Quasinormal Modes of Black Holes Embedded in Dark-Matter Halos: Analysis of Dehnen(1, 4, 5/2) and DC14 Density Profiles

Abstract

We investigate how a surrounding dark-matter halo imprints itself on the QNM spectrum of a static, spherically symmetric black hole, considering two physically distinct density profiles: the Dehnen (1,4,5/2) and DC14 profiles. Both follow double-power-law forms, while the shape parameters of the DC14 profile are calibrated to galaxy-formation simulations and vary with the stellar-to-halo mass ratio $X=\log_{10}(M_*/M_{\rm halo})$, which controls the inner density structure of the halo. For both profiles, we compute the halo contribution to the metric function and the resulting effective potential governing axial gravitational ($s=2$) perturbations, and extract the fundamental QNM frequencies for the $\ell=2$ multipole. Our results show that the halo scale radius $r_s$ has the largest influence on the QNM spectrum among the halo parameters explored, reshaping the effective potential barrier near the light ring and shifting both the real and imaginary parts of the QNM frequencies. The density normalization, by contrast, plays a comparatively minor role for the Dehnen profile but is more consequential for DC14, reflecting the different inner structures associated with its varying $X$ values. The Prony-extracted frequencies show excellent agreement with the sixth-order WKB results, with relative differences remaining below $0.04\%$ across the parameter ranges considered. To our knowledge, this constitutes the first QNM analysis of a black hole embedded in a DC14 dark-matter halo. These results demonstrate the sensitivity of the QNM spectrum to the surrounding dark-matter environment and provide a consistent numerical framework for studying such environmental effects.