Dynamical Evolution and Merger Time-scales of LISA Massive Black Hole Binaries in Disk Galaxy Mergers

The Laser Interferometer Space Antenna (LISA) will detect gravitational-wave (GW) signals from merging supermassive black holes (BHs) with masses below $10^7$~M$_{\odot}$. It is thus of paramount importance to understand the orbital dynamics of these relatively light central BHs, which typically reside in disc-dominated galaxies, in order to produce reliable forecasts of merger rates. To this aim, realistic simulations probing BH dynamics in unequal-mass disc galaxy mergers, into and beyond the binary hardening stage, are performed by combining smooth particle hydrodynamics and direct $N$-body codes. The structural properties and orbits of the galaxies are chosen to be consistent with the results of galaxy formation simulations. Stellar and dark matter distributions are triaxial down to the central 100 pc of merger remnant. In all cases, a BH binary forms and hardens on time-scales of at most 100~Myr, coalescing on another few hundred Myr time-scale, depending on the characteristic density and orbital eccentricity. Overall, the sinking of the BH binary takes no more than $\sim$0.5~Gyr after the merger of the two galaxies is completed, but can be much faster for very plunging orbits. Comparing with previous numerical simulations following the decay of BHs in massive early-type galaxies at $z \sim 3$, we confirm that the characteristic density is the most crucial parameter determining the overall BH merging time-scale, despite the structural diversity of the host galaxies. Our results lay down the basis for robust forecasts of LISA event rates in the case of merging BHs.