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

TL;DR

This paper investigates scalar fields with derivative and non-derivative couplings to curvature tensors in both metric and Palatini formulations, analyzing their implications for inflation and differences in resulting actions.

Contribution

It provides a comparative analysis of scalar field couplings in metric and Palatini formalisms, highlighting the equivalences and differences in their inflationary models.

Findings

Couplings to Ricci tensor yield identical results in Palatini regardless of connection constraints.

Including co-Ricci tensor distinguishes physical outcomes between unconstrained and zero torsion Palatini cases.

All actions are reduced to Einstein frame, revealing leading order differences between metric and Palatini formulations.

Abstract

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.