Search templates for gravitational waves from precessing, inspiraling binaries

Searches for gravitational waves with the LIGO-VIRGO-GEO detector network will require families of ‘‘search templates’’ with which to cross correlate the noisy detectors’ output. This paper introduces a fitting factor (FF), as a quantitative measure of how well the best template in a family ‘‘fits’’ a hypothetical gravitational waveform, in the presence of a specific detector noise spectrum. An FF<0.9 corresponds to a 27% reduction in the event rate of the relevant signals; therefore a family of templates that leads to FF’s below 0.9 should be considered inadequate. The FF is used to explore the adequateness of several families as search templates for gravitational waves from compact inspiraling binaries. The binaries are taken to move in circular orbits, and the ‘‘advanced LIGO noise spectrum’’ is assumed for the detectors. We first study the acceptability of the simplest three-parameter template family, the so-called ‘‘Newtonian family.’’ From previous studies by Finn, Królak, Kokotas, Schäfer, Dhurandar, and Balasubramanian, we infer that post-Newtonian effects in the true waveforms of binaries with vanishing spins cause the Newtonian family to have an unacceptable low FF (∼0.6 to 0.8). We then study the influence of waveform modulations caused by spin-induced orbital precession, and we isolate the modulation effects from other post-Newtonian effects by pretending that the true signals are pure Newtonian with modulation. Many different parameters influence the precession and then the waveform modulation.

A wide range of parameter values is explored, and intuition is developed into which parameters most strongly influence the FF. It is shown that the unmodulated Newtonian template family works quite well (FF>0.9 for almost all parameter values) in searches for the modulated Newtonian signal from two 1.4${\mathit{M}}_{\mathrm{\odot }}$ neutron stars (NS’s) with one of them maximally spinning. By contrast, for a maximally spinning 10${\mathit{M}}_{\mathrm{\odot }}$ black hole (BH) with a nonrotating 1.4${\mathit{M}}_{\mathrm{\odot }}$ NS, the Newtonian template family produces FF<0.9 for more than half of all the binaries’ orientations, if the spin and orbital angular momenta are misaligned by 30°. We introduce a new four-parameter template family, which has the form of the nonmodulated ${\mathrm{post}}^1$-Newtonian signal from a zero-spin-binary. Although, there is a substantial improvement of the FF’s for a spin-modulated Newtonian signal, the FF’s for nonmodulated ${\mathrm{post}}^{1.5}$-Newtonian waveforms are still very poor (∼0.5–0.8). Therefore we propose another four-parameter template family that has the same form as a nonmodulated ${\mathrm{post}}^{1.5}$-Newtonian signal with all the spin-related parameters stripped off. This template family works ${\mathrm{post}}^{1.5}$-Newtonian modulated signals quite well. These results suggest that, in a few years, when waveforms have been computed up to ${\mathrm{post}}^3$-Newtonian order, a good template family will be the four-parameter ${\mathrm{post}}^3$-Newtonian waveforms for zero-spin binaries, augmented by some appropriate modulations to deal with misaligned, rapidly spinning BH-NS systems. Finally, we extend our investigations to the space-based low-frequency LISA detector.