Posterior predictive checking for gravitational-wave detection with pulsar timing arrays: I. The optimal statistic

Abstract

A gravitational-wave background can be detected in pulsar-timing-array data as Hellings--Downs correlations among the timing residuals measured for different pulsars. The optimal statistic implements this concept as a classical null-hypothesis statistical test: a null model with no correlations can be rejected if the observed value of the statistic is very unlikely under that model. To address the dependence of the statistic on the uncertain pulsar noise parameters, the pulsar-timing-array community has adopted a hybrid classical--Bayesian scheme (Vigeland et al. 2018) in which the posterior distribution of the noise parameters induces a posterior distribution for the statistic. In this article we propose a rigorous interpretation of the hybrid scheme as an instance of posterior predictive checking, and we introduce a new summary statistic (the Bayesian signal-to-noise ratio) that should be used to accurately quantify the statistical significance of an observation instead of the mean posterior signal-to-noise ratio, which does not support such a direct interpretation. In addition to falsifying the no-correlation hypothesis, the Bayesian signal-to-noise ratio can also provide evidence supporting the presence of Hellings--Downs correlations. We demonstrate our proposal with simulated datasets based on NANOGrav's 12.5-yr data release. We also establish a relation between the posterior distribution of the statistic and the Bayes factor in favor of correlations, thus calibrating the Bayes factor in terms of hypothesis-testing significance.

12 pages, 8 figures

References in corpus (18)

• The NANOGrav 15-year Data Set: Evidence for a Gravitational-Wave Background
• Search for an isotropic gravitational-wave background with the Parkes Pulsar Timing Array
• Searching for the nano-Hertz stochastic gravitational wave background with the Chinese Pulsar Timing Array Data Release I
• The second data release from the European Pulsar Timing Array III. Search for gravitational wave signals
• The NANOGrav 15-year Data Set: Search for Signals from New Physics
• The NANOGrav 15-year Data Set: Constraints on Supermassive Black Hole Binaries from the Gravitational Wave Background
• The International Pulsar Timing Array second data release: Search for an isotropic Gravitational Wave Background
• On the evidence for a common-spectrum process in the search for the nanohertz gravitational-wave background with the Parkes Pulsar Timing Array
• Common-red-signal analysis with 24-yr high-precision timing of the European Pulsar Timing Array: Inferences in the stochastic gravitational-wave background search
• On measuring the gravitational-wave background using Pulsar Timing Arrays
• The NANOGrav 15-year Data Set: Bayesian Limits on Gravitational Waves from Individual Supermassive Black Hole Binaries
• New advances in the Gaussian-process approach to pulsar-timing data analysis
• Time-domain Implementation of the Optimal Cross-Correlation Statistic for Stochastic Gravitational-Wave Background Searches in Pulsar Timing Data
• A pulsar-based timescale from the International Pulsar Timing Array
• Massive black hole binary systems and the NANOGrav 12.5 year results
• Common-spectrum process versus cross-correlation for gravitational-wave searches using pulsar timing arrays
• Evaluating the prevalence of spurious correlations in pulsar timing array datasets
• Posterior predictive checking for gravitational-wave detection with pulsar timing arrays: II. Posterior predictive distributions and pseudo Bayes factors

Cited by in corpus (11)

• The NANOGrav 15-year Gravitational-Wave Background Methods
• Pulsar Timing Arrays require hierarchical models
• Spatial and Spectral Characterization of the Gravitational-wave Background with the PTA Optimal Statistic
• Posterior predictive checking for gravitational-wave detection with pulsar timing arrays: II. Posterior predictive distributions and pseudo Bayes factors
• Choosing suitable noise models for nanohertz gravitational-wave astrophysics
• Catalog variance of testing general relativity with gravitational-wave data
• Impact of the observation frequency coverage on the significance of a gravitational wave background detection in PTA data
• The NANOGrav 15 yr data set: Posterior predictive checks for gravitational-wave detection with pulsar timing arrays
• Rapid Construction of Joint Pulsar Timing Array Datasets: The Lite Method
• Six Maxims of Statistical Acumen for Astronomical Data Analysis
• On the calculation of p-values for quadratic statistics in Pulsar Timing Arrays