# Propagation of the gravitational waves in a cosmological background

## TL;DR

This paper explores how gravitational waves propagate in a cosmological setting within spatially covariant gravity, revealing conditions under which their speed can match light and highlighting the effects of parity violation.

## Contribution

It derives the quadratic action for gravitational waves in spatially covariant gravity, systematically introduces parity-violating terms, and identifies theories where gravitational wave speeds equal light speed.

## Key findings

- Gravitational wave polarization modes can propagate at the same speed, possibly differing from light.
- Parity-violating terms can influence the propagation speed of gravitational waves.
- A large class of theories allows gravitational waves to travel at the speed of light.

## Abstract

We investigate the propagation of the gravitational waves in a cosmological background. Based on the framework of spatially covariant gravity, we derive the general quadratic action for the gravitational waves. The spatial derivatives of the extrinsic curvature and the parity-violating terms are systematically introduced. Special attention is paid to the propagation speed of the gravitational waves. We find that it is possible to make the two polarization modes propagate in the same speed, which may differ from that of the light, in the presence of parity-violating terms in the action. In particular, we identify a large class of spatially covariant gravity theories with parity violation, in which both the polarization modes propagate in the speed of light. Our results imply that there are more possibilities in the framework of spatially covariant gravity in light of the propagation speed of the gravitational waves.