Investigating the effect of precession on searches for neutron-star–black-hole binaries with Advanced LIGO

The first direct detection of neutron-star– black-hole binaries will likely be made with gravitational-wave observatories. Advanced LIGO and Advanced Virgo will be able to observe neutron-star– black-hole mergers at a maximum distance of 900 Mpc. To achieve this sensitivity, gravitational-wave searches will rely on using a bank of filter waveforms that accurately model the expected gravitational-wave signal. The emitted signal will depend on the masses of the black hole and the neutron star and also the angular momentum of both components. The angular momentum of the black hole is expected to be comparable to the orbital angular momentum when the system is emitting gravitational waves in Advanced LIGO’s and Advanced Virgo’s sensitive band. This angular momentum will affect the dynamics of the inspiralling system and alter the phase evolution of the emitted gravitational-wave signal. In addition, if the black hole’s angular momentum is not aligned with the orbital angular momentum, it will cause the orbital plane of the system to precess. In this work we demonstrate that if the effect of the black hole’s angular momentum is neglected in the waveform models used in gravitational-wave searches, the detection rate of $(10+1.4)M_{\odot }$ neutron-star– black-hole systems with isotropic spin distributions would be reduced by 33%–37% in comparison to a hypothetical perfect search at a fixed signal-to-noise ratio threshold. The error in this measurement is due to uncertainty in the post-Newtonian approximations that are used to model the gravitational-wave signal of neutron-star– black-hole inspiralling binaries. We describe a new method for creating a bank of filter waveforms where the black hole has nonzero angular momentum that is aligned with the orbital angular momentum. With this bank we find that the detection rate of $(10+1.4)M_{\odot }$ neutron-star– black-hole systems would be reduced by 26%–33%. Systems that will not be detected are ones where the precession of the orbital plane causes the gravitational-wave signal to match poorly with nonprecessing filter waveforms. We identify the regions of parameter space where such systems occur and suggest methods for searching for highly precessing neutron-star– black-hole binaries.

Figure 1

The depth of the physically possible range of ${\xi }_3$ (left) and ${\xi }_4$ (right) values as a function of ${\xi }_1$ and ${\xi }_2$ shown for the TaylorF2 NSBH parameter space. The ${\xi }_i$ coordinates have been scaled such that one unit corresponds to the coverage diameter of a template at 0.97 mismatch. Shown using the zero-detuned, high-power Advanced LIGO sensitivity curve with a 15 Hz lower frequency cutoff and a 1000 Hz upper frequency cutoff.

Figure 2

The depth of the physically possible range of ${\xi }_3$ (left) and ${\xi }_4$ (right) values as a function of ${\xi }_1$ and ${\xi }_2$ shown for the TaylorR2F4 NSBH parameter space. The ${\xi }_i$ coordinates have been scaled such that one unit corresponds to the coverage diameter of a template at 0.97 mismatch. Shown using the zero-detuned, high-power Advanced LIGO sensitivity curve with a 15 Hz lower frequency cutoff and a 1000 Hz upper frequency cutoff.

Figure 3

Fitting factor between a set of aligned-spin NSBH signals and our geometrically placed aligned-spin template bank placed using the TaylorF2 metric. Shown when both templates and signals are generated using the TaylorF2 approximant (gray solid line) and when both are modeled with TaylorT2 (gray dashed line). Also shown when the signals are modeled with TaylorT2 and the templates modeled with TaylorF2 waveforms terminated at ISCO (black dotted line) and TaylorF2 waveforms terminated at MECO (black dot-dashed line). Results obtained using the zero-detuned, high-power Advanced LIGO sensitivity curve with a 15 Hz lower frequency cutoff.

Figure 4

Fitting factor between a set of aligned-spin NSBH signals and our geometrically placed aligned-spin template bank placed using the TaylorR2F4 metric. Shown are comparisons between TaylorT4 waveforms, TaylorR2F4 waveforms including terms to 4.5PN order and TaylorR2F4 waveforms including terms to 6PN order. Results obtained using the zero-detuned, high-power Advanced LIGO sensitivity curve with a 15 Hz lower frequency cutoff.

Figure 5

Fitting factor between a set of aligned-spin NSBH signals modeled with the TaylorT4 approximant and our template bank of aligned-spin signals placed using the TaylorF2 parameter space metric. Shown are the fitting factors when the templates used are modeled using the TaylorF2 approximant (gray solid line), TaylorT2 (gray dashed line) and TaylorT4 (black dotted line). Results obtained using the zero-detuned, high-power Advanced LIGO sensitivity curve with a 15 Hz lower frequency cutoff.

Figure 6

Fitting factor between a set of aligned-spin NSBH signals and a template bank of aligned-spin waveforms for varying values of the upper frequency cutoff used in the construction metric. Shown for template banks placed using the TaylorF2 metric and with both templates and signals modeled using the TaylorF2 approximant (left). Also shown for template banks placed using the TaylorR2F4 metric and with both templates and signals modeled using the TaylorR2F4 approximant (right). The performance of using a stochastically placed template bank with varying upper frequency cutoff is also plotted. Results obtained using the zero-detuned, high-power Advanced LIGO sensitivity curve with a 15 Hz lower frequency cutoff.

