Quasinormal Ringing of a Schwarzschild Black Hole Surrounded by a King Dark Matter Halo: Higher-Order WKB Analysis

We study scalar and electromagnetic perturbations of a Schwarzschild black hole (BH) surrounded by a dark matter halo described by the King density profile. The lapse function can be expressed as $1-2[M+M_{\rm DM}(r)]/r$, where $M_{\rm DM}(r)$ is the halo mass enclosed within radius $r$; thus, the halo affects the geometry through its enclosed-mass function. We compute the quasinormal frequencies using the sixth-order WKB method and its Padé-improved version. Rewriting the lapse function in terms of the inverse hyperbolic sine avoids the severe numerical cancellations that arise when evaluating its high-order derivatives. For the configurations examined, the fundamental frequencies agree with those extracted from time-domain evolution to within $0.2\%$.The halo lowers both the oscillation frequency and the damping rate. The change of the real part is almost entirely accounted for by the halo mass enclosed by the photon sphere, and the additional decrease of the damping rate follows the ratio of the Lyapunov exponent to the angular frequency of the circular null orbit. Since part of the halo lies inside the horizon in this metric, we also place the halo outside the BH by means of an Einstein cluster. The frequencies then decrease by a similar amount, but the real part follows the Schwarzschild value multiplied by the gravitational redshift factor of the halo. A jump in the density at the inner edge of the halo scatters the waves and changes the damping rate by up to about $9\%$. For a continuous density profile, deviations from the redshift relation remain below about $1.5\%$ for $r_{\rm in}=3M$ and below about $0.5\%$ for the cases with $r_{\rm in}\geq4M$. No unstable modes are found within the perturbation sectors and parameter configurations examined.