Bayesian Implications for the Primordial Black Holes from NANOGrav's Pulsar-Timing Data Using the Scalar-Induced Gravitational Waves

Assuming that the common-spectrum process in the NANOGrav 12.5-year dataset has an origin of scalar-induced gravitational waves, we study the enhancement of primordial curvature perturbations and the mass function of primordial black holes, by performing the Bayesian parameter inference for the first time. We obtain lower limits on the spectral amplitude, i.e., $\mathcal{A}\gtrsim10^{-2}$ at 95\% confidence level, when assuming the power spectrum of primordial curvature perturbations to follow a log-normal distribution function with width $\sigma$. In the case of $\sigma\rightarrow0$, we find that the primordial black holes with $2\times10^{-4}-10^{-2}$ solar mass are allowed to compose at least a fraction $10^{-6}$ of dark matter. Such a mass range is shifted to more massive regimes for larger values of $\sigma$, e.g., to a regime of $4\times10^{-3}-0.2$ solar mass in the case of $\sigma=1$. We expect the planned gravitational-wave experiments to have their best sensitivity to $\mathcal{A}$ in the range of $10^{-4}$ to $10^{-7}$, depending on the experimental setups. With this level of sensitivity, we can search for primordial black holes throughout the entire parameter space, especially in the mass range of $10^{-16}$ to $10^{-11}$ solar masses, where they could account for all dark matter. In addition, the importance of multi-band detector networks is emphasized to accomplish our theoretical expectation.