Higgs inflation in the Palatini formulation with kinetic terms for the metric

We consider scalar field inflation in the Palatini formulation of general relativity. The covariant derivative of the metric is then non-zero. From the effective theory point of view it should couple to other fields. We write down the most general couplings between it and a scalar field that are quadratic in derivatives. We consider both the case when the torsion is determined by the field equations and the case when it is assumed to be zero a priori. We find the metric derivative terms can significantly modify inflationary predictions. We specialise to Higgs inflation and terms of only up to dimension 4. Transforming to the Einstein frame, we show that by tuning the coefficients of the new terms, we can generate various effective inflationary potentials, including quadratic, hilltop-type, α-attractor and inflection point. Some of these can give inflation in agreement with observations, including with a large tensor-to-scalar ratio, even if the non-minimal coupling is zero.