Ringing of a black hole in a dark matter halo

Recently, we obtained the simple metrics of a spherically symmetric black hole in a dark matter halo, and extended to the case of rotation. As the characteristic sound of black holes, quasinormal modes (QNMs) are one of the important means to understand black holes currently. Based on these two metrics of a spherically symmetric black hole, we study the QNMs of cold dark matter (CDM) and scalar field dark matter (SFDM) models using the methods of the material field perturbations and the gravitational perturbation, and make comparisons with the Schwarzschild black hole. Our results show that black hole QNMs of CDM and SFDM in a dark matter halo are different from the Schwarzschild black hole, unlike a Schwarzschild black hole with a prominent power-law tail. The different kinds models of dark matter can be distinguished by their QNMs. The time of QNMs ringing and frequencies increase with increasing parameter $l$. The overall QNMs of CDM are stronger than that of SFDM in the same condition, which is easier to be detected. In addition, QNMs frequencies using the sixth-order WKB method and the Prony method are in good agreement.