# Theory of Cosmological Perturbations with Cuscuton

## TL;DR

This paper derives the quadratic action for curvature perturbations in cuscuton gravity, demonstrating its ghost-free nature and analyzing stability, with implications for cosmological models involving scalar fields.

## Contribution

It provides the first derivation of the quadratic action for curvature perturbations in cuscuton gravity and analyzes stability conditions.

## Key findings

- Curvature perturbation $ig$ is conserved on super horizon scales.
- Cuscuton modifications are ghost free around FLRW.
- Most models do not exhibit instabilities, but some may have gradient instabilities depending on parameters.

## Abstract

This paper presents the first derivation of the quadratic action for curvature perturbations, $\zeta$, within the framework of cuscuton gravity. We study the scalar cosmological perturbations sourced by a canonical single scalar field in the presence of cuscuton field. We identify $\zeta$ as comoving curvature with respect to the source field and we show that it retains its conservation characteristic on super horizon scales. The result provides an explicit proof that cuscuton modification of gravity around Friedmann-Lemaitre-Robertson-Walker (FLRW) metric is ghost free. We also investigate the potential development of other instabilities in cuscuton models. We find that in a large class of these models, there is no generic instability problem. However, depending on the details of slow-roll parameters, specific models may display gradient instabilities.