Where post-Newtonian and numerical-relativity waveforms meet

TL;DR

The paper assesses how accurately post-Newtonian waveforms can reproduce numerical-relativity waveforms for an equal-mass, nonspinning binary by directly comparing NR data spanning nine orbits before merger to PN templates (TaylorT1 3.5PN and TaylorT3 3PN). Using high-precision NR simulations, phase is found to be matchable within the numerical uncertainty over a significant frequency range ($M\omega\approx 0.0455$ to $0.1$) while the restricted 3.5PN amplitude overestimates NR by about 6% unless higher-order amplitude terms are included. Incorporating a 2.5PN amplitude reduces discrepancies to the NR uncertainty up to about 11 cycles before merger, and the authors argue that only about 4.5 NR orbits may be sufficient for PN/NR matching in such systems, with implications for generating efficient gravitational-wave templates. The work also highlights how PN dynamics align with full GR coordinate motion up to roughly three orbits before merger, informing both data-analysis workflows and future studies extending to unequal-mass or spinning binaries.

Abstract

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 $Mω= 0.0455$ 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.

Figures (18)

• Figure 1: Convergence of the phase $\phi(t)$. Differences between simulations with $N = 64,72,80$ (see Table \ref{['tab:simulations']}) are scaled assuming sixth-order convergence. The convergence of the phase is shown as both a standard and a logarithmic plot, to demonstrate that good sixth-order convergence is seen throughout the simulation, except after merger, when there is a slight drop in convergence. In the logarithmic plot the solid and dashed lines are so close as to be almost indistinguishable.
• Figure 2: Convergence of the amplitude $A(\phi)$. Differences between simulations with $N = 64,72,80$ (see Table \ref{['tab:simulations']}) are scaled assuming sixth-order convergence. The $x$-axis shows $\phi/(4\pi)$, which gives a rough estimate of the number of orbits the system has completed (at least before merger). The phase $\phi$ is chosen to be zero at $t=0$. The convergence of the amplitude is shown in terms of relative (percentage) errors, to allow easier comparison with later results. A vertical line indicates the point at which we end our PN comparison in Section \ref{['sec:comparison']}. The lower plot zooms into the region that will be used for PN comparison.
• Figure 3: Same as Figure \ref{['fig:Conv2']}, but using $A(t)$ instead of $A(\phi)$. We see that the errors are far larger than for $A(\phi)$; the maximum error is now around 60%, while it was only 8% when we considered $A(\phi)$.
• Figure 4: Error in the Richardson-extrapolated functions $\phi(t)$ and $A(\phi)$. For the range of the simulations that will be compared with PN waveforms, the uncertainty in $\phi(t)$ is below 0.01 radians at most times, and the uncertainty in the amplitude is less than 0.5%. At earlier times ($t < 500M$, which are also nominally included in the PN comparison), these plots are dominated by noise and the uncertainty grows by a factor of ten.
• Figure 5: The wave amplitude $A$ as a function of extraction radius $R_{ex}$, at $\phi = 8\pi$, which corresponds to $t \approx 715M$ for the wave extracted at $R_{ex} = 90M$. The solid line shows a curve fit of the form (\ref{['eqn:AmpFalloff']}). The dashed line shows a curve fit with an extra $1/R_{ex}^3$ term. The horizontal solid and dashed lines show the corresponding $R_{ex} \rightarrow \infty$ limits of the two curve fits; our uncertainty estimate in the extrapolation of the amplitude comprises the difference of these two values.