The Halo Mass Function from Excursion Set Theory. II. The Diffusing Barrier

TL;DR

This paper extends excursion set theory by modeling the collapse barrier as a stochastic diffusing variable, improving the match between theoretical halo mass functions and N-body simulation results.

Contribution

It introduces a diffusing barrier into the excursion set framework, accounting for physical complexities in halo formation and modifying the mass function accordingly.

Findings

The exponential factor in the mass function is modified by a factor 'a' related to barrier diffusion.

The diffusion coefficient D_B influences the shape of the halo mass function.

Comparison with N-body simulations shows improved agreement with the diffusing barrier model.

Abstract

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]