When two galaxies collide, they are likely to stick together, if their relative speed is not too high. Within a few crossing times, the transient ripples and distortions will be smoothed out, and the resulting single galaxy will settle down into a new equilibrium configuration. While all this is going on, the dense cores of both galaxies will spiral in, as a result of dynamical friction, in the central regions of the collision. Finally they, too, merge to form a single core.
Many, if not most, galaxies harbor a massive black hole in their
center. Recently, many such black holes have been detected, with
masses spanning a range from a million to a billion solar masses, up
to 
% of the mass of the parent galaxy [18]. When two galaxies
collide and stick together to form one large merger remnant, the dense
nuclei of the two parent galaxies will spiral in, within the central
region of the newly formed galaxy. These nuclei will merge to form a
single dense nucleus, as soon as they come in contact with each other.
What will happen, if each nucleus contains a black hole, however is
far from clear.
At first, they will keep circling each other, within the single newly formed dense nucleus. Although dynamical friction tends to let them spiral in rapidly at first, this process becomes considerably less efficient by the time the amount of mass in stars between the two holes becomes smaller than the mass of the holes themselves. The stars that initially tend to be most efficient in providing a braking mechanism are scattered into different orbits. As a result, the system may reach a stagnation point, in which little further dynamical friction occurs.
The prediction of this stagnation process was made almost twenty years ago [19], and since then many attempts have been made to check this prediction quantitatively, using large-scale N-body calculations. Until the advent of the first GRAPEs, this problem was completely intractable, even on the largest supercomputers available. One reason that the GRAPE computers are suitable for this type of problem is the intrinsically high dimensionality of the problem. With two black holes in an eccentric orbit around each other, there is no symmetry in either configuration space or velocity space. As a result, the stellar dynamics problem is truly six-dimensional, when seen as a fluid flow in phase space.
In contrast, modeling a globular cluster is often done by assuming spherical symmetry, which leaves only one spatial dimension (radial) and two velocity dimensions (radial and tangential) to worry about. In practice, further simplifications have often been made, in which the distribution function of the stars is assumed to be dependent only on energy, or sometimes on energy and angular momentum. Fokker-Planck methods have therefore been very useful, initially, in modeling globular clusters, especially during the core collapse phase. After core collapse, during the reexpansion phase, the effects of binaries have to be taken into account, an extremely granular process that defies the main Fokker-Planck assumptions of smoothness of the distribution function. However, even so, it has been very useful to compare the full N-body calculations in the post-collapse domain with approximate Fokker-Planck treatments. However, a Fokker-Planck treatment of a six-dimensional system is completely impractical from the outset.
The first attempts to use the GRAPE to tackle this problem, were made
in 1990 [20], using the GRAPE-2, followed by more recent attempts
[21] on the GRAPE-4. Three
important conclusions have emerged from these studies.
(i) When two identical galaxies, each harboring a central black hole, merge,
they will produce a merger remnant with a ratio of core radius 
to half-mass radius 
that is comparable to that of the original
galaxies. In contrast, galaxies without black holes tend to produce
merger remnants in which 
is smaller than in the original
galaxies. In the former case, 
, where 
is the mass of the central black hole,
and 
is the mass of the whole galaxy.
(ii) This `core', formed around the black hole binary after the merging
of the two galaxies, does not have a completely flat density
distribution in the center. In fact, it looks more like the `weak
cusps' observed in many galaxies by the Hubble Space Telescope [22].
The formation mechanism of this cusp is not well understood.
(iii) Whether or not a black hole binary, lurking in the core of a merger
remnant, has had time to spiral in within the current age of the Universe,
and under which circumstances, is still largely an open question.
We expect the continuum limit to be reached for 
.
These calculations will only be feasible with the GRAPE-6 (Table 1).