On the consistency of universally non-minimally coupled $f(R,T,R_{\mu\nu}T^{\mu\nu})$ theories

TL;DR

This paper examines the theoretical consistency of a broad class of modified gravity theories involving non-minimal couplings between curvature and matter, revealing stability issues and limitations in their formulations.

Contribution

It provides a general framework analyzing the higher-order equations of motion in these theories and demonstrates stability constraints through explicit examples.

Findings

Higher-order equations of motion are common in these theories.

Non-minimal couplings often lead to instabilities.

Scalar field models can be stable, but vector field models are inherently unstable.

Abstract

We discuss the consistency of a recently proposed class of theories described by an arbitrary function of the Ricci scalar, the trace of the energy-momentum tensor and the contraction of the Ricci tensor with the energy-momentum tensor. We briefly discuss the limitations of including the energy-momentum tensor in the action, as it is a non fundamental quantity, but a quantity that should be derived from the action. The fact that theories containing non-linear contractions of the Ricci tensor usually leads to the presence of pathologies associated with higher-order equations of motion will be shown to constrain the stability of this class of theories. We provide a general framework and show that the conformal mode for these theories generally has higher-order equations of motion and that non-minimal couplings to the matter fields usually lead to higher-order equations of motion. In order to illustrate such limitations we explicitly study the cases of a canonical scalar field, a K-essence field and a massive vector field. Whereas for the scalar field cases it is possible to find healthy theories, for the vector field case the presence of instabilities is unavoidable.