Retrograde binaries of massive black holes in circum-binary accretion
  discs
  We explore the hardening of a massive black hole binary embedded in a
circum-binary gas disc when the binary and the gas are coplanar and the gas is
counter-rotating. 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. 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. 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.
Different accretion prescriptions result in different disc's surface densities
which alter the black hole's dynamics back. Full 3D SPH 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. Initially circular black hole binaries increase only slightly
their eccentricity 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.
