  The origin of the stochastic gravitational wave (GW) background, recently
discovered from pulsar timing array experiments, 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 Hellings-Downs curve,
describing the expectation value of pulsar timing residual correlations.
However, it introduces a new contribution to the variance of the Hellings-Downs
correlation. Hence, due to the anisotropic distribution of sources, the
measurements of pulsar timing residual correlations in our Universe may differ
from the Hellings-Downs curve. This indicates that the variance of the
Hellings-Downs correlation can be utilized as a new cosmological observable
that might help us to unveil the nature of current background observations in
the nHz band.
