The Halo Mass Function from Excursion Set Theory. II. The Diffusing
  Barrier
  In excursion set theory the computation of the halo mass function is mapped
into a first-passage time process in the presence of a barrier, which in the
spherical collapse model is a constant and in the ellipsoidal collapse model is
a fixed function of the variance of the smoothed density field. However, N-body
simulations show that dark matter halos grow through a mixture of smooth
accretion, violent encounters and fragmentations, and modeling halo collapse as
spherical, or even as ellipsoidal, is a significant oversimplification. We
propose that some of the physical complications inherent to a realistic
description of halo formation can be included in the excursion set theory
framework, at least at an effective level, by taking into account that the
critical value for collapse is not a fixed constant $\delta_c$, as in the
spherical collapse model, nor a fixed function of the variance $\sigma$ of the
smoothed density field, as in the ellipsoidal collapse model, but rather is
itself a stochastic variable, whose scatter reflects a number of complicated
aspects of the underlying dynamics. Solving the first-passage time problem in
the presence of a diffusing barrier we find that the exponential factor in the
Press-Schechter mass function changes from $\exp\{-\delta_c^2/2\sigma^2\}$ to
$\exp\{-a\delta_c^2/2\sigma^2\}$, where $a=1/(1+D_B)$ and $D_B$ is the
diffusion coefficient of the barrier. The numerical value of $D_B$, and
therefore the corresponding value of $a$, depends among other things on the
algorithm used for identifying halos. We discuss the physical origin of the
stochasticity of the barrier and we compare with the mass function found in
N-body simulations, for the same halo definition.[Abridged]
