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

We study non-linear gravitational clustering from cold gaussian power-law initial conditions in a family of scale-free EdS models, characterized by a free parameter $\kappa$ fixing the ratio between the mass driving the expansion and the mass which clusters. As in the “usual” EdS model, corresponding to $\kappa =1$, 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 ${\gamma }_{sc}(n,\kappa )$ 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 $\kappa =1$, 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 ${\gamma }_{sc}(n,\kappa )$. 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 $\gamma_{sc} (n, \kappa) \approx 1$,correspondingtothecase$n=-2$for$\kappa=1$. We conclude, in contrast notably with the results of Smith et al. (2003), 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.