Retrograde binaries of massive black holes in circumbinary accretion discs

Context. We explore the hardening of a massive black hole binary embedded in a circumbinary gas disc under a specific circumstance: when the binary and the gas are coplanar and the gas is counter-rotating. The binary has unequal mass and the interaction of the gas with the lighter secondary black hole is the main cause of the braking torque on the binary that shrinks with time. The secondary black hole, revolving in the direction opposite to the gas, experiences a drag from gas-dynamical friction and from direct accretion of part of it. Aims. In this paper, using two-dimensional (2D) hydrodynamical grid simulations we investigate the effect of changing the accretion prescriptions on the dynamics of the secondary black hole, which in turn affect the binary hardening and eccentricity evolution. Methods. We find that realistic accretion prescriptions lead to results that differ from those inferred assuming accretion of all the gas within the Roche Lobe of the secondary black hole. Results. When considering gas accretion within the gravitational influence radius of the secondary black hole (which is smaller than the Roche Lobe radius) to better describe gas inflows, the shrinking of the binary is slower. In addition, in this case, a smaller amount of accreted mass is required to reduce the binary separation by the same amount. Different accretion prescriptions result in different discs' surface densities, which alter the black hole's dynamics back. Full 3D Smoothed-particle hydrodynamics realizations of a number of representative cases, run over a shorter interval of time, validate the general trends observed in the less computationally demanding 2D simulations. Conclusions. Initially circular black hole binaries increase their eccentricity only slightly, which then oscillates around small values (<0.1) while they harden. By contrast, initially eccentric binaries become more and more eccentric. A semi-analytical model describing the black hole's dynamics under accretion only explores the late evolution stages of the binary in an otherwise unperturbed retrograde disc to illustrate how eccentricity evolves with time in relation to the shape of the underlying surface density distribution