Parameter estimation of gravitational waves from precessing black hole-neutron star inspirals with higher harmonics

Precessing black hole-neutron star (BH-NS) binaries produce a rich gravitational wave signal, encoding the binary’s nature and inspiral kinematics. Using the lalinference_mcmc Markov chain Monte Carlo parameter estimation code, we use two fiducial examples to illustrate how the geometry and kinematics are encoded into the modulated gravitational wave signal, using coordinates well adapted to precession. Extending previous work, we demonstrate that the performance of detailed parameter estimation studies can often be estimated by “effective” studies: comparisons of a prototype signal with its nearest neighbors, adopting a fixed sky location and idealized two-detector network. Using a concrete example, we show that higher harmonics provide nonzero but small local improvement when estimating the parameters of precessing BH-NS binaries. We also show that higher harmonics can improve parameter estimation accuracy for precessing binaries by breaking leading-order discrete symmetries and thus ruling out approximately degenerate source orientations. Our work illustrates quantities gravitational wave measurements can provide, such as the orientation of a precessing short gamma ray burst progenitor relative to the line of sight. More broadly, “effective” estimates may provide a simple way to estimate trends in the performance of parameter estimation for generic precessing BH-NS binaries in next-generation detectors. For example, our results suggest that the orbital chirp rate, precession rate, and precession geometry are roughly independent observables, defining natural variables to organize correlations in the high-dimensional BH-NS binary parameter space.

Figure 1

Coordinate system for the precessing binary. The left coordinate corresponds to the conventional GW radiation frame. ${\theta }_{JN}$ (${\varphi }_{JN}$) is a polar (azimuthal) angle of the total angular momentum ($J$) with respect to the radiation vector ($N$). In the right coordinate ${\beta }_{JL}$ (${\alpha }_{JL}$) is a polar (azimuthal) angle of the orbital angular momentum ($L$) with respect to the total angular momentum ($J$). In the right coordinate, $N$, $J$, and ${\mathrm{x}}^{\prime }$ are coplanar and the shaded region indicates the orbital plane.

Figure 2

Time-dependent geometry. For each of the two fiducial binaries in Table 1, a plot of the time-dependent angles ${\beta }_{JL}(t)$ (top; common to both), $\alpha (t)$ plus an arbitrary integer multiple of $2\pi$ (center; blue=A; black=C), and the two components of the orbital angular momentum direction $\widehat{L}$ projected on the plane of the sky (bottom). For completeness, the first two panels also include dotted lines, corresponding to the trajectory adopted when higher harmonics were included and the initial template frequency was reduced. In the bottom panel, solid colored dots indicate the direction of $\widehat{L}$ at 100 Hz. The binary precesses roughly 7 times between 30 Hz and 500 Hz.

Figure 3

Harmonic amplitude versus opening angle. A plot of ${\rho }_{2m}^2/({\rho }_{22}^{\text{ROT}}{)}^2$ predicted by Eq. (14) for $m=2$ (blue), 1 (red), and 0 (yellow).

Figure 4

Estimating astrophysical parameters (C). For our fiducial binary C, the solid and dotted lines show an estimated 90% confidence interval with and without higher harmonics, respectively; colors indicate different noise realizations; and the (nearly indistinguishable) thick solid and dashed lines show an approximate effective Fisher matrix result, with and without higher harmonics, not accounting for the constraint imposed by ${\chi }_1<1$. Results for case A are qualitatively and quantitatively similar. The different panels show different two-dimensional projections of the astrophysically relevant parameters of a merging BH-NS binary: the binary mass ratio, black hole spin, and degree of spin-orbit misalignment $\kappa \equiv \widehat{L}·{\widehat{S}}_1$. Top, center panels: The masses and spin magnitude of the binary can be measured very reliably, consistent with a single Gaussian distribution in four dimensions. The analytic predictions produced by an effective Fisher matrix agree qualitatively but not quantitatively with our simulations. Bottom panel: To guide the eye, the posterior versus ${\chi }_1$ and $\widehat{L}·{\widehat{S}}_1$ is compared with contours of constant ${\beta }_{JL}={\cos }^{-1}0.65$, 0.7, 0.75 (precession cone opening angle; dotted black) and ${\mathrm{\Omega }}_p$ [Eq. (15)] (precession rate; solid black). The precession rate is relatively well constrained by the presence of several ($\simeq 7$) precession cycles available in data, while the geometry is relatively poorly constrained, relative to the whole ${\chi }_1$ vs $\widehat{L}·{\widehat{S}}_1$ plane.

Figure 5

Source geometry: angular momenta (C, A). For case C (top panels) and case A (bottom panels), the posterior for the precession cone (path of the angular momentum direction), expressed using the precession cone representation. This figure demonstrates that both the path (${\theta }_{JN},{\beta }_{JL}$) and instantaneous orientation (${\alpha }_{JL},\iota$) of the orbital angular momentum can be well determined. As in Fig. 4, colors indicate different noise realizations, solid and dotted lines indicate the neglect or use of higher harmonics, the green point shows the actual value, and the solid gray path shows the trajectory of $L$ over one precession cycle. Left panels: The precession angle ${\alpha }_{JL}$ of $L$ around $J$. For comparison, the green points show the simulated values; when present, the solid blue path shows variables covered in one precession cycle. Roughly speaking, the precession phase can be measured with relative accuracy tens of percent at this signal amplitude $\rho$. Right panels: Illustration that both the opening angle ${\beta }_{JL}$ of the precession cone and the angle ${\theta }_{JN}$ between the line of sight and $\widehat{J}$ can be measured accurately.

Figure 6

Angular momentum direction on the sky (C). Projection of the orbital angular momentum direction ($\widehat{L}$) on the plane of the sky at $f=100\mathrm{Hz}$; compare to Fig. 2. This figure demonstrates that the individual angular momenta can be well constrained to two discrete regions; that higher harmonics allow us to distinguish between the two alternatives; and that the precession cone is well determined, at the accuracy level expected from the number of precession cycles. As in Fig. 4, colors indicate different noise realizations, solid and dotted lines indicate the neglect or use of higher harmonics, and the green point shows the expected solution.

Figure 7

Distance and inclination degeneracy broken (C). Posterior probability contours in distance and inclination.

Figure 8

Estimating astrophysical parameters with advanced detectors. Like the bottom panel of Fig. 4, but using advanced instruments; see Table 6.