Joint approach for reducing eccentricity and spurious gravitational radiation in binary black hole initial data construction

At the beginning of binary black hole simulations, there is a pulse of spurious radiation (or junk radiation) resulting from the initial data not matching astrophysical quasi-equilibrium inspiral exactly. One traditionally waits for the junk radiation to exit the computational domain before taking physical readings, at the expense of throwing away a segment of the evolution, and with the hope that junk radiation exits cleanly. We argue that this hope does not necessarily pan out, as junk radiation could excite long-lived constraint violation. Another complication with the initial data is that they contain orbital eccentricity that needs to be removed, usually by evolving the early part of the inspiral multiple times with gradually improved input parameters. We show that this procedure is also adversely impacted by junk radiation. In this paper, we do not attempt to eliminate junk radiation directly, but instead tackle the much simpler problem of ameliorating its long-lasting effects. We report on the success of a method that achieves this goal by combining the removal of junk radiation and eccentricity into a single procedure. Namely, we periodically stop a low resolution simulation; take the numerically evolved metric data and overlay it with eccentricity adjustments; run it through an initial data solver (i.e. the solver receives as free data the numerical output of the previous iteration); restart the simulation; repeat until eccentricity becomes sufficiently low; and then launch the high resolution “production run” simulation. This approach has the following benefits: (1) We do not have to contend with the influence of junk radiation on eccentricity measurements for later iterations of the eccentricity reduction procedure. (2) We reenforce constraints every time the initial data solver is invoked, removing the constraint violation excited by junk radiation previously. (3) The wasted simulation segment associated with the junk radiation’s evolution is absorbed into the eccentricity reduction iterations. Furthermore, (1) and (2) together allow us to carry out our joint-elimination procedure at low resolution, even when the subsequent “production run” is intended as a high resolution simulation.

Figure 1

The solid black curve represents the evolution of $\omega$ in the standard example simulation. The wobbles signify the presence of eccentricity. For comparison, we also display as a dashed red curve the $\omega (t)$ for this simulation after one stop-and-go operation described in Sec. 4, which contains an eccentricity suppression step.

Figure 2

Reproduced from 41. Visualization of junk radiation as the boundary between unrealistic and realistic multipole structures in ${\theta }^t$ contours.

Figure 3

The ($l=2$, $m=2$) mode of the $r{\Psi }_4$ waveform extracted at various coordinate radii $r_{\mathrm{ex}}\ge D+1\mathrm{M}$, with the thick red curve representing $r_{\mathrm{ex}}\approx ƛ$. Curves with peaks further to the right correspond to larger $r_{\mathrm{ex}}$. The four panels describe simulations that are similar to the standard example but with initial separations of $D=10\mathrm{M}$, 15M, 20M and 25M, and corresponding $ƛ=15.8\mathrm{M}$, 29.0M, 44.7M and 62.5M.

Figure 4

(a) Equatorial plane slice of the computational domain in the standard example simulation (but with a nonoverlapping subdomain decomposition for better visual clarity), warped into the paper according to the $r^t$ value. The impact of JR on the subdomain boundaries is to tear them open in the fashion depicted in panel (c). (b) The subdomain boundaries in (a) are shown as a dense concentration of lines in this panel. Their location matches the tearing. (c) A stylized demonstration that under-resolved high frequency JR (denoted by the blue sinusoidal curve) causes a mismatch of data at the subdomain boundary (vertical dashed line). Red and black curves are the representations of the JR in the left and right subdomains, respectively, constructed from sampling at the underpopulated red and black dots.

Figure 5

The constraint violation and waveform near the start of the standard example simulation. The dashed blue and red lines are $r$ times the real and imaginary parts of the ($l=2$, $m=2$) mode of the ${\Psi }_4$ wave extracted at a coordinate radius of 110M. The solid black curve is the $L_2$ norm of the constraint energy GhCe within a spherical shell extending radially from 99M to 120M (i.e. containing the wave extraction sphere). Comparing these curves, we see that constraint violation remains elevated for a long time after the junk radiation (the sharp features at the beginning of the waveforms) has already passed through the spherical shell.

