Unambiguous Determination of Gravitational Waveforms from Binary Black Hole Mergers

Gravitational radiation is properly defined only at future null infinity (${\mathcal{J}}^{+}$), but in practice it is estimated from data calculated at a finite radius. We have used characteristic extraction to calculate gravitational radiation at ${\mathcal{J}}^{+}$ for the inspiral and merger of two equal-mass nonspinning black holes. Thus we have determined the first unambiguous merger waveforms for this problem. The implementation is general purpose and can be applied to calculate the gravitational radiation, at ${\mathcal{J}}^{+}$, given data at a finite radius calculated in another computation.

Figure 1

Schematic of the CCE algorithm with two spatial dimensions suppressed. Spacelike slices ${\Sigma }_t$ are evolved according to the Cauchy evolution scheme (horizontal lines). Geometrical data are recorded on a worldtube $\Gamma$, which is used as interior boundary data for a characteristic evolution along $u=\mathrm{\text{const}}$ null surfaces, transporting the data to ${\mathcal{J}}^{+}$. The outer boundary of the Cauchy domain $\partial {\Sigma }_t$ is chosen so that it is causally disconnected from $\Gamma$ over the course of the evolution.

Figure 2

Inspiral, merger, and ringdown phase of ${\psi }_4(\ell =2,m=2)$ as obtained from finite-radius extrapolation (red) and at ${\mathcal{J}}^{+}$ (blue). The waveforms are aligned at their peaks. There is a maximum difference of 1.08% in the amplitude and a dephasing of 0.019 rad between the two waves. These differences can introduce systematic errors to parameter estimation of events detected at high SNR.

Figure 3

Differences $\Delta {\psi }_4$ in the amplitude of ${\psi }_4(\ell =2,m=2)$ between two characteristic runs using boundary data from $R_{\Gamma }=100M$ and $R_{\Gamma }=200M$. The red curve shows the difference at resolution $h=0.96M$ while the blue curve shows the difference for $h=0.64M$, scaled so as to line up for second-order convergence. The expected second-order convergence of our code is thus demonstrated.