Probing vector hair of black holes with extreme-mass-ratio inspirals

The bumblebee gravity model, with a vector field nonminimally coupled to gravity, is a natural extension of the Einstein-Maxwell theory. In this theory, a black hole can carry a vector hair, making the metric deviate from the Schwarzschild metric. To investigate the detectability of the vector hair, we consider an extreme-mass-ratio inspiral (EMRI) system, where a stellar-mass black hole inspiraling into a supermassive black hole. We find that, with a one-year observation of an EMRI by a space-based gravitational-wave detector, we can probe the vector charge as small as $Q\ensuremath{\sim}{10}^{\ensuremath{-}3}$ in the bumblebee gravity model, which is about three orders of magnitude tighter comparing to current EHT observations.

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