In physical cosmology, the Sheth–Tormen approximation is a halo mass function. It is named after Ravi K. Sheth and Giuseppe Tormen, who proposed their function in 1999.^[1]

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: $${\displaystyle f(\sigma _{r})=A{\sqrt {\frac {2a}{\pi }}}\left[1+\left({\frac {\sigma _{r}^{2}}{a\delta _{c}^{2}}}\right)^{0.3}\right]{\frac {\delta _{c}}{\sigma _{r}}}\exp \left(-{\frac {a\delta _{c}^{2}}{2\sigma _{r}^{2}}}\right),}$$where ${\displaystyle \delta _{c}=1.686}$, ${\displaystyle a=0.707}$, and ${\displaystyle A=0.3222}$.^[2]

The parameters were empirically obtained from the five-year release of WMAP.^[3]

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 ${\displaystyle 10}$ for objects with a redshift ${\displaystyle z>10}$, but is accurate at low redshifts.^[4][3]