Orbital Evolution of Extreme-Mass-Ratio Black-Hole Binaries with Numerical Relativity

We perform the first fully nonlinear numerical simulations of black-hole binaries with mass ratios $100:1$. Our technique is based on the moving puncture formalism with a new gauge condition and an optimal choice of the mesh refinement. The evolutions start with a small nonspinning black hole just outside the ISCO that orbits twice before plunging. We compute the gravitational radiation, as well as the final remnant parameters, and find close agreement with perturbative estimates. We briefly discuss the relevance of these simulations for Advanced LIGO, third-generation ground-based detectors, LISA observations, and self-force computations.

Figure 1

An $xy$ projection of the trajectories for the two highest resolutions of the $q=1/100$ configuration. The dotted circle corresponds to the ISCO radius while the small filled-in circle corresponds to the point on the trajectory where a common horizon is first detected. Note the initial “jump” in radius (see Fig. 2) due to the initial data radiation content.

Figure 2

The orbital radius as a function of time and resolution for the $q=1/100$ configuration. Note the initial “jump” in the orbit due to the initial data.

Figure 3

Convergence of the amplitude and phase of the ($\ell =2$, $m=2$) mode of ${\psi }_4$. The phase converges to second order prior to the peak in the amplitude. The vertical line shows the point when $\omega =0.2$. Note the good agreement in amplitude (the curves have been translated). The phase error at $\omega =0.2$ is 0.44 radians. The reduced order of convergence is due to an aggressive choice of the CFL factor.

Figure 4

The remnant spin $a/M_H$ and the radiated mass from infinite separation $\delta M/M$, as a function of the symmetric mass ratio $\eta =q/(1+q{)}^2$, for $q=1/10$, $1/15$, $1/100$, as well as the empirical formula prediction.