 Solution

Let ${\delta }_{\mathrm{sc}}\simeq 1.686$ be the linear spherical-collapse threshold and let $D(t)$ be the linear growth factor, normalized to one at the epoch to which the random density field has been extrapolated. The time-dependent barrier is ${\delta }_c(t)={\delta }_{\mathrm{sc}}/D(t)$. The smoothed matter density variance ${\sigma }^2(M)$ is the variance of the linear density contrast smoothed with a spherical top-hat window function whose comoving mass is $M$. Both quantities must refer to the same linear normalization; equivalently use $\nu ={\delta }_{\mathrm{sc}}/\sigma (M,t)$.

Let $N$ denote the comoving number density and write $\mathcal{N}(M,t)=dN/dM$ for the positive differential Press-Schechter halo mass function. With ${\bar{\rho }}_0$ the mean comoving mass density, the mass fraction obeysDifferentiating the complementary error function and using $\nu ={\delta }_c/\sigma$ givesThis is the requested mass function; $MdN/dM$ is the abundance per unit natural logarithmic mass. For physical rather than comoving number density, multiply by $(1+z{)}^3$. The normalization in the Press-Schechter formalism includes its factor of two relative to the positive Gaussian tail, so $f(>M)$ tends to one as the smoothing mass tends to zero when $\sigma$ diverges. It is a collapse prescription, not the probability that an arbitrary already nonlinear density field stays Gaussian.