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 Padé 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 40M (sources for ground-based detectors) are brought in band and can be detected with signal-to-noise up to 15 at 100 Mpc, whereas for binaries with total mass larger than 2 10 6 M (sources for space-based detectors) the SNR can be 10 4 at 1 Gpc.

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Numerical simulations of 15 orbits of an equal-mass binary black-hole system are presented. Gravitational waveforms from these simulations, covering more than 30 cycles and ending about 1.5 cycles before merger, are compared with those from quasicircular zero-spin post-Newtonian (PN) formulae. The cumulative phase uncertainty of these comparisons is about 0.05 radians, dominated by effects arising from the small residual spins of the black holes and the small residual orbital eccentricity in the simulations. Matching numerical results to PN waveforms early in the run yields excellent agreement (within 0.05 radians) over the first 15 cycles, thus validating the numerical simulation and establishing a regime where PN theory is accurate. In the last 15 cycles to merger, however, generic time-domain Taylor approximants build up phase differences of several radians. But, apparently by coincidence, one specific post-Newtonian approximant, TaylorT4 at 3.5PN order, agrees much better with the numerical simulations, with accumulated phase differences of less than 0.05 radians over the 30-cycle waveform. Gravitational-wave amplitude comparisons are also done between numerical simulations and postNewtonian, and the agreement depends on the post-Newtonian order of the amplitude expansion: the amplitude difference is about 6%-7% for zeroth order and becomes smaller for increasing order. A newly derived 3.0PN amplitude correction improves agreement significantly ( < 1% amplitude difference throughout most of the run, increasing to 4% near merger) over the previously known 2.5PN amplitude terms.

We define and extensively test a set of boundary conditions that can be applied at black hole excision surfaces when the Hamiltonian and momentum constraints of general relativity are solved within the conformal thin-sandwich formalism. These boundary conditions have been designed to result in black holes that are in quasiequilibrium and are completely general in the sense that they can be applied with any conformal three-geometry and slicing condition. Furthermore, we show that they retain precisely the freedom to specify an arbitrary spin on each black hole. Interestingly, we have been unable to find a boundary condition on the lapse that can be derived from a quasiequilibrium condition. Rather, we find evidence that the lapse boundary condition is part of the initial temporal gauge choice. To test these boundary conditions, we have extensively explored the case of a single black hole and the case of a binary system of equal-mass black holes, including the computation of quasi-circular orbits and the determination of the inner-most stable circular orbit. Our tests show that the boundary conditions work well.

Initial data are the starting point for any numerical simulation. In the case of numerical relativity, Einstein’s equations constrain our choices of these initial data. We will examine several of the formalisms used for specifying Cauchy initial data in the 3 + 1 decomposition of Einstein’s equations. We will then explore how these formalisms have been used in constructing initial data for spacetimes containing black holes and neutron stars. In the topics discussed, emphasis is placed on those issues that are important for obtaining astrophysically realistic initial data for compact binary coalescence.

The construction of initial data for black-hole binaries usually involves the choice of free parameters that define the spins of the black holes and essentially the eccentricity of the orbit. Such parameters must be chosen carefully to yield initial data with the desired physical properties. In this paper, we examine these choices in detail for the quasiequilibrium method coupled to apparent-horizon/quasiequilibrium boundary conditions. First, we compare two independent criteria for choosing the orbital frequency, the ''Komar-mass condition'' and the ''effective-potential method,'' and find excellent agreement. Second, we implement quasilocal measures of the spin of the individual holes, calibrate these with corotating binaries, and revisit the construction of nonspinning black-hole binaries. Higher-order effects, beyond those considered in earlier work, turn out to be important. Without those, supposedly nonspinning black holes have appreciable quasilocal spin; furthermore, the Komar-mass condition and effective-potential method agree only when these higher-order effects are taken into account. We compute a new sequence of quasicircular orbits for nonspinning black-hole binaries, and determine the innermost stable circular orbit of this sequence.

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