Investigating the effect of in-plane spin directions for precessing binary black hole systems

In gravitational-wave observations of binary black holes (BBHs), theoretical waveform models are used to infer the black-hole properties. There are several sources of potential systematic errors in these measurements, including due to physical approximations in the models. One standard approximation is to neglect a small asymmetry between the $+m$ and $-m$ spherical-harmonic modes; this is the effect that leads to emission of linear momentum perpendicular to the orbital plane, and can result in large recoils of the final black hole. The asymmetry is determined by both the magnitude and direction of the spin components that lie in the orbital plane. We investigate the validity of this approximation by comparing numerical relativity (NR) simulations of single-spin NR systems with varying in-plane spin directions and magnitudes (including several “superkick” configurations). We find that the mode asymmetry will impact measurements at signal-to-noise ratios (SNRs) between 15 and 80, which is well within current observations. In particular, mode asymmetries are likely to impact measurements at comparable SNRs to those at which we might hope to make the first unambiguous measurements of orbital precession. We therefore expect that models will need to include mode-asymmetry effects to make unbiased precession measurements.

Figure 1

The left, middle, and right columns show ${\rho }_c$ for signals $\mathrm{q}1\mathrm{a}08\mathrm{p}{180}_{\mathrm{sk}}$, q2a07p180, and q4a08p180 as seen by templates $\mathrm{q}1\mathrm{a}08\mathrm{p}{0}_{\mathrm{sk}}$, q2a07p0, and q4a08p0 respectively, across the signal $\theta$ space. For each signal $\theta$, the match is computed with the template at $\theta +\pi$, over a range of $\varphi$ values, and the black, dashed-red and blue lines show the minimum, mean and maximum match (${\rho }_c$) across the $\varphi$ space. We observe a larger variation of ${\rho }_c$ for the $q=2$ and $q=4$ cases as compared to the $q=1$ due to presence of nonzero subdominant modes, $(l,m)=(2,1)$, (2,0) and (2,-1).

Figure 2

Top panel: difference in the $\beta$ Euler-angle (in radians) between the ${\varphi }_{\mathrm{Sn}}=0$, $\pi /2$ (Blue) and ${\varphi }_{\mathrm{Sn}}=0,\pi$ (Red) configurations of the $q=2$, $\chi =0.7$ system. Bottom panel: the same as above, but for the $q=4$, $\chi =0.8$ system. The legend gives the mass-ratio and spin of signal waveform and the parameter varied between the signal and template waveform. For both systems, $\mathrm{\Delta }\beta$ is small during late-inspiral, with the majority of differences arising near merger.

Figure 3

The left and right panels show the contour plot of ${\rho }_c$ for the signal q2a07p90 and q4a08p90 as seen by template q2a07p0 and q4a08p0 respectively over the signal $(\theta ,\varphi )$ values with the label on the right of each plot showing the values of ${\rho }_c$. See text for further discussion.

Figure 4

$\mathrm{\Gamma }(\rho )$ as defined in Eq. (11). Top panel: $q=1$, 2, 4 systems with ${\varphi }_{\mathrm{Sn}}$ differences of $\pi /2$. Lower panel: $q=2$, 4 systems with different spin values. The results for $q=1$, $q=2$ and $q=4$ are shown in red, black, and blue respectively. The solid lines show the results for systems with ${\varphi }_{\mathrm{Sn}}$ differences of $\pi /2$ and the same ${\chi }_p$, the dashed-lines show the results for large ${\chi }_p$ differences (0.3 for $q=2$ and 0.4 for $q=4$) with the same ${\varphi }_{\mathrm{Sn}}$, and the dotted-dashed for small ${\chi }_p$ differences (0.1 for $q=2$) with the same ${\varphi }_{\mathrm{Sn}}$. See text for further details.

Figure 5

This plot shows the ${\rho }_c$ computed from the match ($\mathcal{M}$) between the symmetrized QA frame waveforms for the $q=2$, $q=4$ systems (solid-black, solid-blue respectively) and symmetrized $q=1$ waveforms (solid-red), with varying values of the upper cutoff frequency in $f_{\max }$ for the match calculation. The legend follows the same naming convention as Fig: 4.

Figure 6

In the top [bottom] panel, we plot the q2a07p0 (blue) and q2a07p90 (black) time [frequency] domain QA frame symmetrized waveforms. For the top panel, the dashed lines show the time at which the waveform has a specific frequency used as $f_{\max }$ value for Fig: 5. For the bottom panel, the dashed lines show the position of that frequency with respect to the frequency domain waveform. Frequency values of (50, 100, 200, 300, 400) are given in dashed (red, blue, black, green, gray) lines respectively.

Figure 7

$\mathrm{\Gamma }(\rho )$ from matching waveforms symmetrized in the QA-frame and then transformed back to the inertial frame. The systems q2a07p90 (solid-black), q2a08p0 (dashed-dotted -black), q2a04p0 (dashed-black) are matched with the q2a07p0 proxy template. The q4a08p90 (solid-blue) and q4a04p0 (dashed-blue) systems are matched with q4a08p0 template. The legend shows the mass-ratio and spin of signal and the difference in the relevant parameter with respect to the template.

Figure 8

$\mathrm{\Gamma }(\rho )$ from matching signal waveforms with both precession and mode-asymmetry against symmetrized template waveforms. Here, the full waveform ${\varphi }_{\mathrm{Sn}}=\pi /2$ signal and ${\varphi }_{\mathrm{Sn}}=0$ signal as seen by the symmetrized ${\varphi }_{\mathrm{Sn}}=0$ template are shown by the dashed and dotted lines respectively. The legend shows the mass-ratio and spin of signal and the difference in the relevant parameter with respect to the template. Where no parameter difference is mentioned, it shows the result for the signal with both precession and mode-asymmetry against its symmetrized self.