Probing black holes in a dark matter spike of M87 using quasinormal modes

Dark matter density can be significantly enhanced by the supermassive black hole at the galactic center, leading to a structure called dark matter spike. Dark matter spike may change the spacetime properties of black holes that constitute deviations from GR black holes. Based on these interesting background, we construct a set of solutions of black holes in a dark matter spike under the Newtonian approximation and full relativity. Combining the mass model of M87, we study the quasinormal modes of black holes in the scalar field and axial gravitational perturbation, then compared them with Schwarzschild black hole. Besides, the impacts of dark matter on the quasinormal mode of black holes have been studied in depth. In particular, in the axial gravitational perturbation, our results show that the impacts of dark matter spike on the quasinormal mode of black holes can reach up to $10^{-4}$. These new features from quasinormal mode of black holes under the Newtonian approximation and full relativity may provide some help for the establishment of the final dark matter model, and provide a new thought for the indirect detection of dark matter.