Scalar-induced gravitational waves in spatially covariant gravity

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–Lemaître–Robertson–Walker background, focusing on polynomial-type SCG Lagrangians up to $$d=3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> , 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.