Quasinormal modes of dark matter core-black hole spacetime

In the galactic core, when the scale of dark matter is small, the distribution of dark matter is that of a constant density dark matter core.Considering the case of a supermassive black hole coupled to a constant density dark matter core, we study the quasinormal modes of the black hole in the constant density dark matter core black hole system and calculate the quasinormal modes frequency of the black hole using the third order WKB approximation and the prony method.In addition, we study the effect of the constant density dark matter core parameter $r_0$ on the quasinormal modes of black holes in the vicinity of black holes.As the angular quantum number increases, the ringdown process becomes closer and closer to the case of the ringdown process of a schwarzschild black hole.The presence of a constant density dark matter core affects the quasinormal modes of the black hole, with relative deviations on the order of $10^{-15}-10^{-13}$ with respect to the detector.These features suggest that with future improvements in detector accuracy, we can use them for the detection of gravitational waves in the spacetime of constant density dark matter core-black hole systems, which in turn opens up the possibility of understanding the behavior of dark matter in the vicinity of black holes.