Analytic distribution of the optimal cross-correlation statistic for stochastic gravitational-wave-background searches using 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 rho = A2gw/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 rho is well approximated by a zero-mean unit-variance Gaussian distribution. Empirical calculations show that the null distribution of rho has "tails" which differ significantly from those for a Gaussian distribution but which follow (exactly) a GX2 distribution. Thus, a correct analytical assessment of the statistical significance of a potential detection requires the use of a GX2 distribution.