9. Results, other applications and further commentaries
We 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 C with ( 24 ), and I believe it will be useful for obtaining a parameter from others that are already known in the distribution: the mean velocity from a complete knowledge of the correlation functions, or an unknown parameter in the correlation function from the rest of the data. We could even obtain more than one parameter: two or three (in my opinion, more than three are too many) that follow the equality between the right- and left-hand sides of Eq. ( 21 ).
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.2
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 ), the force equation, is a tool for looking for the kind of interaction in a system following Hypotheses 1, 2 and 3 (as purposed in Goldman et al. 1992). Once we have derived all the parameters of the distribution we can fit a shape for V, a two-body interaction potential that fits the equality, with or without expansion. We could even obtain other unknown parameters (C for example).
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