Black Holes as Frequency-Dependent Filters of Stochastic Gravitational Waves

Black holes ring when perturbed, whereas their response to a stationary gravitational-wave background is a real-frequency scattering problem, not a source of additional quasi-normal-mode lines. We make this standard distinction quantitative by treating a black hole as a frequency-, angle-, and polarization-dependent filter. For an isotropic stationary background around Schwarzschild holes, elastic scattering produces no net monopole signal, so horizon absorption is the only population-level spectral distortion. We calculate this transfer function for Schwarzschild holes in detail, identify its absorptive and phase-delay signatures, and connect the stationary response to the causal ringdown excited by a finite wave packet. We then promote the single-hole result to an angular and polarization transport kernel for a cosmological population. The resulting optical depth is negligible for realistic black-hole populations, including asteroid-mass primordial black holes comprising all dark matter. We then extend the analysis to Kerr holes, for which superradiance allows genuine amplification in selected co-rotating modes, but isotropic incidence and random spin orientations strongly dilute the diffuse signal. Observable effects are therefore more likely in rare, nearby, aligned, rapidly spinning, or transiently illuminated systems than through cumulative cosmological propagation.