Stability of non-degenerate Ricci-type Palatini theories

TL;DR

This paper analyzes the stability of Ricci-type Palatini theories with arbitrary algebraic dependence on Riemann tensor traces, identifying conditions for ghost-free theories and revealing that projective invariance alone does not ensure stability.

Contribution

It systematically examines the stability and degrees of freedom in Ricci-type Palatini theories, highlighting the conditions under which theories are healthy or contain ghosts.

Findings

Most theories with new degrees of freedom are unstable (ghosts).

Two known cases with a single new vector are stable.

Projective invariance alone does not guarantee ghost-freedom.

Abstract

We study the stability of theories where the gravitational action has arbitrary algebraic dependence on the three first traces of the Riemann tensor: the Ricci tensor, the co-Ricci tensor, and the homothetic curvature tensor. We collectively call them Ricci-type tensors. We allow arbitrary coupling to matter. We consider the case when the connection is unconstrained, and the cases when either torsion or non-metricity is assumed to vanish. We find which combinations of Ricci-type tensors lead to new degrees of freedom around Minkowski and FLRW space, and when there are ghosts. None of the theories with new degrees of freedom are healthy, except for two previously known cases where there is a single new vector. We find that projective invariance is not a sufficient condition for a theory to be ghost-free.