MODERN COSMOLOGY

Galaxy surveys 371

13.7.1 Angular samples

ξ(r)can be obtained from two-dimensional data, by means of the angular two-
point functionw(θ).ξ(r)is reconstructed using the luminosity function, which
is derived assuming homogeneneity in the sample [1]. No independent check
is usually performed on this assumption. The procedure is currently considered
one of the best estimates of three-dimensional clustering properties of galaxies,
at least on a small scale (≤ 20 h−^1 Mpc) [9, 10]. Such a claim is considered to
be justified by the great quantity of available data in angular catalogues with
respect to three-dimensional surveys and by the absence of redshift distortions
in the two-dimensional data. The main conclusion obtained by this approach is
that the galaxy correlation (more precisely for optical selected galaxies)ξgg(r)is
quite close to a power law in the range 10 h−^1 kpc–(10–20h−^1 ) Mpc and more
precisely [9, 10]:

ξgg(r)=

(r

0

r

)− 1. 77

(13.9)

with acorrelation length r 0 ≈ 4. 5 ± 0. 5 h−^1 Mpc.
This is considered to give the ‘canonical shape and parameter values’ ofξ(r)
and is a well-established result in cosmology [1, 10–14].

13.7.2 Redshift samples

13.7.2.1 ML samples

An ML sample is simply the wholeredshift catalogue. By construction, any
ML sample is incomplete in the distribution of galaxies. At larger distances, it
contains fewer and fewer galaxies, as more and more galaxies fall beyond the
threshold of detectability. To account for such an effect, the galaxies in the sample
are weighted, according the luminosity function [1].
The value ofs 0 in different ML catalogues is found to span from 4.5–
8 h−^1 Mpc [10, 13].
ξ(s)does not appear to bea power law. According to Guzzo [15], the shape
ofξ(s)at very small scales (< 3 h−^1 Mpc) is well fitted by a power law with
exponentγ=−1.

13.7.2.2 VL samples

It is possible to extract subsamples from the ML catalogues, which are unaffected
by the aforementioned incompleteness. Such samples are calledVL samples[16].
The main result ofξ(s)analysis is that different VL samples have different values
for the correlation lengths 0. The general trend is that deeper and brighter samples
show largers 0 (figure 13.1) [6, 15, 19–23]. Again,ξ(s)isnot a power lawin
the whole observed range of scale (≈1–50h−^1 Mpc). This has been recognized
by several authors, who have performed the fit with the power law in a limited
range of scales. The value of the exponentγ(see equation (13.9)) is in the range