Gas-induced perturbations on the gravitational wave in-spiral of live post-Newtonian LISA massive black hole binaries

We investigate the effect of dynamically coupling gas torques with gravitational wave (GW) emission during the orbital evolution of an equal-mass massive black hole binary (MBHB). We perform hydrodynamical simulations of eccentric MBHBs with total mass $M=10^6~{\rm M}_\odot$ embedded in a prograde locally isothermal circumbinary disk (CBD). We evolve the binary from $55$ to $49$ Schwarzschild radii separations using up to 2.5 post-Newtonian (PN) corrections to the binary dynamics, which allow us to follow the GW-driven in-spiral. For the first time, we report the measurement of gas torques onto a live binary a few years before the merger, with and without concurrent GW radiation. We also report the gas-induced orbital dephasing $δφ_{\rm orb}\sim-0.007$ rad over $278$ orbital cycles that is likely driven mainly by disc-induced precession and LISA should be able to detect it at redshift $z=1$. Our results show how GWs alone can be used to probe the astrophysical properties of CBDs and have important implications for multi-messenger strategies aimed at studying the environments of MBHBs.