Limitations of the Press-Schechter formalism

The Press-Schechter (PS) formalism for mass functions of the collapsed objects is reanalyzed. The factor of 2 in the Press-Schechter formalism is shown to be correct in the sharp k-space filter when we use the novel approach proposed by Jedamzik for the ''cloud-in-cloud'' problem which was first discussed by Peacock & Heavens and Bond et al. The spatial correlation of the density fluctuations, however, was either neglected or treated in an approximate manner in the previous works dealing with the cloud-in-cloud problem. In this paper the effect of spatial correlations is analyzed by using the Jedamzik formalism, taking explicitly into account the two-point correlations of density fluctuations, and it is found that this effect alters the PS mass function on some mass scales. For example, in the case of the initial power index n = 0, the difference between the modified mass function and the PS mass function is a factor of 1.4 at M = 0.1M* and is 3 orders of magnitude at M = 10M*. With our new derivation of the mass function, we find that the probability of the effective overlap of the collapsed objects can be neglected either on very small or on large mass scales (for example, in the case of the initial power index n = 0, M less than or similar to 0.1M* or M greater than or similar to 10M*, where M* is the characteristic mass scale on which the variance of density fluctuations equals unity), while it cannot be neglected on intermediate mass scales.