  In this work, we explore the effects of surrounding dark matter featuring
different equations of state on the axial gravitational quasinormal modes of
supermassive black holes situated at the center of galaxies. Our attention
primarily rests on dark matter exhibiting a spike structure, originating from
relativistic Bondi accretion through an adiabatic process, which diminishes at
a certain distance from the black hole. We analyze how varying the equation of
state of the dark matter influences the properties of the spacetime in the
black hole's vicinity. Our findings reveal that different states of dark matter
spikes correspondingly affect the black hole's quasinormal modes. In
particular, we identify deviations in both the ringing frequency and damping
time, reaching magnitudes of up to $10^{-3}$ for certain parameter values.
These variations can potentially be detected by upcoming space-borne detectors.
Our findings thus indicate the feasibility of discerning and limiting the
essential properties of dark matter surrounding supermassive black holes using
future gravitational wave detections, particularly in the case of extreme mass
ratio inspiral systems.
