Two-point correlation function

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The two-point correlation function is one of the main statistics used for description of the galaxy distribution. The standard statistical analysis assumes that the objects can be regarded as point particles and these particles are assumed to be distributed homogeneously on a suﬃciently large scale within the sample boundaries. This means that we can meaningfully assign an average number density to the distribution. Therefore, we can characterize the galaxy distribution in terms of the extent of the departures from uniformity on various scales [1].
According to Peebles [2] the two-point correlation function$\xi (r)$ determines a probability $dP$ to find simultaneously two objects on a distance $r$ from each other within two volume elements $\delta V_{1}$ and $\delta V_{2}$ in a sample with number density $n$ as
This correlation function (when one speaks about a distribution of objects of one type) is also known as autocorrelation function, in contrast to cross-correlation function.
Another definition of autocorrelation function (see e.g. [2, 3]) can be given in terms of density contrast $\delta (r)$ when considering the objects distribution as a continuous function of density $\rho (r)$, the mean value of which is $\langle \rho (r)\rangle =n$. In this case:
or
where the angle brackets indicate an averaging over the sample volume $V$.
In practice the correlation function can be estimated from a sample of objects counting the pairs of objects with different separations $r$. There are four different estimators known in the literature. These are:
• Peebles & Hauser [4] estimator: $\xi _{PH}(r)=\frac {N_{R}}{N}\frac {DD(r)}{RR(r)}-1;$
• Davis & Peebles [5] estimator: $\xi _{DP}(r)=\frac {2N_{r}}{N-1}\frac {DD(r)}{DR(r)}-1;$
• Hamilton [6] estimator: $\xi _{H}(r)=\frac {4NN_{r}}{(N-1)(N_{r}-1)}\frac {DD(r}{DR(r)}\frac {RR(r)}{DR(r)}-1;$
• Landy & Szalay [7] estimator: $\xi _{LS}(r)=\frac {N_{r}(N_{r}-1)}{N(N-1)}\frac {DD(r)}{RR(r)}-\frac {2(N_{r}-1)}{N}\frac {DR(r)}{RR(r)}+1.$
Here $DD$, $RR$ and $DR$ are the numbers of pairs in the initial catalogue (data-data), in the so called random catalogue (random-random) and the number of cross-pairs from both catalogues (data-random) correspondingly. The random catalogue is a manually constructed catalogue represinting a random distribution of objects in the same volume. Usually to reduce a noise the random catalogue is several times greater than the initial one. Thus the normalizing coefficients containing the numbers of objects in the initial, $N$, and random, $N_{r}$, catalogues are included in these estimators. These estimators are nearly equivalent on small scales.
References:
• [1]^ Yadav J. K. Nature of Clustering of Large Scale Structures, PhD thesis, arXiv:0910.0808
• [2]^ab Peebles P. J. E. The Large-Scale Structure of the Universe, Princeton University Press, Princeton, New Jersey, 1980
• [3]^ Peacock J. A. Cosmological Physics, Cambridge University Press, Cambridge, UK, 2007
• [4]^ Peebles P. J. E. & Hauser M. J., 1974, ApJSS, 28, 19, 1974ApJS...28...19P
• [5]^ Davis M. & Peebles P. J. E., 1983, ApJ, 267, 465, 1983ApJ...267..465D
• [6]^ Hamilton A. J. S., 1993, ApJ, 417, 19, 1993ApJ...417...19H
• [7]^ Landy S. D. & Szalay A. S., 1993, ApJ, 412, L64, 1993ApJ...412...64L