THE CONFLUENT SYSTEM FORMALISM .1. THE MASS FUNCTION OF OBJECTS IN THE PEAK MODEL

This is the first paper of a series of two devoted to develop a practi cal method to describe the growth history of bound virialized objects in the gravitational instability scenario without resorting to N-body simulations. Here we present the basic tool of this method, ''the conf luent system formalism,'' which allows us to follow the filtering evol ution of peaks in a random Gaussian field of density fluctuations. Thi s is applied to derive the theoretical mass function of objects within the peak model framework. Along the process followed for the derivati on of this function, we prove that the Gaussian window is the only one consistent with the peak model Ansatz. We also give a well-justified derivation of the density of peaks with density contrast upcrossing a given threshold in infinitesimal ranges of scale and correct this scal e function for the cloud-in-cloud effect. Finally, we characterize the form of the mass versus scale and the critical overdensity versus col lapse time relations that are physically consistent with the peak mode l in an Einstein-de Sitter universe with density field endowed with di fferent power spectra. The result is a fully justified semianalytical mass function that is close to the Press and Schechter one, giving goo d fits to N-body simulations. But the interest of the confluent system formalism is not merely formal. It allows us to distinguish between a ccretion and merger events, which is essential for the detailed modeli ng of the clustering process experienced by objects.