Population synthesis of black hole binary mergers from star clusters

Black hole (BH) binary mergers formed through dynamical interactions in dense star clusters are believed to be one of the main sources of gravitational waves for Advanced LIGO and Virgo. Here we present a fast numerical method for simulating the evolution of star clusters with BHs, including a model for the dynamical formation and merger of BH binaries. Our method is based on H\'{e}non's principle of balanced evolution, according to which the flow of energy within a cluster must be balanced by the energy production inside its core. Because the heat production in the core is powered by the BHs, one can then link the evolution of the cluster to the evolution of its BH population. This allows us to construct evolutionary tracks of the cluster properties including its BH population and its effect on the cluster and, at the same time, determine the merger rate of BH binaries as well as their eccentricity distributions. The model is publicly available and includes the effects of a BH mass spectrum, mass-loss due to stellar evolution, the ejection of BHs due to natal and dynamical kicks, and relativistic corrections during binary-single encounters. We validate our method using direct $N$-body simulations, and find it to be in excellent agreement with results from recent Monte Carlo models of globular clusters. This establishes our new method as a robust tool for the study of BH dynamics in star clusters and the modelling of gravitational wave sources produced in these systems. Finally, we compute the rate and eccentricity distributions of merging BH binaries for a wide range of cluster initial conditions, spanning more than two orders of magnitude in mass and radius.