Cosmology in one dimension: Symmetry role in dynamics, mass oriented approaches to fractal analysis

Abstract

The distribution of visible matter in the universe, such as galaxies and galaxy clusters, has its origin in the week fluctuations of density that existed at the epoch of recombination. The hierarchical distribution of the universe, with its galaxies, clusters and super-clusters of galaxies indicates the absence of a natural length scale. In the Newtonian formulation, numerical simulations of a one-dimensional system permit us to precisely follow the evolution of an ensemble of particles starting with an initial perturbation in the Hubble flow. The limitation of the investigation to one dimension removes the necessity to make approximations in calculating the gravitational field and, on the whole, the system dynamics. It is then possible to accurately follow the trajectories of particles for a long time. The simulations show the emergence of a self-similar hierarchical structure in both the phase space and the configuration space and invites the implementation of a multifractal analysis. Here, after showing that symmetry considerations leads to the construction of a family of equations of motion of the one-dimensional gravitational system, we apply four different methods for computing generalized dimensions $D_q$ of the distribution of particles in configuration space. We first employ the conventional box counting and correlation integral methods based on partitions of equal size and then the less familiar nearest-neighbor and k-neighbor methods based on partitions of equal mass. We show that the latter are superior for computing generalized dimensions for indices $q<-1$ which characterize regions of low density.