[Paper Review] Searching for gravitational wave burst in PTA data with piecewise linear functions

Heling Deng, B. Bécsy+3 more|arXiv (Cornell University)|Jun 29, 2023

Pulsars and Gravitational Waves Research4 citations

TL;DR

This paper proposes a fully Bayesian framework for detecting gravitational wave bursts in pulsar timing array (PTA) data by modeling the unknown burst waveform with piecewise linear functions, enabling analytical marginalization over waveform parameters. The method reduces detection to three key parameters (sky location and signal strength), allowing efficient inference and accurate waveform reconstruction when a signal is present, as demonstrated on simulated data with SNR ≈ 14.7 and ≈ 6.5.

ABSTRACT

Transient gravitational waves (aka gravitational wave bursts) within the nanohertz frequency band could be generated by a variety of astrophysical phenomena such as the encounter of supermassive black holes, the kinks or cusps in cosmic strings, or other as-yet-unknown physical processes. Radio-pulses emitted from millisecond pulsars could be perturbed by passing gravitational waves, hence the correlation of the perturbations in a pulsar timing array can be used to detect and characterize burst signals with a duration of $\mathcal{O}(1 ext{-}10)$ years. We propose a fully Bayesian framework for the analysis of the pulsar timing array data, where the burst waveform is generically modeled by piecewise straight lines, and the waveform parameters in the likelihood can be integrated out analytically. As a result, with merely three parameters (in addition to those describing the pulsars' intrinsic and background noise), one is able to efficiently search for the existence and the sky location of {a burst signal}. If a signal is present, the posterior of the waveform can be found without further Bayesian inference. We demonstrate this model by analyzing simulated data sets containing a stochastic gravitational wave background {and a burst signal generated by the parabolic encounter of two supermassive black holes.

Motivation & Objective

• To develop an efficient Bayesian method for detecting transient gravitational wave bursts in pulsar timing array (PTA) data with unknown waveforms.
• To model the burst signal using piecewise linear functions in the time domain, enabling analytical integration over waveform parameters.
• To reduce the number of free parameters to just three (sky location and signal strength) by marginalizing over the waveform, improving computational efficiency.
• To enable direct reconstruction of the burst waveform without additional Bayesian inference if a signal is detected.
• To validate the method on simulated PTA data containing a stochastic gravitational wave background and a burst from a supermassive black hole parabolic encounter.

Proposed method

• The burst waveform is modeled as two piecewise linear functions in the time domain, one for each polarization (plus and cross).
• The waveform parameters are assigned a Gaussian prior with covariance controlled by a single hyperparameter q, enabling analytical marginalization of the likelihood.
• The marginalized likelihood depends only on three parameters: q (signal strength), θ and φ (sky location), in addition to intrinsic and common noise parameters.
• Markov Chain Monte Carlo (MCMC) sampling is used to explore the posterior of q, θ, and φ, with the full waveform reconstructed from the posterior samples.
• The method is tested on three simulated data sets: two with bursts (SNR ≈ 14.7 and ≈ 6.5) and one with no burst, using a realistic PTA setup including a stochastic gravitational wave background.
• Model comparison via Bayes factor is used to assess evidence for a burst signal versus a noise-only model.

Experimental results

Research questions

• RQ1 Can a Bayesian framework efficiently detect gravitational wave bursts in PTA data with unknown waveforms using piecewise linear modeling?
• RQ2 Can the waveform parameters be analytically marginalized, reducing the inference problem to only three parameters?
• RQ3 How well can the method recover the sky location and waveform of a burst signal with moderate to high signal-to-noise ratio?
• RQ4 How does the method perform in distinguishing a burst signal from background noise when the signal is weak?
• RQ5 Can the model be extended to adaptively choose the number and spacing of grid points for better waveform resolution?

Key findings

• For a strong burst signal with SNR ≈ 14.7, the model strongly favors the presence of a signal over noise-only, with the posterior of the sky location parameters θ and φ peaking near the true values.
• The reconstructed waveform from the MCMC samples closely matches the true injected waveform, demonstrating accurate signal recovery.
• For a weaker signal with SNR ≈ 6.5, the sky location posteriors still peak near the true values, indicating detectability despite waveform ambiguity.
• In the absence of a burst, the Bayes factor between the burst model and the noise-only model is ≈ 1, confirming the model's consistency with noise-only data.
• The method successfully recovers the parameters of the stochastic gravitational wave background, showing compatibility with true values in all simulated cases.
• The framework is computationally efficient due to analytical marginalization, enabling fast inference without iterative waveform fitting.