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\title{Cross-correlation function by Ganna Ivashchenko}

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The \textbf{cross-correlation function} $\xi_{ab}$ is a subclass of the \dref[two-point correlation function]{two-point_correlation_function_by_Ganna_Ivashchenko} 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
$$dP=n_{a}n_{b}[1+\xi_{ab}(r)]\delta V_{1}\delta V_{2}.$$

Another definition (see e.g. \cite{peebles,peacock}) 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:
$$\langle\rho_{a}(x)\rho_{b}(r+x)\rangle=n_{a}n_{b}[1+\xi_{ab}(r)],$$
or 
$$\xi_{ab}(r)=\langle\delta_{a}(x)\delta_{b}(x+r)\rangle,$$
where the angle brackets indicate an averaging over the sample volume $V$.

The \textbf{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. \cite{peacock}).


\begin{thebibliography}{10}
\bibitem{peebles} Peebles P. J. E. \textit{The Large-Scale Structure of the Universe}, Princeton University Press, Princeton, New Jersey, 1980
\bibitem{peacock} Peacock J. A. \textit{Cosmological Physics}, Cambridge University Press, Cambridge, UK, 2007
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