Minimal Higgs inflation with an $R^2$ term in Palatini gravity
  It has recently been suggested that the Standard Model Higgs boson could act
as the inflaton while minimally coupled to gravity - given that the gravity
sector is extended with an $\alpha R^2$ term and the underlying theory of
gravity is of Palatini, rather than metric, type. In this paper, we revisit the
idea and correct some shortcomings in earlier studies. We find that in this
setup the Higgs can indeed act as the inflaton and that the tree-level
predictions of the model for the spectral index and the tensor-to-scalar ratio
are $n_s\simeq 0.941$, $r\simeq 0.3/(1+10^{-8}\alpha)$, respectively, for a
typical number of e-folds, $N=50$, between horizon exit of the pivot scale
$k=0.05\, {\rm Mpc}^{-1}$ and the end of inflation. Even though the
tensor-to-scalar ratio is suppressed compared to the usual minimally coupled
case and can be made compatible with data for large enough $\alpha$, the result
for $n_s$ is in severe tension with the Planck results. We briefly discuss
extensions of the model.
