Non-linear gravitational clustering: smooth halos, substructures and scaling exponents

Within the framework of hierarchical clustering scenarios, we investigate t he consequences for the properties of virialized halos of the constraints p rovided by numerical simulations on the first few correlation functions. Th us, we show that the density field cannot be described by a collection of s mooth halos with a universal density profile. This implies that substructur es within larger objects play an important role (but a mean spherically ave raged density profile may exist). In particular, a possible interpretation is that collapsed objects can be divided into an infinite hierarchy of smal ler objects with increasingly large densities (these substructures might al so be continuously destroyed and created by the long-range action of gravit y). Finally, we present multifractal models (restricted to non-linear scale s) which can describe in a natural way such non-linear density fields with increasingly large fluctuations at smaller scales. We relate their properti es to the correlation functions and present a few constraints they are expe cted to satisfy, using theoretical considerations as well as constraints fr om numerical simulations. Thus, the simplest realistic model is the bifract al model described in Balian & Schaeffer (1989a). Moreover, we show that it should provide (at least) a very good approximation of the multifractal pr operties of the actual non-linear density field, hence of the probability d istribution of the density contrast. The implications of this model (e.g. f or galaxies) are detailed in other studies.