Reheating in $R^2$ Palatini inflationary models
  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 non-minimally 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} \,
GeV$, when reheating is instantaneous. However, this can be only attained at
the cost of a delicate fine-tuning of couplings. Deviations from this
fine-tuned values can still yield predictions compatible with the cosmological
data, for couplings that lie in very tight range, giving lower reheating
temperatures.
