PRIMORDIAL FRACTAL DENSITY PERTURBATIONS AND STRUCTURE FORMATION IN THE UNIVERSE: ONE-DIMENSIONAL COLLISIONLESS SHEET MODEL

The two-point correlation function of galaxy distribution shows that structure in the present universe is scale-free up to a certain scale (at least several tens of Mpc), which suggests that a fractal structure may exist. If small primordial density Ñuctuations 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 an EinsteinÈde Sitter universe, and study how the perturbation evolves and what kind of nonlinear structure will result. 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