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

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.