  Stellar-mass binary black holes (BBHs) embedded in active galactic nucleus
(AGN) discs offer a promising dynamical channel to produce black-hole mergers
that are detectable by LIGO/Virgo. Modeling the interactions between the disc
gas and the embedded BBHs is crucial to understand their orbital evolution.
Using a suite of 2D high-resolution simulations of prograde equal-mass circular
binaries in local disc models, we systematically study how their hydrodynamical
evolution depends on the equation of state (EOS; including the $\gamma$-law and
isothermal EOS) and on the binary mass and separation scales (relative to the
supermassive BH mass and the Hill radius, respectively). We find that binaries
accrete slower and contract in orbit if the EOS is far from isothermal such
that the surrounding gas is diffuse, hot, and turbulent. The typical orbital
decay rate is of the order of a few times the mass doubling rate. For a fixed
EOS, the accretion flows are denser, hotter, and more turbulent around more
massive or tighter binaries. The torque associated with accretion is often
comparable to the gravitational torque, so both torques are essential in
determining the long-term binary orbital evolution. We carry out additional
simulations with non-accreting binaries and find that their orbital evolution
can be stochastic and is sensitive to the gravitational softening length, and
the secular orbital evolution can be very different from those of accreting
binaries. Our results indicate that stellar-mass BBHs may be hardened
efficiently under ideal conditions, namely less massive and wider binaries
embedded in discs with a non-isothermal EOS.
