Press–Schechter formalism

The Press–Schechter formalism is a mathematical model for predicting the number of objects (such as galaxies, galaxy clusters or dark matter halos[1]) of a certain mass within a given volume of the Universe. It was described in an academic paper by William H. Press and Paul Schechter in 1974.[2]

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 ${\displaystyle M}$ and ${\displaystyle M+dM}$ is:

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

where ${\displaystyle n}$ is the index of the power spectrum of the fluctuations in the early universe ${\displaystyle P(k)\propto k^{n}}$, ${\displaystyle {\bar {\rho }}}$ is the mean (baryonic and dark) matter density of the universe at the time the fluctuation from which the object was formed had gravitationally collapsed, and ${\displaystyle M^{*}}$ is a cut-off mass below which structures will form. Its value is:

${\displaystyle M^{*}=\left({\frac {{\bar {\rho }}^{1-{\frac {n}{3}}}}{2\sigma ^{2}}}\right)^{\frac {3}{3+n}}=\left({\frac {{\bar {\rho }}_{0}^{1-{\frac {n}{3}}}}{2\sigma _{0}^{2}}}\right)^{\frac {3}{3+n}}\cdot {\frac {R_{0}^{2}}{R^{2}}}}$

${\displaystyle \sigma }$ is the standard deviation per unit volume of the fluctuation from which the object was formed had gravitationally collapsed, at the time of the gravitaional collapse, and R is the scale of the universe at that time. Parameters with subscript 0 are at the time of the initial creation of the fluctuations (or any later time before the gravitational collapse).

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.

The case of a scale-free power spectrum, n=0 (or, equivalentlty, a scalar spectral index of 1), is very close to the spectrum of the current standard cosmological model. In this case, ${\displaystyle dn}$ has a simpler form. Written in mass-free units:

${\displaystyle M{\frac {dn}{dM}}={\frac {1}{\sqrt {\pi }}}{\frac {\bar {\rho }}{M}}\left({\frac {M}{M^{*}}}\right)^{1/2}e^{-M/M^{*}}}$

Assumptions and derivation sketch

The Press–Schechter formalism is derived by assuming that each object is formed by gravitational collapse of a density fluctuation. Furthermore, the fluctuations are assumed to be small at some early cosmological time, and are treated with a linear approximation, even though the eventual collapse is itself a non-linear process.

The density fluctuations are normally distributed, and their variance is: ${\displaystyle \delta _{*}^{2}={\frac {\Sigma ^{2}}{M^{2}}}={\frac {V\cdot \sigma ^{2}}{M^{2}}}={\frac {\sigma ^{2}}{M\cdot \rho }}}$ Where ${\displaystyle \Sigma }$ is the mass standard deviation in the volume of the fluctuation and ${\displaystyle M}$, is its mass.

A fractional fluctuation ${\displaystyle \delta }$; at some cosmological time reaches gravitational collapse after the universe has expanded by a factor of 1/δ since that time. Using this, the normal distribution of the fluctuations, written in terms of the ${\displaystyle M}$, ${\displaystyle \rho }$, and ${\displaystyle \sigma }$ gives the Press-Shechter formula.

Generalizations

A number of generalizations of the Press–Schechter formula exist, such as Sheth-Tormen.[3]

References

1. Dark Matter Halos, Mass Functions, and Cosmology: a Theorist’s View
2. Formation of Galaxies and Clusters of Galaxies by Self-Similar Gravitational Condensation, W.H. Press, P. Schechter, 1974
3. Sheth, R. K., & Tormen, G. (1999). Large-scale bias and the peak background split. Monthly Notices of the Royal Astronomical Society, 308(1), 119-126.