Quasinormal Modes of Schwarzschild Black Holes in the Dehnen-(1, 4, 5/2) Type Dark Matter Halos

Abstract: The Dehnen - type dark matter density distribution model is mainly used for dwarf galaxies. In recent years, researchers have speculated that black holes may exist in this dark matter model and have given the black hole metric solutions. On this basis, this paper conducts a systematic study on the quasinormal modes of a Schwarzschild black hole in a Dehnen - (1,4, 5/2) dark matter halo, revealing the influences of dark matter distribution and perturbation field types on the black hole's quasinormal modes.The research uses the shadow radius data of the M87$^{\ast}$ black hole. Through the geodesic equation, two sets of dark matter halo parameter values of $\rho_{\rm s}$ and $r_{\rm s}$ are determined, and the specific numerical values of the black hole's event horizon radius, photon sphere radius, and shadow radius under the corresponding conditions are obtained. The wave equations and effective potentials of the black hole under the perturbations of the scalar field, electromagnetic field, and axial gravitational were analyzed. It was found that the larger the values of $\rho_{\rm s}$ or $r_{\rm s}$, the smaller the peak value of the effective potential, and the wave function oscillation slows down with a lower frequency. The black hole remains stable under perturbations. These studies provide relevant data for the quasinormal modes of the Schwarzschild black hole in the Dehnen-(1,4, 5/2) type dark matter halo. They also offer crucial evidence for understanding the interaction mechanism between the black hole and the dark matter halo.