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 $\gamma_{sc} (n, \kappa)
\approx 1$, corresponding to the case $n=-2$ 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.
