Calculation of p -values for quadratic statistics in pulsar timing arrays
Year: 2025
Type: article
Abstract: Pulsar Timing Array (PTA) projects have reported various lines of evidence suggesting the presence of a stochastic gravitational wave (GW) background in their data. One key line of evidence involves a detection statistic sensitive to interpulsar correlations, such as those induced by GWs. A p -value is then calculated to assess how unlikely it is for the observed signal to arise under the null hypothesis H 0 , purely by chance. However, PTAs cannot empirically draw samples from H 0 . As a workaround, various techniques are used in the literature to approximate p -values under H 0 . One such technique, which has been characterized as a model-independent method, is the use of “scrambling” transformations that modify the data to cancel out pulsar correlations, thereby simulating realizations from H 0 . In this work, scrambling methods and the detection statistic are investigated from first principles. The p -value methodology that is discussed is general, but the discussions regarding a specific detection statistic apply to the detection of a stochastic background of gravitational waves with PTAs. All methods in the literature to calculate p -values for such a detection statistic are rigorously analyzed, and analytical expressions are derived for the distribution of the detection statistic and the corresponding p -values. All this leads to the conclusion that scrambling methods are model independent and thus not completely empirical. With a single realization of data our results are necessarily always model dependent, which any analysis will need to accept. Instead of scrambling approaches, rigorous Bayesian and Frequentist p -value calculation methods are advocated, the evaluation of which depend on the generalized χ 2 distribution. This view is consistent with the posterior predictive p -value approach that is already in the literature. Efficient expressions are derived to evaluate the generalized χ 2 distribution of the detection statistic on real data. It is highlighted that no Frequentist p -values have been calculated correctly in the PTA literature to date.
DOI: 10.1103/b3y4-yfr1
Author: Rutger van Haasteren
Language: English
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: hybrid
OpenAlex ID: W4415966749