Adapted gauge to a quasilocal measure of the black holes recoil

We explore different gauge choices in the moving puncture formulation in order to improve the accuracy of a linear momentum measure evaluated on the horizon of the remnant black hole produced by the merger of a binary. In particular, motivated by the study of gauges in which $m\eta$ takes on a constant value, we design a gauge via a variable shift parameter $m\eta (\overrightarrow{r}(t))$. This new parameter takes a low value asymptotically, as well as at the locations of the orbiting punctures, and then takes on a value of approximately 2 at the final hole horizon. This choice then follows the remnant black hole as it moves due to its net recoil velocity. We find that this choice keeps the accuracy of the binary evolution. Furthermore, if the asymptotic value of the parameter $m\eta$ is chosen about or below 1.0, it produces more accurate results for the recoil velocity than the corresponding evaluation of the radiated linear momentum at infinity, for typical numerical resolutions. Detailed studies of an unequal mass $q=m_1/m_2=1/3$ nonspinning binary are provided and then verified for other mass ratios $(q=1/2,1/5)$ and spinning ($q=1$) binary black hole mergers.

Figure 1

The horizon measure of the mass and spin after merger of a $q=1/3$ nonspinning binary versus time for $m\eta =2$ at resolutions n100, n120, and n140.

Figure 2

The Hamiltonian constraint behavior versus time for $m\eta =2$ at resolutions n100, n120, n140 in the top panel, and in the bottom panels, the (2,2) waveform as seen by an observer at $R=113m$ for the $q=1/3$ nonspinning binary from an initial separation $D=9m$, the amplitude difference between resolutions, $\mathrm{\Delta }A/A$, and the phase difference (in radians), $\mathrm{\Delta }\phi$, with respect to the infinite resolution extrapolation $(n_{\infty })$.

Figure 3

The horizon measure of the linear momentum (in km/s) after merger of a $q=1/3$ nonspinning binary for the three resolutions n100 (dotted), n120 (dashed), and n140 (solid) for $\eta =2/m$ (blue), $1/m$ (red), and $0.5/m$ (green). The reference value of $V_f$ (black solid line) is found by extrapolation to infinite resolution of the radiated linear momentum.

Figure 4

The horizon measure of the linear momentum (in km/s) after merger of a $q=1/3$ nonspinning binary lowering values of $\eta =2\to 0$ at resolution n140. The reference value of $V_f$ (black solid line) is found by extrapolation to infinite resolution of the radiated linear momentum.

Figure 5

Comparative results of the ${\partial }_t$ gauge (solid) and ${\partial }_0$ gauge (dashed) for the horizon measure of the linear momentum (in km/s) after merger of a $q=1/3$ nonspinning binary for the n140 resolution for $\eta =2/m$ (blue), $1/m$ (green), and $0.5/m$ (red). The reference value of $V_f$ is found by extrapolation to infinite resolution of the radiated linear momentum.

Figure 6

The horizon measure of the linear momentum (in km/s) after merger of a $q=1/3$ nonspinning binary for the three resolutions labeled as n100, n120, and n140 in the ${\partial }_0$ gauge for $\eta =2/m$. The reference value of $V_f$ is found by extrapolation to infinite resolution of the radiated linear momentum in this ${\partial }_0$ gauge.

Figure 7

The horizon measure of the linear momentum after merger of a $q=1/3$ nonspinning binary for the three resolutions labeled as n100 (dotted), n120 (dashed), and n140 (solid) for $\eta =$ N12 (red) and N10 (blue), top to bottom, respectively. The reference value of $V_f$ is found by extrapolation to infinite resolution of the radiated linear momentum.

Figure 8

${\eta }_{\psi }$ profile for ($m=m_1+m_2=1$ here) $m_1=1/4$, $m_2=3/4$; $x_1=-7.5$, $x_2=2.5$; $a=1$, $b=2$; $A=1$, $B=1$.

Figure 9

${\eta }_G$ profile for ($m=m_1+m_2=1$ here) $m_1=1/4$, $m_2=3/4$; $x_1=-7.5$, $x_2=2.5$; $A=1$, $B=1$, $C=1$; $s_1=2m_1$, $s_2=2m_2$.