Extrapolating gravitational-wave data from numerical simulations

Two complementary techniques are developed for obtaining the asymptotic form of gravitational-wave data at large radii from numerical simulations, in the form of easily implemented algorithms. It is shown that, without extrapolation, near-field effects produce errors in extracted waveforms that can significantly affect LIGO data analysis. The extrapolation techniques are discussed in the context of Newman-Penrose data applied to extrapolation of waveforms from an equal-mass, nonspinning black-hole binary simulation. The results of the two methods are shown to agree within error estimates. The various benefits and deficiencies of the methods are discussed.

Figure 1

Data-analysis mismatch between finite-radius waveforms and the extrapolated waveform for initial LIGO. This plot shows the mismatch between extrapolated waveforms and waveforms extracted at several finite radii, scaled to various values of the total mass of the binary system, using the initial-LIGO noise curve. The waveforms are shifted in time and phase to find the optimal match. Note that the data used here are solely numerical, with no direct post-Newtonian contribution. Thus, for masses below $40M_{\odot }$, these data represent only a portion of the physical waveform.

Figure 2

Data-analysis mismatch between finite-radius waveforms and the extrapolated waveform for advanced LIGO. This plot shows the mismatch between extrapolated waveforms and waveforms extracted at several finite radii, scaled to various values of the total mass of the binary system, using the advanced-LIGO noise curve. The waveforms are shifted in time and phase to find the optimal match. Note that the data used here are solely numerical, with no direct post-Newtonian contribution. Thus, for masses below $110M_{\odot }$, these data represent only a portion of the physical waveform.

Figure 3

Convergence of the amplitude of the extrapolated ${\Psi }_4$, with increasing order of the extrapolating polynomial, $N$. This figure shows the convergence of the relative amplitude of the extrapolated Newman-Penrose waveform, as the order $N$ of the extrapolating polynomial is increased. [See Eq. (16).] That is, we subtract the amplitudes of the two waveforms, and normalize at each time by the amplitude of the second waveform. We see that increasing the order tends to amplify the apparent noise during the early and late parts of the waveform. Nonetheless, the broad (low-frequency) trend is towards convergence. Note that the differences decrease as the system nears merger; this is a first indication that the extrapolated effects are due to near-field influences. Also note that the horizontal axis changes in the right part of the figure, which shows the point of merger, and the ringdown portion of the waveform. After the merger, the extrapolation is nonconvergent, though the differences grow slowly with the order of extrapolation.

Figure 4

Convergence of the phase of the extrapolated ${\Psi }_4$, with increasing order of the extrapolating polynomial, $N$. This figure is the same as Fig. 3, except that it shows the convergence of phase. Again, increasing the extrapolation order tends to amplify the noise during the early and late parts of the waveform, though the broad (low-frequency) trend is towards convergence. The horizontal-axis scale changes just before merger.

Figure 5

Convergence of the phase of ${\Psi }_4$, extrapolated with no correction for the dynamic lapse. This figure is the same as Fig. 4, except that no correction is made to account for the dynamic lapse. [See Eq. (14) and surrounding discussion.] Observe that the convergence is very poor after merger (at roughly $t_{\mathrm{ret}}/M_{\mathrm{irr}}=3954$). This corresponds to the time after which sharp features in the lapse are observed. We conclude from this graph and comparison with the previous graph that the correction is crucial to convergence of ${\Psi }_4$ extrapolation through merger and ringdown.

Figure 6

Comparison of extrapolation of ${\Psi }_4$ using different sets of extraction radii. This figure compares the phase of waveforms extrapolated with various sets of radii. All comparisons are with respect to the data set used elsewhere in this paper, which uses extraction radii $R/M_{\mathrm{irr}}=\{75,85,100,110,120,\dots ,200,210,225\}$. The order of the extrapolating polynomial is $N=3$ in all cases.

Figure 7

Convergence of the amplitude of ${\Psi }_4$ extrapolated using the wave phase, with increasing order $N$ of the extrapolating polynomial. This figure shows the convergence of the relative amplitude of the extrapolated Newman-Penrose waveform extrapolated using the wave phase, as the order $N$ of the extrapolating polynomial is increased. [See Eq. (18).] Increasing the extrapolation order tends to amplify the apparent noise during the early and late parts of the waveform, but it improves convergence. The vertical axis at $t_{\mathrm{ret}}/M_{\mathrm{irr}}\approx 3950$ denotes merger.

Figure 8

Convergence of the phase of ${\Psi }_4$ as a function of time extrapolated using the wave phase, with increasing order $N$ of the extrapolating polynomial. Again, increasing the extrapolation order tends to amplify the apparent noise during the early and late parts of the waveform, though convergence is improved significantly.

Figure 9

Relative difference in the amplitude of the two extrapolation methods as we increase the order of extrapolation. The best agreement between both methods is at low orders of extrapolation, for which the relative difference in the amplitude is less than 0.1% during most of the evolution.

Figure 10

Phase difference of the two extrapolation methods as we increase the order of extrapolation. This figure shows the phase difference between waveforms extrapolated using each of the two methods. The best agreement between the methods is at orders $N=2$ and 3. The relative difference in the phase is less than 0.02 radians during most of the evolution.

Figure 11

Difference between the filtered and unfiltered amplitude and phase of the waveform with third-order extrapolation. The upper panel shows the relative amplitude difference between the filtered and unfiltered waveforms; the lower panel shows the phase difference.

Figure 12

The filtered version of Fig. 4. We filtered the extrapolated waveforms and redid Fig. 4, which shows the phase difference between waveforms extrapolated at various orders. This plot shows much smaller high-frequency components at early times.