Comparing gravitational waveform extrapolation to Cauchy-characteristic extraction in binary black hole simulations

We extract gravitational waveforms from numerical simulations of black hole binaries computed using the Spectral Einstein Code. We compare two extraction methods: direct construction of the Newman-Penrose (NP) scalar ${\Psi }_4$ at a finite distance from the source and Cauchy-characteristic extraction (CCE). The direct NP approach is simpler than CCE, but NP waveforms can be contaminated by near-zone effects—unless the waves are extracted at several distances from the source and extrapolated to infinity. Even then, the resulting waveforms can in principle be contaminated by gauge effects. In contrast, CCE directly provides, by construction, gauge-invariant waveforms at future null infinity. We verify the gauge invariance of CCE by running the same physical simulation using two different gauge conditions. We find that these two gauge conditions produce the same CCE waveforms but show differences in extrapolated-${\Psi }_4$ waveforms. We examine data from several different binary configurations and measure the dominant sources of error in the extrapolated-${\Psi }_4$ and CCE waveforms. In some cases, we find that NP waveforms extrapolated to infinity agree with the corresponding CCE waveforms to within the estimated error bars. However, we find that in other cases extrapolated and CCE waveforms disagree, most notably for $m=0$ “memory” modes.

Figure 1

Spacetime diagram illustrating CCE, with two spatial dimensions suppressed. The Cauchy evolution code advances its solution of Einstein’s equations on successive spatial hypersurfaces $\Sigma$ bounded by the outer boundary $R_B$. The wavy, dashed line on the left represents the small-radius part of the Cauchy simulation, whose details are not important here. The characteristic code advances its solution of Einstein’s equations on successive null hypersurfaces (labeled $u=\mathrm{const}$). It uses data from the Cauchy code on the inner boundary (the world tube labeled by $R_{\Gamma }$) to produce a solution that is valid all the way to ${\mathcal{I}}^{+}$, where gravitational radiation is well defined. The characteristic code requires initial data on the null surface $u_0$.

Figure 2

Phase differences in the extrapolated ${\Psi }_4^{2,2}$ mode between successive Cauchy resolutions for simulation 2 of Table , using extrapolation order $N=5$. The waveforms at different resolutions have been aligned over the interval $[1000M,2000M]$. The maximum amplitude occurs at $t_{\mathrm{ret}}\approx 3952M$, shown here as the dotted vertical line.

Figure 3

Phase differences in the high-resolution, extrapolated ${\Psi }_4^{2,2}$ waveform between different extrapolation orders $N$, for simulation 2 of Table . In the left panel, no alignment has been done; in the right, each pair of waveforms has been aligned over $[1000M,2000M]$. Each curve is the phase difference between a waveform with order $N+1$ and an otherwise identical waveform with order $N$. The maximum amplitude occurs at $t_{\mathrm{ret}}\simeq 3952M$, shown in each plot as a vertical dotted line. Note the difference in vertical scales. Because the frequency is greatest near the merger, small time shifts in the alignment window produce large phase differences in the plot on the right. The two noisy regions in the left panel between $t_{\mathrm{ret}}\approx 3400$ and 3700 correspond to gauge or grid changes in the Cauchy simulation.

Figure 4

Top: Phase differences between ${\Psi }_4^{2,2}$ near the merger, computed using different CCE resolutions (as labeled in Table ), for simulation 2 in Table . Bottom: Ratio of the phase differences from the top panel. We find roughly first-order convergence, i.e. a ratio of about 0.5. All waveforms use the same high-resolution Cauchy data and a world tube radius of $R=385M$. Waveforms are aligned in the interval $[1000M,2000M]$. The maximum amplitude occurs at $t_{\mathrm{ret}}\simeq 3952M$, denoted here by the vertical dotted line.

Figure 5

Phase differences as a function of inverse world tube radius $R$. Each of the 28 circles (both open and closed) is the phase difference, evaluated at $t_{\mathrm{ret}}\simeq 2600M$, between the CCE waveform ${\Psi }_4^{2,2}$ computed with world tube radius $385M$ and the same waveform computed with world tube radius $R$. Solid curves are polynomial fits of different orders in powers of $1/R$. The four closed circles represent the typical world tube radii used in most of the simulations, and the dashed curve shows the second-order fit to just these four points. All waveforms are from the high-resolution run of simulation 2 in Table , using CCE resolution $r2$. Waveform alignment is done using the interval $[1000M,2000M]$.

Figure 6

The CCE $rM{\Psi }_4^{3,2}$ modes computed from four different world tube radii, plotted in the complex plane for times in the approximate interval $[5495M,5565M]$, for Case 4 in Table . The trajectory of each waveform is traversed in a clockwise direction with time, entering on the right and leaving on the bottom. The maximum amplitude [of the (2, 2) mode] occurs at $t_{\mathrm{ret}}\simeq 6722M$. The waveform from each radius has been aligned over $[1000M,2000M]$ with the waveform from the largest radius ($455M$). A jump of $2\pi$ in the phase differences between the waveforms occurs because only the waveform from $R=100M$ encircles the origin (shown as $+$ in the figure).

Figure 7

Cauchy-resolution dependence of ${\Delta }_{\ell ,m}$ [cf. Eq. (19)] between CCE and extrapolated ${\Psi }_4^{2,2}$, shown near peak amplitude at $\simeq 3952M$ for simulation 2 in Table . At each lower Cauchy resolution, the extrapolated waveform is aligned with the high-resolution extrapolated waveform. Then the CCE waveform for each Cauchy resolution is aligned with the corresponding extrapolated waveform and ${\Delta }_{\ell ,m}$ is computed. The differences are nearly independent of resolution. Also shown (labeled “Cauchy error”) is the difference between the extrapolated ${\Psi }_4^{2,2}$ waveforms from the high and medium resolutions.

