Scalar fields with derivative coupling to curvature in the Palatini and the metric formulation

We study models where a scalar field has derivative and non-derivative couplings to the Ricci tensor and the co-Ricci tensor with a view to inflation. We consider both the metric formulation and the Palatini formulation. In the Palatini case, the couplings to the Ricci tensor and the Ricci scalar give the same result regardless of whether the connection is unconstrained or the non-metricity or the torsion is assumed to vanish. When the co-Ricci tensor is included, the unconstrained case and the zero torsion case are physically different. We reduce all the actions to the Einstein frame with minimally coupled matter, and find the leading order differences between the metric case and the Palatini cases.