Gravitational waves (GWs) create correlations in the arrival times of pulses
from different pulsars. The expected correlation $\mu(\gamma)$ as a function of
the angle $\gamma$ between the directions to two pulsars was calculated by
Hellings and Downs for an isotropic and unpolarized GW background, and several
pulsar timing array (PTA) collaborations are working to observe these. We ask:
given a set of noise-free observations, are they consistent with that
expectation? To answer this, we calculate the expected variance
$\sigma^2(\gamma)$ in the correlation for a single GW point source, as pulsar
pairs with fixed separation angle $\gamma$ are swept around the sky. We then
use this to derive simple analytic expressions for the variance produced by a
set of discrete point sources uniformly scattered in space for two cases of
interest: (1) point sources radiating GWs at the same frequency, generating
confusion noise, and (2) point sources radiating GWs at distinct
non-overlapping frequencies. By averaging over all pulsar sky positions at
fixed separation angle $\gamma$, we show how this variance may be cleanly split
into cosmic variance and pulsar variance, also demonstrating that measurements
of the variance can provide information about the nature of GW sources. In a
series of technical appendices, we calculate the mean and variance of the
Hellings-Downs correlation for an arbitrary (polarized) point source, quantify
the impact of neglecting pulsar terms, and calculate the pulsar and cosmic
variance for a Gaussian ensemble. The mean and variance of the Gaussian
ensemble may also be obtained from the discrete-source confusion noise model in
the limit of a high density of weak sources.
