  We examine the electromagnetic (EM) and gravitational wave (GW) signatures of
stellar-mass compact objects (COs) spiraling into a supermassive black hole
(extreme mass-ratio inspirals or EMRIs), embedded in a thin, radiation-pressure
dominated, accretion disk. At large separations, the tidal effect of the
secondary CO clears a gap. We show that the gap refills during the late
GW-driven phase of the inspiral, leading to a sudden EM brightening of the
source. The accretion disk leaves an imprint on the GW through its angular
momentum exchange with the binary, the mass increase of the binary members due
to accretion, and its gravity. We compute the disk-modified GWs both in an
analytical Newtonian approximation and in a numerical effective-one-body
approach. We find that disk-induced migration provides the dominant
perturbation to the inspiral, with weaker effects from the mass accretion onto
the CO and hydrodynamic drag. Depending on whether a gap is present, the
perturbation of the GW phase is between 10 and 1000 radians per year,
detectable with the future Laser Interferometer Space Antenna (LISA) at high
significance. The Fourier transform of the disk-modified GW in the stationary
phase approximation is sensitive to disk parameters with a frequency trend
different from post-Newtonian vacuum corrections. Our results suggest that
observations of EMRIs may place new sensitive constraints on the physics of
accretion disks.
