Efficient asymptotic frame selection for binary black hole spacetimes using asymptotic radiation

Previous studies have demonstrated that gravitational radiation reliably encodes information about the natural emission direction of the source (e.g., the orbital plane). In this paper, we demonstrate that these orientations can be efficiently estimated by the principal axes of $⟨L_{(a}{L}_{b)}⟩$, an average of the action of rotation group generators on the Weyl tensor at asymptotic infinity. Evaluating this average at each time provides the instantaneous emission direction. Further averaging across the entire signal yields an average orientation, closely connected to the angular components of the Fisher matrix. The latter direction is well-suited to data analysis and parameter estimation when the instantaneous emission direction evolves significantly. Finally, in the time domain, the average $⟨L_{(a}{L}_{b)}⟩$ provides fast, invariant diagnostics of waveform quality.

Figure 1

Recovering a synthetic rotation: Starting with a equal-mass nonspinning ($a=0.0$) binary waveform (bottom panel: ${\psi }_{4lm}$ for $\ell \le 2$, extracted at $r=60M$), we apply a rotation taking $\widehat{z}$ to the polar angles $(\theta ,\varphi )$, where $\theta (t)={\theta }_o(1+0.1\cos 2\pi t/P)$ and $\varphi (t)=2\pi t/P$ for $P=80M$ and ${\theta }_o=\pi /20$. The bottom panel shows ${\psi }_{4lm}$ after (dashed lines) and before (solid lines) this rotation, for the modes $l=2$ and $m=2$ (red line), 1 (blue line), and 0 (green line), from top to bottom. Using $⟨L_{(a}{L}_{b)}{⟩}_t$, we recover an orientation $\widehat{Z}(t)$ from the rotated waveform that agrees with the imposed rotation to within $5\times {10}^{-6}\mathrm{deg}$ [top panel: $\theta (Z(t))-\theta (t)$]. Applying this rotation recovers the unperturbed waveform. A vertical dotted line indicates the peak $l=|m|=2$ emission in both figures.

Figure 2

Aligning a precessing binary: Starting with a precessing equal-mass binary [$a_1=0.6\widehat{z}$, $a_2=0.6(\widehat{x}+\widehat{z})/\sqrt{2}$; top panel)], we derive the principal axis orientation from $⟨L_{(a}{L}_{b)}{⟩}_t$, restricted to $l=2$ modes. The bottom panel shows the transformed harmonics. The orientation transformation is derived and applied at all times shown. To illustrate this method’s robustness, in this figure we retain initial transients and late-time errors.