Gravitational waves from black hole-neutron star binaries: Effective Fisher matrices and parameter estimation using higher harmonics

Inspiralling black hole-neutron star binaries emit a complicated gravitational wave signature, produced by multiple harmonics sourced by their strong local gravitational field and further modulated by the orbital plane’s precession. Some features of this complex signal are easily accessible to ground-based interferometers (e.g., the rate of change of frequency), others less so (e.g., the polarization content), and others still are unavailable (e.g., features of the signal out of band). For this reason, an ambiguity function (a diagnostic of dissimilarity) between two such signals varies on many parameter scales and ranges. In this paper, we present a method for computing an approximate, effective Fisher matrix from variations in the ambiguity function on physically pertinent scales which depend on the relevant signal-to-noise ratio. As a concrete example, we explore how higher harmonics improve parameter measurement accuracy. As previous studies suggest, for our fiducial black hole-neutron star binaries and for plausible signal amplitudes, we see that higher harmonics at best marginally improve our ability to measure parameters. For nonprecessing binaries, these Fisher matrices separate into intrinsic (mass, spin) and extrinsic (geometrical) parameters; higher harmonics principally improve our knowledge about the line of sight. For the precessing binaries, the extra information provided by higher harmonics is distributed across several parameters. We provide concrete estimates for measurement accuracy, using coordinates adapted to the precession cone in the detector’s sensitive band.

Figure 1

Coordinate system for the precessing binary. The left coordinate corresponds to the conventional GW radiation frame. ${\theta }_{NJ}$ (${\varphi }_{NJ}$) is a polar (azimuthal) angle of the total angular momentum ($J$) with respect to the radiation vector ($N$). In the left 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 ${\mathrm{x}}_2$ are coplanar and the shaded region indicates the orbital plane.

Figure 2

One-dimensional ambiguity function for the nonspinning binary, showing variability on multiple scales. Dotted line shows $P$ calculated from Eq. (9) for the leading-order waveforms, where we compare our fiducial simulation with slightly offset analogs, changing just the chirp mass. For comparison, the two solid curves show quadratic fits to the ambiguity function, on scales of $P>0.99$ (top solid) and $P>0.999$ (bottom solid). Note that the upper panel is computed with the reference frequency at 40 Hz, a default choice that accentuates this scale dependence. In the lower panel, we adopt a reference frequency at 100 Hz to mitigate the change in scale; see Appendix . In this paper we adopt a reference frequency at 100 Hz.

Figure 3

Two methods for fitting the ambiguity function: Illustration of differences between our default fitting technique (top solid), which fixes $P_{\max }=1$, and an alternate fitting technique (bottom solid) that lets $P_{\max }\ne 1$, shown here for the $P>0.999$ scale (dotted curve). These two methods produce comparable global fit functions, but the fitting parameters for $\Gamma$ differ by tens of percent; see, e.g., Table . While numerical error can produce fluctuations in the overlap (e.g., due to insufficient sampling rate; we use 65536 Hz), the change in shape shown here is resolved.

Figure 4

Comparison of the ambiguity contours between the leading-order (solid line) and higher-order (dotted line) waveforms for the nonspinning binaries. $\delta \lambda$ is defined by a difference from the fiducial value in Table . The lines correspond to $P=0.99$ (top panel) and $P=0.999$ (bottom panel). Higher-order waveforms only marginally reduce the ambiguity contours.

Figure 5

Comparison of the ambiguity surfaces between the leading-order (top panel) and higher-order (bottom panel) waveforms for the nonspinning binaries. $\delta \lambda$ is defined by a difference from the fiducial value in the Table . Higher-order waveforms change the ambiguity surface and break degeneracies related to the inclination and orbital phase so these parameters can be marginally observable at the scale of $P>0.99$.

Figure 6

Ambiguity function for aligned-spin: Points in $M_{\mathrm{c}}$, $\eta$, $\chi$ with $P\in [0.99,0.991]$ for the leading-order (outer ellipsoid) and higher-order (inner ellipsoid) waveforms. The two long, narrow surfaces (ambiguity ellipsoids) these points cover illustrates gravitational wave measurements can constrain one combination of $M_{\mathrm{c}}$, $\eta$, $\chi$ tightly (e.g., $M_{\mathrm{c}}$); one less so (e.g., $\eta$); and one almost not at all. The two surfaces nearly agree, with the most significant change being a slight reduction in the least-well-determined direction. The close agreement between these two surfaces shows higher harmonics provide fairly little additional information to break this degeneracy.

Figure 7

Symmetry can produce highly nonquadratic behavior: Plot of the ambiguity function when changing only ${\theta }_{NJ}$ and ${\alpha }_{JL}$ (top panel) and ${\beta }_{JL}$, ${\alpha }_{JL}$ (bottom panel), for the high-symmetry binary parameter set “case 2.”

Figure 8

Comparison of the ambiguity functions between a real and complex overlaps for the extrinsic parameters. Dotted line is calculated by changing $\iota$ where the fiducial value is 0 and other parameters are the same as in Table . Dashed line is calculated by changing $\varphi$ where $\iota =\pi /2$ and other parameters are the same as in Table . The ambiguity surface for the real overlap is flat.

Figure 9

Comparison of the ambiguity functions between a real and complex overlaps for the intrinsic parameters. Parameter values are summarized in Table . Thick lines are calculated by changing $M_{\mathrm{c}}$, others by changing $\eta$.

Figure 10

Examples for a dependence of the ambiguity function on the reference frequency using higher-order waveforms.

Figure 11

Examples for a dependence of the ambiguity function on the reference frequency. Left: higher-order waveforms, $\iota =\pi /4$. Right: leading-order waveforms, $\iota =\pi /2$.

Figure 12

One dimensional ambiguity function with various reference frequencies for $\iota =\pi /4$, using leading-order waveforms.