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 formali sm 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 sp atial correlation of the density fluctuations, however, was either neg lected or treated in an approximate manner in the previous works deali ng with the cloud-in-cloud problem. In this paper the effect of spatia l correlations is analyzed by using the Jedamzik formalism, taking exp licitly into account the two-point correlations of density fluctuation s, and it is found that this effect alters the PS mass function on som e 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 tha t the probability of the effective overlap of the collapsed objects ca n be neglected either on very small or on large mass scales (for examp le, in the case of the initial power index n = 0, M less than or simil ar to 0.1M or M greater than or similar to 10M*, where M* is the char acteristic mass scale on which the variance of density fluctuations eq uals unity), while it cannot be neglected on intermediate mass scales.