We consider $R^2$ inflation in the Palatini gravity assuming the existence of scalar fields coupled to gravity in the most general manner. 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 popular inflationary models, whose potentials are monomials, $V\sim h^n$, with $n$ a positive even integer. We also study the Higgs model nonminimally coupled to gravity. Although these have been recently studied, in the framework of the Palatini approach, we show that the scalar power spectrum severely constrains these models. Although we do not propose a particular reheating mechanism, we show that the quadratic $\sim h^2$ and the Higgs model can survive these constraints with a maximum reheating temperature as large as $\sim {10}^{15}\mathrm{GeV}$ when reheating is instantaneous. However, this can be attained only at the cost of a delicate fine-tuning of couplings. Deviations from these fine-tuned values can still yield predictions compatible with the cosmological data for couplings that lie in very tight range, giving lower reheating temperatures.

• Received 5 December 2019
• Accepted 11 March 2020

DOI:https://doi.org/10.1103/PhysRevD.101.084007