Higgs inflation with loop corrections in the Palatini formulation
  We compare Higgs inflation in the metric and Palatini formulations of general
relativity, with loop corrections are treated in a simple approximation. We
consider Higgs inflation on the plateau, at a critical point, at a hilltop and
in a false vacuum. In the last case there are only minor differences. Otherwise
we find that in the Palatini formulation the tensor-to-scalar ratio is
consistently suppressed, spanning the range $1\times10^{-13}<r<7\times10^{-5}$,
compared to the metric case result $2\times10^{-5}<r<0.2$. Even when the values
of $n_s$ and $r$ overlap, the running and running of the running are different
in the two formulations. Therefore, if Higgs is the inflaton, inflationary
observables can be used to distinguish between different gravitational degrees
of freedom, in this case to determine whether the connection is an independent
variable. Non-detection of $r$ in foreseeable future observations would not
rule out Higgs inflation, only its metric variant. We conclude that in order to
fix the theory of Higgs inflation, not only the particle physics UV completion
but also the gravitational degrees of freedom have to be explicated.