Figure 6

The junk radiation in the $r{\Psi }_4$ waveforms extracted at coordinate radii $r_{\mathrm{ex}}=100\mathrm{M}$ and 380M, denoted by dashed and solid curves, respectively. The expected arrival times, as $r_{\mathrm{ex}}$ minus black hole initial separation, are plotted as vertical lines. The peak of JR stays relatively consistent in its time lag from the expected arrival time, but the superluminally moving leading edge runs further ahead in the case where $r_{\mathrm{ex}}$ is larger.

Figure 7

The dependence on $T_{\min }$ of eccentricity $e$ (left vertical axis) estimated by Eq. (21) and $(\delta {\dot{a}}_0,\delta {\Omega }_0)$ (right vertical axis) suggested by Eq. (24), for the first traditional approach eccentricity reduction iteration (see Sec. 5 below) of the standard example simulation. Even though the direct impact of junk radiation can be avoid by setting $T_{\min }>100\mathrm{M}$, the parameters do not settle down even with $T_{\min }$ at 600M.

Figure 8

Comparison between $ds/dt$ and $d\omega /dt$ for the same simulation as shown in Fig. 7.

Figure 9

The ($l=2$, $m=2$) mode $r{\Psi }_4$ waveforms extracted at a coordinate radius of 380M, from the evolutions using initial data generated with the traditional (left panel) and joint-elimination (right panel) approaches.

Figure 10

Left: Two contours of ${\theta }^t$ in the simulation starting from the traditional approach initial data, after eccentricity has been removed. There is an obvious junk pulse moving outwards. Right: Two contours of ${\theta }^t$ in the simulation starting from the joint-elimination approach initial data. The only visible new junk radiation is a pulse of small disturbance moving from the outer boundary inwards, as highlighted by the arrows. The inset in the black frame shows a continuation of a clean simulation (after initial JR has exited), but with the outer boundary shifted inwards. A new pulse of JR resulting from the mismatch between bulk data and the evolution boundary condition can be seen moving inwards.

Figure 11

The $d\omega /dt$ curves for the initial and subsequent eccentricity reduction iterations, with both the traditional approach and the joint-elimination approach shown. With both approaches, the oscillations in $d\omega /dt$ become dominated by frequencies higher than the orbital one during the second iteration.

Figure 12

Comparison between the dependence on $T_{\min }$ of $e$ (left panel), $\delta \Omega$ (middle panel) and $\delta \dot{a}$ (right panel) for the traditional and joint-elimination approaches. These parameters are extracted during the first eccentricity reduction iteration. For better visual comparison between curves, the time axis for the joint-elimination approach has been shifted such that it appears to start at $t=0$. The simulations in both approaches are evolved for a sufficiently long time to accommodate $T_{\max }$ for all $T_{\min }$ used in this figure.

Figure 13

The $L_2$ norm of the GhCe constraint measurement within individual subdomains, calculated at the beginning of the first iteration in the joint-elimination approach. For comparison, we also show the constraint violation just before the stop-and-go operation, i.e. before the initial data solver is invoked. The horizontal axis is the index for the subdomains. The sectors separated by vertical lines contain, from left to right, inner spherical shells around one of the black holes extending from the excision boundary to 4.5M, similar shells around the other black hole, some cylindrical shapes filling up the region in between and surrounding the inner spherical shells, and finally outer spherical shells extending from 45M to 480M that enclose all of the aforementioned subdomains.

Figure 14

The evolution of $L_2(\mathrm{GhCe})$ broken down to individual subdomains, for the joint-elimination (left panel) and traditional (right panel) approaches. In both cases, the constraint violation asymptotes to the levels sustainable by the (shared) resolution of the simulations. During the intermediate time, constraint violation is smaller for the joint-elimination case due to the reduction in JR. The curves near the bottom of the plots correspond to outer spherical shell subdomains.