The investigation of primordial non-Gaussianities holds immense importance in testing the inflation paradigm and shedding light on the physics of the early universe. In this study, we conduct the first complete analysis of scalar-induced gravitational waves (SIGWs) by simultaneously incorporating the local-type non-Gaussianities $f_{\mathrm{NL}}$ and $g_{\mathrm{NL}}$. To achieve this, we develop a 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 meticulously analyze all the relevant Feynman-like diagrams, systematically 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 that arise from the coupling between short-wavelength and long-wavelength modes due to primordial non-Gaussianities. These inhomogeneities give rise to anisotropies in SIGWs. Our analysis reveals that this spectrum exhibits a typical multipole dependence, characterized by $\tilde{C}_{\ell}\propto[\ell(\ell+1)]^{-1}$. This dependence plays a crucial role in distinguishing between different sources of gravitational waves. Moreover, depending on the model parameters, significant anisotropies of $\tilde{C}_{\ell}\sim10^{-3}$ can be achieved. Additionally, we demonstrate that the degeneracies in the model parameters can be broken. The findings of our study underscore the power of this spectrum as a robust probe for investigating primordial non-Gaussianities and exploring the physics of the early universe through gravitational-wave observations. Furthermore, the theoretical predictions derived from our research can be experimentally tested using space-borne gravitational-wave detectors and pulsar timing arrays.
Comment: 48 pages, 16 figures, 3 appendices