Figure 7

Fitting factor between a set of generic, precessing, NSBH signals and a template bank of aligned-spin waveforms. Shown when both templates and signals are generated using the TaylorT2 approximant (black solid line) and the TaylorT4 approximant (black dashed line). Also shown when the templates are modeled using TaylorT2 and the signals are modeled using TaylorT4 (black dotted line). For comparison the same results using a template bank of nonspinning waveforms are also plotted in gray. Plotted over the full range of fitting factors (left) and zoomed in to show only fitting factors greater than 0.9 (right). The distribution that the NSBH signals are drawn from is described in Sec. 2. The template bank construction is described in Sec. 6. Results obtained using the zero-detuned, high-power Advanced LIGO sensitivity curve with a 15 Hz lower frequency cutoff.

Figure 8

Average fitting factor between a set of generic, precessing, NSBH signals and a template bank of nonspinning waveforms as a function of the component masses (left) and as a function of the mass ratio and the black-hole dimensionless spin magnitude (right). Both the signals and the template waveforms are modeled using the TaylorT4 approximant. The distribution that the NSBH signals are drawn from is described in Sec. 2. Results obtained using the zero-detuned, high-power Advanced LIGO sensitivity curve with a 15 Hz lower frequency cutoff.

Figure 9

Average fitting factor between a set of generic, precessing, NSBH signals and a template bank of nonspinning waveforms as a function of the mass ratio and the black-hole dimensionless spin magnitude (right). Shown when both the template waveforms and signals are modeled with TaylorT2 (left) and when the template waveforms are modeled with TaylorT2 and the signals are modeled with TaylorT4 (right). The results in these plots are almost identical to each other and to the right panel of Fig. 8. The distribution that the NSBH signals are drawn from is described in Sec. 2. Results obtained using the zero-detuned, high-power Advanced LIGO sensitivity curve with a 15 Hz lower frequency cutoff.

Figure 10

The signal recovery fraction obtained for a set of generic, precessing, NSBH signals and a template bank of nonspinning waveforms as a function of the mass ratio and the black-hole dimensionless spin. Shown when both the template waveforms and the signals are modeled with TaylorT2 (left) and when both the template waveforms and the signals are modeled with TaylorT4 (right). The distribution of the signal recovery fraction over the mass space is very similar to the distribution of average fitting factors shown in Figs. 8 and 9. The distribution that the NSBH signals are drawn from is described in Sec. 2. Results obtained using the zero-detuned, high-power Advanced LIGO sensitivity curve with a 15 Hz lower frequency cutoff.

Figure 11

Average fitting factor between a set of generic, precessing, NSBH signals and a template bank of aligned-spin waveforms as a function of the component masses (top left) and as a function of the mass ratio and the black-hole dimensionless spin magnitude (top right). Also plotted is the minimum fitting factor (bottom left) and the signal recovery fraction (bottom right) as a function of the mass ratio and the black-hole dimensionless spin magnitude. Both signals and template waveforms are modeled using the TaylorT4 approximant. The distribution that the NSBH signals are drawn from is described in Sec. 2. The template bank construction is described in Sec. 6. Results obtained using the zero-detuned, high-power Advanced LIGO sensitivity curve with a 15 Hz lower frequency cutoff.

Figure 12

The distribution of precessing NSBH signals that are recovered with fitting factors $<0.7$ when searching with an aligned-spin template bank. We use $\widehat{J}$ to denote the initial total angular momentum of the system, $\widehat{n}$ denotes the line of sight towards the observer and $\widehat{L}$ denotes the orbital angular momentum when the gravitational-wave frequency is 60 Hz (at which point approximately half of the signal power has accumulated). Both signals and template waveforms are modeled using the TaylorT4 approximant. The distribution that the NSBH signals are drawn from is described in Sec. 2. The template bank construction is described in Sec. 6. Results obtained using the zero-detuned, high-power Advanced LIGO sensitivity curve with a 15 Hz lower frequency cutoff.

Figure 13

Average fitting factor between a set of generic, precessing, NSBH signals and a template bank of aligned-spin waveforms as a function of the mass ratio and the black-hole dimensionless spin magnitude. Shown when both the template waveforms and signals are modeled with TaylorT2 (left) and when the template waveforms are modeled with TaylorT2 and the signals are modeled with TaylorT4 (right). The results in these plots are almost identical to each other and to the top right panel of Fig. 11. The distribution that the NSBH signals are drawn from is described in Sec. 2. The template bank construction is described in Sec. 6. Results obtained using the zero-detuned, high-power Advanced LIGO sensitivity curve with a 15 Hz lower frequency cutoff.

Figure 14

Average fitting factor between a set of generic, precessing, NSBH signals and a template bank of aligned-spin waveforms as a function of the mass ratio and the neutron-star dimensionless spin magnitude. Both signals and template waveforms are modeled using the TaylorT4 approximant. The distribution that the NSBH signals are drawn from is described in Sec. 2. The template bank construction is described in Sec. 6. Results obtained using the zero-detuned, high-power Advanced LIGO sensitivity curve with a 15 Hz lower frequency cutoff.

Figure 15

The fractional increase in the number of recovered signals when searching for generic, precessing, NSBH signals between a template bank of aligned-spin waveforms and a template bank of nonspinning waveforms. Both signals and template waveforms are modeled using the TaylorT4 approximant. The distribution that the NSBH signals are drawn from is described in Sec. 2. The template bank construction is described in Sec. 6. Results obtained using the zero-detuned, high-power Advanced LIGO sensitivity curve with a 15 Hz lower frequency cutoff.