Analytic Expressions for Quasinormal Modes of a Regular Black Hole Sourced by a Dehnen-Type Halo

Using an expansion beyond the eikonal regime, we derive relatively compact and accurate analytic expressions for the gravitational quasinormal modes of an asymptotically flat black hole supported by a Dehnen-type dark-matter halo. The spacetime admits a simple analytic metric describing a supermassive black hole embedded in a galactic environment, with the lapse function \ (f (r) =1-2 M r^2 (r+a) ^{3}. \) The parameter \ (a\) sets the characteristic scale of the surrounding halo and controls the regularization of the central region. The axial gravitational sector splits into two distinct channels, referred to as the “up” and “down” perturbations, which are not isospectral.