  A new theory for determining the mass function of cosmic structures is
presented. It relies on a realistic treatment of collapse dynamics.
Gravitational collapse is analyzed in the Lagrangian perturbative framework.
Lagrangian perturbations provide an approximation of truncated type, i.e.
small-scale structure is filtered out. The collapse time is suitably defined as
the instant at which orbit crossing takes place. The convergence of the
Lagrangian series in predicting the collapse time of a homogeneous ellipsoid is
demonstrated; it is also shown that third-order calculations are necessary in
predicting collapse. Then, the Lagrangian prediction, with a correction for
quasi-spherical perturbations, can be used to determine the collapse time of a
homogeneous ellipsoid in a fast and precise way. Furthermore, ellipsoidal
collapse can be considered as a particular truncation of the Lagrangian series.
Gaussian 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 collapse is found to poorly
reproduce the collapse time, even in a statistical sense. Analytical fits of
the distribution functions of the inverse collapse times, as predicted by the
ellipsoid model and by third-order Lagrangian theory, are given. These will be
necessary for a determination of the mass function, which will be given in
paper II.
