Calculation of the (3D) real-space two-point correlation function $\xi (r)$ is complicated due to redshift-space distortions, caused by the fact that the observed redshifts (used for distance determination) are contaminated by the peculiar velocities of galaxies. But it is possible to avoid these complications dealing with the projected correlation function, $w_{p}$, which can be obtained [1] when integrating along the line-of-sight:
where $\sigma$ and $\pi$ and transverse and parallel (to the line-of-sight) separations correspondingly. In practice the upper limit on the second integral is set to some finite value $\pi _{max}$ defined as a scale on which the correlation function is close to zero.
In principle, this statistics can be used to recover the real-space (3D) two-point correlation function by using the inverse relation for the Abel integral equation (see e.g. [2]), but in practice, in the case of power-low form of the correlation function, $\xi (r)=\left (\frac {r_{0}}{r}\right )^{\gamma }$ (which is a fair assumption but can be regarded as true on limited scales) the first equation can be integrated analytically: which allows to obtain the correlation length $r_{0}$ and the slope $\gamma$ of the real-space two-point correlation function; here $\Gamma (x)$ is the Gamma function.
In contrast to $\xi (r)$, projected correlation function is not dimensionless, but has dimensions of length, that is why the quantity $w_{p}/\sigma$ is usually used in practice as the projected analogue of $\xi (r)$.