7. How to obtain the correlations from the distribution
To this end, we have to use Eq. ( 8 ) as defined previously. However, when we have a discrete number of points instead of a continuum distribution, perhaps it might be better to use other equivalent expressions.
When homogeneity is given, one method discussed by Rivolo (1986) is to use the estimator
where is the number of particles lying in a shell of thickness from the i th particle, is the volume of the shell lying within the sample volume and is the average density in a macroscopically homogeneous system ( is independent of position).
The evaluation of and would be:
where and is the mass lying in a shell of thickness from the i th particle. Knowledge of is sometimes problematic and its solution differs in each case. In the case of galaxies or stars information is required about the mass-luminosity relationship as well as data on magnitudes and distances. This could also be provided with a knowledge of the mass distribution function according to different zones, ( would be the average of the different mass distribution functions in the entire space) and by randomly assigning a mass to each particle following these distribution functions that would give us not the real mass distribution but an equivalent one. In any case, the problem of assigning mass is different in each case and generally requires tailor-made solutions for each one.
The three-point correlation function is also obtainable by counting groups of three particles with different distances between them. Usually, in the isotropic case, this is approximated as a function of different two-point correlation functions. For example, in liquid theory the so-called superposition approximation is commonplace (see for example March & Tosi 1976 and first formulation of it in Kirkwood 1935), and applied to the mass-mass-particle case would take the form
From the large-scale distribution of galaxies in the Universe, we can extract statistical information (as in Saunders et al. 1991). When we observe the projected distribution onto a 2-dimensional surface, i.e. we do not know the distance of the objects, we can obtain the correlation on the 2-dimensional surface (angular correlation) and relate it to the 3-dimensional distribution correlations by means of Limber's equation (Peebles 1980). Maddox et al. (1990) obtained the two-point angular correlation function on large scales, so we can derive . To assign the masses in order to achieve and is again troublesome and depends on the data available on the masses of the galaxies.
For the mass of galaxies in the large-scale structure, future data will become available with the help of the DENIS project (Mamon 1995), a near-infrared sky survey. The near-infrared is thought to be a better tracer of the stellar mass in the galaxies, and if the stellar mass content follows the total mass content (including dark matter), near-infrared surveys should be the best way of obtaining the distribution of matter in the Universe.
Also, on other astronomical scales we find point-particle distributions and we may obtain their correlations. Borgani et al. (1991) obtained the correlation functions for scales between 3 and 350 kpc, and this could be done for smaller scales as well. Another problem is posed by considering our hypotheses as valid, especially Hypothesis 1.
An application to any other distribution is also possible.
© European Southern Observatory (ESO) 1997
Online publication: October 15, 1997