Pulsar timing array source ensembles

The stochastic gravitational wave background for pulsar timing arrays is often modeled by a Gaussian ensemble which is isotropic and unpolarized. However, the Universe has a discrete set of polarized gravitational wave sources at specific sky locations. Can we trust that the Gaussian ensemble is an accurate description? To investigate this, we explicitly construct an ensemble containing $N$ individual binary sources with circular orbits. The orbital inclination angles are randomly distributed, hence the individual sources are elliptically polarized. We then compute the first two moments of the Hellings and Downs correlation, as well as the pulsar-averaged correlation mean and (cosmic) variance. The first moments are the same as for a previously studied ensemble of circularly polarized sources. However, the second moments, and hence the variances, are different for the two ensembles. While neither discrete source model is exactly described by a Gaussian ensemble, we show that in the limit of large $N$, the differences are small.