Equivalence of inflationary models between the metric and Palatini formulation of scalar-tensor theories

TL;DR

This paper establishes a classification framework showing when inflationary models in scalar-tensor theories are equivalent in metric and Palatini formulations, using invariant quantities under conformal transformations and field redefinitions.

Contribution

It introduces a systematic method to identify and construct inflationary models with equivalent observational predictions across both geometric formalisms.

Findings

Identifies conditions for model equivalence in metric and Palatini formalisms.

Provides a classification principle based on invariant quantities.

Outlines a construction method for equivalent models.

Abstract

With a scalar field non-minimally coupled to curvature, the underlying geometry and variational principle of gravity - metric or Palatini - becomes important and makes a difference, as the field dynamics and observational predictions generally depend on this choice. In the present paper we describe a classification principle which encompasses both metric and Palatini models of inflation, employing the fact that inflationary observables can be neatly expressed in terms of certain quantities which remain invariant under conformal transformations and scalar field redefinitions. This allows us to elucidate the specific conditions when a model yields equivalent phenomenology in the metric and Palatini formalisms, and also to outline a method how to systematically construct different models in both formulations that produce the same observables.