The final parsec problem (or last parsec problem) is an astrophysical problem regarding the merger of a binary SMBH, e.g., after a galaxy merger. When the supermassive black holes are extremely close, they merge because gravitational waves significantly bleed their orbital energy. Beyond such extreme closeness, close encounters with stars can remove orbital energy, tightening the orbit, but the odds of encountering a star decreases as the orbit decreases, to a point where the black holes would remain in virtually the same orbit, i.e., the timescale for further decrease is far too long (longer than the age of the universe). Assuming they sometimes do merge, it is not clear how their orbits continue to become smaller through the final parsec: there is a gap within which neither process is sufficient to continue to tighten the orbit.
The problem is often described as an inability to sufficiently refill the loss cone (a region within the phase space of the galaxy's stars such that they can have the opportunity for encounters with the orbiting black holes that absorb energy and angular momentum from the orbit) to close the gap between black holes within a timescale of interest.
If there is a mechanism that regularly overcomes the final parsec problem, then it is presumed there are as many SMBH mergers as there are galaxy mergers. How the black holes gain their large masses is also a mystery, and such mergers could be a contributor to a solution.
The problem has been of interest, and approached a number of ways. A current theory is that the distribution of stars in the galaxy core after a merger differs from models developed from/for simpler galaxies, in effect, allowing dynamical friction to remain significant. The distribution of star trajectories within the center of a merged galaxy is likely to differ from assumptions that suggested insufficient stars in the vicinity to slow the orbiting black holes. Such galaxies are likely to be triaxial (differing "widths" in the three dimensions) rather than a symmetric disk such as a spiral galaxy. Another theory is that a circumbinary accretion disk some distance out from accretion disks around the individual black holes, accreting material onto those inner disks (disk coupling) has a means of drawing out orbital angular momentum into ejected material.