The competing effect of gas and stars in massive black hole binaries
  evolution
  Massive black hole binaries are predicted to form during the hierarchical
assembly of cosmic structures and will represent the loudest sources of
low-frequency gravitational waves (GWs) detectable by present and forthcoming
GW experiments. Before entering the GW-driven regime, their evolution is driven
by the interaction with the surrounding stars and gas. While stellar
interactions are found to always shrink the binary, recent studies predict the
possibility of binary outspiral mediated by the presence of a gaseous disk,
which could endlessly delay the coalescence and impact the merger rates of
massive binaries. Here we implement a semi-analytical treatment that follows
the binary evolution under the combined effect of stars and gas. We find that
binaries may outspiral only if they accrete near or above their Eddington limit
and only until their separation reaches the gaseous disk self-gravitating
radius. Even in case of an outspiral, the binary eventually reaches a large
enough mass for GW to take over and drive it to coalescence. The combined
action of stellar hardening, mass growth and GW-driven inspiral brings binaries
to coalescence in few hundreds Myr at most, implying that gas-driven expansion
will not severely affect the detection prospects of upcoming GW facilities.
