Kerr black holes with vector dark matter hair

We consider a massive vector-field perturbation on a Kerr black hole background. We fix the field's density at the edge of the black hole sphere of influence to model a bath of wave cold dark matter, from which the black hole can accrete endlessly. We employ the approach of Lunin, Frolov, Krtou\v{s}, Kubiz\v{n}\'{a}k, and Santos to separate the equations, and solve them with a mix of analytical and numerical techniques. We find that the field forms a density spike around the black hole with profile $\rho \sim r^{-3/2}$, consistent with dark matter spikes studied in the literature, which is not significantly distorted by spacetime rotation, except near the black hole's ergoregion. For any nonzero spin, an additional superradiant regime appears, in which co-rotating modes extract mass and angular momentum from the black hole. We find a mass \textit{extraction} rate as high as $10\, M_\odot/\text{yr}$ in the superradiant regime for a $10^9\, M_\odot$ black hole. For large field masses, the rate at which the black-hole mass grows behaves similarly to the Schwarzschild case, reaching $\sim 100\, M_\odot/\text{yr}$ for a $10^9\, M_\odot$ black hole. We compare these results to their scalar-field counterparts and discuss how they mix with the superradiant instability.