Analytic distribution of the optimal cross-correlation statistic for stochastic gravitational-wave-background searches using pulsar timing arrays
Jeffrey S. Hazboun, P. M. Meyers, Joseph D. Romano, Xavier Siemens, Anne M. Archibald
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 $\stackrel{^}{\ensuremath{\rho}}\ensuremath{\equiv}{\stackrel{^}{A}}_{\mathrm{gw}}^{2}/{\ensuremath{\sigma}}_{0}$ 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 $\stackrel{^}{\ensuremath{\rho}}$ is well approximated by a zero-mean unit-variance Gaussian distribution. Empirical calculations show that the null distribution of $\stackrel{^}{\ensuremath{\rho}}$ 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.