Self-similarity and stable clustering in a family of scale-free cosmologies

Abstract

We study non-linear gravitational clustering from cold Gaussian power-law initial conditions in a family of scale-free Einstein de Sitter (EdS) models, characterized by a free parameter κ fixing the ratio between the mass driving the expansion and the mass which clusters. As in the ‘usual’ EdS model, corresponding to κ = 1, self-similarity provides a powerful instrument to delimit the physically relevant clustering resolved by a simulation. Likewise, if stable clustering applies, it implies scale-free non-linear clustering. We derive the corresponding exponent γsc(n, κ) of the two-point correlation function. We then report the results of extensive N-body simulations, of comparable size to those previously reported in the literature for the case κ = 1, and performed with an appropriate modification of the gadget2 code. We observe in all cases self-similarity in the two-point correlations, down to a lower cut-off which decreases monotonically in time in comoving coordinates. The self-similar part of the non-linear correlation function is fitted well in all cases by a single power-law with an exponent in good agreement with γsc(n, κ). Our results thus indicate that stable clustering provides an excellent approximation to the non-linear correlation function over the resolved self-similar scales, at least down to γsc(n, κ) ≈ 1, corresponding to the case n = −2 for κ = 1. We conclude, in contrast notably with the results of Smith et al., that a clear identification of the breakdown of stable clustering in self-similar models – and the possible existence of a ‘universal’ region in which non-linear clustering becomes independent of initial conditions – remains an important open problem, which should be addressed further in significantly larger simulations.