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Cosmic inhomogeneities are usually described by the
fluctuation field

(7) 
where is the density of the Universe at the point .
A popular statistical characterization of the inhomogeneities in the
distribution of cosmic structures is provided by
the
twopoint correlation function, , which describes the expected excess
fluctuations with respect to a uniform distribution:

(8) 
where the symbol
indicates the average over all the pairs of points at separation .
Its Fourier transform is the power spectrum

(9) 
If clustering is isotropic, then and depend only on
the modulus of the separation and wavevectors, respectively. The
power spectrum of the observed galaxy and cluster distributions
contains formally the same amount of information as , although
the practical computation of the two quantities is differently
affected by the systematics of redshift surveys.
Another important statistical characterisation is provided by the
variance, , of the mass density fluctuations within a
sphere of radius :

(10) 
Galaxy surveys show that for normal galaxies the variance in the
galaxy number counts is about unity for
[7,21]. For this reason, it is customary to specify the
normalisation of the power spectrum at this scale, so that
provides the
socalled galaxy biasing with respect to the dark matter
distribution. Therefore, by definition, the galaxy biasing provides a
measure of the difference between the distribution of the observable
light and the distribution of the unseen cosmic dark matter.
Observations of the largescale distribution of cosmic structures
shows that the twopoint correlation function for both galaxies and
galaxy clusters can be well approximated by a power law,

(11) 
Although the slope
is quite simila
i.e. the correspondence between background and physical spacetime)
in which fluctuation measurements are performed. In this case, a
proper gaugeinvariant description of fluctuation evolution is
required and several prescriptions have been proposed by different
authors (see ref.[24] for a recent review on this
subject). At the present time, all the scales relevant for clustering
studies are are well inside the horizon. However, since perturbation
wavelengths scale with the redshift as
and the horizon as
,
the typical scale of galaxy clustering,
, crosses
the horizon at , roughly corresponding to the epoch of
matterradiation equality. Thus, a gauge invariant treatment is
required when studying the formation of density fluctuations relevant
to the present observed clustering. Here we adopt a purely Newtonian
treatment, which is adequate to follow the perturbation evolution
after recombination, when structure formation starts.
Passing to the Fourier representation of the density fluctuation
field, the system () becomes
Eqs. (12) and () can be equivalently used as the
nonlinear systems describing the evolution of cosmic density fluctuations.
In the following we will describe some approximations that
have been proposed in order to get hints about the nature of gravitational
clustering. In the next Chapter we will discuss the use of cosmological
Nbody simulations as a tool to follow in detail the nonlinear dynamics
implied by eqs.(12).
Subsections
Next: The linear approximation
Up: THE LARGESCALE STRUCTURE OF
Previous: The standard cosmological model
Waleska Aldana Segura
20010116