The Cosmological Mass Distribution Function in the Zel’dovich Approximation

An analytic estimation of the mass function for gravitationally bound objects is presented. We use the Zel’dovich approximation to extend the Press-Schecter analysis to nonspherical collapse. For the Zel’dovich approximation, the gravitational collapse along all three directions which will eventually lead to the formation of real virialized objects clumps occurs in the regions where the lowest eigenvalue λ3 of the deformation tensor is positive. In this paper, first we derive analytically the individual probability density distribution of each eigenvalue, λ1, λ2, λ3 of the deformation tensor. Then we calculate the conditional probability of λ3 > 0 as a function of the linearly extrapolated density contrast δ and the distribution of δ under the condition of λ3 > 0. These two conditional probability distributions demonstrate explicitly the underdense regions (δ < δc) can also participate in the collapse process and the most probable density δ in the collapsed regions (λ3 > 0) is roughly 1.5 at the characteristic mass scale M∗. Finally the analytic mass function of clumps is derived with a help of one simple ansatz which is employed to approach the multistream regions beyond the validity of the Zel’dovich approximation. Our mass function is shown to be different from the Press-Schecter one, having a lower peak and predicting more small-mass objects. Subject headings: cosmology:theory – large-scale structure of universe