Reheating constraints and consistency relations of the Starobinsky model and some of its generalizations

Building on the success of the Starobinsky model in describing the inflationary period of the universe, we investigate two simple generalizations of this model and their constraints imposed by the reheating epoch. The first generalization takes the form $R^{2p}$, while the second is the $\alpha$-Starobinsky model. We first focus on the case where $p=1$ or equivalently, $\alpha=1$, which corresponds to the original Starobinsky model. We derive exact consistency relations between observables and cosmological quantities, without neglecting any terms, and impose the reheating condition $0 < \omega_{re} < 0.25$, where $\omega_{re}$ is the equation of state parameter at the end of reheating. This allows us to obtain new bounds for $n_s$ and $r$ that satisfy this condition and apply them to other observables and cosmological quantities. We repeat this process for the cases where $p \neq 1$ and $\alpha \neq 1$ and find that these generalizations only result in minor modifications of the Starobinsky model, including the potential and the bounds on observables and cosmological quantities.