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In this work, we explore the effects of surrounding dark matter featuringdifferent equations of state on the axial gravitational quasinormal modes ofsupermassive black holes situated at the center of galaxies. Our attentionprimarily rests on dark matter exhibiting a spike structure, originating fromrelativistic Bondi accretion through an adiabatic process, which diminishes ata certain distance from the black hole. We analyze how varying the equation ofstate of the dark matter influences the properties of the spacetime in theblack hole's vicinity. Our findings reveal that different states of dark matterspikes correspondingly affect the black hole's quasinormal modes. Inparticular, we identify deviations in both the ringing frequency and dampingtime, 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 theessential properties of dark matter surrounding supermassive black holes usingfuture gravitational wave detections, particularly in the case of extreme massratio inspiral systems.

Influence of dark matter equation of state on the axial gravitational ringing of supermassive black holes