Quasinormal modes of a Kerr black hole embedded in a King-type cored dark matter halo

We investigate scalar quasinormal modes of a Kerr black hole immersed in a King-type cored dark matter halo within the slow-rotation approximation. Starting from a static King density profile, we construct the rotating spacetime via the Newman–Janis algorithm and derive the effective potential. The quasinormal mode frequencies are computed using the WKB approximation up to sixth order.We find that both the black hole spin and the dark matter halo significantly modify the quasinormal spectrum. The halo deforms the effective potential, shifting its peak and changing its height and width, leading to corrections in both the real and imaginary parts of the frequencies. Rotation introduces an m -dependent splitting via frame-dragging, while the overtone number n governs the decay hierarchy.We further perform time-domain simulations and extract the ringdown signal. Multi-mode fitting shows that although the dominant mode is well captured by a single frequency, additional modes are necessary for an accurate description of intermediate and early-time behavior.A comparison of different WKB orders shows good convergence for low-lying modes, with the sixth-order approximation providing stable results for the fundamental mode and first overtone. A Fisher matrix analysis indicates that the spin and halo density can be tightly constrained, while the halo scale parameter suffers from larger uncertainties due to degeneracies.Our results suggest that dark matter halos leave observable imprints on quasinormal mode spectra and gravitational wave signals, offering a potential probe for future space-based detectors such as LISA.