Where post-Newtonian and numerical-relativity waveforms meet

We analyze numerical-relativity (NR) waveforms that cover nine orbits (18 gravitational-wave cycles) before merger of an equal-mass system with low eccentricity, with numerical uncertainties of 0.25 radians in the phase and less than 2% in the amplitude; such accuracy allows a direct comparison with post-Newtonian (PN) waveforms. We focus on one of the PN approximants that has been proposed for use in gravitational-wave data analysis, the restricted 3.5PN TaylorT1'' waveforms, and compare these with a section of the numerical waveform from the second to the eighth orbit, which is about one and a half orbits before merger. This corresponds to a gravitational-wave frequency range of $M\omega = 0.0455$ to 0.1. Depending on the method of matching PN and NR waveforms, the accumulated phase disagreement over this frequency range can be within numerical uncertainty. Similar results are found in comparisons with an alternative PN approximant, 3PN TaylorT3’'. The amplitude disagreement, on the other hand, is around 6%, but roughly constant for all 13 cycles that are compared, suggesting that only 4.5 orbits need be simulated to match PN and NR waves with the same accuracy as is possible with nine orbits. If, however, we model the amplitude up to 2.5PN order, the amplitude disagreement is roughly within numerical uncertainty up to about 11 cycles before merger.