Scalar-induced gravitational waves in spatially covariant gravity

TL;DR

This work computes scalar-induced gravitational waves within spatially covariant gravity, a Lorentz-violating framework that preserves only spatial diffeomorphisms. By deriving the cubic action for one tensor and two scalar modes and constructing the kernel function on a flat FLRW background, the authors isolate the luminal, physically propagating SIGW signal while projecting out potentially divergent non-luminal contributions. They implement a power-law time dependence for SCG coefficients and enforce a luminal tensor speed, enabling explicit results for representative parameter choices and two primordial scalar spectra (monochromatic and log-normal). The findings reveal notable deviations from GR in both amplitude and spectral shape of the SIGW background, suggesting that future stochastic GW observations could probe SCG operator structures and Lorentz-violating gravity. The work lays a foundation for further exploration of parity-violating, higher-derivative, or multi-field extensions within the SIGW phenomenology.

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 $d=3$, 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.

Figures (8)

• Figure 1: Energy density of SIGWs with $\beta_{1}=-\beta_{2}=\beta_{3}=y$. The left and right panels correspond to cases of $c_{s}^{2}=1/3$ and $c_{s}^{2}=1$, respectively.
• Figure 2: Energy density of SIGWs with $\beta_{3}=y,C_{4}^{(3)}=-2C_{5}^{(3)}=-y$. The left and right panels correspond to cases of $c_{s}^{2}=1/3$ and $c_{s}^{2}=1$, respectively.
• Figure 3: Energy density of SIGWs with $\beta_{1}=-\beta_{2}=y$, $\beta_{3}=Y=1/18$, $C_{4}^{(3)}=-2C_{5}^{(3)}=y-Y$. The left and right panels correspond to cases of $c_{s}^{2}=1/3$ and $c_{s}^{2}=1$, respectively.
• Figure 4: Energy density of SIGWs with $C_{3}^{(4)}=-2C_{3}^{(5)}=y$, $\frac{1}{2}C_{3}^{(1)}=-\frac{1}{3}C_{3}^{(2)}=C_{3}^{(3)}=Y=-\frac{1}{3}y$. The left and right panels correspond to cases of $c_{s}^{2}=1/3$ and $c_{s}^{2}=1$, respectively.
• Figure 5: Energy density of SIGWs with $\beta_{1}=-\beta_{2}=\beta_{3}=y$. The left and right panels correspond to cases of $c_{s}^{2}=1/3$ and $c_{s}^{2}=1$, respectively.