A physical template family for gravitational waves from precessing binaries of spinning compact objects: Application to single-spin binaries

TL;DR

This work introduces a physically grounded template family (ST extsubscript{N}) for detecting gravitational waves from precessing binaries with a single significant spin in the adiabatic PN regime. By separating intrinsic binary parameters from extrinsic detector-related parameters, the authors develop a fast, two-stage matched-filtering scheme that automatically maximizes over many extrinsic degrees of freedom, while constraining the intrinsic bank to a manageable dimensionality. They demonstrate that a 3–D reduced bank, informed by a projected metric and dimensional reduction curves, can cover the signal space with a minimum match near 0.98, requiring roughly 7.6×10^4 templates for LIGO-I sensitivity. The approach yields favorable false-alarm statistics and a practical template-count, with extensions to more general spin configurations discussed for companion work and potential adaptation to space-based detectors like LISA.

Abstract

The detection of the gravitational waves (GWs) emitted by precessing binaries of spinning compact objects is complicated by the large number of parameters (such as the magnitudes and initial directions of the spins, and the position and orientation of the binary with respect to the detector) that are required to model accurately the precession-induced modulations of the GW signal. In this paper we describe a fast matched-filtering search scheme for precessing binaries, and we adopt the physical template family proposed by Buonanno, Chen, and Vallisneri [Phys.Rev.D 67, 104025 (2003)] for ground-based interferometers. This family provides essentially exact waveforms, written directly in terms of the physical parameters, for binaries with a single significant spin, and for which the observed GW signal is emitted during the phase of adiabatic inspiral (for LIGO-I and VIRGO, this corresponds to a total mass M < 15Msun). We show how the detection statistic can be maximized automatically over all the parameters (including the position and orientation of the binary with respect to the detector), except four (the two masses, the magnitude of the single spin, and the opening angle between the spin and the orbital angular momentum), so the template bank used in the search is only four-dimensional; this technique is relevant also to the searches for GW from extreme--mass-ratio inspirals and supermassive blackhole inspirals to be performed using the space-borne detector LISA. Using the LIGO-I design sensitivity, we compute the detection threshold (~10) required for a false-alarm probability of 10^(-3)/year, and the number of templates (~76,000) required for a minimum match of 0.97, for the mass range (m1,m2)=[7,12]Msun*[1,3]Msun.

Figures (11)

• Figure 1: Ending frequency (instantaneous GW frequency at the MECO) as a function of $\eta$, evaluated from Eq. (\ref{['s1']}) at 2PN order for $M=15 M_\odot$, $\chi_1 = 1$, and for different values of $\kappa_1$.
• Figure 2: Plot of $\epsilon \equiv (\dot{\omega}/\omega^2)/(96/5 \eta (M\,\omega)^{5/3})$ as a function of $f_{\rm GW} = \omega/\pi$, evaluated from Eq. \ref{['omegadot']} at different PN orders for a $(10 + 1.4)\,M_\odot$ binary. We do not show the 3.5PN curves, which are very close to the 3PN curves.
• Figure 3: Plot of $\epsilon \equiv (\dot{\omega}/\omega^2)/(96/5 \eta (M\,\omega)^{5/3})$ as a function of $f_{\rm GW} = \omega/\pi$, evaluated from Eq. \ref{['omegadot']} at different PN orders for a $(1.4 + 1.4)\,M_\odot$ NS--NS binary. We do not show the 2.5PN, 3PN ($\widehat{\theta}=0$), and 3.5PN curves, which are very close to the 2PN curves. Note the change in scale with respect to Fig. \ref{['Fig1']}.
• Figure 4: Ratio between the unconstrained (${\rho'_{\Xi^{\alpha}}}$) and constrained (${\rho_{\Xi^{\alpha}}}$) maximized overlaps, as a function of ${\rho_{\Xi^{\alpha}}}$. Each point corresponds to one out of $20 \times 50$ sets of intrinsic parameters for target signal and template, and is averaged over 100 sets of extrinsic parameters for the target signal. The error bars show the standard deviations of the sample means (the standard deviations of the samples themselves will be 10 times larger, since we sample 100 sets of extrinsic parameters). The two panels show results separately for $(10+1.4)M_\odot$ (left) and $(7+3)M_\odot$ target systems (right). The few points scattered toward higher ratios and lower ${\rho_{\Xi^{\alpha}}}$ are obtained when the first set of extrinsic parameters happens to yield a high ${\rho'_{\Xi^{\alpha}}}$ that is not representative of most other values of the extrinsic parameters; then the magnitude of the intrinsic-parameter deviation is set too high, and the comparison between ${\rho'_{\Xi^{\alpha}}}$ and ${\rho_{\Xi^{\alpha}}}$ is done at low ${\rho_{\Xi^{\alpha}}}$, where the unconstrained maximized overlap is a poor approximation for its constrained version.
• Figure 5: Inner product between target-signal source direction $\hat{\mathbf{N}}_\mathrm{true}$ and ${\rho_{\Xi^{\alpha}}}$-maximizing source direction $\hat{\mathbf{N}}_\mathrm{max}$, as a function of ${\rho_{\Xi^{\alpha}}}$. Each point corresponds to one out of $20 \times 50$ sets of intrinsic parameters for target signal and template, and is averaged over 100 sets of extrinsic parameters for the target signal. Standard deviations of the sample means are shown as error bars, as in Figure \ref{['ratiovscon']}. The two panels show separately $(10+1.4)M_\odot$ target systems (left) and $(7+3)M_\odot$ target systems (right).