[Paper Review] The Need For Speed: Rapid Refitting Techniques for Bayesian Spectral Characterization of the Gravitational Wave Background Using PTAs

William G. Lamb, Stephen R. Taylor+1 more|arXiv (Cornell University)|Mar 27, 2023

Pulsars and Gravitational Waves Research4 citations

TL;DR

This paper introduces rapid Bayesian refitting techniques that accelerate spectral characterization of the gravitational wave background in pulsar timing arrays (PTAs) by operating on pre-computed posterior estimates of timing power spectra. The method achieves sub-10% Hellinger distance from full pipeline results while being 100–10,000× faster, enabling efficient model selection and parameter estimation on large-scale PTA data sets.

ABSTRACT

Pulsar timing arrays (PTAs) have recently found evidence for a nanohertz-frequency stochastic gravitational-wave background (SGWB). Constraining its spectral characteristics will reveal its origins. To achieve this, we must understand how data and modeling conditions in each pulsar influence the precision and accuracy of SGWB spectral recovery, typically requiring many Bayesian analyses on real data sets and large-scale simulations that are slow and computationally taxing. To combat this, we have developed several new rapid approaches that operate on intermediate SGWB analysis products. These techniques refit SGWB spectral models to previously computed Bayesian posterior estimates of the timing power spectra. We test our new techniques on simulated PTA data sets and the NANOGrav 12.5-year data set, where in the latter our refit posterior achieves a Hellinger distance -- bounded between 0 for identical distributions and 1 for zero overlap -- from the current full production-level pipeline that is < 0.1. Our techniques are ~ $10^2$--$10^4$ times faster than the production-level likelihood and scale much more favorably (sub-linearly) as a PTA is expanded with new pulsars or observations. Our techniques also allow us to demonstrate conclusively that SGWB spectral characterization in PTA data sets is driven by the longest-timed pulsars and the best-measured power spectral densities. Indeed, the common-process spectral properties found in the NANOGrav 12.5-year data set are given by analyzing only the ~14 longest-timed pulsars out of the full 45 pulsar array, and we find that the 'shallowing' of the common-process power-law model occurs when gravitational-wave frequencies higher than ~50 nanohertz are included. The implementation of our techniques is openly available as a software suite to allow fast and flexible PTA SGWB spectral characterization and model selection.

Motivation & Objective

• To address the computational bottleneck of Bayesian spectral characterization in pulsar timing arrays (PTAs), which currently requires repeated full likelihood evaluations on large data sets.
• To develop faster, scalable alternatives to full Bayesian inference that preserve accuracy in spectral model fitting.
• To identify which pulsars and data characteristics most influence SGWB spectral recovery precision and accuracy.
• To enable rapid, flexible model selection for astrophysical and cosmological inference from PTA data.
• To provide an open-source software suite for efficient spectral analysis of gravitational wave backgrounds.

Proposed method

• The method operates on intermediate Bayesian posterior estimates of timing power spectra, treating them as sufficient statistics for refitting spectral models.
• It uses kernel density estimation (KDE) with optimized bandwidths (via Sheather-Jones algorithm) to represent posterior distributions of power spectra.
• Posterior probabilities derived from KDEs are used to construct fast, approximate likelihoods for spectral parameter estimation.
• The approach employs MCMC (PTMCMC) and nested sampling (UltraNest) for parameter estimation and model selection, respectively.
• The technique is validated using simulated PTA data and the NANOGrav 12.5-year data set, comparing results to production-level pipelines.
• The software suite, ceffyl, integrates with enterprise and PTArcade, and supports efficient posterior comparison via ChainConsumer.

Experimental results

Research questions

• RQ1 What is the computational cost and accuracy trade-off of refitting spectral models to pre-computed posterior estimates of timing power spectra?
• RQ2 Which pulsars and data characteristics most significantly influence the precision and accuracy of SGWB spectral recovery?
• RQ3 Can rapid refitting techniques achieve comparable accuracy to full Bayesian pipelines while reducing computation time by orders of magnitude?
• RQ4 How does the inclusion of higher-frequency gravitational-wave components affect the inferred spectral shape of the common-process signal?
• RQ5 What conditions enable the detection and characterization of sub-dominant cosmological gravitational wave backgrounds beneath a dominant astrophysical signal?

Key findings

• The rapid refitting technique achieves a Hellinger distance of ≤0.1 from the full production-level pipeline on the NANOGrav 12.5-year data set, indicating near-identical posterior distributions.
• The method is 10² to 10⁴ times faster than the full likelihood pipeline and scales sub-linearly with increasing PTA size, enabling efficient scaling to larger arrays.
• Spectral characterization is dominated by the longest-timed pulsars and best-measured power spectral densities, not by cross-correlation across many pulsars.
• The NANOGrav 12.5-year data set's common-process spectral shape is fully reproduced using only ~14 of the 45 pulsars, primarily the longest-timed ones.
• The 'shallowing' of the common-process power-law model occurs when gravitational-wave frequencies above ~50 nHz are included, indicating spectral curvature from high-frequency contributions.
• The open-source software suite ceffyl enables fast, flexible, and reproducible spectral characterization and model selection for future PTA analyses.