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We analyze the energy density spectrum of scalar induced gravitational waves (SIGWs) using the NANOGrav 15-year data set, thereby constraining the primordial non-Gaussian parameter f₍₋. For the first time, we calculate the seventeen missing two-loop diagrams proportional to f₍₋A_^3 that correspond to the two-point correlation function ⟨h₊^, (3) h₊^{'}^{^', (2) }⟩ for local-type primordial non-Gaussianity. The total energy density spectrum of SIGWs can be significantly suppressed by these two-loop diagrams. If SIGWs dominate the stochastic gravitational wave backgrounds observed in pulsar timing array experiments, the parameter interval f₍₋-5, -1 is notably excluded based on NANOGrav 15-year data set. After taking into account abundance of primordial black holes and the convergence of the cosmological perturbation expansion, we find that the only possible parameter range for f₍₋ might be -1f₍₋<0.
Chang et al. (Fri,) studied this question.