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We have studied extreme mass-ratio inspirals (EMRIs) in spacetimes containing a rotating black hole and a non-self-gravitating torus with a constant distribution of specific angular momentum. We have found that the dissipative effect of the hydrodynamic drag exerted by the torus on the satellite is much smaller than the corresponding one due to radiation reaction, for systems such as those generically expected in active galactic nuclei and at distances from the central supermassive black hole (SMBH) which can be probed with the Laser Interferometer Space Antenna (LISA). However, given the uncertainty on the parameters of these systems, namely, on the masses of the SMBH and of the torus, as well as on its size, there exist configurations in which the effect of the hydrodynamic drag on the orbital evolution can be comparable to the radiation reaction one in phases of the inspiral which are detectable by the Laser Interferometer Space Antenna. This is the case, for instance, for a 106M[sun] SMBH surrounded by a corotating torus of comparable mass and with radius of 103–104 gravitational radii, or for a 105M[sun] SMBH surrounded by a corotating 104M[sun] torus with radius of 105 gravitational radii. Should these conditions be met in astrophysical systems, EMRI-gravitational waves could provide a characteristic signature of the presence of the torus. In fact, while radiation reaction always increases the inclination of the orbit with respect to the equatorial plane (i.e., orbits evolve towards the equatorial retrograde configuration), the hydrodynamic drag from a torus corotating with the SMBH always decreases it (i.e., orbits evolve towards the equatorial prograde configuration). However, even when initially dominating over radiation reaction, the influence of the hydrodynamic drag decays very rapidly as the satellite moves into the very strong-field region of the SMBH (i.e., p<~5M), thus allowing one to use pure-Kerr templates for the last part of the inspiral. Although our results have been obtained for a specific class of tori, we argue that they will be qualitatively valid also for more generic distributions of the specific angular momentum.
