Implications for primordial black holes from cosmological constraints on scalar-induced gravitational wave

Sufficiently large scalar perturbations in the early Universe can create over-dense regions that collapse into primordial black holes (PBH). This process is accompanied by the emission of scalar-induced gravitational waves (SIGW) that behave like an extra radiation component, thus contributing to the relativistic degrees of freedom ($N_{\mathrm{e}\mathrm{f}\mathrm{f}}$). We show that the cosmological constraints on $N_{\mathrm{e}\mathrm{f}\mathrm{f}}$ can be used to pose stringent limits on PBHs created from this particular scenario as well as the relevant small-scale curvature perturbation (${\mathcal{P}}_{\mathcal{R}}(k)$). We show that the combination of cosmic microwave background (CMB), baryon acoustic oscillation (BAO) and Big-Bang nucleosynthesis (BBN) datasets can exclude supermassive PBHs with peak mass $M_{\bullet }\in [5\times {10}^5,5\times {10}^{10}]{\mathrm{M}}_{\odot }$ as the major component of dark matter, while the detailed constraints depend on the shape of the PBHs mass distribution. The future CMB mission like CMB-S4 can broaden this constraint window to a much larger range $M_{\bullet }\in [8\times {10}^{-5},5\times {10}^{10}]{\mathrm{M}}_{\odot }$, covering sub-stellar masses. These limits on PBH correspond to a tightened constraint on ${\mathcal{P}}_{\mathcal{R}}$ on scales of $k\in [10,{10}^{22}]\mathrm{M}\mathrm{p}{\mathrm{c}}^{-1}$, much smaller than those probed by direct CMB and large-scale structure power spectra.