  The advent of gravitational wave astronomy has seen a huge influx of new
predictions for potential discoveries of beyond the Standard Model fields. The
coupling of all fundamental fields to gravity, together with its dominance on
large scales, makes gravitational physics a rich laboratory to study
fundamental physics. This holds especially true for the search for the elusive
dark photon, a promising dark matter candidate. The dark photon is predicted to
generate instabilities in a rotating black hole spacetime, birthing a
macroscopic Bose-Einstein condensate. These condensates can especially form
around super massive black holes, modifying the dynamical inspiralling process.
This then opens another window to leverage future space-borne gravitational
wave antennas to join the hunt for the elusive dark matter particle. This study
builds a preliminary model for the gravitational waveform emitted by such a
dressed extreme mass-ratio inspiral. Comparing these waveforms to the vacuum
scenario allows projections to the potential constrainability on the dark
photon mass by space-borne gravitational wave antennas. The superradiant
instability of a massive vector field on a Kerr background is calculated and,
under reasonable approximations, the backreaction on the orbit of an
inspiralling solar mass-scale compact object due to the secular evolution of
the resulting boson cloud is determined. The end result is the projection that
the LISA mission should be able to constrain the dark photon mass using extreme
mass ratio inspirals in the range $[1.8 \times 10^{-17}, 4.47 \times 10^{-16}]$
eV.
