Inspiral, merger and ring-down of equal-mass black-hole binaries

TL;DR

This work uses numerical relativity to simulate equal-mass binary black-hole mergers from inspiral through ring-down, and conducts first-order comparisons with post-Newtonian and effective-one-body analyses. It demonstrates that the inspiral is approximately quasi-circular, followed by a blurred plunge into merger and a ring-down dominated by Kerr quasi-normal modes, with the final black hole characterized by M_f ≈ 0.95 M and a_f/M_f ≈ 0.73 across cases. By fitting QNM content and leveraging EOB-style matching, the authors explore how NR results inform analytical templates and the detectability of signals by ground- and space-based detectors, showing that merger and ring-down extend the observable bandwidth and influence SNR, particularly for high-mass systems. The study highlights the potential for NR-PN hybrids and warns of gauge and initial-data artifacts that limit precision, outlining clear paths for improving waveform models for gravitational-wave astronomy.

Abstract

We investigate the dynamics and gravitational-wave (GW) emission in the binary merger of equal-mass black holes as obtained from numerical relativity simulations. Results from the evolution of three sets of initial data are explored in detail, corresponding to different initial separations of the black holes. We find that to a good approximation the inspiral phase of the evolution is quasi-circular, followed by a "blurred, quasi-circular plunge", then merger and ring down. We present first-order comparisons between analytical models of the various stages of the merger and the numerical results. We provide comparisons between the numerical results and analytical predictions based on the adiabatic Newtonain, post-Newtonian (PN), and non-adiabatic resummed-PN models. From the ring-down portion of the GW we extract the fundamental quasi-normal mode and several of the overtones. Finally, we estimate the optimal signal-to-noise ratio for typical binaries detectable by GW experiments.

Figures (34)

• Figure 1: (left) The orbital motion of one BH from each of the three cases: the dot-dashed (blue) line is $d=13$, the solid (black) $d=16$ and the dashed (red) $d=19$. The position of the BH is defined as the center of its AH, and the curve ends once an encompassing horizon is found. The eccentricity present in the initial data is particularly evident for the $d=19$ case, though in part this is due to numerical error---see Fig. \ref{['d19_coord_e']}. To aid in the comparison each trajectory was rotated by a constant phase so that they coincide at a coordinate separation of $3M$. (right) The orbital motion of the BH's from the d=16 simulation, showing the coordinate shapes of the AHs at several key moments. Also shown is the location of the co-rotating light ring of the final BH---see the discussion in Sec. \ref{['secmerger']}.
• Figure 2: The ${\rm Re}[_{-2}C_{22}]$ component of the $d=13$ and $d=16$ waveforms, unshifted (top) and shifted (bottom). All resolutions are shown to demonstrate the size of numerical errors in the simulation, and data such as this was used to compute the errors for waveform quantities listed in Table \ref{['tab_res']}.
• Figure 3: In the left panel we show the the ${\rm Re}[_{-2}C_{22}]$ component of the $d=19$ waveform, unshifted (top) and shifted (bottom). The lower resolution data for the $d=19$ case was accidentally deleted. In the right panel we zoom in for a close-up of the inspiral part of the unshifted waveform. From these two results alone it would appear as if the $d=19$ simulations have anomalously good convergence behavior (compare to Fig. \ref{['d13-16_wave']}). However, this is not the case---refer to the discussion in Sec. \ref{['sec_num_res_and_errors']}, and see Figs. \ref{['d19_am']} and \ref{['d19_coord_e']} for other estimators of the convergence of the solution.
• Figure 4: Sum of AH masses (left panel), and the Kerr angular momentum parameter of the final BH (right panel) for the $d=19$ simulations. The angular momentum was estimated using the ratio of polar to equatorial proper circumference of the horizon brandt_seidel; the dynamical horizons estimate ashtekar_krishnan gives similar results modulo the oscillations about the mean. Except near the time of merger the sum of AH masses in the spacetime should be conserved, and similarly at late times for the Kerr parameter. As resolution increases we see the expected trends in these quantities. Note that the "jaggies" in the AH mass estimate is a reflection of AH finder problems in the code, and not irregularities in the underlying solution FP3.
• Figure 5: In the left panel we show the coordinate separation of the BHs as a function of time for the $d=19$ simulations. This plot highlights the eccentricity within the orbit and it also reflects the phasing behavior in the waveform for the $3/4 h$ and $1/2 h$ cases---see Fig. \ref{['d19_wave']}. In the right panel we estimate the eccentricity for the $d=19$ case: shown is a plot of the left hand side of Eq. (\ref{['e1_def']}) together with a fit of the form $a_0+a_1 t + e\,\cos(a_2 t + a_3)$ to the early time behavior of this function. We estimate the eccentricity to be the amplitude of the sinusoidal part of the fitting function.