Press-Schechter formalism
Encyclopedia
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..
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.
and 
is:
where
is the mean (baryonic and dark) matter density of the universe, 
is the index of the power spectrum of the flucuations in the early universe 
, and 
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.
Mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences and engineering disciplines A mathematical model is a...
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
Research paper
Research paper may refer to:* Academic paper , which is published in academic journals and contains original research results or reviews existing results* Term paper, written by high school or college students...
by William H. Press
William H. Press
William H. Press is an astrophysicist, theoretical physicist, and computational biologist. He is a member of the United States National Academy of Sciences, the American Academy of Arts and Sciences, and the Council on Foreign Relations. Other honors include the 1981 Helen B...
and Paul Schechter
Paul L. Schechter
Paul L. Schechter is an astrophysicist and observational cosmologist. He is the William A. M. Burden Professor of Astrophysics at MIT.Schechter received his bachelor's degree from Cornell in 1968, and his Ph.D. degree from Caltech in 1975...
in 1974..
Background
In the context of cold dark matterDark matter
In astronomy and cosmology, dark matter is matter that neither emits nor scatters light or other electromagnetic radiation, and so cannot be directly detected via optical or radio astronomy...
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
Physical cosmology
Physical cosmology, as a branch of astronomy, is the study of the largest-scale structures and dynamics of the universe and is concerned with fundamental questions about its formation and evolution. For most of human history, it was a branch of metaphysics and religion...
).
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 and is:where
is the mean (baryonic and dark) matter density of the universe, is the index of the power spectrum of the flucuations in the early universe , and 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
Luminosity function (astronomy)
In astronomy, the luminosity function gives the number of stars or galaxies per luminosity interval. Luminosity functions are used to study the properties of large groups or classes of objects, such as the stars in clusters or the galaxies in the Local Group....
,
and are now known as Schechter luminosity functions. The Press-Schechter
formalism provided the first quantitative model for how such functions might
arise.