A unified approach to ${\chi }^2$ discriminators for searches of gravitational waves from compact binary coalescences

Abstract

We describe a general mathematical framework for ${\chi }^2$ discriminators in the context of the compact binary coalescence search. We show that with any ${\chi }^2$ is associated a vector bundle over the signal manifold, that is, the manifold traced out by the signal waveforms in the function space of data segments. The ${\chi }^2$ is then defined as the square of the $L_2$ norm of the data vector projected onto a finite dimensional subspace (the fibre) of the Hilbert space of data trains and orthogonal to the signal waveform - any such fibre leads to a ${\chi }^2$ discriminator and the full vector bundle comprising the subspaces and the base manifold constitute the ${\chi }^2$ discriminator. We show that the ${\chi }^2$ discriminators used so far in the CBC searches correspond to different fiber structures constituting different vector bundles on the same base manifold, namely, the parameter space. The general formulation indicates procedures to formulate new ${\chi }^2$s which could be more effective in discriminating against commonly occurring glitches in the data. It also shows that no ${\chi }^2$ with a reasonable degree of freedom is foolproof. It could also shed light on understanding why the traditional ${\chi }^2$ works so well. As an example, we propose a family of ambiguity ${\chi }^2$ discriminators that is an alternative to the traditional one. Any such ambiguity ${\chi }^2$ makes use of the filtered output of the template bank, thus adding negligible cost to the overall search. We test the performance of ambiguity ${\chi }^2$ on simulated data using spinless TaylorF2 waveforms. We show that the ambiguity ${\chi }^2$ essentially gives a clean separation between glitches and signals. Finally, we investigate the effects of mismatch between signal and templates on the ${\chi }^2$ and also further indicate how the ambiguity ${\chi }^2$ can be generalized to detector networks for coherent observations.

21 pages, 5 figure, abstract is shortened to comply with the arXiv's 1920 characters limitation, v2: accepted for publication in PRD

References in corpus (1)

• Comparison of post-Newtonian templates for compact binary inspiral signals in gravitational-wave detectors

Cited by in corpus (8)

• First gravitational-wave search for intermediate-mass black hole mergers with higher order harmonics
• SiGMa-Net: Deep learning network to distinguish binary black hole signals from short-duration noise transients
• An optimal ${\chi }^2$ discriminator against modelled noise-transients in interferometric data in searches for binary black-hole mergers
• False positives for gravitational lensing: the gravitational-wave perspective
• Improved binary black hole searches through better discrimination against noise transients
• Multi-detector null-stream-based ${\chi }^2$ statistic for compact binary coalescence searches
• A ${\chi }^2$ statistic for the identification of strongly lensed gravitational waves from compact binary coalescences
• Improved Binary Black Hole Search Discriminator from the Singular Value Decomposition of Non-Gaussian Noise Transients