The last orbit of binary black holes

TL;DR

This work advances numerical relativity for binary black hole mergers by employing a moving-puncture approach within the BSSN formulation to evolve equal-mass binaries without excision. It demonstrates that the last orbit lasts about $1.33$ cycles before a common horizon forms and provides high-precision measurements of radiated energy ($E_{rad}/M \approx 3.3\%$) and angular momentum ($J_{rad}/J \approx 24.7\%$), consistent between horizon geometry and waveform extraction. The plunge part of the waveform closely resembles the ISCO case, and the remnant black hole has spin $\tilde{a} = J_H/M_H^2 \approx 0.673$–$0.688$, with a horizon mass $M_H \approx 0.952 M$ and irreducible mass $M_{irr} \approx 0.8848 M$. These results, supported by fourth-order convergence and robust against boundary placement, enhance gravitational-wave template accuracy for detectors and deepen understanding of horizon dynamics during merger.

Abstract

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.

Figures (5)

• 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 non-ideal (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.
• 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 around $t=26M$.
• 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 $\Delta t/M = 95$
• 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 that 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 from the plot. The high frequency violations near the numerical coordinate $y=\pm8$ are due to the extreme fisheye deresolution near the boundary, and converge with resolution.