We study scalar field inflation in F (R) gravity in the Palatini formulation of general relativity. Unlike in the metric formulation, in the Palatini formulation F (R) gravity does not introduce new degrees of freedom. However, it changes the relations between existing degrees of freedom, including the inflaton and spacetime curvature. Considering the case F (R) = R + αR 2 , we find that the R 2 term decreases the height of the effective inflaton potential. By adjusting the value of α, this mechanism can be used to suppress the tensorto-scalar ratio r without limit in any scalar field model of inflation without affecting the spectrum of scalar perturbations.

We study inflation with the non-minimally coupled Standard Model Higgs in the case when quantum corrections generate a hilltop in the potential. We consider both the metric and the Palatini formulation of general relativity. We investigate hilltop inflation in different parts of the Higgs potential and calculate predictions for CMB observables. We run the renormalization group equations up from the electroweak scale and down from the hilltop, adding a jump in-between to account for unknown corrections in the intermediate regime.Within our approximation, no viable hilltop inflation is possible for small field values, where the non-minimal coupling has no role, nor for intermediate field values. For large field values, hilltop inflation works. We find the spectral index to be n s ≤ 0.96 in both the metric and the Palatini formulation, the upper bound coinciding with the tree-level result. The tensorto-scalar ratio is r ≤ 1.2 × 10 −3 in the metric case and r ≤ 2.2 × 10 −9 in the Palatini case. Successful inflation is possible even when the renormalization group running is continuous with no jumps. In the metric formulation, r is smaller than in Higgs inflation on the treelevel plateau or at the critical point, making it possible to distinguish hilltop inflation from these scenarios with next-generation CMB experiments.

We investigate the dependency of Higgs inflation on the non-renormalisable matching between the low energy Standard Model limit and the inflationary regime at high energies. We show that for the top mass range m t 171.8 GeV the scenario robustly predicts the spectral index n s 0.97 and the tensor-to-scalar ratio r 0.003. The matching is however non-trivial, even the best-fit values m h = 125.09 GeV and m t = 173.21 GeV require a jump δλ ∼ 0.01 in the Higgs coupling below the inflationary scale. For m t 171.8 GeV, the matching may generate a feature in the inflationary potential. In this case the predicted values of n s and r vary but the model is still falsifiable. For example, a detection of negative running of spectral index at level α s −0.01 would rule out Higgs inflation.