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Building on the success of the Starobinsky model in describing theinflationary period of the universe, we investigate two simple generalizationsof this model and their constraints imposed by the reheating epoch. The firstgeneralization takes the form $R^{2p}$, while the second is the$\alpha$-Starobinsky model. We first focus on the case where $p=1$ orequivalently, $\alpha=1$, which corresponds to the original Starobinsky model.We derive exact consistency relations between observables and cosmologicalquantities, without neglecting any terms, and impose the reheating condition $0< \omega_{re} < 0.25$, where $\omega_{re}$ is the equation of state parameterat 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 andcosmological quantities. We repeat this process for the cases where $p \neq 1$and $\alpha \neq 1$ and find that these generalizations only result in minormodifications of the Starobinsky model, including the potential and the boundson observables and cosmological quantities.

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