Exploring the outer limits of numerical relativity

We perform several black-hole binary evolutions using fully nonlinear numerical relativity techniques at separations large enough that low-order post-Newtonian expansions are expected to be accurate. As a case study, we evolve an equal-mass nonspinning black-hole binary from a quasicircular orbit at an initial coordinate separation of $D=100M$ for three different resolutions. We find that the orbital period of this binary (in the numerical coordinates) is $T=6422M$. The orbital motion agrees with post-Newtonian predictions to within 1%. However, we find that the time derivative of the coordinate separation is dominated by a purely gauge effect leading to an apparent contraction and expansion of the orbit at twice the orbital frequency. Based on these results, we improved our evolution techniques and studied a set of black hole binaries in quasicircular orbits starting at $D=20M$, $D=50M$, and $D=100M$ for $\sim 5$, 3, and 2 orbits, respectively. We then find good agreement between the numerical results and post-Newtonian predictions for the orbital frequency and radial decay rate, radiated energy and angular momentum, and waveform amplitude and phases. The results are relevant for the future computation of long-term waveforms to assist in the detection and analysis of gravitational waves by the next generation of detectors as well as the long-term simulations of black-hole binaries required to accurately model astrophysically realistic circumbinary accretion disks.

Figure 1

The orbital trajectory for the $D=100M$ generation 0 simulations at different resolutions.

Figure 2

The orbital frequency $d{\varphi }_{\mathrm{orbit}}/dt$ versus orbital separation $r$, as well as the Newtonian and 3.5PN predictions for the $D=100M$ generation 0 simulations. The mapping between $r$ and $\Omega$ is not unique because $r$ is not a monotonic function of $t$.

Figure 3

The orbital separation versus time at three resolutions, as well as the 3.5PN prediction, for the $D=100M$ generation 0 simulations.

Figure 4

The orbital frequency versus time at three resolutions, as well as the 3.5PN prediction, for the $D=100M$ generation 0 simulations.

Figure 5

The horizon mass of the individual BHs versus time of the generation 0 simulations for the three resolutions. The secular growth with time for all resolutions and the growth in mass as the gridspacing is reduced indicate a loss of accuracy at later times for these generation 0 simulations.

Figure 6

(Left) Convergence plots for the orbital phase (top left) and separation (bottom left) showing stronger than eighth-order convergence. If a quantity $F(h)$ is eighth-order convergent, then $F(h_0)-F(h_1)=4.29982[F(h_1)-F(h_2)]$. Here we see hyperconvergence, which almost certainly means that the errors are oscillating with $h$ rather than behaving monotonically. The top-right plot shows the phase error of the coarsest resolution run (assuming eighth-order convergence and comparing the Richardson extrapolated value), while the bottom-right plot shows the orbital separation error.

Figure 7

Comparing the $D=100M$ mass conservation for the generation 0 and generation 1 runs. Note that the generation 1 simulation shows no late time growth. The conservation of the mass at levels of below 1 part in ${10}^5$ is crucial for long term evolutions.

Figure 8

The orbital frequency $d{\varphi }_{\mathrm{orbit}}/dt$ versus orbital separation $r$ and versus shifted semiproper distance $\tilde{s}$, as well as the Newtonian and 3.5PN predictions for the $D=20M$, $50M$, $100M$ generation 1 runs. Here $\tilde{s}=s-(s_0-r_0)$, where $s_0$ is the initial SPD and $r_0$ is the initial PN orbital separation.

Figure 9

Proper distance (minus a constant) and coordinate distance comparison for the $D=20M$, $50M$, $100M$ generation 1 runs. The $D=100M$ plot also shows the coordinate separation versus time for the generation 0 runs.

Figure 10

The orbital frequency for the generation 0 and generation 1 runs versus time ($D=100M$) and the PN prediction. Note how the generation 1 appears to outperform the highest resolution generation 0 run.

Figure 11

Simple proper distance decay rate versus post-Newtonian prediction for the $D=20M$, $D=50M$, and $D=100M$ generation 1 runs. Note that the periods of oscillations roughly match those of the orbital period, $T/M=598$, 2281, 6369, respectively.

Figure 12

The orbital trajectory obtained using the coordinate orbital phase and the proper orbital separation along the coordinate line separating the two BHs for the generation 1 simulations. The zoomed plots show the tightness of the inspiral.

Figure 13

The waveform extracted at different observer locations for the $D=20M$ generation 1 run. Here the waveforms have been extrapolated to $\infty$ using Eq. (12). While we see that the extraction at $R_{\mathrm{obs}}=100$ has large phase errors, the phases and amplitudes stabilize for $R_{\mathrm{obs}}\ge 300$, i.e. one wavelength distance from the sources.

Figure 14

Comparison of $r{\psi }_4$ and the characteristic variable $\Psi$ for the $D=50M$ simulation. The boxes indicate regions where the fast Fourier transform smoothing operation on the CCE data distorted the waveform.

Figure 15

Comparison of 3.5PN and full numerical waveforms for the $D=20M$, $50M$, $100M$ generation 1 simulations. Note the excellent agreement in both amplitude and phase and the very small scale of the amplitudes.