Hellings and Downs correlation of an arbitrary set of pulsars

Pulsar timing arrays (PTAs) detect gravitational waves (GWs) via the correlations they induce in the arrival times of pulses from different pulsars. We assume that the GWs are described by a Gaussian ensemble. The mean correlation $h^2 \mu_{\rm u}(\gamma)$ as a function of the angle $\gamma$ between the directions to two pulsars was predicted by Hellings and Downs (HD) in 1983. The variance $\sigma^2_{\rm tot}(\gamma)$ in this correlation was recently calculated by Allen[11] for a single noise-free pulsar pair at angle $\gamma$, which shows that after averaging over many pairs, the variance reduces to an intrinsic cosmic variance $\sigma^2_{\rm cos}(\gamma)$. Here, we extend this to an $arbitrary$ set of pulsars at specific sky locations, with pulsar pairs binned by $\gamma$. We derive the linear combination of pulsar-pair correlations which is the optimal estimator of the HD correlation for each bin, illustrating our methods with plots of the expected range of variation away from the HD curve, for the sets of pulsars monitored by three active PTA collaborations. We compute the variance of and the covariance between these binned estimates, and show that these reduce to the cosmic variance and covariance $s(\gamma,\gamma')$ respectively, in the many-pulsar limit. The likely fluctuations away from the HD curve $\mu_{\rm u}(\gamma)$ are strongly correlated/anticorrelated in the three angular regions where $\mu_{\rm u}(\gamma)$ is successively positive, negative, and positive. We also construct the optimal estimator of the squared strain $h^2$. When there are very many pulsar pairs, this determines $h^2$ with arbitrary precision because PTAs probe an infinite set of GW modes. To assess observed deviations away from the HD curve, we characterize several $\chi^2$ goodness-of-fit statistics. We also show how pulsar noise and measurement noise can be included.