We present * two scale invariant models of inflation in which the addition of quadratic in curvature terms in the usual Einstein-Hilbert action, in the context of Palatini formulation of gravity, manages to reduce the value of the tensor-to-scalar ratio. In both models the Planck scale is dynamically generated via the vacuum expectation value of the scalar fields.

We study the effect of primordial black holes on the classical rate of nucleation of AdS regions within the standard electroweak vacuum at high temperature. We base our analysis on the assumption that, at temperatures much higher than the Hawking temperature, the main effect of the black hole is to distort the Higgs configuration dominating the transition to the new vacuum. We estimate the barrier for the transition by the ADM mass of this configuration, computed through the temperature-corrected Higgs potential. We find that the exponential suppression of the nucleation rate can be reduced significantly, or even eliminated completely, in the black-hole background if the Standard Model Higgs is coupled to gravity through the renormalizable term ξ Rh 2 .

We consider R 2 inflation in the Palatini gravity assuming the existence of scalar fields, coupled to gravity in the most general manner. These theories, in the Einstein frame, and for one scalar field h, share common features with K -inflation models. We apply this formalism for the study of popular inflationary models, whose potentials are monomials, V ∼ h n , with n a positive even integer. We also study the Higgs model non-minimally coupled to gravity. Although these have been recently studied, in the framework of the Palatini approach, we show that the scalar power spectrum severely constrains these models. Although we do not propose a particular reheating mechanism, we show that the quadratic ∼ h 2 and the Higgs model can survive these constraints with a maximum reheating temperature as large as ∼ 10 15 GeV , when reheating is instantaneous. However, this can be only attained at the cost of a delicate fine-tuning of couplings. Deviations from this fine-tuned values can still yield predictions compatible with the cosmological data, for couplings that lie in very tight range, giving lower reheating temperatures.