Primordial Fractal Density Perturbations and Structure Formation in the Universe: 1-Dimensional Collisionless Sheet Model

Abstract

Two-point correlation function of galaxy distribution shows that the structure in the present Universe is scale-free up to a certain scale (at least several tens Mpc), which suggests that a fractal structure may exist. If small primordial density fluctuations have a fractal structure, the present fractal-like nonlinear structure below the horizon scale could be naturally explained. We analyze the time evolution of fractal density perturbations in Einstein-de Sitter universe, and study how the perturbation evolves and what kind of nonlinear structure will come out. We assume a one-dimensional collisionless sheet model with initial Cantor-type fractal perturbations. The nonlinear structure seems to approach some attractor with a unique fractal dimension, which is independent of the fractal dimensions of initial perturbations. A discrete self-similarity in the phase space is also found when the universal nonlinear fractal structure is reached.Subject headings: cosmology: theory -large-scale structure -fractal -1-dimensional sheet model

Mathematical Geosciences: Local Singularity Analysis of Nonlinear Earth Processes and Extreme Geo-Events

“…For example, the term "fractal density" has been used to refer the number of fractals per area (Hou and Wu 1989) which does not mean the same as the concept introduced in the current paper. Tatekawa and Maeda (2001) analyzed time evolution of fractal density perturbations in the Einstein-de Sitter universe, in which the emphasis is on how the perturbation evolves and what kind of nonlinear structure will come out. Similarly, Federrath et al (2009) has used fractal density structure in supersonic isothermal turbulence when referring to density structure.…”

Natural Genesis. An Introduction to the World Wide Discovery of a Creative Organic Universe

“…A research paper in The Astrophysical Journal entitled 'Primordial Fractal Density Perturbations and Structure Formation in the Universe' describes a dynamic, scale-free cosmic evolution (Tatekawa, Maeda 2001). On an infinitesimal but special planet orbiting a star among billions in a galaxy, amongst billions of galaxies, sentient beings suddenly appear who can explore and comprehend the sidereal reaches from which they arose.…”