Non-Linear Moving Barrier and the Ordinary and Conditional Mass Function

We study the problem of the determination of a high precision cosmological mass function, which is of fundamental importance in several problems of astrophysics, like the building up of an halo model. The determination of a mass function through numerical simulation is time demanding, and restrict to particular cosmological parameters. Analytical methods, allow to determine high precision mass functions, without the limitations of the simulations. We improve on previous analytical models, reaching high level of precision. In particular, we improve the scheme of Del Popolo presented in older and more recent papers, based on the excursion set approach. We further improve the mass function with respect to the quoted papers, using an higher order first crossing distribution, and an improved barrier with respect previous papers. As a result, we can obtain a mass function with a precision of 1%, in the mass range ≃ (109−1015) h−1M☉, and in the redshift range 0 < z s< 10. The paper also shows how the use of the quoted improved first-crossing distribution, improves the agreement between the conditional mass function, and simulations