Extracting the orbital axis from gravitational waves of precessing binary systems

We present a new method for extracting the instantaneous orbital axis only from gravitational wave strains of precessing binary systems observed from a particular observer direction. This method enables us to reconstruct the coprecessing frame waveforms only from observed strains for the ideal case with the high signal-to-noise ratio. Specifically, we do not presuppose any theoretical model of the precession dynamics and coprecessing waveforms in our method. We test and measure the accuracy of our method using the numerical relativity simulation data of precessing binary black holes taken from the SXS Catalog. We show that the direction of the orbital axis is extracted within $\approx 0.07\mathrm{rad}$ error from gravitational waves emitted during the inspiral phase. The coprecessing waveforms are also reconstructed with high accuracy; the mismatch (assuming white noise) between them and the original coprecessing waveforms is typically a few times ${10}^{-3}$ including the merger-ringdown phase, and can be improved by an order of magnitude focusing only on the inspiral waveform. In this method, the coprecessing frame waveforms are not only the purely technical tools for understanding the complex nature of precessing waveforms but also direct observables.

Figure 1

The definitions of the angles in the source frame. The unit vector, $\widehat{\mathbf{N}}$, denotes the direction from the source to the observer. $\widehat{\mathit{\theta }}$ and $\widehat{\mathit{\varphi }}$ denote unit vectors in the directions of $(\partial /\partial \theta {)}^i$ and $(\partial /\partial \varphi {)}^i$, respectively, where those two angles describe the sky position of the source.

Figure 2

The mode spectra of the gravitational waveforms from a precessing binary (SXS:BBH:0058). We used the waveforms observed from the direction in which ${\theta }_{\mathrm{L}}=\pi /2$ and ${\phi }_{\mathrm{L}}=0$ are satisfied at the initial time of the simulation. The curves “${\mathrm{\Phi }}^{\mathrm{QA}}$” and “${\mathrm{\Phi }}_{\text{fit}}$” show the mode spectra using ${\mathrm{\Phi }}^{\mathrm{QA}}$ and ${\mathrm{\Phi }}_{\text{fit}}$ as the orbital phase in Eq. (8), respectively. The curve “extracted $(|m|=2)$” shows the mode spectra to which a window function Eq. (16) is applied.

Figure 3

The comparison of orbital frequencies obtained by several methods. The curves “$d{\mathrm{\Phi }}^{\prime }/dt$,” “$d{\mathrm{\Phi }}^{\mathrm{QA}}/dt$,” and “$d{\mathrm{\Phi }}_{\text{fit}}/dt$” show orbital frequencies obtained by the time derivative of ${\mathrm{\Phi }}^{\prime }$, ${\mathrm{\Phi }}^{\mathrm{QA}}$, and ${\mathrm{\Phi }}_{\text{fit}}$, respectively.

Figure 4

The comparison of the original and extracted waveforms in the time domain. The upper panel shows the real part of the original complex waveform strain, $h$, and sum of $m=\pm 2$ wave components extracted from the mode spectrum with respect to ${\mathrm{\Phi }}_{\text{fit}}(t)$, and the middle panel shows the difference between those two waveforms. The bottom panel shows the real part of $m=2$ and $m=-2$ mode wave components.

Figure 5

The comparison of ${\theta }_{\mathrm{L}}$ and ${\phi }_{\mathrm{L}}$ obtained from the original QA method and the ones obtained from the extraction procedure using the mode decomposition.

Figure 6

The comparison of the coprecessing frame waveforms obtained by the QA method and the ones reconstructed from extracted waveforms using the mode decomposition. The top panel compares of the coprecessing frame amplitude of the $(l,m)=(2,2)$ mode. The middle panel shows the phase difference between two waveforms for the case that the mismatch for $(t_{\mathrm{i}},t_{\mathrm{f}})=(1000M,\infty )$ is the minimum [see Eq. (22)]. The bottom panel shows the mismatch between the two waveforms as a function of the upper bound of integral, $t_{\mathrm{f}}$. The lower band of the integral, $t_{\mathrm{i}}$, is always set to be $1000M$. We note that we take the average of the $(l,m)=(2,\pm 2)$ modes for the QA waveforms to impose the equatorial symmetry. The vertical dashed line denotes the peak time of the amplitude.

Figure 7

The same as Fig. 5 but for ${\theta }_{\mathrm{L}}(0)=\pi /4$ of SXS:BBH:0164 (top panel) and SXS:BBH:0037 (bottom panel). For the comparison, we shift the extracted result of ${\phi }_{\mathrm{L}}$ by $\pi /2$ and restrict its value to $[-\pi ,\pi ]$ due to its uncertainty in the extraction [see Eq. (20)].

Figure 8

The same as Fig. 6 but for ${\theta }_{\mathrm{L}}(0)=\pi /4$ of SXS:BBH:0164 (top panel) and SXS:BBH:0037 (bottom panel).