Single-spin precessing gravitational waveform in closed form

In coming years, gravitational-wave detectors should find black hole–neutron star (BH-NS) binaries, potentially coincident with astronomical phenomena like short gamma ray bursts. These binaries are expected to precess. Gravitational-wave science requires a tractable model for precessing binaries, to disentangle precession physics from other phenomena like modified strong field gravity, tidal deformability, or Hubble flow; and to measure compact object masses, spins, and alignments. Moreover, current searches for gravitational waves from compact binaries use templates where the binary does not precess and are ill-suited for detection of generic precessing sources. In this paper we provide a closed-form representation of the single-spin precessing waveform in the frequency domain by reorganizing the signal as a sum over harmonics, each of which resembles a nonprecessing waveform. This form enables simple analytic calculations of the Fisher matrix for use in template bank generation and coincidence metrics, and jump proposals to improve the efficiency of Markov chain Monte Carlo sampling. We have verified that for generic BH-NS binaries, our model agrees with the time-domain waveform to 2%. Straightforward extensions of the derivations outlined here (and provided in full online) allow higher accuracy and error estimates.

Figure 1

Faithfulness of our model: Fraction of $4\times {10}^4$ simulated binaries with overlap greater than the specified threshold. The simulated binaries are each $10+1.4M_{\odot }$ BH-NS binaries, with random orientations and random BH spins with magnitudes in [0.5,1]. The overlap is calculated as the inner product [Eq. (1) and below], maximized over time and phase, between signals with identical physical parameters, drawn from the SpinTaylorF2SingleSpin and SpinTaylorT2 models, respectively, using a fiducial advanced LIGO noise curve. The agreement is better that 5% except for a small number of cases (2% with match less than 0.95) with significant spin-orbit misalignment $\widehat{L}·\widehat{S}\lesssim -0.5$ and unfavorable viewing orientations. This figure can be reproduced using lalsimulation and the scripts provided at Supplemental Material Ref. [22].