A detailed theoretical analysis describes how a dark matter halo with a King density profile modifies the quasinormal frequencies of a Schwarzschild black hole. By studying scalar and electromagnetic perturbations, the authors find that the halo lowers both the oscillation frequency and the damping rate of the signal. The change in the real part is almost entirely accounted for by the halo mass enclosed within the photon sphere, while the additional decrease in damping follows the ratio of the Lyapunov exponent to the angular frequency of the circular null orbit.

The work addresses an open question in black hole physics: how would a dark matter environment leave its imprint on the gravitational-wave signal emitted during a black hole merger. Because part of the halo falls inside the horizon in this metric, the authors repeat the calculation placing the halo entirely outside the black hole by means of an Einstein cluster. In that configuration the frequencies decrease by a similar amount, but the real part follows the Schwarzschild value multiplied by the gravitational redshift factor of the halo.

To obtain the quasinormal frequencies, the authors use 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. The fundamental frequencies agree with those extracted from time-domain evolution to within 0.2 %.

The results also show that a jump in density at the inner edge of the halo scatters the waves and changes the damping rate by up to about 9 %. For continuous density profiles, deviations from the redshift relation remain below about 1.5 % for r_in = 3M, and below about 0.5 % for the cases with r_in ≥ 4M. No unstable modes are found in any of the perturbation sectors or parameter configurations examined, suggesting that these halos are compatible with black hole stability.