Gravitational waves of quasicircular, inspiraling black hole binaries in an ultralight vector-dark-matter environment

The gravitational waves emitted by massive black hole binaries can be affected by a variety of environmental effects, which, if detected, could inform astrophysics and cosmology. We here study how gravitational waves emitted by black holes in quasicircular orbits are affected by the presence of an ultralight, vector field, dark-matter environment that is minimally coupled to the binary. This dark-matter environment induces oscillatory gravitational potentials that perturb the orbit of the binary, leaving an imprint in the binary's binding energy, and thus on the gravitational waves emitted. We here compute the effect of this environment on the gravitational wave phase using the stationary-phase approximation within the post-Newtonian formalism. We then perform a Fisher analysis to estimate the detectability of this environmental effect with a four-year Laser Interferometric Space Antenna (LISA) observation, focusing on vector fields with ultralight masses in the $({10}^{\ensuremath{-}19},{10}^{\ensuremath{-}16})\text{ }\text{ }\mathrm{eV}$ range. We conclude that the observation of such gravitational waves with spaceborne interferometers, like LISA, could yield a measurement or constraint on local, vector dark-matter environments, provided the dark-matter density is larger than roughly ${10}^{14}{M}_{\ensuremath{\bigodot}}/{\mathrm{pc}}^{3}$.