  One dimensional versions of cosmological N-body simulations have been shown
to share many qualitative behaviours of the three dimensional problem. They can
resolve a large range of time and length scales, and admit exact numerical
integration. We use such models to study how non-linear clustering depends on
initial conditions and cosmology. More specifically, we consider a family of
models which, like the 3D EdS model, lead for power-law initial conditions to
self-similar clustering characterized in the strongly non-linear regime by
power-law behaviour of the two point correlation function. We study how the
corresponding exponent \gamma depends on the initial conditions, characterized
by the exponent n of the power spectrum of initial fluctuations, and on a
single parameter \kappa controlling the rate of expansion. The space of initial
conditions/cosmology divides very clearly into two parts: (1) a region in which
\gamma depends strongly on both n and \kappa and where it agrees very well with
a simple generalisation of the so-called stable clustering hypothesis in three
dimensions, and (2) a region in which \gamma is more or less independent of
both the spectrum and the expansion of the universe. We explain the observed
location of the boundary in (n, \kappa) space dividing the "stable clustering"
region from the "universal" region. We compare and contrast our findings to
results in three dimensions, and discuss in particular the light they may throw
on the question of "universality" of non-linear clustering in this context.
