and Cosmology

8. Cosmology III: The Cosmological Parameters
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for red galaxies is larger by about a factor 1.4 than for
blue galaxies. This result is not completely unexpected,
because red galaxies are located preferentially in clusters
of galaxies, whereas blue galaxies are rarely found
in massive clusters. Hence, red galaxies seem to follow
the density concentrations of the dark matter much
more closely than blue galaxies.
8.1.4 Effect of Peculiar Velocities
Redshift Space. The relative velocities of galaxies in
the Universe are not only due to the Hubble expansion
but, in addition, galaxies have peculiar velocities. The
peculiar velocity of the Milky Way is measurable from
the CMB dipole (see Fig. 1.17). Owing to these peculiar
velocities, the observed redshift of a source is the
superposition of the cosmic expansion velocity and its
peculiar velocity v along the line-of-sight,
cz= H 0 D + v. (8.6)
The measurement of the other two spatial coordinates
(the angular position on the sky) is not affected by
the peculiar velocity. The peculiar velocity therefore
causes a distortion of galaxy positions in wedge diagrams,
yielding a shift in the radial direction relative to
their true positions. Since, in general, only the redshift
is measurable and not the true distance D, the observed
three-dimensional position of a source is specified by
the angular coordinates and the redshift distance
s = cz
H 0
= D + v H 0
. (8.7)
The space that is spanned by these three coordinates
is called redshift space. In particular, we expect that
the correlation function of galaxies is not isotropic in
redshift space.
Galaxy Distribution in Redshift Space. The best
known example of this effect is the “Fingers of God”. To
understand their origin, we consider galaxies in a cluster.
They are situated in a small region in space, all at
roughly the same distance D and within a small solid
angle on the sphere. However, due to the high velocity
dispersion, the galaxies span a broad range in s,
which is easily identified in a wedge diagram as a highly
stretched structure pointing towards us, as can be seen
in Fig. 7.2.
Mass concentrations of smaller density have the opposite
effect: galaxies that are closer to us than the center
of this overdensity move towards the concentration,
hence away from us. Therefore their redshift distance s
is larger than their true distance D. Conversely, the peculiar
velocity of galaxies behind the mass concentration
is pointing towards us, so their s is smaller than their true
distance. If we now consider galaxies that are located
on a spherical shell around this mass concentration, this
sphere in physical space becomes an oblate ellipsoid
with symmetry axis along the line-of-sight in redshift
space. This effect is illustrated in Fig. 8.6.
Hence, the distortion between physical space and
redshift space is caused by peculiar velocities which
manifest themselves in the transformation (8.7) of the
radial coordinate in space (thus, the one along the lineof-sight).
Due to this effect, the correlation function of
galaxies is not isotropic in redshift space. The reason for
this is the relation between the density field and the corresponding
peculiar velocity field. Specializing (7.49)
to the current epoch and using the relation (8.1) between
the density fields of matter and of galaxies, we obtain
u(x) = β H ∫
0
4π
d 3 y δ g ( y)
where we defined the parameter
β := Ω0.6 m
b
y − x
|y − x| 3 , (8.8)
. (8.9)
This relation between the density field of galaxies and
the peculiar velocity is valid in the framework of linear
perturbation theory under the assumption of linear
biasing. The anisotropy of the correlation function is
now caused by this correlation between u(x) and δ g (x),
and the degree of anisotropy depends on the parameter
β. Since the correlation function is anisotropic, this
likewise applies to the power spectrum.
Cosmological Constraints. Indeed, the anisotropy of
the correlation function can be measured, as is shown in
Fig. 8.7 for the 2dFGRS (where in this figure, the usual