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 a collisionless pressureless fluid out of a power-law spectrum of primeval Gaussian mass density fluctuations, P(k) ∝ kn, in an Einstein-de Sitter cosmological model. The self-similar position two-point correlation function, ξ(r), seems to be well established. The renormalization solutions for ξ(r) show a satisfying insensitivity to the parameters in the method, and at n = -1 and n = 0 they are quite close to the Hamilton et al. formula for interpolation between the large-scale perturbative limit and stable small-scale clustering. Self-similar behavior is tested by the comparison of the mean relative peculiar velocity ⟨vij⟩ of particle pairs (ij) and the velocity derived from ξ(r) 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 does better at small separations. Other comparisons of renormalization and conventional solutions are more demanding and the results much less satisfactory. Maps of the particle positions in redshift space in the renormalization solutions show more nearly empty voids and fewer prominent walls than do comparison conventional N-body solutions. The rms relative velocity dispersion is systematically smaller in the renormalization solution; the difference approaches a factor of 2 on small scales. This is related to substantial differences in the frequency distributions of clump masses in the renormalization and conventional solutions. The third moment S3 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 S3 is roughly consistent with hierarchical clustering, but is heavily affected by shot noise.