Last orbit of binary black holes

We have used our new technique for fully numerical evolutions of orbiting black-hole binaries without excision to model the last orbit and merger of an equal-mass black-hole system. We track the trajectories of the individual apparent horizons and find that the binary completed approximately one and a third orbits before forming a common horizon. Upon calculating the complete gravitational radiation waveform, horizon mass, and spin, we find that the binary radiated $3.2\%$ of its mass and 24% of its angular momentum. The early part of the waveform, after a relatively short initial burst of spurious radiation, is oscillatory with increasing amplitude and frequency, as expected from orbital motion. The waveform then transitions to a typical “plunge” waveform; i.e. a rapid rise in amplitude followed by quasinormal ringing. The plunge part of the waveform is remarkably similar to the waveform from the previously studied “ISCO” configuration. We anticipate that the plunge waveform, when starting from quasicircular orbits, has a generic shape that is essentially independent of the initial separation of the binary.

Figure 1

The puncture trajectories and apparent horizon profiles on the $xy$ plane for the $M/21$ run. The solid and dotted spirals are the puncture trajectories, the solid and dotted ellipsoids are the individual apparent horizons (at every $10M$ of evolution), the dot-dash “peanut shaped” figure is the first detected common horizon, and the dashed spiral is the puncture trajectory for the $M/27$ run. The initial growth of the individual apparent horizon is due to the nonideal (vanishing) initial data for the shift. Note that we track the puncture positions throughout the evolution. The period of the last orbit is around $62M$. The last orbit begins when the punctures are located at $2.6M$ from the origin (in these coordinates).

Figure 2

The $(\ell =2,m=2)$ mode of ${\psi }_4$ at $r=20M$. The top plot shows the waveforms for central resolutions of $M/21$, $M/24$, and $M/27$. The bottom plot shows the differences ${\psi }_4(M/21)-{\psi }_4(M/24)$ and ${\psi }_4(M/24)-{\psi }_4(M/27)$, with the latter rescaled by 1.879 to demonstrate fourth-order convergence. Note the spurious radiation from $t=17M$ to $t=40M$.

Figure 3

The real part of the $(\ell =2,m=2)$ mode of the ${\psi }_4$ at $r=20M$ from this “last orbit” configuration and from the ISCO configuration. Note the near perfect overlap once the ISCO waveform has been translated by $\Delta t/M=94$

Figure 4

The $(\ell =2,m=2)$ component of the strain. Both the $+$ and $\times$ mode are shown. The early time strain is dominated by spurious radiation (from the initial data) up to $t=55M$. Afterwords, the strain shows a gradual transition from orbital motion to a plunge waveform. This transition is less distinct than that in ${\psi }_4$.

Figure 5

The Hamiltonian constraint violation at $t=70M$ along the $y$-axis for the $M/24$ and $M/27$ runs (the latter rescaled by $(27/24{)}^4$. The punctures crossed the $y$-axis for the second time at $t=64M$. Note the near perfect fourth-order convergence. Points contaminated by boundary errors have been excluded. The high frequency violations near the numerical coordinate $y/M=\pm 8$ are due to the extreme fisheye deresolution near the boundary, and converge with resolution.