Analytic distribution of the optimal cross-correlation statistic for pulsar timing arrays

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) 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 the S/N is well-approximated by a zero-mean unit-variance Gaussian distribution. The "tails" of this simple distribution which differ significantly from those for the GX2distribution, which taken by itself would lead to an incorrect assessment of the statistical significance of a potential detection.