Cosmology in One Dimension: Fractal Geometry, Power Spectra and Correlation

TL;DR

This paper investigates the development of fractal geometry and power spectra in one-dimensional cosmological models, revealing how initial conditions influence hierarchical clustering and self-similar evolution.

Contribution

It introduces explicit solutions for gravitational potential in 1D, develops a method for scale-free initial conditions, and analyzes fractal and spectral properties during evolution.

Findings

Power spectra accurately describe the evolution from linear to nonlinear regimes.

Hierarchical clustering depends on initial conditions and model specifics.

A relation between power spectrum, correlation function, and fractal dimension is confirmed in nonlinear regime.

Abstract

Concentrations of matter, such as galaxies and galactic clusters, originated as very small density fluctuations in the early universe. The existence of galaxy clusters and super-clusters suggests that a natural scale for the matter distribution may not exist. A point of controversy is whether the distribution is fractal and, if so, over what range of scales. One-dimensional models demonstrate that the important dynamics for cluster formation occur in the position-velocity plane. Here the development of scaling behavior and multifractal geometry is investigated for a family of one-dimensional models for three different, scale-free, initial conditions. The methodology employed includes: 1) The derivation of explicit solutions for the gravitational potential and field for a one-dimensional system with periodic boundary conditions (Ewald sums for one dimension); 2) The development of a procedure for obtaining scale-free initial conditions for the growing mode in phase space for an arbitrary power-law index; 3) The evaluation of power spectra, correlation functions, and generalized fractal dimensions at different stages of the system evolution. It is shown that a simple analytic representation of the power spectra captures the main features of the evolution, including the correct time dependence of the crossover from the linear to nonlinear regime and the transition from regular to fractal geometry. A possible physical mechanism for understanding the self-similar evolution is introduced. It is shown that hierarchical cluster formation depends both on the model and the initial power spectrum. Under special circumstances a simple relation between the power spectrum, correlation function, and correlation dimension in the highly nonlinear regime is confirmed.