Sub-Planckian $\phi^{2}$ Inflation in the Palatini Formulation of
  Gravity with an $R^2$ term
  In the context of the Palatini formalism of gravity with an $R^{2}$ term, a
$\phi^{2}$ potential can be consistent with the observed bound on $r$ whilst
retaining the successful prediction for $n_{s}$. Here we show that the Palatini
$\phi^{2} R^2$ inflation model can also solve the super-Planckian inflaton
problem of $\phi^{2}$ chaotic inflation, and that the model can be consistent
with Planck scale-suppressed potential corrections. If $\alpha \gtrsim
10^{12}$, where $\alpha$ is the coefficient of the $R^2$ term, the inflaton in
the Einstein frame, $\sigma$, remains sub-Planckian throughout inflation. In
addition, if $\alpha \gtrsim 10^{20}$ then the predictions of the model are
unaffected by Planck-suppressed potential corrections in the case where there
is a broken shift symmetry, and if $\alpha \gtrsim 10^{32}$ then the
predictions are unaffected by Planck-suppressed potential corrections in
general. The value of $r$ is generally small, with $r \lesssim 10^{-5}$ for
$\alpha \gtrsim 10^{12}$. We calculate the maximum possible reheating
temperature, $T_{R\;max}$, corresponding to instantaneous reheating. For
$\alpha \approx 10^{32}$, $T_{R\; max}$ is approximately $10^{10}$ GeV, with
larger values of $T_{R\;max}$ for smaller $\alpha$. For the case of
instantaneous reheating, we show that $n_{s}$ is in agreement with the 2018
Planck results to within 1-$\sigma$, with the exception of the $\alpha \approx
10^{32}$ case, which is close to the 2-$\sigma$ lower bound. Following
inflation, the inflaton condensate is likely to rapidly fragment and form
oscillons. Reheating via inflaton decays to right-handed neutrinos can easily
result in instantaneous reheating. We determine the scale of unitarity
violation and show that, in general, unitarity is conserved during inflation.
