Variance of the Hellings-Downs Correlation

Understand

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.