Gravitational Waves from Primordial Black Holes: Connecting Low-Frequency Scalar-Induced Signatures to High-Frequency Binary Mergers

Abstract

Formation of primordial black holes (PBHs) requires a significant enhancement of curvature perturbations. This mechanism leaves a twofold gravitational-wave (GW) signature: a \emph{low-frequency} stochastic background of scalar-induced GWs (SIGWs) and a distinct \emph{high-frequency} signal from subsequent PBH binary mergers. We leverage this shared origin to establish a consistent, \emph{model-independent} connection between these two observables for a monochromatic PBH mass function. Using PBH abundance constraints on the primordial curvature power spectrum, we evaluate the stochastic SIGW background for spherical and ellipsoidal collapse models, demonstrating that the ellipsoidal scenario yields a significantly stronger signal. Furthermore, we analyze the GW signal from PBH binary mergers and find a direct correspondence between the SIGW frequency and the innermost stable circular orbit (ISCO) frequency of the binaries. Because GW emission is nearly maximal near the ISCO, we additionally show that the peak of the full merger GW spectrum relates to the ISCO frequency via $f_{\text{peak}} = 1.79 \, f_{\text{ISCO}}$, a relation that is independent of the binary masses. Remarkably, this unified framework connects these distinct GW channels, enabling the same primordial fluctuations to be probed across widely separated frequency bands.