Press–Schechter formalism

The Press–Schechter formalism is a mathematical model for predicting the number of objects (such as galaxies or galaxy clusters) of a certain mass within a given volume of the Universe. It was described in a famous paper by William H. Press and Paul Schechter in 1974.^[1]

Background

In the context of cold dark matter cosmological models, perturbations on all scales are imprinted on the universe at very early times, for example by quantum fluctuations during an inflationary era. Later, as radiation redshifts away, these become mass perturbations, and they start to grow linearly. Only long after that, starting with small mass scales and advancing over time to larger mass scales, do the perturbations actually collapse to form (for example) galaxies or clusters of galaxies, in so-called hierarchical structure formation (see Physical cosmology).

Press and Schechter observed that the fraction of mass in collapsed objects more massive than some mass M is related to the fraction of volume samples in which the smoothed initial density fluctuations are above some density threshold. This yields a formula for the mass function (distribution of masses) of objects at any given time.

Result

The Press–Schechter formalism predicts that the number of objects with mass between $M$ and $M+dM$ is:

$N(M)dM = \frac{1}{\sqrt{\pi}}\left(1+\frac{n}{3}\right)\frac{\bar{\rho}}{M^2}\left(\frac{M}{M^*}\right)^{\left(3+n\right)/6}\exp\left(-\left(\frac{M}{M^*}\right)^{\left(3+n\right)/3}\right)dM$

where $\bar{\rho}$ is the mean (baryonic and dark) matter density of the universe, $n$ is the index of the power spectrum of the fluctuations in the early universe $P(k)\propto k^n$, and $M^*$ is a critical mass above which structures will form.

Qualitatively, the prediction is that the mass distribution is a power law for small masses, with an exponential cutoff above some characteristic mass that increases with time. Such functions had previously been noted by Schechter as observed luminosity functions, and are now known as Schechter luminosity functions. The Press-Schechter formalism provided the first quantitative model for how such functions might arise.

References

1. Formation of Galaxies and Clusters of Galaxies by Self-Similar Gravitational Condensation, W.H. Press, P. Schechter, 1974