Analytic distribution of the optimal cross-correlation statistic for stochastic gravitational-wave-background searches using pulsar timing arrays

TL;DR

This paper demonstrates that the optimal cross-correlation statistic for pulsar timing array searches of the gravitational-wave background follows a generalized chi-squared distribution, which is crucial for accurate significance testing of potential detections.

Contribution

It provides an analytical and numerical demonstration that the null distribution of the statistic is a generalized chi-squared distribution, correcting previous Gaussian assumptions.

Findings

The OS statistic follows a GX2 distribution, not Gaussian.

Accurate significance assessment requires using the GX2 distribution.

Empirical results confirm the theoretical distribution.

Abstract

We show via both analytical calculation and numerical simulation that the optimal cross-correlation statistic (OS) for stochastic gravitational-wave-background (GWB) searches using data from pulsar timing arrays follows a generalized chi-squared (GX2) distribution-i.e., a linear combination of chi-squared distributions with coefficients given by the eigenvalues of the quadratic form defining the statistic. This observation is particularly important for calculating the frequentist statistical significance of a possible GWB detection, which depends on the exact form of the distribution of the OS signal-to-noise ratio (S/N) $\hat\rho \equiv \hat A_{\rm gw}^2/\sigma_0$ in the absence of GW-induced cross correlations (i.e., the null distribution). Previous discussions of the OS have incorrectly assumed that the analytic null distribution of $\hat\rho$ is well-approximated by a zero-mean unit-variance Gaussian distribution. Empirical calculations show that the null distribution of $\hat\rho$ has "tails" which differ significantly from those for a Gaussian distribution, but which follow (exactly) a GX2 distribution. So, a correct analytical assessment of the statistical significance of a potential detection requires the use of a GX2 distribution.