Perturbative extraction of gravitational waveforms generated with numerical relativity

We derive an analytical expression for extracting the gravitational waveforms at null infinity using the Weyl scalar ${\psi }_4$ measured at a finite radius. Our expression is based on a series solution in orders of $1/r$ to the equations for gravitational perturbations about a spinning black hole. We compute this expression to order $1/r^2$ and include the spin parameter $a$ of the Kerr background. We test the accuracy of this extraction procedure by measuring the waveform for a merging black-hole binary at ten different extraction radii (in the range $r/M=75–190$ and for three different resolutions in the convergence regime. We find that the extraction formula provides a set of values for the radiated energy and momenta that at finite extraction radii converges towards the expected values with increasing resolution, which is not the case for the “raw” waveform at finite radius. We also examine the phase and amplitude errors in the waveform as a function of observer location and again observe the benefits of using our extraction formula. The leading corrections to the phase are $\mathcal{O}(1/r)$ and to the amplitude are $\mathcal{O}(1/r^2)$. This method provides a simple and practical way of estimating the waveform at infinity, and may be especially useful for scenarios such as well separated binaries, where the radiation zone is far from the sources, that would otherwise require extended simulation grids in order to extrapolate the raw waveform to infinity. Thus this method saves important computational resources and provides an estimate of errors.

Figure 1

The energy radiated (adding up to $\ell =6$) as a function of the observer location $M/R_{\mathrm{obs}}=1/75$, $1/80.4$, $1/86.7$, $1/94.0$, $1/102.6$, $1/113.0$, $1/125.7$, $1/141.7$, $1/162.3$, $1/190.0$ for the directly extracted waveform, labeled as “Raw” (left) and for the analytically extrapolated waveform, labeled as “Perturbative $1/r$” (center) and “Perturbative $1/r^2$” (right).

Figure 2

The angular momentum radiated (adding up to $\ell =6$) as a function of the observer location $M/R_{\mathrm{obs}}=1/75$, $1/80.4$, $1/86.7$, $1/94.0$, $1/102.6$, $1/113.0$, $1/125.7$, $1/141.7$, $1/162.3$, $1/190.0$ for the directly extracted waveform, labeled as Raw (left) and for the analytically extrapolated waveform, labeled as Perturbative $1/r$ (center) and Perturbative $1/r^2$ (right).

Figure 3

The linear momentum radiated (adding up to $\ell =6$) as a function of the observer location $M/R_{\mathrm{obs}}=1/75$, $1/80.4$, $1/86.7$, $1/94.0$, $1/102.6$, $1/113.0$, $1/125.7$, $1/141.7$, $1/162.3$, $1/190.0$ for the directly extracted waveform, labeled as Raw (left) and for the analytically extrapolated waveform, labeled as Perturbative $1/r$ (center) and Perturbative $1/r^2$ (right).

Figure 4

“$\infty$” is the extrapolated NR waveform (phase and amplitude) at highest resolution using 10 radii equally spaced between 75 and $190M$. We compare the above best extrapolated waveform, $\infty$, with that directly extracted at $R_{\mathrm{obs}}=190M$, 1st order, $(1/r)$, perturbative extrapolation (PE), and 2nd order, $(1/r^2)$, PE. In all cases the mode $(\ell ,m)=(2,2)$ is displayed. The first row is the Weyl scalar $rM{\mathrm{\Psi }}_4^{22}$ and the second row is the gravitational strain $r{h}^{22}/M$.

Figure 5

Comparison of the best extrapolated waveform, $\infty$, with that directly extracted at $R_{\mathrm{obs}}=190M$, 1st order, $(1/r)$, PE, and 2nd order, $(1/r^2)$, PE, calculating the absolute value of the difference ($\infty$-waveform) for the phase and amplitude. The mode $(\ell ,m)=(2,2)$ is displayed.

Figure 6

$(\ell ,m)=(2,2)$ mode of the raw (NR) and first (EX1) and second (EX2) perturbative extrapolated waveforms as a function of radius versus the waveform extrapolated to infinity. Displayed is the mean of the % difference between perturbative and extrapolated for each radii for the amplitude and phase for the times between $400M$ and $800M$.