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The detection of nanoHertz gravitational waves through pulsar timing arrayshinges on identifying a common stochastic process affecting all pulsars in acorrelated way across the sky. In the presence of other deterministic andstochastic processes affecting the time-of-arrival of pulses, a detection claimmust be accompanied by a detailed assessment of the various physical orphenomenological models used to describe the data. In this study, we proposeposterior predictive checks as a model-checking tool that relies on thepredictive performance of the models with regards to new data. We derive andstudy predictive checks based on different components of the models, namely theFourier coefficients of the stochastic process, the correlation pattern, andthe timing residuals. We assess the ability of our checks to identify modelmisspecification in simulated datasets. We find that they can accurately flag astochastic process spectral shape that deviates from the common power-law modelas well as a stochastic process that does not display the expected angularcorrelation pattern. Posterior predictive likelihoods derived under differentassumptions about the correlation pattern can further be used to establishdetection significance. In the era of nanoHertz gravitational wave detectionfrom different pulsar-timing datasets, such tests represent an essential toolin assessing data consistency and supporting astrophysical inference.

Posterior predictive checking for gravitational-wave detection with  pulsar timing arrays: II. Posterior predictive distributions and pseudo Bayes  factors