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Massive black hole binaries are predicted to form during the hierarchicalassembly of cosmic structures and will represent the loudest sources oflow-frequency gravitational waves (GWs) detectable by present and forthcomingGW experiments. Before entering the GW-driven regime, their evolution is drivenby the interaction with the surrounding stars and gas. While stellarinteractions are found to always shrink the binary, recent studies predict thepossibility of binary outspiral mediated by the presence of a gaseous disk,which could endlessly delay the coalescence and impact the merger rates ofmassive binaries. Here we implement a semi-analytical treatment that followsthe binary evolution under the combined effect of stars and gas. We find thatbinaries may outspiral only if they accrete near or above their Eddington limitand only until their separation reaches the gaseous disk self-gravitatingradius. Even in case of an outspiral, the binary eventually reaches a largeenough mass for GW to take over and drive it to coalescence. The combinedaction of stellar hardening, mass growth and GW-driven inspiral brings binariesto coalescence in few hundreds Myr at most, implying that gas-driven expansionwill not severely affect the detection prospects of upcoming GW facilities.

The competing effect of gas and stars in massive black hole binaries evolution