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We study the impact of gas accretion on the orbital evolution of black-holebinaries initially at large separation in the band of the planned LaserInterferometer Space Antenna (LISA). We focus on two sources:(i)~stellar-origin black-hole binaries~(SOBHBs) that can migrate from the LISAband to the band of ground-based gravitational-wave observatories withinweeks/months; and (ii) intermediate-mass black-hole binaries~(IMBHBs) in theLISA band only. Because of the large number of observable gravitational-wavecycles, the phase evolution of these systems needs to be modeled to greataccuracy to avoid biasing the estimation of the source parameters. Accretionaffects the gravitational-wave phase at negative ($-4$) post-Newtonian order,and is therefore dominant for binaries at large separations. If accretion takesplace at the Eddington or at super-Eddington rate, it will leave a detectableimprint on the dynamics of SOBHBs. In optimistic astrophysical scenarios, amultiwavelength strategy with LISA and a ground-based interferometer can detectabout $10$ (a few) SOBHB events for which the accretion rate can be measured at$50\%$ ($10\%$) level. In all cases the sky position can be identified withinmuch less than $0.4\,{\rm deg}^2$ uncertainty. Likewise, accretion at $\gtrsim10\%$ ($\gtrsim 100\%$) of the Eddington rate can be measured in IMBHBs up toredshift $z\approx 0.1$ ($z\approx 0.5$), and the position of these sources canbe identified within less than $0.01\,{\rm deg}^2$ uncertainty. Altogether, adetection of SOBHBs or IMBHBs would allow for targeted searches ofelectromagnetic counterparts to black-hole mergers in gas-rich environmentswith future X-ray detectors (such as Athena) and radio observatories (such asSKA).

Gravitational-wave detection and parameter estimation for accreting black-hole binaries and their electromagnetic counterpart