Sheth–Tormen approximation

The Sheth–Tormen approximation is a halo mass function.

Background
The Sheth–Tormen approximation extends the Press–Schechter formalism by assuming that halos are not necessarily spherical, but merely elliptical. The distribution of the density fluctuation is as follows: $$f(\sigma_r)=A\sqrt{\frac{2a}{\pi}}[1+(\frac{\sigma_r^2}{a\delta_c^2})^{0.3}]\frac{\delta_c}{\sigma_r}\exp(-\frac{a\delta_c^2}{2\sigma_r^2})$$, where $$\delta_c=1.686$$, $$a=0.707$$, and $$A=0.3222$$. The parameters were empirically obtained from the five-year release of WMAP.

Discrepancies with simulations
In 2010, the Bolshoi Cosmological Simulation predicted that the Sheth–Tormen approximation is inaccurate for the most distant objects. Specifically, the Sheth–Tormen approximation overpredicts the abundance of haloes by a factor of $$10$$ for objects with a redshift $$z>10$$, but is accurate at low redshifts.