  We perform a series of high-resolution 2D hydrodynamical simulations of
equal-mass binary black holes (BBHs) embedded in active galactic nucleus (AGN)
accretion disks to study whether these binaries can be driven to merger by the
surrounding gas. We find that the gravitational softening adopted for the BBH
has a profound impact on this result. When the softening is less than ten
percent of the binary separation, we show that, in agreement with recent
simulations of isolated equal-mass binaries, prograde BBHs expand in time
rather than contract. Eventually, however, the binary separation becomes large
enough that the tidal force of the central AGN disrupts them. Only when the
softening is relatively large do we find that prograde BBHs harden. We
determine through detailed analysis of the binary torque, that this dichotomy
is due to a loss of spiral structure in the circum-single disks orbiting each
BH when the softening is a significant fraction of the binary separation.
Properly resolving these spirals -- both with high resolution and small
softening -- results in a significant source of binary angular momentum. Only
for retrograde BBHs do we find consistent hardening, regardless of softening,
as these BBHs lack the important spiral structure in their circum-single disks.
This suggests that the gas-driven inspiral of retrograde binaries can produce a
population of compact BBHs in the gravitational-wave-emitting regime in AGN
disks, which may contribute a large fraction to the observed BBH mergers.
