Sheth–Tormen approximation

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.

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