Exploring the inflation of $F(R)$ gravity

Abstract: The early time expansion of the space-time, namely inflation, is introduced to solve some cosmological problems. $F(R)$ gravity is a simple extension of the general relativity to induce the inflationary expansion. The precise observation of the Cosmic Microwave Background gives us the information to inspect the model of the inflation. By the Weyl transformation, $F(R)$ gravity can be transformed to the Einstein-Hilbert term with a scalar field that plays the role of the inflaton. In this paper we obtain general formulae to derive the inflationary parameters including the models with a non-canonical kinetic term. The inflationary parameters are described as functions of the inflation potential and its derivatives. We apply the procedure to $F(R)=R+\gamma R^n$ and $R(1+\gamma R^n)^{1/n}$ models and show the allowed parameters space.