Gravitational ringing and superradiant instabilities of the Kerr-like black holes in a dark matter halo

Supermassive black holes from the center of galaxy may be immersed in a dark matter halo. This dark matter halo may form a "cusp" structure around the black hole and disappear at a certain distance from the black hole. Based on this interesting physical background, we use the continued fraction method to study gravitational ringring of the Kerr-like black holes immersed in a dark matter halo, i.e., quasinormal modes (QNM) and quasibound states (QBS). We consider these gravitational ringring of black holes both in cold dark matter (CDM) model and scalar field dark matter (SFDM) model at the LSB galaxy, and compare them with Kerr black hole. By testing the states of QNM/QBS frequencies with different parameters $l,m,a$, we confirm the existence of the superradiant instabilities when the black holes both in CDM model and SFDM model. Besides, we also study the impacts of dark matter parameters on the QNM/QBS of black holes at the specific circumstances. In the future, these results may be used for gravitational wave detection of supermassive black holes, and may provide an effective method for detecting the existence of dark matter.