Issues in Palatini ${\cal{R}}^2$ inflation: Bounds on the Reheating Temperature

We consider ${\cal{R}}^2$-inflation in Palatini gravity, in the presence of scalar fields coupled to gravity. These theories, in the Einstein frame, and for one scalar field $h$, share common features with $K$ - inflation models. We apply this formalism for the study of single-field inflationary models, whose potentials are monomials, $ V \sim h^{n} $, with $ n $ a positive even integer. We also study the Higgs model non-minimally coupled to gravity. With ${\cal{R}}^2$-terms coupled to gravity as $\sim \alpha {\cal{R}}^2 $, with $\alpha$ constant, the instantaneous reheating temperature $T_{ins}$, is bounded by $ T_{ins} \leq { 0.290 \, m_{Planck}} / {\, \alpha^{1/4}} $, with the upper bound being saturated for large $\alpha$. For such large $\alpha$ need go beyond slow-roll to calculate reliably the cosmological parameters, among these the end of inflation through which $T_{ins}$ is determined. In fact, as inflaton rolls towards the end of inflation point, the quartic in the velocity terms, unavoidable in Palatini gravity, play a significant role and can not be ignored. The values of $\alpha$, and other parameters, are constrained by cosmological data, setting bounds on the inflationary scale $M_{s} \sim 1/\sqrt{\alpha}$ and the reheating temperature of the Universe.