Inspiral, merger, and ring-down of equal-mass black-hole binaries

Abstract

We investigate the dynamics and gravitational-wave (GW) emission in the binary merger of equal-mass black holes as obtained from numerical relativity simulations. The simulations were performed with an evolution code based on generalized harmonic coordinates developed by Pretorius, and used quasiequilibrium initial-data sets constructed by Cook and Pfeiffer. Results from the evolution of three sets of initial data are explored in detail, corresponding to different initial separations of the black holes, and exhibit between 2--8 GW cycles before coalescence. We find that to a good approximation the inspiral phase of the evolution is quasicircular, followed by a ``blurred, quasicircular plunge'' lasting for about 1--1.5 GW cycles. After this plunge the GW frequency decouples from the orbital frequency, and we define this time to be the start of the merger phase. Roughly 10--15 M separates the time between the beginning of the merger phase and when we are able to extract quasinormal ring-down modes from gravitational waves emitted by the newly formed black hole. This suggests that the merger lasts for a correspondingly short amount of time, approximately 0.5--0.75 of a full GW cycle. We present first-order comparisons between analytical models of the various stages of the merger and the numerical results---more detailed and accurate comparisons will need to await numerical simulations with higher accuracy, better control of systemic errors (including coordinate artifacts), and initial configurations where the binaries are further separated. During the inspiral, we find that if the orbital phase is well modeled, the leading order Newtonian quadrupole formula is able to match both the amplitude and phase of the numerical GW quite accurately until close to the point of merger. We provide comparisons between the numerical results and analytical predictions based on the adiabatic post-Newtonian (PN) and nonadiabatic resummed-PN models (effective-one-body and Pad\'e models). For all models considered, 3PN and 3.5PN orders match the inspiral numerical data the best. From the ring-down portion of the GW, we extract the fundamental quasinormal mode and several of the overtones. Finally, we estimate the optimal signal-to-noise ratio (SNR) for typical binaries detectable by GW experiments. We find that, when the merger and ring-down phases are included, binaries with total mass larger than $40{M}_{\ensuremath{\bigodot}}$ (sources for ground-based detectors) are brought in band and can be detected with signal-to-noise up to $\ensuremath{\approx}15$ at 100 Mpc, whereas for binaries with total mass larger than $2\ifmmode\times\else\texttimes\fi{}{10}^{6}{M}_{\ensuremath{\bigodot}}$ (sources for space-based detectors) the SNR can be $\ensuremath{\approx}{10}^{4}$ at 1 Gpc.