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 2% in the amplitude; such accuracy allows a direct comparison with post-Newtonian (PN) waveforms. We focus on waveforms predicted by one of the PN approximants that has been proposed for use in gravitational-wave data analysis, restricted 3.5PN TaylorT1, 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 for the purpose of producing “hybrid waveforms,” 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, numerical and post-Newtonian amplitude disagreement is close to the numerical error of 2%.

Figure 1

Convergence of the phase $\varphi (t)$. Differences between simulations with $N=64$, 72, 80 (see Table ) 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 $A(\varphi )$. Differences between simulations with $N=64$, 72, 80 (see Table ) are scaled assuming sixth-order convergence. The $x$-axis shows $\varphi /(4\pi )$, which gives a rough estimate of the number of orbits the system has completed (at least before merger). The phase $\varphi$ is chosen to be in the interval $[-\pi ,\pi ]$ 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 Sec. 5. The plot on the right zooms into the region that will be used for PN comparison.

Figure 3

Same as Fig. 2, but using $A(t)$ instead of $A(\varphi )$. We see that the differences are far larger than for $A(\varphi )$; the maximum difference is now around 60%, while it was only 8% when we considered $A(\varphi )$.

Figure 4

Error in the Richardson-extrapolated functions $\varphi (t)$ and $A(\varphi )$. For the range of the simulations that will be compared with PN waveforms, the uncertainty in $\varphi (t)$ 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 10.

Figure 5

The wave amplitude $A$ as a function of extraction radius $R_{\mathrm{ex}}$, at $\varphi =8\pi$, which corresponds to $t\approx 715M$ for the wave extracted at $R_{\mathrm{ex}}=90M$. The solid line shows a curve fit of the form (4). The dashed line shows a curve fit with an extra $1/R_{\mathrm{ex}}^3$ term. The horizontal solid and dashed lines show the corresponding $R_{\mathrm{ex}}\to \infty$ limits of the two curve fits; our uncertainty estimate in the extrapolation of the amplitude comprises the difference of these two values.

Figure 6

Error in the $R_{\mathrm{ex}}\to \infty$ extrapolated function $A(\varphi )$. For the range of the simulations that will be compared with PN waveforms, the uncertainty in the amplitude is less than 2%.

Figure 7

Final waveforms from the D10, D11, and D12 runs, produced by the method discussed in the text and shifted in time so that their amplitude maxima occur at the same time. The three lines are not individually labeled; the main point is that their differences are almost indistinguishable, except in the last cycle before merger; see text. The phase disagreement with the D12 simulation is shown in the lower panel.

Figure 8

Numerical (solid line) and TaylorT1 3.5PN (dashed line) waveforms $r{\Psi }_{4,22}$ for equal-mass inspiral.

Figure 9

The wave frequency $\omega$ as a function of time for the D12 NR and TaylorT1 3.5PN waveforms. The frequencies agree at $t=347.4M$, when $M\omega =0.0455$. The percentage disagreement between the two is shown in the panel on the right. Also shown is the frequency disagreement when matching is done at $M\omega =0.1$, $t=1772M$.

Figure 10

The disagreement in the phase between NR waveforms and PN waveforms constructed with the TaylorT1 and TaylorT3 approximants. In the left plot the phase and frequency are matched at $t=1772M$, $M\omega =0.1$. In the right plot they are matched at $t=347.4M$, $M\omega =0.0455$. We see that the relative merits of the two approximants can appear quite different depending on the matching time.

Figure 11

The disagreement in the phase between NR waveforms and PN waveforms constructed with the TaylorT1 and TaylorT3 approximants, but now shown as a function of GW frequency $M\omega$.

Figure 12

NR and restricted PN amplitudes of $r{\Psi }_{4,22}$. The amplitudes are plotted versus $M\omega$, for which the restricted PN amplitude is unique, i.e., independent of the choice of approximant for the phase evolution.

Figure 13

Percentage disagreement between the NR wave amplitude and that of the PN waves with the restricted (quadrupole) and 2.5PN amplitude treatments. At low frequencies the disagreement between the 2.5PN and NR amplitudes is close to the uncertainty in the numerical value. The figure gives the absolute value of the disagreement: for reference, $A_{\mathrm{PN},\mathrm{restricted}}>A_{\mathrm{NR}}$ and $A_{2.5\mathrm{PN}}<A_{\mathrm{NR}}$.

Figure 14

The percentage difference in amplitude between the D12 simulation and the D9, D10, and D11 simulations. The D9, D10, and D11 amplitudes show small oscillations around the D12 value, but recall that the disagreement between the D12 and restricted 3.5PN amplitudes was 6%.

Figure 15

The same quantities as in Fig. 9, this time comparing both the D12 and QC12 wave phase with that of the TaylorT1 3.5PN waves. The disagreement between the 3.5PN and QC phases displays clear oscillations, presumably due to the eccentricity in the QC12 simulation.

Figure 16

Percentage disagreement between restricted PN and NR wave amplitudes, for both D12 and QC12 simulations. The disagreement between the QC12 and 3.5PN wave amplitudes is clearly dominated by eccentricity. The low-eccentricity D12 simulation is necessary to identify the error in the restricted PN wave amplitude.

Figure 17

Orbital coordinate motion of the D12 numerical-relativity evolution compared with a PN evolution with the same initial parameters. In both panels the PN evolution is drawn as a dashed line. Left panel: the separation of the black holes (the puncture position in the full NR case). Right panel: the coordinate angular velocity.

Figure 18

Orbital coordinate motion of the D12 numerical-relativity evolution compared with a PN evolution with the same initial parameters. Dashed line: $({\omega }_{\mathrm{NR}}-{\omega }_{\mathrm{PN}})/{\omega }_{\mathrm{PN}}$, full line: $(D_{\mathrm{NR}}-D_{\mathrm{PN}})/D_{\mathrm{PN}}$.