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Fractal geometry in an expanding, one-dimensional, Newtonian universe

Bruce N. Miller*

Jean-Louis Rouet

Emmanuel Le Guirriec

  • Department of Physics and Astronomy, Texas Christian University, Fort Worth, Texas 76129, USA

  • Institut des Sciences de la Terre d’Orléans, (ISTO)—UMR 6113 CNRS/Université d’Orléans, 1A, rue de la Férollerie, 45071 Orléans Cedex 2, France and Laboratoire de Mathématique, Applications et Physique Mathématique—UMR 6628 CNRS/Université d’Orléans, UFR des Sciences, F-45067 Orléans Cedex 2, France

  • Laboratoire de Mathématique, Applications et Physique Mathématique—UMR 6628 CNRS/Université d’Orléans, UFR des Sciences, F-45067 Orléans Cedex 2, France

  • *b.miller@tcu.edu; URL: http://personal.tcu.edu/∼bmiller
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Phys. Rev. E 76, 036705 – Published 14 September, 2007

DOI: https://doi.org/10.1103/PhysRevE.76.036705

Abstract

Observations of galaxies over large distances reveal the possibility of a fractal distribution of their positions. The source of fractal behavior is the lack of a length scale in the two body gravitational interaction. However, even with new, larger, sample sizes from recent surveys, it is difficult to extract information concerning fractal properties with confidence. Similarly, three-dimensional N-body simulations with a billion particles only provide a thousand particles per dimension, far too small for accurate conclusions. With one-dimensional models these limitations can be overcome by carrying out simulations with on the order of a quarter of a million particles without compromising the computation of the gravitational force. Here the multifractal properties of two of these models that incorporate different features of the dynamical equations governing the evolution of a matter dominated universe are compared. For each model at least two scaling regions are identified. By employing criteria from dynamical systems theory it is shown that only one of them can be geometrically significant. The results share important similarities with galaxy observations, such as hierarchical clustering and apparent bifractal geometry. They also provide insights concerning possible constraints on length and time scales for fractal structure. They clearly demonstrate that fractal geometry evolves in the μ (position, velocity) space. The observed patterns are simply a shadow (projection) of higher-dimensional structure.

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Phys. Rev. E 76, 036705– Published 14 September, 2007
  • Received 18 December 2006

© 2007 American Physical Society

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