Simulations of unequal-mass black hole binaries with spectral methods

This paper presents techniques and results for simulations of unequal-mass, nonspinning binary black holes with pseudospectral methods. Specifically, we develop an efficient root-finding procedure to ensure the black hole initial data have the desired masses and spins; we extend the dual coordinate frame method and eccentricity removal to asymmetric binaries. Furthermore, we describe techniques to simulate mergers of unequal-mass black holes. The second part of the paper presents numerical simulations of nonspinning binary black holes with mass ratios 2, 3, 4, and 6, covering between 15 and 22 orbits, merger and ringdown. We discuss the accuracy of these simulations, the evolution of the (initially zero) black hole spins, and the remnant black hole properties.

Figure 1

Residual $|\underline{F}|$ of the initial-data root-finding procedure (top panel) vs iteration number. The lower panel indicates the numerical resolution of each elliptic constraint solve, with 5 being highest resolution (each increase of this integer corresponds to adding a certain number of basis functions, cf. Fig. 6 of Ref. 47).

Figure 2

Eccentricity removal for mass ratio $q=4$. The main panel shows proper separation as a function of time, the inset radial velocity $ds/dt$. Initial data parameters based on TaylorT3 post-Newtonian approximation result in the black dotted line with $e\approx 0.008$. The red dashed and the solid blue lines represent two iterations of eccentricity removal, for a final eccentricity of $e\approx 4\times {10}^{-5}$.

Figure 3

Domain decomposition used for the plunge and merger for mass ratio $q=2$. The thick blue lines represent subdomain boundaries in the $z=0$ plane. The region $z>0$ is not shown. Also not shown is the additional deformation of the grid near the black holes that matches the shape of the excision spheres to the apparent horizons.

Figure 4

Orbital trajectories for mass ratio $q=2$, 3, 4, 6. For all mass ratios, the trajectory of the larger hole is represented by a dashed blue line, and that of the smaller hole by a solid red line.

Figure 5

Convergence of the irreducible mass. Plotted is $\delta M_{\mathrm{irr}}\equiv [M_{\mathrm{irr}}(t)-M_{\mathrm{irr}}(0)]/M_{\mathrm{irr}}(0)$ for three different numerical resolutions ($N=3$, 4, 5).

Figure 6

Phase convergence of RWZ-h (2,2) modes for inspiral-merger-ringdown waveforms. Shown are the phase differences between a given resolution and the highest resolution.

Figure 7

As Fig. 6, but for the subdominant modes (3,3) and (2,1).

Figure 8

Effect of the outer boundary location. Shown are phase differences of $h_{22}$ between simulations with outer boundary radii given in Table . The solid lines give the differences between “normal” and “far” boundaries; the dashed lines give the differences between “close” and far boundaries. For clarity, $q=2$, 3, 4 are offset vertically by multiples of 0.02 rad, and $q=2$, 3 are offset horizontally by multiples of $300M$. $R$ denotes the extraction radius of each simulation.

Figure 9

Gravitational waveforms for $q=1$, 2, 3, 4, 6 that have been extrapolated to infinity. Shown are the $h_{22}$ mode (black, large amplitude), $h_{33}$ mode (red, small amplitude and short wavelength), and the $h_{21}$ mode (blue, small amplitude and long wavelength). Only the real parts have been plotted. The $x$ axis has been time-shifted so that 0 indicates the merger, as determined by the peak of the extrapolated $h_{22}$ mode waveform, for each mass ratio. The left-hand panels show the full coalescence: inspiral, merger, ringdown. The right-hand panels show a close up of the merger and ringdown. Only the $h_{22}$ mode is shown for $q=1$, since the odd-m modes do not appear here.

Figure 10

Amplitudes of $h_{33}$ (top) and $h_{21}$ (bottom) modes, normalized by the amplitude of the leading $h_{22}$ mode, for mass ratios $q=2$, 3, 4, 6. Relative amplitudes are plotted vs the frequency of the $h_{22}$ mode. The filled circles indicate the frequencies where the amplitude of the $h_{22}$ mode peaks.

Figure 11

Convergence test of the dimensionless black hole spins $\chi =S/M^2$. The left panel shows data for the more massive black hole, the right panel for the less massive black hole. For each mass ratio, three resolutions are shown, labeled $N=3,4,5$. The spin of the more massive black hole, ${\chi }_1$ is convergently resolved and is monotonically growing during the simulation. The spin of the smaller black hole ${\chi }_2$, is consistent with ${\chi }_2=0$ within numerical errors.

Figure 12

The coupling coefficient $f$ which determines the magnitude of the change of the spin ${\chi }_1$ during the inspiral as a function of the mass ratio $q=M_1/M_2$. The red circles denote the coefficients for the large black hole for the mass ratios simulated here. The crosses denote the coefficient for the small black hole, which can be obtained from the same plot at the inverse mass ratio.

Figure 13

Dimensionless spin ${\chi }_1$ of the larger black hole as a function of the orbital frequency $M\Omega$. Plotted is the numerical data, and three fits to the data, fitted in the interval $M\Omega \le 0.055$.

Figure 14

$M_f/M$ as function of $q$. Also shown are the results from the fitting formula of Tichy and Marronetti [100], the analytical prediction of Berti et al. [99], and the fit of Buonanno et al. [17] to numerical data.

Figure 15

$S_f/M_f^2$ as function of symmetric mass ratio $\eta$. Also shown are the results of fitting formulas and estimates from Barausse and Rezzolla [101], from Buonanno, Kidder and Lehner [102], and from Tichy and Marronetti [100]. The inset shows the difference ${\chi }_{\mathrm{fit}}-{\chi }_{\mathrm{NR}}$ between fitting formula and our numerical results.

Figure 16

$|v_{\mathrm{kick}}|$ as function of $q$. Also shown are the results of fitting formulas and estimates from Baker et al. [103] and from Gonzalez et al. [14].

Figure 17

Schematic geometry of the spherical regions of the grid geometry. The outer radii of the regions around the excision boundaries covered by spherical grid is indicated by arrows. The excision boundaries themselves are marked by circles drawn with dashed line.

Figure 18

Schematic diagram of the projection map used to connect various surfaces with spheres. Given a surface ${\mathcal{S}}_L$, a point $P_W$, and a sphere ${\mathcal{S}}_U^3$, the projection $Q_U$ of a point $Q_L\in {\mathcal{S}}_L$ is defined by the intersection of the line crossing $P_W$, $Q_L$ and the sphere ${\mathcal{S}}_U^3$, such that $Q_L$ is between $P_W$ and $Q_U$.

Figure 19

Left: Schematic diagram indicating the $x=\mathrm{const}$ plane separating the regions around the two excision boundaries. Right: Schematic diagram of the spheres ${\mathcal{S}}_{EA}^3$, ${\mathcal{S}}_{EE}^3$, ${\mathcal{S}}_{EB}^3$.

Figure 20

Schematic geometry of the touching grid geometry. In a typical simulation, we surround the shown grid geometry by about 20 further spherical shells on the outside, which are not shown in this diagram.