and Cosmology

8.1 Redshift Surveys of Galaxies
ies, and dD specifies the physical distance interval
corresponding to a redshift interval dz,
dD =−c dt =− c da
aH
⇒
dD
dz = c
(1 + z) H(z) .
Long before extensive redshift surveys were performed,
the correlation w(θ) had been measured. Since
it is linearly related to ξ g , and since ξ g in turn is related
to the power spectrum of the matter fluctuations and to
the bias factor, the measured angular correlation function
could be compared to cosmological models. For
some time, such analyses have hinted at a small value
for the shape parameter Γ = Ω m h of about 1/4 (see
Fig. 8.4), which is incompatible with an Einstein–de
Sitter model. Figure 8.8 shows w(θ) for four magnitude
intervals measured from the SDSS. We see that
w(θ) follows a power law over a wide angular range,
which we would also expect from (8.12) and from the
fact that ξ(r) follows a power law. 3 In addition, the figure
shows that w(θ) becomes smaller the fainter the
galaxies are, because fainter galaxies have a higher
redshift on average and they define a broader redshift
distribution.
Fig. 8.8. The angular correlation function w(θ) in the four
magnitude intervals 18 < r ∗ < 19, 19 < r ∗ < 20, 20 < r ∗ <
21, and 21 < r ∗ < 22, as measured from the first photometric
data of the SDSS, together with a power law fit to the data in
the angular range 1 ′ ≤ θ ≤ 30 ′ ; the slope in all cases is very
close to θ −0.7
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8.1.6 Cosmic Peculiar Velocities
The relation (8.8) between the density field of galaxies
and the peculiar velocity can also be used in a different
context. To see this, we assume that the distance of galaxies
can be determined independently of their redshift.
In the relatively local Universe this is possible by using
secondary distance measures (such as, e.g., the scaling
relations for galaxies that were discussed in Sect. 3.4).
With the distance known, we are then able to determine
the radial component of the peculiar velocity by means
of the redshift,
v = cz − H 0 D .
To measure values of v of order ∼ 500 km/s, D needs
to be determined with a relative accuracy of
v
cz .
3 It is an easy exercise to show that a power law ξ(r) ∝ r −γ implies an
angular correlation function w(θ) ∝ θ −(γ −1) .
With the distance measurements being accurate to about
10%, the distance to which this method can be applied
is limited to cz/H 0 ∼ 100 Mpc, corresponding to an
expansion velocity cz ∼ 6000 km/s. Thus, the peculiar
velocity field can be determined only relatively locally.
In order to measure D, one typically uses the Tully–
Fisher relation for spirals, and the fundamental plane or
the D n –σ relation for ellipticals. In most cases, these
measurements are carried out for groups of galaxies
which then all have roughly the same distance; in this
way, the measurement accuracy of their common (or
average) distance is improved.
Equation (8.8) now allows us to predict the peculiar
velocity field from the measured density field of galaxies,
which can then be compared with the measured
peculiar velocities – where this relation depends on β
(see Eq. 8.9). Therefore, we can estimate β from this
comparison. The inverse of this method is also possible:
to derive the density distribution from the peculiar velocity
field, and then to compare this with the observed