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 compare Higgs inflation in the metric and Palatini formulations of general relativity, with loop corrections 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 1×10 −13 < r < 7×10 −5 , compared to the metric case result 2×10 −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. arXiv:1709.07853v2 [astro-ph.CO]

We consider Higgs inflation with an αR 2 term. This term adds a new scalar degree of freedom, which leads to a two-field model of inflation. We do a complete slow-roll analysis of the three-dimensional parameter space of the R 2 coefficient α, the non-minimal coupling ξ and the Higgs self-coupling λ. We find three classes of inflationary solutions. We also find that pure Higgs inflation is impossible when the