# Varying the Horndeski Lagrangian within the Palatini approach

## TL;DR

This paper investigates the effects of varying the Horndeski Lagrangian within the Palatini approach, revealing conditions under which the resulting metric-affine theories are ghost-free and classifying their cosmological solutions.

## Contribution

It provides a detailed analysis of metric-affine Horndeski theories, identifying ghost-free subclasses and exploring their cosmological implications.

## Key findings

- Ghost-free subclass identified for linear-in-connection theories.
- Classified homogeneous and isotropic cosmologies within these theories.
- Non-linear connection terms generally introduce ghosts, but some can be mitigated.

## Abstract

We analyse what happens when the Horndeski Lagrangian is varied within the Palatini approach by considering the metric and connection as independent variables. Assuming the connection to be torsionless, there can be infinitely many metric-affine versions $L_{\rm P}$ of the original Lagrangian which differ from each other by terms proportional to the non-metricity tensor. After integrating out the connection, each $L_{\rm P}$ defines a metric theory, which can either belong to the original Horndeski family, or it can be of a more general DHOST type, or it shows the Ostrogradsky ghost. We analyse in detail the subclass of the theory for which the equations are linear in the connection and find that its metric-affine version is ghost-free. We present a detailed classifications of homogeneous and isotropic cosmologies in these theories. Taking into consideration other pieces of the Horndeski Lagrangian which are non-linear in the connection leads to more complex metric-affine theories which generically show the ghost. In some special cases the ghost can be removed by carefully adjusting the non-metricity contribution, but it is unclear if this is always possible. Therefore, the metric-affine generalisations of the Horndeski theory can be ghost-free, but not all of them are ghost-free, neither are they the only metric-affine theories for a gravity-coupled scalar field which can be ghost-free.