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Abstract

Pulsar timing arrays (PTAs) detect gravitational waves (GWs) via the
correlations that the waves induce in the arrival times of pulses from
different pulsars. The mean correlation $\mu_{\rm u}(\gamma)$ as a function of
the angle $\gamma$ between the directions to two pulsars was calculated by
Hellings and Downs in 1983. The variance $\sigma^2_{\rm tot}(\gamma)$ in this
correlation was recently calculated for a single pulsar pair at angle $\gamma$.
Averaging over many such pairs, uniformly distributed on the sky, reduces this
to an intrinsic cosmic variance $\sigma^2_{\rm cos}(\gamma)$. We extend that
analysis to an arbitrary finite set of pulsars, distributed at specific sky
locations, for which the pulsar pairs are grouped into finite-width bins in
$\gamma$. Given (measurements or calculations of) the correlations for any set
of pulsars, we find the best way to estimate the mean in each bin. The optimal
estimator of the correlation takes into account correlations among all of the
pulsars that contribute to that angular bin. We also compute the variance in
the binned estimate. For narrow bins, as the number of pulsar pairs grows, the
variance drops to the cosmic variance. For wider bins, by sacrificing angular
resolution in $\gamma$, the variance can even be reduced below the cosmic
variance. Our calculations assume that the GW signals are described by a
Gaussian ensemble, which provides a good description of the confusion noise
produced by expected PTA sources. We illustrate our methods with plots of the
GW variance for the sets of pulsars currently monitored by several PTA
collaborations. The methods can also be applied to future PTAs, where the
improved telescopes will provide larger pulsar populations and higher-precision
timing.