  Investigation of primordial non-Gaussianity holds immense importance in
testing the inflation paradigm and shedding light on the physics of the early
Universe. In this study, we conduct the complete analysis of scalar-induced
gravitational waves (SIGWs) by incorporating the local-type non-Gaussianity
$f_{\mathrm{NL}}$ and $g_{\mathrm{NL}}$. We develop Feynman-like diagrammatic
technique and derive semi-analytic formulas for both the energy-density
fraction spectrum and the angular power spectrum. For the energy-density
fraction spectrum, we analyze all the relevant Feynman-like diagrams,
determining their contributions to the spectrum in an order-by-order fashion.
As for the angular power spectrum, our focus lies on the initial
inhomogeneities, giving rise to anisotropies in SIGWs, that arise from the
coupling between short- and long-wavelength modes due to primordial
non-Gaussianity. Our analysis reveals that this spectrum exhibits a typical
multipole dependence, characterized by
$\tilde{C}_{\ell}\propto[\ell(\ell+1)]^{-1}$, which plays a crucial role in
distinguishing between different sources of gravitational waves. Depending on
model parameters, significant anisotropies can be achieved. We also show that
the degeneracies in model parameters can be broken. The findings of our study
underscore the angular power spectrum as a robust probe for investigating
primordial non-Gaussianity and the physics of the early Universe. Moreover, our
theoretical predictions can be tested using space-borne gravitational-wave
detectors and pulsar timing arrays.
