Black hole mergers: can gas discs solve the `final parsec' problem?

We compute the effect of an orbiting gas disc in promoting the coalescence of a central supermassive black hole binary. Unlike earlier studies, we consider a finite mass of gas with explicit time dependence: we do not assume that the gas necessarily adopts a steady state or a spatially constant accretion rate, i.e. that the merging black hole was somehow inserted into a pre--existing accretion disc. We consider the tidal torque of the binary on the disc, and the binary's gravitational radiation. We study the effects of star formation in the gas disc in a simple energy feedback framework. The disc spectrum differs in detail from that found before. In particular, tidal torques from the secondary black hole heat the edges of the gap, creating bright rims around the secondary. These rims do not in practice have uniform brightness either in azimuth or time, but can on average account for as much as 50 per cent of the integrated light from the disc. This may lead to detectable high--photon--energy variability on the relatively long orbital timescale of the secondary black hole, and thus offer a prospective signature of a coalescing black hole binary. We also find that the disc can drive the binary to merger on a reasonable timescale only if its mass is at least comparable with that of the secondary black hole, and if the initial binary separation is relatively small, i.e. $a_0 \lesssim 0.05$ pc. Star formation complicates the merger further by removing mass from the disc. In the feedback model we consider, this sets an effective limit to the disc mass. As a result, binary merging is unlikely unless the black hole mass ratio is $\la 0.001$. Gas discs thus appear not to be an effective solution to the `last parsec' problem for a significant class of mergers.