  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.
