  Massive black hole binaries (BHBs) are expected to be one of the most
powerful sources of gravitational waves (GWs) in the frequency range of the
pulsar timing array and of forthcoming space-borne detectors. They are believed
to form in the final stages of galaxy mergers, and then harden by slingshot
ejections of passing stars. However, evolution via the slingshot mechanism may
be ineffective if the reservoir of interacting stars is not readily
replenished, and the binary shrinking may come to a halt at roughly a parsec
separation. Recent simulations suggest that the departure from spherical
symmetry, naturally produced in merger remnants, leads to efficient loss cone
refilling, preventing the binary from stalling. However, current N-body
simulations able to accurately follow the evolution of BHBs are limited to very
modest particle numbers. Brownian motion may artificially enhance the loss cone
refilling rate in low-N simulations, where the binary encounters a larger
population of stars due its random motion. Here we study the significance of
Brownian motion of BHBs in merger remnants in the context of the final parsec
problem. We simulate mergers with various particle numbers (from 8k to 1M) and
with several density profiles. Moreover, we compare simulations where the BHB
is fixed at the centre of the merger remnant with simulations where the BHB is
free to random walk. We find that Brownian motion does not significantly affect
the evolution of BHBs in simulations with particle numbers in excess of one
million, and that the hardening measured in merger simulations is due to
collisionless loss cone refilling.
