A Lagrangian dynamical theory for the mass function of cosmic structures — II. Statistics

The statistical tools needed to obtain a mass function from realistic collapse-time estimates are presented. Collapse dynamics has been dealt with in Paper I of this series by means of the powerful Lagrangian perturbation theory and the simple ellipsoidal collapse model. The basic quantity considered here is the inverse collapse time F; it is a non-linear functional of the initial potential, with a non-Gaussian distribution. In the case of sharp k-space smoothing, it is demonstrated that the fraction of collapsed mass can be determined by extending to the F process the diffusion formalism introduced by Bond et al. The problem is then reduced to that of a random walk with a moving absorbing barrier, and numerically solved; an accurate analytical fit, valid for small and moderate resolutions, is found. For Gaussian smoothing, the F trajectories are strongly correlated in resolution. In this case, an approximation proposed by Peacock & Heavens can be used to determine the mass functions. Gaussian smoothing is preferred, as it optimizes the performances of dynamical predictions and stabilizes the F trajectories. The relation between resolution and mass is treated at a heuristic level, and the consequences of this approximation are discussed. The resulting mass functions, compared with the classical Press & Schechter one, are shifted toward large masses (confirming the findings of Monaco), and tend to give more intermediate-mass objects at the expense of smallmass objects. However, the small-mass part of the mass function, which depends on uncertain dynamics and is likely to be affected by uncertainties in the resolution-mass relation, is not considered a robust prediction of this theory.