Document Type


Publication Date



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, which models the confusion noise produced by expected PTA sources. The mean correlation h2μu(γ) as a function of the angle γ between the directions to two pulsars was predicted by Hellings and Downs in 1983. The variance σ2tot(γ) in this correlation was recently calculated [B. Allen, Variance of the Hellings-Downs correlation, Phys. Rev. D 107, 043018 (2023)] for a single noise-free pulsar pair at angle γ, which shows that after averaging over many pairs, the variance reduces to an intrinsic cosmic variance σ2cos(γ). Here, we extend this to an arbitrary set of pulsars at specific sky locations, with pulsar pairs binned by γ. We derive the linear combination of pulsar-pair correlations which is the optimal estimator of the Hellings and Downs correlation for each bin, illustrating our methods with plots of the expected range of variation away from the Hellings and Downs 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(γ,γ′) respectively, in the many-pulsar limit. The likely fluctuations away from the Hellings and Downs curve μu(γ) are strongly correlated/anticorrelated in the three angular regions where μu(γ) is successively positive, negative, and positive. We also construct the optimal estimator of the squared strain h2 from pulsar-pair correlation data. Remarkably, when there are very many pulsar pairs, this determines h2 with arbitrary precision because (in contrast to LIGO-like GW detectors) PTAs probe an infinite set of GW modes. To assess if observed deviations away from the Hellings and Downs curve are consistent with predictions, we propose and characterize several χ2 goodness-of-fit statistics. While our main focus is ideal noise-free data, we also show how pulsar noise and measurement noise can be included. Our methods can also be applied to future PTAs, where the improved telescopes will provide larger pulsar populations and higher-precision timing.


Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Open access publication funded by the Max Planck Society.

Creative Commons License

Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.

Publication Title

Physical Review D





To view the content in your browser, please download Adobe Reader or, alternately,
you may Download the file to your hard drive.

NOTE: The latest versions of Adobe Reader do not support viewing PDF files within Firefox on Mac OS and if you are using a modern (Intel) Mac, there is no official plugin for viewing PDF files within the browser window.