Evolutions of unequal mass, highly spinning black hole binaries

We evolve a binary black hole system bearing a mass ratio of $q=m_1/m_2=2/3$ and individual spins of $S_1^z/m_1^2=0.95$ and $S_2^z/m_2^2=-0.95$ in a configuration where the large black hole has its spin antialigned with the orbital angular momentum, $L^z$, and the small black hole has its spin aligned with $L^z$. This configuration was chosen to measure the maximum recoil of the remnant black hole for nonprecessing binaries. We find that the remnant black hole recoils at just above $500\mathrm{km}/\mathrm{s}$, the largest recorded value from numerical simulations for aligned spin configurations. The remnant mass, spin, and gravitational waveform peak luminosity and frequency also provide a valuable point in parameter space for source modeling.

Figure 1

The convergence of the $L^2$ norm of the Hamiltonian and momentum constraint residuals of the initial data solver versus the number of collocation points $(N\times N\times 40)$. The $L^2$ norm is measured with respect $10^6$ randomly chosen points lying outside the two horizons and inside a sphere with radius equal to the separation of the two black holes. The observed power law of $\sim N^{-7}$ for the convergence rate is consistent with the $C^6$ smoothness of the attenuation function $g$ [see Eq. (11)].

Figure 2

The trajectories of the two black holes, as well as the time dependence of the orbital separation (coordinate and simple proper distance) and phase.

Figure 3

The real (red) and imaginary (blue) parts of the (2,2) mode of ${\psi }_4$ (left), its amplitude and phase (middle), and the reconstructed strain $h_{22}$ (right) as measured by an observer at location $r=102.6M$. The strain is extrapolated to infinite observer location using the analytic perturbative extrapolation described in [46].

Figure 4

(left) The evolution in velocity space of the recoil vector during the inspiral. (middle) The cumulative recoil versus time. (right) The instantaneous radiated power and cumulative radiated energy versus time. All calculated at infinite observer location.

Figure 5

The dimensionless spin (top) and horizon (Christodoulou) mass (bottom) for the two horizons in the binary and the final remnant black hole. The $y$-scale for the mass and spins is set by black hole 2 and is kept the same for the other two black holes.

Figure 6

The coordinate radii (minimum, maximum and average) of the three horizons versus time for the full simulation. Note that there is an extremely rapid evolution of the horizon size and shape during the first few $M$ of evolution.

Figure 7

$L^2$ norm of the Hamiltonian and momentum constraints versus time. Note the rapid growth during the first 2M of evolution. The CCZ4 damping parameters ${\kappa }_{1,2}$ managed to suppress the constraint growths during the evolution down to merger and afterwards.

Figure 8

The constraints versus time for three resolutions: N100 in red, N120 in blue, and N144 in green. The bottom panel in each figure shows the convergence order, $d$, versus time. The data is sampled every 48M and starts at $t=150M$ after 1 orbit of evolution.