Scalar-induced gravitational waves in spatially covariant gravity with two tensorial degrees of freedom

We study scalar-induced gravitational waves (SIGWs) in a spatially covariant gravity model with two tensorial degrees of freedom, minimally coupled to a $k$-essence scalar field. For the parameter choice considered during radiation domination, the quadratic tensor action coincides with that of general relativity (GR). We derive the scalar transfer functions at finite $σ\equiv C_2\bar N$ and in the large-$|σ|$ limit. We then construct the source function of SIGWs. We further examine a cubic extrinsic-curvature interaction that modifies the source function of SIGWs without affecting the background or quadratic actions. We calculate the fractional energy density of SIGWs for monochromatic and log-normal primordial curvature power spectra. We find that, for the positive values of $σ$ studied, finite-$σ$ effects suppress the spectrum over a broad range of scales relative to the result in the $σ\rightarrow \infty$ limit. In this limit, the cubic interaction suppresses the spectrum at small wavenumbers and enhances it at larger wavenumbers. These results show that modified scalar dynamics and non-linear interactions can alter the fractional energy density of SIGWs even when the quadratic tensor action is identical to that of GR.