Sheth–Tormen approximation

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: ${\displaystyle f(\sigma _{r})=A{\sqrt {\frac {2a}{\pi }}}[1+({\frac {\delta _{r}^{2}}{a\delta _{c}^{2}}})^{0.3}]{\frac {\delta _{c}}{\sigma _{r}}}\exp(-{\frac {a\delta _{c}^{2}}{2\sigma _{r}^{2}}})}$, where ${\displaystyle \delta _{c}=1.686}$, ${\displaystyle a=0.707}$, and ${\displaystyle A=0.3222}$.[1] The parameters were empirically obtained from the five-year release of WMAP.[2]

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.[3][2]