## Cross-correlation function

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The cross-correlation function$\xi _{ab}$ is a subclass of the two-point correlation function used in case of two different classes of objects (e.g. two types of galaxies from two different surveys in the same volume) $a$ and $b$. It defines a (symmetric) probability to find a pair in which $\delta V_{a}$ is occupied by an object from a first catalogue and $\delta V_{2}$ by one from the second as

Another definition (see e.g. [1, 2]) can be given in terms of density contrast $\delta (r)$ when considering the objects distributions as continuous functions of density $\rho _{a}(r)$ and $\rho _{b}(r)$, the mean values of which are $\langle \rho _{a}(r)\rangle =n_{a}$ and $\langle \rho _{b}(r)\rangle =n_{b}$. In this case:
or
where the angle brackets indicate an averaging over the sample volume $V$.
The cross-correlation function gives information about the density profile around objects, e.g. $\xi _{gc}$ between galaxies and clusters measures the average galaxy density profile around clusters, at least out to radii where clusters overlap (see e.g. [2]).
References:
• [1]^ Peebles P. J. E. The Large-Scale Structure of the Universe, Princeton University Press, Princeton, New Jersey, 1980
• [2]^ab Peacock J. A. Cosmological Physics, Cambridge University Press, Cambridge, UK, 2007