Mathematical model
The Press–Schechter formalism is a mathematical model for predicting the number of objects (such as galaxies, galaxy clusters or dark matter halos) 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.
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 formula_1 and formula_2 is:
formula_3
where formula_4 is the index of the power spectrum of the fluctuations in the early universe formula_5, formula_6 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 formula_7 is a cut-off mass below which structures will form. Its value is:
formula_8
formula_9 is the standard deviation per unit volume of the fluctuation from which the object was formed had gravitationally collapsed, at the time of the gravitational 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, equivalently, a scalar spectral index of 1), is very close to the spectrum of the current standard cosmological model. In this case, formula_10 has a simpler form. Written in mass-free units:
formula_11
Assumptions and derivation sketch.
The Press–Schechter formalism is derived through three key assumptions:
In other words, fluctuations are small at some early cosmological time, and grow until they cross a threshold ending in gravitational collapse into a halo. These perturbations are modeled linearly, even though the eventual collapse is itself a non-linear process.
We introduce the smoothed density field formula_16 given by formula_17 averaged over a sphere with center formula_18 and mass formula_1 contained inside (i.e., formula_12 is convolved with a top-hat window function). The sphere radius is of order formula_21 Then if formula_22 a halo exists at formula_18 with mass at least formula_24
Since perturbations formula_25 are Gaussian distributed with an average 0 and variance formula_26 we can directly compute the probability of halos forming with masses at least formula_1 as
formula_28
Implicitly, formula_29 and formula_25 depend on redshift, so the above probability does as well. The variance given in the 1974 paper is
formula_31
where formula_32 is the mass standard deviation in the volume of the fluctuation.
Note, that in the limit of large perturbations formula_33 we expect all matter to be contained in halos such that formula_34 However, the above equation gives us the limit formula_35 One can make an ad-hoc argument and say that negative perturbations are not contributing in this scheme so that we are mistakenly leaving out half of the mass. And so, the Press-Schechter ansatz is 
formula_36
the fraction of matter contained in halos of mass formula_37
A fractional fluctuation formula_12; 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 formula_1, formula_40, and formula_9 gives the Press-Schechter formula.
Generalizations.
A number of generalizations of the Press–Schechter formula exist, such as the Sheth–Tormen approximation.
References.
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