Issues in Palatini R2 inflation: Bounds on the reheating temperature

Abstract

We consider ${\mathcal{R}}^{2}$ 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, $V\ensuremath{\sim}{h}^{n}$, with $n$ a positive even integer. We also study the Higgs model nonminimally coupled to gravity. With ${\mathcal{R}}^{2}$ terms coupled to gravity as $\ensuremath{\sim}\ensuremath{\alpha}{\mathcal{R}}^{2}$, with $\ensuremath{\alpha}$ constant, the instantaneous reheating temperature ${T}_{ins}$, is bounded by ${T}_{ins}\ensuremath{\le}0.290{m}_{\text{Planck}}/{\ensuremath{\alpha}}^{1/4}$, with the upper bound being saturated for large $\ensuremath{\alpha}$. For such large $\ensuremath{\alpha}$ 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 the inflaton rolls towards the end of the inflation point, the quartic in the velocity terms, unavoidable in Palatini gravity, play a significant role and cannot be ignored. The values of $\ensuremath{\alpha}$, and other parameters, are constrained by cosmological data, setting bounds on the inflationary scale ${M}_{s}\ensuremath{\sim}1/\sqrt{\ensuremath{\alpha}}$ and the reheating temperature of the Universe.