Constant-roll inflation in F(R) gravity

In this paper, we study constant-roll inflation in F(R) gravity. We take two different approaches, one that relates F(R) gravity to well-known scalar models of constant-roll and a second that examines the constant-roll condition in F(R) gravity directly. With regard to the first approach, by using well-known techniques, we find the F(R) gravity that realizes a given constant-roll evolution in the scalar-tensor theory. We also perform a conformal transformation in the resulting F(R) gravity and find the Einstein frame counterpart theory. As we demonstrate, the resulting scalar potential is different in comparison to the original scalar constant-roll case, and the same applies for the corresponding observational indices. Moreover, we discuss how cosmological evolutions that can realize constant-roll to constant-roll era transitions in the scalar-tensor description can be realized by vacuum F(R) gravity. With regard to the second approach, we examine the effects of the constant-roll condition on the inflationary dynamics of vacuum F(R) gravity directly. We present in detail the formalism of constant-roll F(R) gravity inflationary dynamics and we discuss the inflationary indices for this case. We use two well-known F(R) gravities in order to illustrate our findings: the R-2 model and a power-law F(R) gravity in vacuum. As we demonstrate, in both cases the parameter space is enlarged in comparison to the slow-roll counterparts of the models and, in effect, the models can also be compatible with the observational data. Finally, we briefly address the graceful exit issue.