Perturbative Effects of Dark Matter Environments on Black Hole Shadows

Constructing spacetime solutions that describe black holes embedded in dark matter environments is a crucial step toward probing the properties of dark matter in the strong-field regime of gravity. At present, however, there is no unique or systematic framework to model such configurations, and several commonly adopted approaches raise methodological ambiguities. Motivated by these challenges, we build upon a perturbative framework to describe deformations of static, spherically symmetric black holes induced by a surrounding dark matter distribution. Within this framework, we compute the leading-order corrections to both the photon-sphere radius and the radius of the black hole shadow, assuming a generic dark matter halo profile. We then apply the formalism to physically motivated density profiles, including the Hernquist and Navarro-Frenk-White models, obtaining closed-form analytical expressions for the perturbed metric functions and for the critical impact parameter in the Schwarzschild background. Using these results, we obtain quantitative estimates for the corresponding shadow deviations and find that they lie well beyond the current observational bounds set by Keck and VLTI measurements. As a consistency check, we further estimate the total dark matter mass enclosed within the orbital radius of the S2 star and show that it remains well below the 0.1% upper limit reported by the GRAVITY collaboration. Overall, this approach offers a systematic avenue to investigate perturbative effects of dark matter on black hole phenomenology, including potential implications for gravitational wave observations.