Probing small-scale primordial power spectra with tensor-scalar induced gravitational waves

Large-scale primordial perturbations have been well constrained by current cosmological observations, but the properties of small-scale perturbations remain elusive. This study focuses on second-order tensor-scalar induced gravitational waves (TSIGWs) generated by large-amplitude primordial scalar and tensor perturbations on small scales. We provide the analytical expressions for the kernel functions and the corresponding energy density spectra of second-order TSIGWs. By combining observations of SGWB across different scales, TSIGWs can be used to constrain small-scale primordial curvature perturbations and primordial gravitational waves. Furthermore, we discuss the feasibility of TSIGWs dominating the current PTA observations under various primordial power spectra scenarios. Our results indicate that TSIGWs generated by monochromatic primordial power spectra might be more likely to dominate the current PTA observations.