Implications for Primordial Black Holes from Cosmological Constraints on Scalar-induced Gravitational Waves

Sufficiently large scalar perturbations in the early universe can create overdense regions that collapse into primordial black holes (PBHs). This process is accompanied by the emission of scalar-induced gravitational waves that behave like an extra radiation component, thus contributing to the relativistic degrees of freedom ( N _eff ). We show that the cosmological constraints on N _eff 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, and Big Bang nucleosynthesis data sets can exclude supermassive PBHs with peak mass M _• ∈ [5 × 10 ^5 , 5 × 10 ^10 ] M _⊙ as the major component of dark matter, while the detailed constraints depend on the shape of the PBHs’ mass distribution. A future CMB mission such as CMB-S4 can greatly broaden this constraint window to M _• ∈ [8 × 10 ^−5 , 5 × 10 ^10 ] M _⊙ , covering substellar masses. These limits on PBHs correspond to a tightened constraint on ${{ \mathcal P }}_{{ \mathcal R }}$ on scales of k ∈ [10, 10 ^22 ] Mpc ^−1 , much smaller than those probed by direct CMB and large-scale structure power spectra.