Imprints of core/cusp dark matter distributions on black hole signatures in galaxies

Abstract

In galactic environments, a host dark matter (DM) halo can imprint weak but coherent corrections on black hole (BH) strong-field observables. We construct an exact family of static and spherically symmetric BH spacetimes sourced by a generic core/cusp DM halo, described by an anisotropic stress-energy tensor with nonvanishing radial pressure. The resulting geometry is determined self-consistently from the Einstein equations for a broad $\{α,β,γ\}$ density profile, including the NFW, Moore, Hernquist, Jaffe, and core/cusp Dehnen models as special cases. We discuss the asymptotic structure, horizon location, curvature scale, and energy conditions of the corresponding geometries, emphasizing that the inner logarithmic slope $γ$ controls the amount of DM probed by the relativistic region. We then obtain perturbative analytic estimates for the characteristic circular geodesics, demonstrating that the leading strong-field corrections are controlled by the dimensionless compactness $q_γ$, rather than by the total halo mass alone. To leading order in $q_γ$, we derive analytic expressions for the light ring radius, angular frequency, critical impact parameter, Lyapunov exponent, innermost stable circular orbit (ISCO) radius, and ISCO frequency. For cuspy profiles, the light ring and ISCO are displaced outward, while the corresponding orbital frequencies are redshifted; for cored profiles, the light ring radius is unchanged at this order, although its frequency and capture impact parameter still carry finite environmental corrections. We also investigate the weak- and strong-deflection angles and their dependence on the halo's inner structure. We further discuss how these environmental corrections may affect ringdown physics, in particular through the perturbative imprint of the halo on quasinormal-mode redshifts and late-time wave propagation.