The fractal structure of the universe : a new field theory approach
  While the universe becomes more and more homogeneous at large scales, statistical analysis of galaxy catalogs have revealed a fractal structure at small-scales (\lambda < 100 h^{-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. We develop a field theoretical approach to compute the galaxy distribution, assuming them to be in quasi-isothermal equilibrium. 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 thermo- dynamic 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 nu and eta and predict the fractal dimension D = 1/nu 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^{-(1 + eta)/2}. That is, r^{-0.5} for mean field or r^{- 0.519} 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 leq i leq N . There are no free parameters in this theory. 
H.J. de Vega, N. S\'anchez and F. Combes