  Within the framework of hierarchical clustering scenarios, we investigate the
consequences for the properties of virialized halos of the constraints provided
by numerical simulations on the first few correlation functions. Thus, we show
that the density field cannot be described by a collection of smooth halos with
a universal density profile. This implies that substructures within larger
objects play an important role (but a mean spherically averaged density profile
may exist). In particular, a possible interpretation is that collapsed objects
can be divided into an infinite hierarchy of smaller objects with increasingly
large densities (these substructures might also be continuously destroyed and
created by the long-range action of gravity). Finally, we present multifractal
models (restricted to non-linear scales) which can describe in a natural way
such non-linear density fields with increasingly large fluctuations at smaller
scales. We relate their properties to the correlation functions and present a
few constraints they are expected to satisfy, using theoretical considerations
as well as constraints from numerical simulations. Thus, the simplest realistic
model is the bifractal model described in Balian & Schaeffer (1989a). Moreover,
we show that it should provide (at least) a very good approximation of the
multifractal properties of the actual non-linear density field, hence of the
probability distribution of the density contrast. The implications of this
model (e.g. for galaxies) are detailed in other studies.
