The Impact of Large-Scale Galaxy Clustering on the Variance of the Hellings-Downs Correlation: Theoretical Framework

While pulsar timing array experiments have recently found evidence for the existence of a stochastic gravitational wave (GW) background, its origin is still unclear. If this background is of astrophysical origin, we expect the distribution of GW sources to follow the one of galaxies. Since galaxies are not perfectly isotropically distributed at large scales, but follow the cosmological large-scale structure, this would lead to an intrinsic anisotropy in the distribution of GW sources. In this work, we develop a formalism to account for this anisotropy, by considering a Gaussian ensemble of sources in each realization of the universe and then taking ensemble averages over all such realizations. We find that large-scale galaxy clustering has no impact on the expectation value of pulsar timing residual correlations, described by the Hellings-Downs curve. However, it introduces a new contribution to the variance of the Hellings-Downs correlation. Hence, the anisotropic distribution of sources contributes to the amount by which the measurements of pulsar timing residual correlations, in our single realization of the universe, may differ from the Hellings-Downs curve.