Dynamical evolution of quasicircular binary black hole data

We study the fully nonlinear dynamical evolution of binary black hole data, whose orbital parameters are specified via the effective potential method for determining quasicircular orbits. The cases studied range from the Cook-Baumgarte innermost stable circular orbit (ISCO) to significantly beyond that separation. In all cases we find the black holes to coalesce (as determined by the appearance of a common apparent horizon) in less than half an orbital period. The results of the numerical simulations indicate that the initial holes are not actually in quasicircular orbits, but that they are in fact nearly plunging together. The dynamics of the final horizon are studied to determine physical parameters of the final black hole, such as its spin, mass, and oscillation frequency, revealing information about the inspiral process. We show that considerable resolution is required to extract accurate physical information from the final black hole formed in the merger process, and that the quasinormal modes of the final hole are strongly excited in the merger process. For the ISCO case, by comparing physical measurements of the final black hole formed to the initial data, we estimate that less than 3% of the total energy is radiated in the merger process.

Figure 1

Convergence in Hamiltonian constraint for the QC-0 case shown at the time of formation of the common AH, $t=17.76M$. The resolutions are $dx=0.08M,0.06M$ and $0.04M$, with the evaluated constraint scaled appropriately so that the lines would lie on top of each other in the case of perfect second order convergence. The lower scale shows the physical distance along the $x$ axis, while the upper scale shows the grid coordinate $r_{\mathrm{grid}}$ defined by (4.1). The vertical dotted line shows the apparent-horizon position. Note that although we do not obtain second order convergence near the outer boundary, there is good second order convergence in the neighborhood of the horizons.

Figure 2

The time to appearance of a common AH for each of the QC models. Filled circles indicate the results of numerical experiments using $dx=0.06$, from initial proper spatial separations indicated along the $x$-axis. The upper line indicates the expected orbital period, based on the initial angular velocity (Table ) and assuming a Newtonian circular orbit. Empty circles indicate the fraction of an orbit before common AH formation. The error bars show $\pm$ the difference between $dx=0.06$ and $dx=0.08$ evolutions; the effects of variations in gauge parameters and outer-boundary position are generally smaller than these.

Figure 3

The time to common AH formation is compared with various head-on collision results. The upper line (filled circles) repeats the curve of Fig. 2, showing common AH formation times for each of the QC models. The dashed line indicates the Newtonian collision time for a pair of particles falling together from the given distance with zero initial velocity. The corresponding relativistic head-on collisions were simulated and give the results plotted as filled diamonds. Finally, the lower curve (open boxes) refers to the same QC simulations as the upper (filled circle) curve, but uses the proper time at the origin at the time of common AH appearance (see text).

Figure 4

Evolution of the irreducible mass $M_{\mathrm{irr}}$ of the AH as a function of time. Part (a) shows the mass for each QC-N model, for the low resolution $dx=0.08$ evolutions, showing both individual- and common-horizon masses on the same scale. (For the QC-4 model, the individual AHs are briefly “lost” due to the proximity of the excision region, somewhat before a common horizon is found.) Part (b) shows the mass for the QC-0 model for the $dx=0.08M$, $0.06M$, and $0.048M$ evolutions, with individual-horizon masses in the left subplot and common-horizon masses in the right subplot. For the $dx=0.08M$ QC-0 case, results from both apparent-horizon finders are shown, AHFinderDirect as a solid line, AHFinder as points.

Figure 5

Event and apparent horizons for an evolution of the QC-1 model ($dx=0.08M$), showing horizons on the initial slice ($t=0.0M$), when the EH first merges ($t=13.0M$), at the appearance of the first common AH ($t=18.1M$), and at a late time slice ($t=30.0M$) when the system has essentially settled and the two contours cannot be distinguished anymore.

Figure 6

Comparison of event- to apparent-horizon geometry for case QC-1. The ratio of the polar and equatorial horizon circumferences, $C_r=C_p/C_e$, after formation of common horizons, measured in the xz plane, is shown for the low resolution ($dx=0.08M$) case. Both the AH and the EH oscillate at the QNM frequency of the final Kerr BH, as confirmed by the fit shown (see text).

Figure 7

Comparison of $a/m$ measurements for case QC-0. We show the circumference ratio $C_r$, measured on the AH, for 3 different resolutions for the two different polar circumferences (computed in the $xz$ and $yz$ plane). We also show the results of the fitting procedure to determine an offset value of $C_r$ (straight lines) for the average of the measurements of circumference ratios as listed in Table , showing good agreement with the curves. The offsets determined from the two individual fits are used to compute the error bars at the left side of the figure. Also plotted is the result from the isolated horizon computation converted into a circumference measure for comparison. We note that the value from the isolated horizon has developed a plateau between $t=20-25M$, which agrees relatively well with the fits to $C_r$ for the geometric measurements.