On the mass radiated by coalescing black hole binaries

We derive an analytic phenomenological expression that predicts the final mass of the black hole (BH) remnant resulting from the merger of a generic binary system of BHs on quasi-circular orbits. Besides recovering the correct test-particle limit for extreme mass-ratio binaries, our formula reproduces well the results of all the numerical-relativity simulations published so far, both when applied at separations of a few gravitational radii and when applied at separations of tens of thousands of gravitational radii. These validations make our formula a useful tool in a variety of contexts ranging from gravitational-wave (GW) physics to cosmology. As representative examples, we first illustrate how it can be used to decrease the phase error of the effective-one-body waveforms during the ringdown phase. Second, we show that, when combined with the recently computed self-force correction to the binding energy of nonspinning BH binaries, it provides an estimate of the energy emitted during the merger and ringdown. Finally, we use it to calculate the energy radiated in GWs by massive BH binaries as a function of redshift, using different models for the seeds of the BH population.