Scale Symmetry in the Universe

Jose Gaite

doi:10.3390/sym12040597

Abstract

Scale symmetry is a fundamental symmetry of physics that seems however not to be fully realized in the universe. Here, we focus on the astronomical scales ruled by gravity, where scale symmetry holds and gives rise to a truly scale invariant distribution of matter, namely it gives rise to a fractal geometry. A suitable explanation of the features of the fractal cosmic mass distribution is provided by the nonlinear Poisson–Boltzmann–Emden equation. An alternative interpretation of this equation is connected with theories of quantum gravity. We study the fractal solutions of the equation and connect them with the statistical theory of random multiplicative cascades, which originated in the theory of fluid turbulence. The type of multifractal mass distributions so obtained agrees with results from the analysis of cosmological simulations and of observations of the galaxy distribution.

Highlights

The symmetry of the physical laws is probably the essential foundation of our current understanding of physics and the universe [1]
We show that there is a considerable range of scales in the universe in which scale symmetry is effectively realized, that is to say, the mass distribution is a self-similar multifractal, with identical appearance and properties at any scale
This symmetry is a consequence of the absence of any intrinsic length scale in Newtonian gravitation, which is the theory that rules the mass distribution on scales beyond the size of galaxies but small compared to the Hubble length

Summary

Introduction

The symmetry of the physical laws is probably the essential foundation of our current understanding of physics and the universe [1]. This situation is surely a consequence of the different definitions used and the limitations of the current methods of observation At any rate, this mainly observational issue is not crucial for us and we are content to study the fractal structure of the universe without worrying about the definitive value of the scale of transition to homogeneity. Deep studies of the fractal geometry of mass distributions related to the Poisson–Boltzmann–Emden equation have been carried out in the theory of stochastic processes, namely in the theory of random multiplicative cascades [22,23,24]. The fractal geometry of the large scale structure of the universe is produced by three-dimensional gravity, but the study of lower dimensional models has interest in cosmology. We take a further step in the attempt to describe analytically the fractal geometry of the universe, by means of simple models and taking advantage of the power of the scale symmetry

Multifractal Geometry

Gravity and Scale Symmetry

Random Cascades

Experimental and Observational Results

Discussion