Eccentric binary black-hole mergers: The transition from inspiral to plunge in general relativity

We study the transition from inspiral to plunge in general relativity by computing gravitational waveforms of nonspinning, equal-mass black-hole binaries. We consider three sequences of simulations, starting with a quasicircular inspiral completing 1.5, 2.3 and 9.6 orbits, respectively, prior to coalescence of the holes. For each sequence, the binding energy of the system is kept constant and the orbital angular momentum is progressively reduced, producing orbits of increasing eccentricity and eventually a head-on collision. We analyze in detail the radiation of energy and angular momentum in gravitational waves, the contribution of different multipolar components and the final spin of the remnant, comparing numerical predictions with the post-Newtonian approximation and with extrapolations of point-particle results. We find that the motion transitions from inspiral to plunge when the orbital angular momentum $L=L_{\mathrm{crit}}\simeq 0.8M^2$. For $L<L_{\mathrm{crit}}$ the radiated energy drops very rapidly. Orbits with $L\simeq L_{\mathrm{crit}}$ produce our largest dimensionless Kerr parameter for the remnant, $j=J/M^2\simeq 0.724\pm 0.13$ (to be compared with the Kerr parameter $j\simeq 0.69$ resulting from quasicircular inspirals). This value is in good agreement with the value of 0.72 reported in [I. Hinder, B. Vaishnav, F. Herrmann, D. Shoemaker, and P. Laguna, Phys. Rev. D 77, 081502 (2008).]. These conclusions are quite insensitive to the initial separation of the holes, and they can be understood by extrapolating point-particle results. Generalizing a model recently proposed by Buonanno, Kidder and Lehner [A. Buonanno, L. E. Kidder, and L. Lehner, Phys. Rev. D 77, 026004 (2008).] to eccentric binaries, we conjecture that (1) $j\simeq 0.724$ is close to the maximal Kerr parameter that can be obtained by any merger of nonspinning holes, and (2) no binary merger (even if the binary members are extremal Kerr black holes with spins aligned to the orbital angular momentum, and the inspiral is highly eccentric) can violate the cosmic censorship conjecture.

Figure 1

The PN eccentricity parameters $e_t$, $e_r$ and $e_{\varphi }$ for an equal-mass binary with binding energy $E_{\mathrm{b}}$ are shown as functions of the orbital angular momentum $L/M^2$ for ADM-type coordinates and harmonic coordinates. The left panel shows the result for a binding energy corresponding to our sequence 1, the right panel that obtained for a much smaller binding energy.

Figure 2

Upper panels: convergence analysis of the $P/M=0.1247$, quasicircular model of sequence 2, using resolutions $h=1/48$, $1/44$ and $1/40$. The left panel shows the differences in the ($\ell =2$, $m=2$) multipole of $M_{\mathrm{ADM}}r{\psi }_{22}$, the right panel the total radiated energy and $z$-component of the angular momentum. In both cases, the differences between the higher resolution simulations have been rescaled by $Q_4$ for the expected fourth-order convergence. Middle panels: same, but for the $P/M=0.08$, eccentric model of sequence 1. Lower panels: same, but for the unequal-mass model listed at the bottom of Table . For clarity, we present the convergence of ${\Psi }_4$ using phase and amplitude instead of the real part. For this configuration, we observe second-order convergence. The corresponding convergence factors for the resolutions used here are $Q_4=1.58$ and $Q_2=0.72$.

Figure 3

Modulus of the real part of the ($\ell =2$, $m=2$), ($\ell =2$, $m=0$) and ($\ell =4$, $m=4$) components of the waveforms. We show waveforms from four representative simulations: sequence 2 runs with $P/M=0$ (head-on limit), $P/M=0.02$, $P/M=0.04$ and $P/M=0.1247$ (quasicircular case).

Figure 4

Trajectories of the models with $P/M=0.02$, 0.06, 0.10 and 0.1247 of sequence 2. The trajectory of one hole only is shown in each case. The positions of the respective second holes follow from symmetry across the origin. The $+$ denote the locations of the individual holes at the time of common apparent horizon formation.

Figure 5

Frequency and phase obtained from the $\ell =2$, $m=2$ multipole of the gravitational radiation as well as the puncture trajectory for models $P/M=0.06$ and 0.1247 of sequence 2 (upper panels) and $P/M=0.06$ and 0.0850 of sequence 3 (lower panels). The dotted vertical lines mark the formation of the apparent horizon.

Figure 6

$\ell =m=2$ components of the low-$L$ (or low-$P$) waveforms and their polarization for various models of sequence 2. For $L=0$ the imaginary part of the waveform is zero within the noise level (i.e., the cross component is zero for symmetry reasons). As $L$ increases, the polarization becomes circular. Spikes in $P_E$ at early times are due to the initial data burst, and spikes at late times are due to boundary reflection noise and low strength of the signal.

Figure 7

Multipolar energy distribution as a function of the initial momentum $P$ for sequence 1 (upper left), sequence 2 (upper right) and sequence 3 (lower). $E_{\ell }$ denotes the energy radiated in all multipoles with indices $\ell$ and $m=-\ell ,\dots ,\ell$. We remove the initial data burst by ignoring all data with $t<r_{\mathrm{ext}}+50M$.

Figure 8

Final angular momentum as a function of the initial momentum $P$ for sequence 1 (solid line, crosses), 2 (dashed line, plus) and 3 (dotted line, diamonds).

Figure 9

Left panel: predicted angular momentum by point-particle extrapolation of results from perturbation theory. Solid, dashed and dash-dotted lines assume a point-particle eccentricity $e_p=0$, 0.5 and 0.9, respectively (see appendix for a definition of this eccentricity parameter, not to be confused with the Newtonian eccentricity). Numbers next to each set of lines denote different values of $j_i=j_1=j_2$, and the initial spin on each hole is assumed to be aligned with the orbital angular momentum. Right panel: we assume the initial spin on each hole to be antialigned with the orbital angular momentum ($j_i=j_1=j_2<0$). For various eccentricities (as indicated in the legend) we show the functional dependence between mass ratio and $j_i$ needed to produce a final nonspinning hole.

Figure 10

Prony estimates for the quality factor $Q$ (left) and dimensionless oscillation frequency $M{\omega }_R$ (right) of the fundamental mode with $\ell =m=2$. Different line styles (see the legend in the left panel) refer to different models in sequence 1.

Figure 11

Angular momentum estimate $j_{\mathrm{fin}}$ from Prony fits. Horizontal lines are estimates of the final Kerr parameter from energy balance arguments (see Table ).