To study clustering of galaxies and clusters, one makes use of the two-point correlation function. However, we can see that all the estimators of $\xi(r)$ use a random catalogue.
What is the random catalogue used for?
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The point of the two-point correlation function (pun not intended) is to describe how clustered the galaxies in the universe are. Astronomers want to know if they're all bunched up in tight bundles with huge voids, or maybe they're more or less uniformly distributed.
The two-point correlation function is a way of measuring the distribution of these galaxies compared to a random distribution. In other words, if your distribution of galaxies is not uniformly random (which it isn't for our universe - our universe has a distinct non-random structure of clusters and voids) then your correlation function will indicate the degree to which it isn't random. Let's take a look at the last equation listed in your link.
The most commonly-used estimator is from Landy & Szalay (1993)
$$\xi(r) = \frac{1}{RR(r)}\Big[DD(r)\Big(\frac{n_R}{n_D}\Big)^2 - 2DR(r)\Big(\frac{n_R}{n_D}\Big)+RR(r)\Big]$$
The symbols in this equation are defined as follows:
- $\xi(r)$ - The two-point correlation function.
- $r$ - The separation between two galaxies.
- $DD$,$DR$,$RR$ - The counds of pairs of galaxies as a function of separation in the data catalogue, between the data and random catalogue, and in the random catalogue, respectively.
- $n_D$, $n_R$ - The mean number density of galaxies in the data and random catalogues, respectively.
If your data catalogue happens to be completely random, then $DD=DR=RR$ and you find that $\xi(r)=0$ (assuming also that $n_D=n_R$), showing your data catalogue is indeed indistinguishable from a random distribution. If, however, your data catalogue is not defined with completely random galaxies, you'll find $\xi(r)\ne0$, and the degree to which it is non-zero indicates how non-random, or organized your system is.
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$\begingroup$ To find out if the clustering is not random, we can simply do so by calculating the mean density of the data catalogue within circles of a given radius at different points as the centre. Why do we need to create something called as a random catalogue? $\endgroup$ Commented Oct 1, 2016 at 21:12
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$\begingroup$ @ThePredator I don't see how calculating the mean density as a function of radius from some (arbitrary) point tells you anything about the nature of the distribution. You need a random catalogue so you can compare your catalogue to that random catalogue and determine how different it is. How can you compare A to B if you have no B to compare to? $\endgroup$– zephyrCommented Oct 1, 2016 at 22:38
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$\begingroup$ Since clusters of galaxies are not random, if we are going to calculate the density at several points as the centre inside the catalogue, we will be finding different values at different points. Doesn't this tell us that it is different from a random catalogue (as in a random catalogue we will find the density to be almost even at any point). $\endgroup$ Commented Oct 2, 2016 at 7:40
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1$\begingroup$ @ThePredator If I understand you correctly, you want to calculate the radial density from several points to compare them. This won't show that it isn't random though. This will only show that your catalogue is homogeneous, i.e., the same everywhere. But it could be random and the same, or clustered and the same. You have to compare your catalogue to a random one to determine if it is random. Comparing it to itself tells you nothing about the clustering. $\endgroup$– zephyrCommented Oct 3, 2016 at 12:38