Imprints of dark matter on gravitational ringing of supermassive black holes

Gravitational waves emitted from the gravitational ringing of supermassive black holes are important targets to test general relativity and probe the matter environment surrounding such black holes. The main components of the ringing waveform are black hole quasi-normal modes. In this paper, we study the effects of the dark matter halos with three different density profiles on the gravitational polar (even-parity) perturbations of a supermassive black hole. For this purpose, we first consider modified Schwarzschild spacetime with three different dark matter profiles and derive the equation of motion of the polar perturbations of the supermassive black hole. It is shown that by ignoring the dark matter perturbations, a Zerilli-like master equation with a modified potential for the polar perturbation can be obtained explicitly. Then we calculate the complex frequencies of the quasi-normal modes of the supermassive black hole in the dark matter halos. The corresponding gravitational wave spectra with the effects of the dark matter halos and their detectability have also been discussed.