New constraints on primordial non-Gaussianity from missing two-loop contributions of scalar induced gravitational waves

We analyze the energy density spectrum of \acp{SIGW} using the NANOGrav 15-year data set, thereby constraining the primordial non-Gaussian parameter $f_{\mathrm{N}\mathrm{L}}$. For the first time, we calculate the seventeen missing two-loop diagrams proportional to $f_{\mathrm{N}\mathrm{L}}{A}_{\zeta }^3$ that correspond to the two-point correlation function $⟨h_{\mathbf{k}}^{\lambda ,(3)}{h}_{{\mathbf{k}}^{\mathrm{\prime }}}^{{\lambda }^{\mathrm{\prime }},(2)}⟩$ for local-type primordial non-Gaussianity. The total energy density spectrum of \acp{SIGW} can be significantly suppressed by these two-loop diagrams. If \acp{SIGW} dominate the \acp{SGWB} observed in \ac{PTA} experiments, the parameter interval $f_{\mathrm{N}\mathrm{L}}\in [-5,-1]$ is notably excluded based on NANOGrav 15-year data set. After taking into account abundance of \acp{PBH} and the convergence of the cosmological perturbation expansion, we find that the only possible parameter range for $f_{\mathrm{N}\mathrm{L}}$ might be $-1\le f_{\mathrm{N}\mathrm{L}}<0$.