  The evolution of Supermassive Black Holes (SMBHs) initially embedded in the
centres of merging galaxies realised with a stellar mass function (SMF) is
studied from the onset of galaxy mergers till coalescence. We performed a large
set of direct N-body simulations with three different slopes of the central
stellar cusp and different random seeds. Post Newtonian terms up to order 3.5
are used to drive the SMBH binary evolution in the relativistic regime. The
impact of a SMF on the hardening rate and the coalescence time is investigated.
We find that SMBH binaries coalesce well within one billion years when our
models are scaled to galaxies with a steep cusp at low redshift. Here higher
central densities provide larger supply of stars to efficiently extract energy
from the SMBH binary orbit and shrink it to the phase where gravitational wave
(GW) emission becomes dominant leading to the coalescence of the SMBHs. Mergers
of models with shallow cusps that are representative for giant elliptical
galaxies having central cores result in less efficient extraction of binary
orbital energy due to the lower stellar densities in the centre. However, high
values of eccentricity witnessed for SMBH binaries in such galaxy mergers
ensure that the GW emission dominated phase sets in earlier at larger values of
the semi-major axis. This helps to compensate for the less efficient energy
extraction during the phase dominated by stellar encounters resulting in
mergers of SMBHs in about one Gyr after the formation of the binary.
Additionally, we witness mass segregation in the merger remnant resulting in
enhanced SMBH binary hardening rates. We show that at least the final phase of
the merger in cuspy low mass galaxies would be observable with the GW detector
eLISA.
