Detecting vector charge with extreme mass ratio inspirals onto Kerr black holes

Extreme mass ratio inspirals (EMRIs) are excellent sources for space-based observatories to explore the properties of black holes and test no-hair theorems. We consider EMRIs with a charged compact object inspiralling onto a Kerr black hole in quasi-circular orbits. Using the Teukolsky and generalized Sasaki-Nakamura formalisms for the gravitational and vector perturbations about a Kerr black hole, we numerically calculate the energy fluxes for both gravitational and vector perturbations induced by a charged particle moving in equatorial circular orbits. With one-year observations of EMRIs, we apply the Fisher information matrix method to estimate the charge uncertainty detected by space-based gravitational wave detectors such as the Laser Interferometer Space Antenna, TianQin, and Taiji, and we find that it is possible to detect vector charge as small as $q\sim 0.0049$. The results show that EMRIs composed of a Kerr black hole with a higher spin $a$ and lighter mass $M$, and a secondary charged object with more vector charge give smaller relative error on the charge, thus constrain the charge better. The positive spin of the Kerr black hole can decrease the charge uncertainty by about one or two orders of magnitude.