The Ringdown and the Tide: Fingerprints of Dark Matter Halo Profiles
Astrophysical black holes (BH) are not isolated, but embedded in matter supplied by their host galaxies. We study how the shape of a surrounding dark matter (DM) halo modifies the ringdown and tidal response of an asymptotically flat, static, spherically symmetric BH. The halo is modelled as an anisotropic Einstein cluster with vanishing radial pressure and a generalized $(α,β,γ)$ density profile, supplemented by an inner cut-off near the BH and, where required, an outer tidal truncation. We derive the axial gravitational perturbation equation and compute the quasinormal mode (QNM) spectrum using sixth-order Wentzel-Kramers-Brillouin (WKB) methods and time-domain evolutions. The halo redshifts both the oscillation frequency and the damping rate, by an amount set not only by the halo compactness but also by the profile parameters: the inner slope $γ$ dominates for centrally concentrated halos, while $α$ and $β$ give subleading but profile-dependent corrections. In the eikonal limit, the shift is governed by a single redshift integral encoding the halo mass distribution outside the light ring, explaining the close correspondence between the QNM frequencies, the light ring frequency, and the Lyapunov exponent. We also show that different combinations of compactness and profile shape can yield nearly degenerate ringdown spectra. Time-domain evolutions confirm the WKB frequencies and display the expected late-time Price law decay, with an intermediate tail controlled by the outer density falloff for slowly decaying profiles. Finally, we compute the static axial tidal Love number and show that it probes the halo with a radial weighting different from the ringdown sector. The combined ringdown and tidal response therefore provides a possible way to distinguish environmental effects from genuine deviations of the vacuum BH geometry.