Spacetime metrics and ringdown waveforms for galactic black holes surrounded by a dark matter spike

Abstract: Theoretical models suggest the existence of a dark matter spike surrounding the supermassive black holes at the core of galaxies. The spike density is thought to obey a power law that starts at a few times the black hole horizon radius and extends to a distance, $R_\text{sp}$, of the order of a kiloparsec. We use the Tolman-Oppenheimer-Volkoff equations to construct the spacetime metric representing a black hole surrounded by such a dark matter spike. We consider the dark matter to be a perfect fluid, but make no other assumption about its nature. The assumed power law density provides in principle three parameters with which to work: the power law exponent $\gamma_\text{sp}$, the external radius $R_\text{sp}$, and the spike density $\rho_\text{DM}^\text{sp}$ at $R_\text{sp}$. These in turn determine the total mass of the spike. We focus on Sagittarius A* and M87 for which some theoretical and observational bounds exist on the spike parameters. Using these bounds in conjunction with the metric obtained from the Tolman-Oppenheimer-Volkoff equations, we investigate the possibility of detecting the dark matter spikes surrounding these black holes via the gravitational waves emitted at the ringdown phase of black hole perturbations. Our results suggest that if the spike to black hole mass ratio is roughly constant, greater mass black holes require relatively smaller spike densities to yield potentially observable signals. We find that is unlikely for the spike in M87 to be detected via the ringdown waveform with currently available techniques unless its mass is roughly an order of magnitude larger than existing observational estimates. However, given that the signal increases with black hole mass, spikes might be observable for more massive galactic black holes in the not too distant future.