PressSchechter

The Press–Schechter formalism, named after William H. Press and Paul Schechter, is a semi-analytic model in cosmology introduced in 1974 to predict the abundance of dark matter halos as a function of mass and cosmic time from the statistics of the initial density field.

The method starts from a Gaussian random field of initial density fluctuations and considers the fraction of mass in regions where the linearly extrapolated density contrast exceeds a critical threshold for collapse, δ_c, typically around 1.686 for spherical collapse. By smoothing the density field on different mass scales and applying a cloud-in-cloud argument, one derives the halo mass function, which estimates the number density of halos per unit mass.

The standard result can be written as n(M) dM = (ρ_m / M) f(ν) dν, where ν = δ_c / σ(M), σ(M) is the variance of the smoothed density field, and f(ν) = √(2/π) ν exp(-ν^2/2) for the original model. This yields a prediction for how many halos of mass M exist at a given redshift in a given cosmology.

Applications include predicting halo abundances in simulations and analytic models, constructing merger trees, and informing galaxy formation and reionization studies. Limitations include the assumption of spherical collapse, neglect of environmental effects, and discrepancies with N-body simulations in the detailed shape of the mass function. Extensions such as extended Press–Schechter, and, more accurately, Sheth–Tormen and other ellipsoidal-collapse refinements, improve agreement with simulations.