Isotropy, anisotropies and non-Gaussianity in the scalar-induced gravitational-wave background: diagrammatic approach for primordial non-Gaussianity up to arbitrary order

Produced nonlinearly by the enhanced linear cosmological curvature perturbations, the scalar-induced gravitational waves (SIGWs) can serve as a potentially powerful probe of primordial non-Gaussianity (PNG) in the early Universe. In this work, we comprehensively investigate the imprints of local-type PNG on the SIGW background beyond the widely used quadratic and cubic approximations. We extend the diagrammatic approach to simplify the calculation of the SIGW energy density spectrum with high-order PNG, thereby facilitating systematic analysis for PNG up to arbitrary order. Following this approach, we derive semi-analytic formulas for the energy-density fraction spectrum, the angular power spectrum, and the angular bispectrum and trispectrum to describe the isotropic component, anisotropies, and non-Gaussianity of the SIGW background, respectively. Particularly, focusing on PNG up to quartic approximation (parameterized by $f_\mathrm{NL}$, $g_\mathrm{NL}$, and $h_\mathrm{NL}$), we numerically compute all contributions to these SIGW spectra. We find that PNG can significantly alter the magnitude of the SIGW energy-density spectrum, and can generate substantial anisotropies through the initial inhomogeneities in the SIGW distribution. Furthermore, we observe that the SIGW angular bispectrum and trispectrum always vanish when the primordial curvature perturbations are Gaussian; otherwise, they do not, indicating their potential utility as probes of PNG. Therefore, we anticipate that the SIGW background will provide essential information about the early Universe.