# Scalar-induced gravitational waves in spatially covariant gravity

## TL;DR

This paper studies scalar-induced gravitational waves within spatially covariant gravity, deriving kernel functions and analyzing how Lorentz-violating modifications affect the GW spectrum, offering a way to test such theories with future observations.

## Contribution

It extends the formulation of scalar-induced GWs in spatially covariant gravity, providing explicit kernel functions and analyzing their observational signatures.

## Key findings

- Deviations from GR in GW amplitude and spectral shape.
- Scale-dependent modifications in SIGW energy density.
- Potential to probe Lorentz-violating gravity theories with GW data.

## Abstract

We investigate scalar-induced gravitational waves (SIGWs) in the framework of spatially covariant gravity (SCG), a broad class of Lorentz-violating modified gravity theories respecting only spatial diffeomorphism invariance. Extending earlier SCG formulations, we compute the general kernel function for SIGWs on a flat Friedmann-Lema\^itre-Robertson-Walker background, focusing on polynomial-type SCG Lagrangians up to $d=3$, where $d$ denotes the total number of derivatives in each monomial. We derive explicit expressions for the kernel in the case of power-law time evolution of the coefficients, and restrict attention to the subset of SCG operators whose tensor modes propagate at the speed of light, thereby avoiding late-time divergences in the fractional energy density of SIGWs. Instead of the usual Newtonian gauge, the breaking of time reparametrization symmetry in SCG necessitates a unitary gauge analysis. We compute the energy density of SIGWs for representative parameter combinations, finding distinctive deviations from general relativity (GR), including scale-dependent modifications to both the amplitude and the spectral shape. Our results highlight the potential of stochastic GW background measurements to probe spatially covariant gravity and other Lorentz-violating extensions of GR.