Exploring Contradictory Definitions of f(M) & F(M) in Press-Schechter Formalism

[cosmo_boy]

In press-Schechter mass formalism what exactly are f(M) and F(M) ? I am finding two contradictiory definitions.

In some places it has been said that f(M)dM is the number density of collapsed objects in mass range [M,M+dm] and F(M) is the number density of objects
having mass > M.
(Padamanabhan, Structure formation in the universe).

In many places (most of the places), f(M) is defined as the fraction of total mass which lies in collapsed
objects which have mass in range [M,M+dM] and F(M) is the fraction of mass which lies in collapsed objects of
mass >M.

Now if we follow the first definition then the dimensions of f(M) and F(M) should be 1/(mass*volume) and 1/volume respectvely. However, if we follow the second definition the both are dimensional physical quantities.

please give your advice in this matter ...

thanks

 

[lippyka]

The correct definition is that f(M)dM is the fraction of mass which lies in collapsed objects which have mass in range [M,M+dM] and F(M) is the fraction of total mass which lies in collapsed objects of mass >M. This is the most commonly accepted definition and the one that is used in the majority of the literature. It is also the definition that is used in Padmanabhan's Structure Formation in the Universe.