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

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 a time evolution of a 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 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