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Two-point correlation function of galaxy distribution shows that thestructure in the present Universe is scale-free up to a certain scale (at leastseveral tens Mpc), which suggests that a fractal structure may exist. If smallprimordial density fluctuations have a fractal structure, the presentfractal-like nonlinear structure below the horizon scale could be naturallyexplained. We analyze the time evolution of fractal density perturbations inEinstein-de Sitter universe, and study how the perturbation evolves and whatkind of nonlinear structure will come out. We assume a one-dimensionalcollisionless sheet model with initial Cantor-type fractal perturbations. Thenonlinear structure seems to approach some attractor with a unique fractaldimension, which is independent of the fractal dimensions of initialperturbations. A discrete self-similarity in the phase space is also found whenthe universal nonlinear fractal structure is reached.

Primordial Fractal Density Perturbations and Structure Formation in the Universe: One‐dimensional Collisionless Sheet Model