Schwarzschild-like Black Holes Submerged in an Exponential Density Dark Matter Profile

Abstract

We study a class of Schwarzschild black holes embedded in an exponential-spheroidal dark matter halo, modelled by a phenomenological density profile $\rho(r)=\rho_0 e^{-r/r_0}$. By solving the Einstein equations for a static, spherically symmetric spacetime, we obtain an analytic solution for the lapse function that reduces to the Schwarzschild spacetime in the absence of the halo and to a regular halo configuration when the central black hole mass vanishes. Indeed, the two halo parameters, $\rho_0$ and $r_0$, describe the strength and radial extent of the dark matter distribution. We analyse the curvature structure, energy conditions, shadow observables, scalar quasi-normal modes and grey-body bounds of the resulting spacetime. The Ricci scalar and the Ricci square remain finite at the origin, whilst the Kretschmann scalar retains the usual central tidal singularity in the presence of a black hole mass. The weak, null, and dominant energy conditions are satisfied, whilst the strong energy condition is violated on a finite radial interval. We also show that the halo monotonically shifts the photon sphere and the shadow radius, which allows us to derive approximate constraints from the EHT observations of M87* and Sgr A*. For scalar perturbations, the Pad\'{e}-resummed WKB approximation yields stable quasinormal frequencies, while the standard WKB approximation becomes unreliable for higher overtones and strong-halo configurations. Finally, the greybody bounds indicate that the halo weakens transmission through the effective barrier, particularly at low frequencies.