Gravitational-Wave Background as a Probe of the Primordial Black-Hole Abundance

The formation of a significant number of black holes (PBHs) is realized if and only if primordial density fluctuations have a large amplitude, which means that tensor perturbations generated from these scalar perturbations as a second-order effect are also large and comparable to the observational data. We show that pulsar timing data essentially rule out PBHs with ${10}^2–{10}^4{M}_{\odot }$, which were previously considered as a candidate of intermediate-mass black holes, and that PBHs with a mass range of ${10}^{20}$ to ${10}^{26}\mathrm{g}$, which serves as a candidate of dark matter, may be probed by future space-based laser interferometers and atomic interferometers.

Figure 1

Energy density of scalar-induced GWs associated with PBH formation together with current pulsar constraint (thick solid line segment) and sensitivity of various GW detectors (convex curves). Left and right wedge-shaped curves indicate expected power spectra of GWs from two different peaked scalar fluctuations corresponding to $({\Omega }_{\mathrm{PBH}}{h}^2,M_{\mathrm{PBH}},g_{*p})=({10}^{-5},6\times {10}^2{M}_{\odot },10.75)$ (left) and (${10}^{-1},3\times {10}^{22}\mathrm{g},106.75$) (right), respectively. The red dotted (green broken) line shows an envelope curve, $(16/27)A_{\mathrm{GW}}$, corresponding to ${\Omega }_{\mathrm{PBH}}={10}^{-1}$ (${10}^{-5}$) obtained by moving $k_p$ and $\mathcal{A}$, which depend on the frequency logarithmically except for the discontinuities due to the change of the relativistic degrees of freedom at the QCD phase transition and the electron-positron pair annihilation.

Figure 2

New constraints on the mass spectrum of PBHs imposed by scalar-generated GWs. Dotted line represents the mass range to be constrained by future GW detectors.