Perturbation theory and excursion set estimates of the probability distribution function of dark matter, and a method for reconstructing the initial distribution function

Nonlinear evolution can sometimes be modelled by a deterministic mapping from initial to final of the local smoothed overdensity. Perturbation theory methods base on this deterministic and local mapping and ignore the ‘cloud-in-cloud’ effect, while the excursion set approach methods take this nonlocality into account. We compared these methods using the spherical collapse mapping and showed that, on scales where the rms fluctuation is small, both models give similar results and they are in good agreement with numerical simulations. If the deterministic mapping depends on quantities other than overdensity, this will also manifest as stochasticity if the other quantities are ignored. We considered the Zeldovich approximation and Ellipsoidal Collapse model, both include the tidal field in the evolution. Our anaylsis shows that the change in cell shape effect should be included on scales where the rms is of order of unity or larger. On scales where the rms is less than 2 methods based on the spherical collapse model allow a rather accurate reconstruction of the shape of the initial distribution from the nonlinear field. This can be used as the basis for constraining the statistical properties of the initial fluctuation field. (Abridge)