A Numerical Renormalization Solution for Self-Similar Cosmic Structure Formation

We present results of a numerical renormalization approximation to the self- similar growth of clustering of pressureless dust out of a power-law spectrum of primeval Gaussian mass density fluctuations (index n) in an Einstein-de Sitter cosmological model. The self-similar two-point correlation function, xi, seems to be well established. The renormalization solutions for xi show a satisfying insensitivity to the parameters in the method, and at n=-1 and 0 are close to the Hamilton et al. formula for interpolation between the large-scale perturbative limit and stable small-scale clustering. The solutions are tested by comparing the mean relative peculiar velocity of particle pairs and the velocity derived from xi under the assumption of self-similar evolution. Both the renormalization and a comparison conventional N-body solution are in reasonable agreement with the test, although the conventional approach does slightly better at large separations and the renormalization approach slightly better at small separations. Other comparisons of renormalization and conventional solutions are more demanding and the results less satisfactory. Maps of particle positions in redshift space in the renormalization solutions show more nearly empty voids and less prominent walls than do the conventional solutions. The rms relative velocity dispersion is systematically smaller in the renormalization solution. There also are sizeable differences in the frequency distributions of clump masses in the renormalization and conventional solutions. The third moment S_3 from the distribution of mass within cells is in reasonable agreement with second-order perturbation theory on large scales, while on scales less than the clustering length S_3 is roughly consistent with hierarchical clustering but is heavily affected by shot noise.