On the cosmological mass function theory

This paper provides, from one side, a review of the theory of the cosmological mass function from a theoretical point of view, starting from the seminal paper of Press & Shechter (1974) to the last developments (Del Popolo & Gambera (1998, 1999), Sheth & Tormen 1999 (ST), Sheth, Mo & Tormen 2001 (ST1), Jenkins et al. 2001 (J01), Shet & Tormen 2002 (ST2), Del Popolo 2002a, Yagi et al. 2004 (YNY)), and from another side some improvements on the multiplicity function models in literature. For what concerns this second aspect, I compare the numerical multiplicity function given in Yahagi, Nagashima & Yoshii (2004), with the theoretical multiplicity function obtained in the present paper by means of the excursion set model and an improved version of the barrier shape obtained in Del Popolo & Gambera (1998), which implicitly takes account of total angular momentum acquired by the proto-structure during evolution and of a non-zero cosmological constant. I show that the multiplicity function obtained in the present paper, is in better agreement with Yahagi, Nagashima & Yoshii (2004) simulations than other previous models (Sheth & Tormen 1999; Sheth, Mo & Tormen 2001; Sheth & Tormen 2002; Jenkins et al. 2001). The multiplicity function of the present paper gives a good fit to simulations results as the fit function proposed by Yahagi, Nagashima & Yoshii (2004), but differently from that it was obtained from a sound theoretical background. Then, I calculate the mass function evolution in a ΛCDM model by means of the previous model. I compare the result with Reed et al. (2003) (R03), who used a high resolution ΛCDM numerical simulation to calculate the mass function of dark matter haloes down to the scale of dwarf galaxies, back to a redshift of fifteen. I show that the mass function obtained in the present paper, gives similar predictions to the Sheth & Tormen mass function but it does not show the overprediction of extremely rare objects shown by the Sheth and Tormen mass function. The results confirm previous findings that the simulated halo mass function can be described solely by the variance of the mass distribution, and thus has no explicit redshift dependence. I moreover show that the PS-like approach together with the ellipsoidal model introduced in Del Popolo (2002b) gives a better description of the theoretical mass function. Subject headings: clusters of galaxies — cosmology: theory — dark matter — galaxies: clustering — largescale structure of the universe