Inflation and cosmological dynamics in $f(R)$ gravity

We reanalyze inflation and subsequent cosmological dynamics in $f(R)=R+\alpha R^n-\beta R^{2-n}$ model, which was proposed by Artimowski and Lalak to explain both inflation and dark energy. First, we reevaluate the condition for $n$ by applying the latest data from Planck satellite. We find a new condition, $1.97<n<2$, which is completely different from the previous result based on BICEP2 data. Next, we employ the viability conditions for subsequent cosmological evolution, which were derived by Amendola et al. for the present model. We find that the model satisfies all the conditions to realize radiation era, matter era, and dark energy era. We also evaluate the parameter $\beta$ from the observational data for the late-time acceleration.

Figure 1

The number of e-folds (13) in the model (7). Sufficient inflation is realized in the regime $F>1.36\times {10}^2$. In this graph we choose $n=1.98$.

Figure 2

Theoretical relations between $r,n_s$ and $n$ and observational constraint on $n$ [25]. The solid line and the dotted line correspond to $N=50$, 60, respectively. The ranges (i) and (ii) correspond to parameter values constrained by Planck [9] and BICEP2 [26], respectively. From our new result (i) we find $1.97<n<2$.

Figure 3

The Einstein frame potential in the model (7). In this graph, we chose $n=1.98$ and $\alpha ={10}^7$.

Figure 4

The Einstein frame potential in the model (14). In this graph, we chose $n=1.98,\alpha ={10}^7,\beta ={10}^{-10}$.

Figure 5

Inflationary dynamics in the phase space ($\varphi$, $\dot{\varphi }$), where $\dot{\varphi }=\frac{d\varphi }{dt}$. The initial values we can choose are $H$ and $\dot{H}$. (a) We set $H_i=1$, 5, 10 for fixed $\dot{H_i}=0$. (b) We set ${\dot{H}}_i=0$, 1, 2 for fixed $H_i=1$.