Optimal reconstruction of the Hellings and Downs correlation

Pulsar timing arrays (PTAs) detect gravitational waves (GWs) via the correlations they create in the arrival times of pulses from different pulsars. The mean correlation, a function of the angle between the directions to two pulsars, was predicted in 1983 by Hellings and Downs (HD). Observation of this angular pattern is the ``smoking gun'' that GWs are present, so PTAs ``reconstruct the HD curve'' by estimating the correlation using pulsar pairs separated by similar angles. Several studies have examined the amount by which this curve is expected to differ from the HD mean. The variance arises because (a) a finite set of pulsars at specific sky locations is used, (b) the GW sources interfere, and (c) the data are contaminated by noise. Here, for a Gaussian ensemble of sources, we predict that variance using an optimal estimator of the HD correlation, taking into account the pulsar sky locations and the power spectrum of the GWs. The variance is a ratio: the numerator depends upon the pulsar sky locations, and the denominator is the number of frequency bins for which the GW signal dominates the noise. In effect, after suitable combination, each frequency bin gives an independent estimate of the HD correlation.