THE MULTIPLICITY FUNCTION OF GALAXIES, CLUSTERS AND VOIDS

We calculate the multiplicity function of matter condensations with gi ven mass and defined by an arbitrary density threshold which can be a function of the mass of the objects, by directly considering the actua l, deeply non-linear density field, which we compare to the popular Pr ess-Schechter approximation. This comparison is made possible owing to an analytic description of the non-linear mass distribution, but also thanks to a modellization of the evolution of the two-body correlatio n function from its linear to non-linear behaviour based on Feeble's s pherical collapse picture. We show the mass function is a function of a unique parameter that contains all the dependence on the mass and ra dius of the object. We compare this new, still simple, analytic model to a more involved formulation that should be even closer to the resul ts obtained using the standard density threshold algorithms. This give s some hindsight into the ''cloud-in-cloud'' problem, for both Press-S chechter and nonlinear prescriptions, and it enables us to derive the mass function of general astrophysical objects (in addition to just-vi rialized halos) which may be defined by any density threshold, that ma y even vary with the mass of the object. This is beyond the reach of u sual formulations based on the initial gaussian field and gives a clea r illustration of the advantages of our approach. We explain why numer ical tests seem to favor both Press-Schechter and non-linear prescript ions even though the two approximations differ by their scaling as a f unction of mass as well as a function of redshift. We argue that numer ical simulations will be closer to the non-linear predictions and shou ld be reexamined in the light of our findings. The difference between the two is due to the fact that the Press-Schechter prescription assum es that present-day mass fluctuations can be recognized in the early l inear universe, and that their number is conserved, while the non-line ar approach takes into account their evolution which leads to an incre ase of the number of highly non-linear objects (very large or small ma sses). This difference is seen to be of the same magnitude as the diff erence obtained by varying the initial spectrum of density fluctuation s. Earlier conclusions drawn about the relevance of the latter using a nalytical approximations to the mass function must thus be reexamined.