Halo Mass Definition and Multiplicity Function
  Comparing the excursion set and CUSP formalisms for the derivation of the
halo mass function, we investigate the role of the mass definition in the
properties of the multiplicity function of cold dark matter (CDM) haloes. We
show that the density profile for haloes formed from triaxial peaks that
undergo ellipsoidal collapse and virialisation is such that the ratio between
the mean inner density and the outer local density is essentially independent
of mass. This causes that, for suited values of the spherical overdensity
$\Delta$ and the linking length $b$, SO and FoF masses are essentially
equivalent to each other and the respective multiplicity functions are
essentially the same. The overdensity for haloes having undergone ellipsoidal
collapse is the same as if they had formed according to the spherical top-hat
model, which leads to a value of $b$ corresponding to the usual virial
overdensity, $\Delta_{vir}$, equal to $\sim$ 0.2. The multiplicity function
resulting from such mass definitions, expressed as a function of the top-hat
height for spherical collapse, is very approximately universal in all CDM
cosmologies. The reason for this is that, for such mass definitions, the
top-hat density contrast for ellipsoidal collapse and virialisation is close to
a universal value, equal to $\sim$ 0.9 times the usual top-hat density contrast
for spherical collapse.
