A LAGRANGIAN DYNAMICAL THEORY FOR THE MASS FUNCTION OF COSMIC STRUCTURES .1. DYNAMICS

A new theory for determining the mass function of cosmic structures is presented. It relies on a realistic treatment of collapse dynamics. G ravitational collapse is analysed in the Lagrangian perturbative frame work. Lagrangian perturbations provide an approximation of truncated t ype, i.e. small-scale structure is filtered out. The collapse time is suitably defined as the instant at which orbit crossing takes place. T he convergence of the Lagrangian series in predicting the collapse tim e of a homogeneous ellipsoid is demonstrated; it is also shown that th ird-order calculations are necessary in predicting collapse. Then, the Lagrangian prediction, with a correction for quasi-spherical perturba tions, can be used to determine the collapse time of a homogeneous ell ipsoid in a fast and precise way. Furthermore, ellipsoidal collapse ca n be considered as a particular truncation of the Lagrangian series. G aussian fields with scale-free power spectra are then considered. The Lagrangian series for the collapse time is found to converge when the collapse time is not large. In this case, ellipsoidal collapse gives a fast and accurate approximation of the collapse time; spherical colla pse is found to poorly reproduce the collapse time, even in a statisti cal sense. Analytical fits of the distribution functions of the invers e collapse times, as predicted by the ellipsoid model and by third-ord er Lagrangian theory, are given. These will be necessary for a determi nation of the mass function, which will be given in Paper II.