  Understanding the interaction of massive black hole binaries with their
gaseous environment is crucial since at sub-parsec scales the binary is too
wide for gravitational wave emission to take over and to drive the two black
holes to merge. We here investigate the interaction between a massive black
hole binary and a self-gravitating circumbinary disc using 3D smoothed particle
hydrodynamics simulations. We find that, when the disc self-gravity regulates
the angular momentum transport, the binary semi-major axis decreases regardless
the choice of disc masses and temperatures, within the range we explored. In
particular, we find that the disc initial temperature (hence the disc aspect
ratio) has little effect on the evolution of the binary since discs with the
same mass self-regulate towards the same temperature. Initially warmer discs
cause the binary to shrink on a slightly shorter timescale until the disc has
reached the self-regulated equilibrium temperature. More massive discs drive
the binary semi-major axis to decrease at a faster pace compared to less
massive discs and result in faster binary eccentricity growth even after the
initial-condition-dependent transient evolution. Finally we investigate the
effect that the initial cavity size has on the binary-disc interaction and we
find that, in the self-gravitating regime, an initially smaller cavity leads to
a much faster binary shrinking, as expected. Our results are especially
important for very massive black hole binaries such as those in the PTA band,
for which gas self gravity cannot be neglected.
