  In this study of the `Resolving supermAssive Black hole Binaries In galacTic
hydrodynamical Simulations' (RABBITS) series, we focus on the hardening and
coalescing process of supermassive black hole (SMBH) binaries in galaxy
mergers. For simulations including different galaxy formation processes (i.e.
gas cooling, star formation, SMBH accretion, stellar and AGN feedback), we
systematically control the effect of stochastic eccentricity by fixing it to
similar values during the SMBH hardening phase. We find a strong correlation
between the SMBH merger time-scales and the presence of nuclear star formation.
Throughout the galaxy merging process, gas condenses at the centre due to
cooling and tidal torques, leading to nuclear star formation. These recently
formed stars, which inherit low angular momenta from the gas, contribute to the
loss cone and assist in the SMBH hardening via three-body interactions.
Compared to non-radiative hydrodynamical runs, the SMBH merger time-scales
measured from the runs including cooling, stellar and SMBH physical processes
tend to be shortened by a factor of ${\sim}1.7$. After fixing the eccentricity
to the range of $e \sim 0.6$--$0.8$ during the hardening phase, the simulations
with AGN feedback reveal merger time-scales of ${\sim} 100$--$500$ Myr for disc
mergers and ${\sim} 1$--$2$ Gyr for elliptical mergers. With a semi-analytical
approach, we find that the torque interaction between the binary and its
circumbinary disc has minimal impact on the shrinking of the binary orbit in
our retrograde galaxy merger. Our results are useful in improving the modelling
of SMBH merger time-scales and gravitational wave event rates.
