Comparison of numerical and post-Newtonian waveforms for generic precessing black-hole binaries

We compare waveforms and orbital dynamics from the first long-term, fully nonlinear, numerical simulations of a generic black-hole binary configuration with post-Newtonian (PN) predictions. The binary has mass ratio q∼0.8 with arbitrarily oriented spins of magnitude S1/m12∼0.6 and S2/m22∼0.4 and orbits 9 times prior to merger. The numerical simulation starts with an initial separation of r≅11M and orbital parameters determined by 2.5 PN and 3.5 PN evolutions of a quasi-circular binary starting from r=50M. The resulting binaries have very little eccentricity according to the 2.5 PN and 3.5 PN systems, but show eccentricities of e∼0.01-0.02 and e∼0.002-0.005 in the respective numerical simulations, thus demonstrating that 3.5 PN significantly reduces the eccentricity of the binary compared to 2.5 PN. We perform three numerical evolutions from r≅11M with maximum resolutions of h=M/48, M/53.3, M/59.3, to verify numerical convergence. We observe a reasonably good agreement between the PN and numerical waveforms, with an overlap of nearly 99% for the first six cycles of the (l=2, m=±2) modes, 91% for the (l=2, m=±1) modes, and nearly 91% for the (l=3, m=±3) modes. The phase differences between numerical and post-Newtonian approximations appear to be independent of the (l,m) modes considered and relatively small for the first 3-4 orbits. An advantage of the 3.5 PN model over the 2.5 PN one seems to be observed, which indicates that still higher PN order (perhaps even 4.0 PN) may yield significantly better waveforms. In addition, we identify features in the waveforms likely related to precession and precession-induced eccentricity.