and Cosmology

8. Cosmology III: The Cosmological Parameters
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Fig. 8.7. The 2-point correlation function ξ g , as measured from
the 2dFGRS, plotted as a function of the transverse separation
σ and the radial separation π in redshift space. Solid
contours connect values of constant ξ g . The dashed curves
show the same correlation function, determined from a cosmological
simulation that accounts for small-scale velocities.
The oblateness of the distribution for large separation and the
Fingers of God are clearly visible
following the cosmic velocity field. Due to small-scale
gravitational interactions they have a velocity dispersion
σ p around the velocity field as predicted by linear
theory. A quantitative interpretation of the anisotropy of
the correlation function needs to account for this effect,
which causes an additional smearing of galaxy positions
in redshift space along the line-of-sight. Therefore, the
derived value of β is related to σ p . It is possible to determine
both quantities simultaneously, by comparing the
observed correlation function with models for different
values of β and σ p . From this analysis, confidence
regions in the β-σ p -plane are obtained, which feature
a distinct minimum in the corresponding χ 2 function
and by which both parameters can be estimated simultaneously.
For the best estimate of these values, the
2dFGRS yielded
β = 0.51 ± 0.05 ; σ p ≈ 520 km/s . (8.10)
8.1.5 Angular Correlations of Galaxies
Measuring the correlation function or the power spectrum
is not only possible with extensive redshift surveys
of galaxies, which have become available only relatively
recently. In fact, the correlation properties of galaxies
can also be determined from their angular positions on
the sphere. The three-dimensional correlation of galaxies
in space implies that their angular positions are
likewise correlated. These angular correlations are easily
visible in the projection of bright galaxies onto the
sphere (see Fig. 6.2).
The angular correlation function w(θ) is defined in
analogy with the three-dimensional correlation function
ξ(r) (see Sect. 7.3.1). Considering two solid angle
element dω at θ 1 and θ 2 , the probability of finding a galaxy
at θ 1 is P 1 = n dω, where n denotes the average
density of galaxies on the sphere (with well-defined
properties like, for instance, a minimum magnitude
limit). The probability of finding a galaxy near θ 1 and
another one near θ 2 is then
P 2 = (n dω) 2 [1 + w(|θ 1 − θ 2 |)] , (8.11)
where we utilize the statistical homogeneity and
isotropy of the galaxy distribution, by which the correlation
function w depends only on the absolute angular
separation. The angular correlation function w(θ) is
of course very closely related to the three-dimensional
correlation function ξ g of galaxies. Furthermore, w(θ)
depends on the redshift distribution of the galaxies considered;
the broader this distribution is, the fewer pairs
of galaxies are found at a given angular separation which
are also located close to each other in three-dimensional
space, and hence are correlated. This means that the
broader the redshift distribution of galaxies, the smaller
the expected angular correlation.
The relation between w(θ) and ξ g (r) is given by the
Limber equation, which can, in its simplest form, be
written as
∫ ∫
w(θ) = dz p 2 (z) d(Δz) (8.12)
⎛√
⎞
( )
× ξ g
⎝
dD 2
[D A (z)θ] 2 + (Δz)
dz
2 ⎠ ,
where D A (z) is the angular diameter distance (4.45),
p(z) describes the redshift distribution of galax-