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3.1. The two-point correlation function

The structure of the universe qualitatively described in the previous section needs to be quantified by means of statistical measures having the capacity of distinguish between different point patterns.

The most popular measure used in this context has been the two-point correlation function [14, 1] xi(r). The first time this quantity was applied to a galaxy catalog was in 1969 by Totsuji and Kihara [15]. Since then, its use has been widely spread. The quantity xi(r) is defined in terms of the probability that a galaxy is observed within a volume dV lying at a distance r from an arbitrary chosen galaxy,

Equation 2 (2)

where n is the average galaxy number density. For a completely random distribution xi(r) = 0. Positive values indicate clustering, negative values indicate anti-clustering or regularity. In this definition, isotropy and homogeneity of the point process is being assumed, otherwise the function xi(r) should depend on a vector quantity.

Several estimators have been used to obtain the two-point correlation function from a given data set [16, 17]. At short distances their results are nearly indistinguishable; at large distances, however, the differences become important. The best performance is reached by the Hamilton [18] and the Landy and Szalay [19] estimators.

To illustrate the kind of information that we can extract from this second-order spatial statistic we can use a point process having an analytic expression for its two-point correlation function. A segment Cox process is generated by randomly placing segments of length ell within a window W. Then, we scatter points on the segments with a given intensity. If the mean number of segments per unit volume is lambdas, the correlation function of the process has the form [20]

Equation 3 (3)

for r leq ell and vanishes for larger r. Note that this expression is independent of the number of points per unit length scattered on each segment.

In Fig. 4 we show a 3-D simulation of this process with parameters lambdas = 0.001 and ell = 10. The correlation function estimate is shown together with the analytical expectation of Eq. 3. Note that, at small scales, xiCox(r) ~ r-gamma with gamma = 2. The strong clustering signal of this point field can be smeared out by applying independent random shifts to each point of the simulation. If the random shifts are performed by a three-dimensional Gaussian distributed vector with sigma = 0.5, the short scale correlations are completely destroyed (see Fig. 4). If the shifts are distributed according to a power-law density probability function d*(r) propto ralpha, the value of gamma is reduced by 2(1 + alpha). In the example alpha = -0.75 and therefore gamma changes from 2 to 1.5. This seems to be a rather general phenomenon [21]. The random shifts affect the correlation function mimicking the way peculiar velocities suppress the short range correlations [22] (for scales r leq 2h-1 Mpc, where h is the Hubble parameter in units of 100 km s-1 Mpc-1).

Figure 4a Figure 4b
Figure 4c Figure 4d Figure 4e

Figure 4. The top left panel shows a segment Cox process simulated on a cube with side-length 100. The top right panel shows the two-point correlation function: The dotted line corresponds to the expected analytical expression (see Eq. 3), that is, in this range of scales, close to a power-law with exponent -2. Solid bullets are the empirically calculated values of xi(r). Open diamonds correspond to the function calculated on the shifted point process with shifts following a power-law distribution function, while crosses correspond to Gaussian shifts. The left bottom panel shows a slice with dimensions 40×40×10 drawn from the unshifted full realization. The same slice is shown after applying power-law shifts (central bottom panel) and Gaussian fits (right bottom panel).

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