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Supermassive black holes at the center of each galaxy may be surrounded bydark matter. Such dark matter admits a spike structure and vanishes at acertain distance from the black hole. This dark matter will impact thespacetime near the black hole and the related ringing gravitational waves canshow distinguished features of the black hole without dark matter. In thepresent work, we focus on the quasi-normal modes of the axial gravitationalperturbation who dominate the ringdown process of the perturbed black holessurrounded by dark matter spikes. The relativistic modification results in lessimpact on QNMs. And the relative ringing frequency difference between the blackholes with and without dark matter can be as large as $10^{-2}$. These featurescan be used in future gravitational wave detection about extremal mass ratioinspiral systems to probe the existence of dark matter around supermassiveblack holes.

The Axial Gravitational Ringing of a Spherically Symmetric Black Hole Surrounded by Dark Matter Spike