Variance of the Hellings-Downs Correlation

TL;DR

This paper derives the variance of the Hellings-Downs correlation for pulsar timing arrays, accounting for different GW source distributions, and explores how variance measurements can reveal GW source characteristics.

Contribution

It provides analytical expressions for the variance of the Hellings-Downs correlation considering discrete GW sources and distinguishes cosmic and pulsar variance components.

Findings

Derived variance expressions for GW point sources and confusion noise.

Separated cosmic variance from pulsar variance in correlation measurements.

Showed how variance analysis informs GW source properties.

Abstract

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 previous discrete-source confusion-noise model in the limit of a high density of weak sources.