Sheth–Tormen approximation explained

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{ (2 a)/(\pi)} }[1+(\frac{\sigma_r^2}{a\delta_c^2})^{0.3}]\frac\exp(-\frac), where \delta _{c=1.686} , a =0 . 707 , and A =0 . 3222 .^[1] The parameters were empirically obtained from the five-year release of WMAP.^[2]

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.^[3]

Notes and References

1. http://sancerre.as.arizona.edu/~fan/Home/AST541_files/SphericalCollapse.pdf AST541 Notes: Spherical Collapse, Press-Schechter Oct/Nov 2018
2. http://physics.ucsc.edu/~joel/10Phys224/10_Wk9-CosmicWeb.pdf Physics 224 - Spring 2010 Origin and Evolution of the Universe
3. Klypin . Anatoly . Dark Matter Halos in the Standard Cosmological Model: Results from the Bolshoi Simulation . etal. The Astrophysical Journal. 2011 . 740 . 2 . 102 . 10.1088/0004-637X/740/2/102 . 1002.3660 . 2011ApJ...740..102K . 16517863 .