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.