We compare waveforms and orbital dynamics from the first long-term , fully non-linear , numerical simulations of a generic black-hole binary configuration with post-Newtonian predictions .
The binary has mass ratio q \sim 0.8 with arbitrarily oriented spins of magnitude S _ { 1 } / m _ { 1 } ^ { 2 } \sim 0.6 and S _ { 2 } / m _ { 2 } ^ { 2 } \sim 0.4 and orbits 9 times prior to merger .
The numerical simulation starts with an initial separation of r \approx 11 M , with orbital parameters determined by initial 2.5PN and 3.5PN post-Newtonian evolutions of a quasi-circular binary with an initial separation of r = 50 M .
The resulting binaries have very little eccentricity according to the 2.5PN and 3.5PN systems , but show significant eccentricities of e \sim 0.01 - 0.02 and e \sim 0.002 - 0.005 in the respective numerical simulations , thus demonstrating that 3.5PN significantly reduces the eccentricity of the binary compared to 2.5PN .
We perform three numerical evolutions from r \approx 11 M 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 ( \ell = 2 ,m = \pm 2 ) modes , 91 % for the ( \ell = 2 ,m = \pm 1 ) modes , and nearly 91 % for the ( \ell = 3 ,m = \pm 3 ) modes .
The phase differences between numerical and post-Newtonian approximations appear to be independent of the ( \ell,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 4.0PN ) may yield significantly better waveforms .
In addition , we identify features in the waveforms likely related to precession and precession-induced eccentricity .