Active Brownian particles (ABPs), when subject to purely repulsive
interactions, are known to undergo activity-induced phase separation broadly
resembling an equilibrium (attraction-induced) gas-liquid coexistence. Here we
present an accurate continuum theory for the dynamics of phase-separating ABPs,
derived by direct coarse-graining, capturing leading-order density gradient
terms alongside an effective bulk free energy. Such gradient terms do not obey
detailed balance; yet we find coarsening dynamics closely resembling that of
equilibrium phase separation. Our continuum theory is numerically compared to
large-scale direct simulations of ABPs and accurately accounts for domain
growth kinetics, domain topologies and coexistence densities
