Analysis of $R^p$ inflationary model as $p\geqslant 2$

We study the $R^p$ inflationary model of [Muller:1989rp] for $p>2$ using the result of Ref. [Motohashi:2014tra]. After reproducing the observable quantities: the power spectral index $n_s$, its corresponding running $\alpha=\frac{dn_s}{d\ln(k)}$ and the tensor to scalar ration $r$ in terms of e-folding number $N$ and $p$, we show that $R^p$ inflation model is still alive as $p$ is from $2$ to $2.02$. In this range, our calculation confirms that $n_s$ and $r$ agree with observations and $\alpha$ is of order $10^{-4}$ which needs more precise observational constraints. We find that, as the value of $p$ increases, all $n_s$, $r$ and $|\alpha|$ decrease. However, the precise interdependence between these observables is such that this class of models can in principle be tested by the next generation of dedicated satellite CMB probes.