Fractal Dimensions and Scaling Laws in the Interstellar Medium and Galaxy Distributions: a new Field Theory Approach

Abstract

We develop a field theoretical approach to the cold interstellar medium (ISM) and large structure of the universe. We show that a non-relativistic self- gravitating gas in thermal equilibrium with variable number of atoms or fragments is exactly equivalent to a field theory of a scalar field phi(x) with exponential self-interaction. We analyze this field theory perturbatively and non-perturbatively through the renormalization group(RG).We show scaling behaviour (critical) for a continuous range of the physical parameters as the temperature. We derive in this framework the scaling relation M(R) \sim R^{d_H} for the mass on a region of size R, and Delta v \sim R^\frac12(d_H -1) for the velocity dispersion. For the density-density correlations we find a power-law behaviour for large distances \sim |r_1 - r_2|^{2D - 6}.The fractal dimension D turns to be related with the critical exponent νby D = 1/ ν. Mean field theory yields ν= 1/2, D = 2. Both the Ising and the mean field values are compatible with the present ISM observational data:1.4\leq D \leq 2. We develop a field theoretical approach to the galaxy distribution considering a gas of self-gravitating masses on the FRW background, in quasi-thermal equi- librium. We show that it exhibits scaling behaviour by RG methods. The galaxy correlations are computed without assuming homogeneity. We find <ρ({\vec r_0})ρ({\vec r_0} + {\vec r}) > \sim r^{D-3} $.Thetheoryallowstocomputethethreeandhigherdensitycorrelatorswithoutanyassumption.WefindthattheconnectedN-pointsdensityscalesas{r}_1^{N(D-3)},when$ r_1 >> r_i, 2\leq i \leq N $. There are no free parameters in this theory.

Latex file, 35 pages, no figures. To appear in the Proceedings of the 6th. Erice Chalonge School on Astrofundamental Physics, N. Sánchez and A. Zichichi eds., Kluwer, 1998, references updated