The cosmological mass function problem is analyzed in full detail in the case
of 1D gravity, with analytical, semi-analytical and numerical techniques. The
extended Press & Schechter theory is improved by detailing the relation between
smoothing radius and mass of the objects. This is done by introducing in the
formalism the concept of a growth curve for the objects. The predictions of the
extended Press & Schechter theory are compared to large N-body simulations of
flat expanding 1D universes with scale-free power spectra of primordial
perturbations. The collapsed objects in the simulations are located with a
clump-finding algorithm designed to find regions that have undergone orbit
crossing or that are in the multi-stream regime (these are different as an
effect of the finite size of the multi-stream regions). It is found that the
semi-analytical mass function theory, which has no free parameters, is able to
recover the properties of collapsed objects both statistically and object by
object. In particular, the predictions of regions in orbit crossing are
optimized by the use of Gaussian filtering, while the use of sharp k-space
filtering apparently allows to reproduce the larger multi-stream regions. The
mass function theory does not reproduce well the clumps found with the standard
friends-of-friends algorithm; however, the performance of this algorithm has
not been thoroughly tested in the 1D cosmology. Our preliminary analyses of the
3D case confirms that the techniques developed in this paper are precious in
understanding the cosmological mass function problem in 3D.Comment: 25 pages, revtex, postscript figures included, in press on Physical
Review
