Intricacies of Cosmological bounce in polynomial metric $f(R)$ gravity for flat FLRW spacetime

In this paper we present the techniques for computing cosmological bounces in polynomial $f(R)$ theories, whose order is more than two, for spatially flat FLRW spacetime. In these cases the conformally connected Einstein frame shows up multiple scalar potentials predicting various possibilities of cosmological evolution in the Jordan frame where the $f(R)$ theory lives. We present a reasonable way in which one can associate the various possible potentials in the Einstein frame, for cubic $f(R)$ gravity, to the cosmological development in the Jordan frame. The issue concerning the energy conditions in $f(R)$ theories is presented. We also point out the very important relationships between the conformal transformations connecting the Jordan frame and the Einstein frame and the various instabilities of $f(R)$ theory. All the calculations are done for cubic $f(R)$ gravity but we hope the results are sufficiently general for higher order polynomial gravity.