## 9. Results, other applications and further commentariesWe have a numerical relationship between the distribution of galaxies in space, which is represented by the correlation functions, and the two-galaxy interaction, which is represented by the potential energy. We have obtained the equality ( 21 ) that must be followed by the distributions under Hypotheses 1 to 5. Also, we have equality ( 19 ) for the correlation functions and any interaction force under Hypotheses 1, 2 and 3 (we could also obtain an expresion like this including the expansion of space by means of the method explained in the subsection dedicated to the expansion). Eq. (
21 ) relates the distribution correlation with
the mean velocity of the galaxies by means of When equality between the two-sides of Eq. (
21 ) is unattainable, this will indicate that
our hypotheses are unsuitable. Probably, the most doubtful hypothesis
is the first, i.e. that of Boltzmann equilibrium, and it is possible
to verify the relaxation using this
equation. In a sufficiently evolved system, Boltzmann equilibrium is achieved because the particle-points are classical particles and the probability of a state in such a case is proportional to the number of different states for each particle that preserves the number of particles and the total energy (the reader is referred to any book dealing with the foundations of statistical mechanics, e.g. Tolman 1938). It is also true that after a long time, the systems become virialized, and many systems are known to be in these conditions, although not all of them. Otherwise, Eq. (
19 ), In order to demonstrate what might be the caveats in the implementation of the force equation, I developed a real example in the previous section where it was used to infer information about the mass-mass correlation function from the galaxy-galaxy correlation function, and the average density and peculiar velocity in the large-scale distribution of galaxies in the Universe. Further improvements are necessary, both in the observations and the theoretical assumptions, to obtain an accurate result, but the method is at least capable of telling us that the mass is more correlated than the galaxies at short distances (Fig. 3) when we assume relaxation on scales greater than Mpc. © European Southern Observatory (ESO) 1997 Online publication: October 15, 1997 |