δ ∼ 10 −6 . However, at the cosmological scale 100Mpc (Cole et al. 2005;Kowalski et al. 2008;Merino 2016), as per the cosmological principle of homogeneity and isotropy, no such rotations are present and we have to infer its value from the accelerated expansion, which is the dominating physical effect at such a scale. Indeed, our calculations provide the clear indication that δ = 0.98 is not ruled out by the present observations of densities ρ m ∼ 10 −48 (GeV) 4 and ρ f ∼ 10 −47 (GeV) 4 (Aghanim et al. 2020) at the cosmological scale. We examined the range of δ that is consistent with the present acceleration by plotting the relation between w and ρ f as given in Figure 8. It shows the range for which w stays close to −1 (scalar field densities ρ f ∼ 10 −48 (GeV) 4 at the cosmological scale). Thus, it turns out that the value of δ = 0.98 obtained for inflationary expansion during the very early epochs is consistent with the dynamics of the accelerating universe later (at the present time) at the cosmological scale. Clearly, it is not appropriate to compare the same value of δ = 0.98 at very early epochs of the universe with the present value of δ at galactic scales for the simple reason that we have the inflationary expansion at very early epochs, while we have no such effect (but only the dark matter effect) at the galactic scale, which gives δ = 10 −6 from the galactic rotational velocity curves.
Thus, we feel that the value of δ at the very early epochs causing the inflationary expansion must be compared only with the value of δ at the present cosmic accelerated expansion scale, because of the similarity in physical effects or the dynamical effects at these two epochs, and not with the value of δ at galactic scales, where we do not observed any dynamical effect of the expansion. Therefore, the dynamical effects of the accelerated expansion are produced in the very early universe as well as in the present universe at large scales with the same value of δ ; 0.98. This is consistent with the observations of n s and r with our paper. If we use δ ; 10 −6 , then slow roll conditions become ε 1 ? 1 and η ? 1. That means this value of δ ; 10 −6 is not compatible with the production of inflation, and observational parameters n s and r are far beyond the observational limits.
At the galactic scale, the energy density ρ m of matter dominates over the energy density ρ f of the scalar field, while in the very early universe ρ m is negligible in comparison to ρ f . It may be another reason for the small value of δ at the galactic scales and the large value of δ at the inflationary epoch.
In this paper, one of our findings is regarding δ ; 0.98. The value of the scalar field at minimum is complex and f m 2 is also complex number at δ < 1 because kf = min [ ( )] d d -3 2 ln 2 1 . If δ > 1, then both are real. We will further analyze these results and implications.
In our future work, we will obtain the reheating temperature via possible processes of reheating. Further investigations may be done to study phase transitions. We would also use these models to examine the evolution of δ, especially up to phase of the late-time cosmic acceleration of the universe and we suspect that some corrections may arise in the form of the potential. It would be also be reasonable to examine the issues related to the value of R c , its broad physical interpretation, and the dependence of δ on the background matter density distribution at the different scales in detail.
Authors are thankful to IUCAA, Pune, for extending the facilities and support under the associateship program where most of the work was done. A.K.S. also thanks Vipin Sharma, Bal Krishna Yadav, Swagat Mishra, and Varun Sahni for the useful discussions on various aspects of inflationary theories.