The Fractal Structure of the Universe: A New Field Theory Approach

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Abstract
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 < 100 h-1 Mpc), with a fractal dimension D = 1.5-2. We study the thermodynamics of a self-gravitating system using the theory of critical phenomena and finite-size scaling, and we show that gravity provides a dynamical mechanism for producing this fractal structure. We develop a field theoretical approach for computing 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 cutoff, below which other physics intervene, and a large-distance cutoff, beyond which the thermodynamic equilibrium is not satisfied. The galaxy ensemble can be considered at critical conditions, with large density fluctuations developing at any scale. From the theory of critical phenomena, we derive the two independent critical exponents ν and η and predict the fractal dimension D = 1/ν to be either 1.585 or 2, depending on whether the long-range behavior 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 behavior of the gravitational potential to be r-(1+η)/2; that is, r-0.5 for mean field or r-0.519 for the Ising fixed point. The theory allows us to compute the 3 and higher density correlators without any assumption or Ansatz. We find that the N-point density scales as r when r1 ri, 2 ≤ i ≤N. There are no free parameters in this theory.
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