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

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.

Included plots of power spectra at a sequence of epochs. Introduced analytic model that captures the central features of the evolution

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• Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation
• Breaking the self-averaging properties of spatial galaxy fluctuations in the Sloan Digital Sky Survey - Data Release Six
• Relaxation of a 1-D gravitational system
• Gravitational Dynamics of an Infinite Shuffled Lattice of Particles
• Ewald Sums for One Dimension
• Gravitational force in an infinite one-dimensional Poisson distribution
• Structure and evolution of Zel'dovich pancakes as probes of dark energy models
• Universality in the distribution of caustics in the expanding Universe

Cited by in corpus (11)

• Nonequilibrium Statistical Mechanics of Systems with Long-Range Interactions: Ubiquity of Core-Halo Distributions
• Cosmological perturbation theory in 1+1 dimensions
• Cosmological structure formation with negative mass
• Continuous wavelet analysis of matter clustering using the Gaussian-derived wavelet
• Non-linear gravitational clustering of cold matter in an expanding universe: indications from 1D toy models
• Exponents of non-linear clustering in scale-free one dimensional cosmological simulations
• Structure formation in a Dirac-Milne universe: comparison with the standard cosmological model
• Cosmology in One Dimension: Vlasov Dynamics
• Cosmology in One Dimension: The Two Component Universe
• Kinetic properties of fractal media
• Fractal Dimension Computation From Equal Mass Partitions