Study of primordial non-Gaussianity $f_{\mathrm{N}\mathrm{L}}$ and $g_{\mathrm{N}\mathrm{L}}$ with the cross-correlations between the scalar-induced gravitational waves and the cosmic microwave background

The stochastic gravitational-wave background originating from cosmic sources contains vital information about the early universe. In this work, we comprehensively study the cross-correlations between the energy-density anisotropies in scalar-induced gravitational waves (SIGWs) and the temperature anisotropies and polarization in the cosmic microwave background (CMB). In our analysis of the angular power spectra for these cross-correlations, we consider all contributions of the local-type primordial non-Gaussianity $f_{\mathrm{N}\mathrm{L}}$ and $g_{\mathrm{N}\mathrm{L}}$ that can lead to large anisotropies. We show that the angular power spectra are highly sensitive to primordial non-Gaussianity. Furthermore, we project the sensitivity of future gravitational-wave detectors to detect such signals and, consequently, measure the primordial non-Gaussianity.