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 atsmall-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-gravitatingsystem with the theory of critical phenomena and finite-size scaling and showthat 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, (althoughlarge), range of scales is involved, between a short-distance cut-off belowwhich other physics intervene, and a large-distance cut-off, where the thermo-dynamic equilibrium is not satisfied. The galaxy ensemble can be considered atcritical conditions, with large density fluctuations developping at any scale. From the theory of critical phenomena, we derive the two independent criticalexponents nu and eta and predict the fractal dimension D = 1/nu to be either1.585 or 2, depending on whether the long-range behaviour is governed by theIsing or the mean field fixed points, respectively. Both set of values arecompatible with present observations. In addition, we predict the scalingbehaviour 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 theoryallows to compute the three and higher density correlators without anyassumption or Ansatz. We find that the N-points density scales asr_1^{(N-1)(D-3)}, when r_1 >> r_i, 2 leq i leq N . There are no free parametersin this theory.Comment: Latex, 20 pages, no figures, to be published in the Astrophysical Journa