Hellings and Downs correlation of an arbitrary set of pulsars

Understand

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 $\sigma^2_{\rm cos}(\gamma)$.