Galaxy two-point correlation function

Remarks

Equivalently, $\xi(r) = \langle \delta(x) \, \delta(x + r) \rangle$, where $\delta = n/\bar n - 1$ is the contrast of the galaxy number density; it is the autocorrelation function of a stationary random field in space. In a survey, $\xi$ is estimated from pair counts of the data (DD), of a random catalogue (RR) and cross pairs (DR). The Landy–Szalay estimator is $(DD - 2DR + RR)/RR$, with normalised counts. The projected version on the sky is the angular correlation function $w(\theta)$.

Example: for a Poisson process, the points in disjoint volumes are independent, so $dP = (\bar n \, dV_1)(\bar n \, dV_2)$ and $\xi(r) = 0$ for all $r$.

Non-example: if all galaxies lay in the half-space $x_1 > 0$, the pair probability would depend on where the volume elements are, not only on their distance. The process is not homogeneous, so it has no correlation function $\xi(r)$ in this sense.