 Correlation function (astronomy)

The cosmological two-point correlation function measures excess pairing or correlated density fluctuations at a separation. For a statistically homogeneous and isotropic mean-zero density contrast, $\xi (r)=⟨\delta (\mathbf{x})\delta (\mathbf{x}+\mathbf{r})⟩$. For distinct points in a discrete galaxy population it instead specifies the factorial pair density, ${\bar{n}}^2[1+\xi (r)]$. The latter excludes self-pairs; an unsmoothed discrete-density covariance also contains a Dirac delta at zero separation. These definitions connect large-scale galaxy clustering with the matter power spectrum while keeping galaxy power spectrum shot noise separate.