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Massive Black Hole binaries in gas-rich galaxy mergers; multiple regimes of orbital decay and interplay with gas inflows

We revisit the phases of the pairing and sinking of BHs in galaxy mergers and circunmunclear disks in light of the results of recent simulations with massive BHs embedded in predominantly gaseous backgrounds. After a general overview we discuss the importance of a fast orbital decay regime dominated by global disk torques rather than by the local dynamical friction wake. This regime can dominate at BH binary separations of a few tens of parsecs and below, following a phase of orbital circularization dominated by local dynamical friction. It is similar to Type-I migration in planetary evolution. It can bring the black holes to separations small enough for gravitational waves to take over on a timescale ranging from less than $\sim 10^7$ yr to up to $10^8$ yr, depending on whether the interstellar medium is smooth or clumpy. Eventual gap opening at sub-pc scale separations slows down but does not interrupt the orbital decay.Subsequently, we discuss a new intriguing connection between the conditions required for rapid orbital decay of massive BH binaries and those required for prominent gas inflows in gas-rich galaxies. We derive a condition for the maximum inflow rate that a circumnuclear disk can host while still maintaining a sufficiently high gas density at large radii to sustain the decay of a BH binary. We find that gas inflows rates exceeding 10 $M_{\odot}$/yr, postulated to form massive BH seeds in some direct collapse models, would stifle the sinking of massive BH binaries in gas-dominated galactic nuclei. Vice-versa, lower inflow rates, below a solar mass per year, as required to feed typical AGNs, are compatible with a fast orbital decay of BH binaries across a wide range of masses.