We analyze the energy density spectrum of \acp{ SIGW} using the NANOGrav 15-year data set, thereby constraining the primordial non-Gaussian parameter $f_{ \mathrm{ NL}}$. For the first time, we calculate the seventeen missing two-loop diagrams proportional to $f_{ \mathrm{ NL}}A_{ \zeta}^3$ that correspond to the two-point correlation function $\langle h^{ \lambda,(3)}_{ \mathbf{ k}} h^{ \lambda',(2)}_{ \mathbf{ k}'} \rangle$ 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 $f_{ \mathrm{ NL}}\in [-5,-1]$ 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 $f_{ \mathrm{ NL}}$ might be $-1\le f_{ \mathrm{ NL}}< 0$.
Categories: astro-ph.CO gr-qc