Two abundance crossings of the Press-Schechter mass function (${\mathcal{N}}_{\ln M}\propto \nu e^{-{\nu }^2/2}$)

At fixed mass, the Press-Schechter halo mass function has abundance per logarithmic interval proportional to $\nu e^{-{\nu }^2/2}$. This increases as $\nu$ falls from infinity to one and then decreases as $\nu$ falls toward zero, reflecting incorporation of small dark-matter haloes into larger ones. A target below the maximum therefore has two mathematical crossings if the linear growth factor can increase without limit. The large-$\nu$ crossing describes first appearance; the small-$\nu$ crossing is late depletion. Late vacuum domination limits growth, so a formal future crossing need not be reached physically.