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 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 an Einstein-de 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.