Accurate halo mass functions from the simplest excursion set theory

Excursion set theory is a powerful and widely used tool for describing the distribution of dark matter haloes, but it is normally applied with simplifying approximations. We use numerical sampling methods to study the mass functions predicted by the theory without approximations. With a spherical top-hat window and a constant $\delta =1.5$ threshold, the theory accurately predicts mass functions with the $M_{200}$ mass definition, both unconditional and conditional, in simulations of a range of matter-dominated cosmologies. For $\mathrm{\Lambda }$CDM at the present epoch, predictions lie between the $M_{200\mathrm{m}}$ and $M_{200\mathrm{c}}$ mass functions. In contrast, with the same window function, a nonconstant threshold based on ellipsoidal collapse predicts uniformly too few haloes. This work indicates a new way to simply and accurately evaluate halo mass functions, clustering bias, and assembly histories for a range of cosmologies. We provide a simple fitting function that accurately represents the predictions of the theory for a wide range of parameters.