Figure 8

Magnitude of phase difference between the CCE and the outermost finite-radius ($R=385M$) ${\Psi }_4^{2,2}$ waveforms, for Case 2 in Table . The error bar for the phase of the CCE waveform and the phase difference between extrapolated and CCE waveforms are also shown. The error bar includes the Cauchy error (measured using CCE waveforms), the CCE truncation error, and the CCE initial-data error. Waveforms are aligned over $[1000M,2000M]$. The maximum amplitude occurs at $t_{\mathrm{ret}}\sim 3952M$, indicated by the dotted vertical line.

Figure 9

Phase errors in the ${\Psi }_4^{2,2}$ waveform at ${\mathcal{I}}^{+}$ from various sources, for simulation 2 in Table . Cauchy errors determined from both extrapolated ($C^E$) and CCE ($C^C$) waveforms are shown (see Sec. 4g2). The outermost extraction radius is $R=385M$, and all waveforms are aligned over $[1000M,2000M]$. The maximum amplitude occurs at $t_{\mathrm{ret}}\sim 3952M$, shown here as the dotted vertical line.

Figure 10

Same as Fig. 9 except showing relative amplitude errors instead of phase errors.

Figure 11

Cauchy errors $C^C$ and $C^E$ (see Sec. 4g2) as a function of $(\ell ,m)$ spherical harmonic mode for different simulations in Table . The vertical axis is the error measure ${\Delta }_{\ell ,m}$ of Eq. (19), time-averaged so that each source of error is described by a single number for each $(\ell ,m)$ mode. The horizontal axis represents the spherical harmonic $m$ index; vertical dashed lines separate $\ell =2$, $\ell =3$, and $\ell =4$ modes, and for each $\ell$, every other value of $m$ is labeled on the horizontal axis. The pink bars represent the Cauchy error $C^E$ in the extrapolated waveforms, and the dark green bars represent the Cauchy error $C^C$ in the CCE waveforms.

Figure 12

Same as Fig. 11, but showing multiple sources of error. The Cauchy error shown here is the average of those computed using CCE and extrapolated waveforms, for each $(\ell ,m)$.

Figure 13

The real part of $rM{\Psi }_4^{2,2}$ for both extrapolated and CCE waveforms, for the first two simulations in Table . Waveforms are aligned in the interval $[1000M,2000M]$. The four curves agree very well. Time-averaged differences between these curves are shown in Figs. 16, 17, 18 below.

Figure 14

The real part of $rM{\Psi }_4^{2,0}$ extracted at multiple radii, before extrapolation, for the first simulation in Table . Waveforms are shown only near peak amplitude because they are very small elsewhere. The waveform for each extraction radius $r$ is plotted versus time, rather than $t_{\mathrm{ret}}$, so that waveforms extracted at larger $r$ reach their peak amplitude later. The increase in amplitude with extraction radius $r$ indicates that ${\Psi }_4^{2,0}$ falls off more slowly than $1/r$. We attribute this slow falloff to the gauge condition used for simulation 1. The other simulations, which use a more robust gauge condition, do not exhibit this behavior.

Figure 15

The real part of $rM{\Psi }_4^{2,0}$ for both extrapolated and CCE waveforms, for the first two simulations in Table . Waveforms are aligned in the interval $[1000M,2000M]$. We show only times near the merger because the waveform is very small elsewhere. Although the difference between CCE and extrapolated waveforms for Case 2 is far smaller than for Case 1, even in Case 2 this difference is several times the numerical error. Note that the time-averaged difference shown below in Fig. 18 for Case 2 is dominated by the inspiral portion of the waveforms.

Figure 16

Fractional differences ${ϵ}_{\ell m}$ [cf. Eqs. (20, 21)] between extrapolated ${\Psi }_4^{\ell ,m}$ from physically equivalent simulations with different gauge conditions (i.e., the first two simulations in Table ), as a function of $(\ell ,m)$. The $(\ell ,m)$ modes are labeled as in Fig. 11. Waveforms are aligned in the interval $[1000M,2000M]$.

Figure 17

Fractional differences ${ϵ}_{\ell m}$ [cf. Eqs. (20, 22)] between CCE waveforms from the same simulations as shown in Fig. 16. Labels are the same as Fig. 16, except here the differences are shown on a linear plot.

Figure 18

Fractional differences between extrapolated and CCE ${\Psi }_4^{\ell ,m}$ for all four cases in Table , as a function of $(\ell ,m)$. The $(\ell ,m)$ modes are labeled as in Fig. 11. Waveforms are aligned in the interval $[1000M,2000M]$.

Figure 19

Merger portion of the real part of the extrapolated $rM{\Psi }_4^{3,2}$ mode for the $q=6$ case (simulation 3 in Table ). Divergence of the extrapolated waveform is evident as extrapolation order is increased. Note that the order $N=2$ extrapolated waveform agrees well with CCE in this case. Maximum amplitude (of $rM{\Psi }_4^{2,2}$) is at $t_{\mathrm{ret}}\simeq 4901M$.

Figure 20

Differences between CCE and extrapolated waveforms integrated over the sky and normalized by error bars, computed according to Eq. (24) for the four cases of Table . Curves have been shifted so that the merger in each case is at $t\simeq 0$. For each case, solid lines are computed using all modes up to $L=8$, while dotted lines are the same but with $m=0$ modes omitted. For Cases 3 and 4, the dotted lines are indistinguishable from the corresponding solid lines.