High accuracy simulations of black hole binaries: Spins anti-aligned with the orbital angular momentum

High-accuracy binary black hole simulations are presented for black holes with spins anti-aligned with the orbital angular momentum. The particular case studied represents an equal-mass binary with spins of equal magnitude $S/m^2=0.43757\pm 0.00001$. The system has initial orbital eccentricity $\sim 4\times {10}^{-5}$, and is evolved through 10.6 orbits plus merger and ringdown. The remnant mass and spin are $M_f=(0.961109\pm 0.000003)M$ and $S_f/M_f{}^2=0.54781\pm 0.00001$, respectively, where $M$ is the mass during early inspiral. The gravitational waveforms have accumulated numerical phase errors of $\lesssim 0.1$ radians without any time or phase shifts, and $\lesssim 0.01$ radians when the waveforms are aligned with suitable time and phase shifts. The waveform is extrapolated to infinity using a procedure accurate to $\lesssim 0.01$ radians in phase, and the extrapolated waveform differs by up to 0.13 radians in phase and about 1% in amplitude from the waveform extracted at finite radius $r=350M$. The simulations employ different choices for the constraint damping parameters in the wave zone; this greatly reduces the effects of junk radiation, allowing the extraction of a clean gravitational wave signal even very early in the simulation.

Figure 1

Irreducible mass (top panel) and spin (bottom panel) of the black holes during the relaxation of the initial data to the equilibrium (steady-state) inspiral configuration. Shown are four different numerical resolutions, N1 (lowest) to N4 (highest), cf. Table . Up to $t\sim 10M$, both mass and spin change by a few parts in ${10}^4$, then they remain approximately constant (as indicated by the dashed horizontal lines) until shortly before merger. These steady-state values are used to define $M$ and $\chi$.

Figure 2

Coordinate trajectories of the centers of the apparent horizons represented by the blue and red curves, up until the formation of a common horizon. The closed curves show the coordinate shapes of the corresponding apparent horizons.

Figure 3

Constraint violations of runs on different resolutions. The top panel shows the $L^2$ norm of all constraints, normalized by the $L^2$ norm of the spatial gradients of all dynamical fields. The bottom panel shows the same data, but without the normalization factor. The $L^2$ norms are taken over the portion of the computational volume that lies outside apparent horizons. Note that the time when we change the gauge before merger, $t_g\sim 2370M$, and the time when we regrid onto a new single-hole domain after merger, $t_m\sim 2400M$, are slightly different for different resolutions.

Figure 4

Dimensionless spins $\chi$ of one black hole in the N4 evolution, evaluated using an approximate Killing vector, a coordinate rotation vector $-{\partial }_{\varphi }$, or the extrema of the instrinsic scalar curvature on the apparent horizon. Bottom panels show detail at early and late times. Also shown are the time of gauge change $t_g$ before merger, and the time $t_m$ that we transition to a single-hole evolution just after merger.

Figure 5

Sum of Christodoulou masses $M(t)$ and sum of irreducible masses $M_{\mathrm{irr}}(t)$ of the two black holes during inspiral. The data is from the N4 evolution, and uses ${\chi }_{\mathrm{AKV}}$ when computing $M(t)$. Insets show detail at late times, and indicate the transition times $t_g$ and $t_m$.

Figure 6

Dimensionless spins $\chi (t)$ of the final black hole in the N4 evolution. The (most reliable) spin diagnostic ${\chi }_{\mathrm{AKV}}$ starts at $\sim 0.4$ and increases to its final value ${\chi }_f=0.54781\pm 0.00001$. The other spin diagnostics are unreliable for the highly distorted black hole shortly after merger, but subsequently approach ${\chi }_{\mathrm{AKV}}$.

Figure 7

The top panel shows the Christodoulou mass $M_f(t)$ of the final black hole in the N4 and N3 runs, computed using ${\chi }_{\mathrm{AKV}}(t)$. The bottom panel shows the irreducible mass $M_{\mathrm{irr},\mathrm{f}}(t)$.

Figure 8

Gravitational waveform extracted at finite radius $r=350M$ for the N4 evolution. The top panel zooms in on the inspiral waveform, and the bottom panel zooms in on the merger and ringdown.

Figure 9

Convergence of gravitational waveforms with numerical resolution. Shown are phase and amplitude differences between numerical waveforms ${\Psi }_4^{22}$ computed using different numerical resolutions. All waveforms are extracted at $r=350M$, and no time shifting or phase shifting is done to align waveforms.

Figure 10

Convergence of gravitational waveforms with numerical resolution. The same as Fig. 9 except all other waveforms are time- shifted and phase shifted to best match the waveform of the N4 run.

Figure 11

Convergence of extrapolation to infinity for extrapolation of order $n$. For each $n$, plotted is the extrapolated waveform from N4 using order $n+1$ minus the extrapolated waveform using order $n$. The top panel shows phase differences, the bottom panel shows amplitude differences. No shifting in time or phase has been done for this comparison.

Figure 12

Late-time phase convergence of extrapolation to infinity. The same as the top panel of Fig. 11, except zoomed to late times. The peak amplitude of the waveform occurs at $t_s-r^{*}=2410.6M$.

Figure 13

Phase and relative amplitude differences between extrapolated and extracted waveforms for N4. The extracted waveform is extracted at coordinate radius $r=350M$. The waveforms are time-shifted and phase-shifted to produce the best least-squares match.

Figure 14

Final waveform, extrapolated to infinity. The top panels show the real part of ${\Psi }_4^{22}$ with a linear y-axis, the bottom panels with a logarithmic y-axis. The right panels show an enlargement of merger and ringdown.

Figure 15

Comparison of waveform extrapolation between the current simulation of counter-rotating black holes (top panel), and the earlier simulation of nonspinning black holes [9, 60]. The noise is significantly reduced in the newer simulation, due to smaller constraint damping parameters in the wave zone.