The matter
power spectrum
In signal processing, the power spectrum S_(f) of a continuous time signal x(t) describes the distribution of Power (physics), power into frequency components f composing that signal. According to Fourier analysis, any physical signal can be ...
describes the density contrast of the universe (the difference between the local density and the mean density) as a function of scale. It is the
Fourier transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
of the matter
correlation function. On large scales,
gravity
In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
competes with
cosmic expansion, and structures grow according to
linear theory. In this regime, the density contrast field is Gaussian, Fourier modes evolve independently, and the power spectrum is sufficient to completely describe the density field. On small scales, gravitational collapse is
non-linear
In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathe ...
, and can only be computed accurately using
N-body simulation
In physics and astronomy, an ''N''-body simulation is a simulation of a dynamical system of particles, usually under the influence of physical forces, such as gravity (see n-body problem, ''n''-body problem for other applications). ''N''-body ...
s. Higher-order statistics are necessary to describe the full field at small scales.
Definition
Let
represent the matter overdensity, a dimensionless quantity defined as:
where
is the average matter density over all space.
The power spectrum is most commonly understood as the Fourier transform of the
autocorrelation function
Autocorrelation, sometimes known as serial correlation in the discrete time case, measures the correlation of a signal with a delayed copy of itself. Essentially, it quantifies the similarity between observations of a random variable at differe ...
,
, mathematically defined as:
for
.
This then determines the easily derived relationship to the power spectrum,
, that is
Equivalently, letting
denote the Fourier transform of the overdensity
, the power spectrum is given by the following average over Fourier space:
(note that
is not an overdensity but the
Dirac delta function
In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
).
Since
has dimensions of (length)
3, the power spectrum is also sometimes given in terms of the dimensionless function:
Development according to gravitational expansion
If the
autocorrelation function
Autocorrelation, sometimes known as serial correlation in the discrete time case, measures the correlation of a signal with a delayed copy of itself. Essentially, it quantifies the similarity between observations of a random variable at differe ...
describes the probability of a galaxy at a distance
from another galaxy, the matter power spectrum decomposes this probability into characteristic lengths,
, and its amplitude describes the degree to which each characteristic length contributes to the total over-probability.
The overall shape of the matter power spectrum is best understood in terms of the linear perturbation theory analysis of the growth of structure, which predicts to first order that the power spectrum grows according to:
Where
is the linear growth factor in the density, that is to first order
, and
is commonly referred to as the ''primordial matter power spectrum''. Determining the primordial
is a question that relates to the physics of inflation.
The simplest
is the Harrison–Zeldovich spectrum (named after
Edward R. Harrison and
Yakov Zeldovich
Yakov Borisovich Zeldovich (, ; 8 March 1914 – 2 December 1987), also known as YaB, was a leading Soviet people, Soviet Physics, physicist of Belarusians, Belarusian origin, who is known for his prolific contributions in physical Physical c ...
), which characterizes
according to a power law,
. More advanced primordial spectra include the use of a transfer function which mediates the transition from the universe being radiation dominated to being matter dominated.
The broad shape of the matter power spectrum is determined by the
growth of large-scale structure, with the turnover (the point where the spectrum goes from increasing with k to decreasing with ''k'') at
, corresponding to
(where ''h'' is the
dimensionless Hubble constant
Hubble's law, also known as the Hubble–Lemaître law, is the observation in physical cosmology that galaxies are moving away from Earth at speeds proportional to their distance. In other words, the farther a galaxy is from the Earth, the faste ...
). The co-moving wavenumber corresponding to the maximum power in the mass power spectrum is determined by the size of the cosmic particle horizon at the time of matter-radiation equality, and therefore depends on the mean density of matter and to a lesser extent on the number of neutrino families (
),
, for
. The
at smaller ''k'' (equivalently, larger scales) corresponds to scales which were larger than the particle horizon at the time of the transition from the regime of radiation dominance to that of matter dominance.
At linear order in perturbations
, the power spectrum's broad shape follows
where
is the scalar spectral index.
References
* {{Cite book, title=Modern Cosmology , last=Dodelson, first=Scott , publisher=Academic Press , year=2003 , isbn=978-0-12-219141-1
Theuns, Physical Cosmology
Physical cosmology