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 scale 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 single scalar field $\phi \left( {\vec x} \right)$ with exponential self-interaction. We analyze this field theory perturbatively and non-perturbatively through the renormalization group approach. We show scaling (critical) behaviour for a continuous range of the temperature and of the other physical parameters. We derive in this framework the scaling relation M(R) ∼ R ^dH for the mass on a region of size R, and Δυ ∼ R ^q for the velocity dispersion where $q=\frac{1}{2}({d_H}- 1)$. For the density-density correlations we find a power-law behaviour for large distances $\sim \left| {{{\vec r}_i}} \right. - {\left. {{{\vec r}_2}} \right|^{2dH - 6}}$ The fractal dimension d _H turns out to be related with the critical exponent v of the correlation length by d _H = 1/v. The renormalization group approach for a single component scalar field in three dimensions states that the long-distance critical behaviour may be governed by the (non-perturbative) Ising fixed point. The Ising values of the scaling exponents are v = 0.631…, d _H = 1.585… and q = 0.293…. Mean field theory yields for the scaling exponents v = 1/2, d _H = 2 and q = 1/2. Both the Ising and the mean field values are compatible with the present ISM observational data: 1.4 ≤ d _H ≤ 2, 0.3 ≤ q≤ 0.6. As typical in critical phenomena, the scaling behaviour and critical exponents of the ISM can be obtained without dwelling into the dynamical (time-dependent) behaviour.

We develop a field theoretical approach to the galaxy distribution. We consider a gas of self-gravitating masses in quasi-thermal equilibrium on the FRW background. We show that it exhibits scaling behaviour by renormalization group methods. The galaxy correlations are first computed assuming homogeneity for very large scales and then without assuming homogeneity. In the first case we find ξ(r)≡< ρ $(\vec r_0 )\rho (\vec r_0 + \vec r)$ >/< ρ >² -1 ~ r^-γ, with γ = 2. In the second case we find Γ(r)=> ρ $(\vec r_0 )\rho (\vec r_0 + \vec r)$ > ~ r^D-3 with with D = 2. while the universe becomes more and more homogeneous at large scales, statistical analysis of galaxy catalogs have revealed a fractal structure at small-scales (λ<100h-1 Mpc), with a fractal dimension D = 1.5-2 (Sylos Labini et al 1996). We study the thermodynamics of a self-gravitating system with the theory of critical phenomena and finite-size scaling and show that gravity provides a dynamical mechanism to produce this fractal structure. Only a limited, (although large), range of scales is involved, between a short-distance cut-off below which other physics intervene, and a large-distance cut-off, where the thermodynamic equilibrium is not satisfied. The galaxy ensemble can be considered at critical conditions, with large density fluctuations developping at any scale. From the theory of critical phenomena, we derive the two independent critical exponents v and η and predict the fractal dimension D = l/v to be either 1.585 or 2, depending on whether the long-range behaviour is governed by the Ising or the mean field fixed points, respectively. Both set of values are compatible with present observations. In addition, we predict the scaling behaviour of the gravitational potential to be $r - \frac{1}{2}(1 + \eta )$. That is, r⁻⁰⁵ for mean field or r^−0.515 for the Ising fixed point. The theory allows to compute the three and higher density correlators without any assumption or Ansatz. We find that the N-points density scales as $r_1^{(N-1)(D-3)}$ when $r_1 > > r_{i,} 2 \le i \le N.$. There are no free parameters in this theory.