Max Planck Institute for Astronomy - Annual Report 2005

The Conditional Luminosity Function
As discussed above, gravity causes hierarchical growth
of structures in the Universe. As a consequence, the
matter distribution in the present day Universe is strongly
clustered. Since galaxies reside in dark matter haloes, it
therefore should not come as a surprise that the distribution
of galaxies is also strongly clustered, as is indeed
confirmed by large galaxy redshift surveys. A more
detailed analysis shows that more luminous galaxies are
more strongly clustered. This luminosity dependence of
the clustering strength provides the information required
to establish a statistical description of the galaxy-dark
matter connection. The reason that this approach works
is that more massive haloes are more strongly clustered:
in terms of the halo bias mentioned above, more mas-
Fig. III.3.1: Comparison of the observed present-epoch galaxy
luminousity function and (spatial) correlation function and the
matched CLF models. In each panel the contours show the 68
and 95 percent confidence limits from our CLF model. Upper
left: The galaxy luminosity function. Upper right: the correlation
lengths as function of luminosity. Lower left: the relation
between light and mass implied by the LCF models matched
to the data (upper panels). Lower right: the implied average
mass-to-light ratio as function of halo mass.
log(�(L) L / (h 3 Mpc –3 ))
log(L / (h –2 L � ))
–2
–4
–6
12
10
8
6
4
7
2dFGRS
8 9
log(L / (h –2 10 11
L� ))
10 12
log(M / (h –1 M � ))
14
III.3 The Galaxy-Dark Matter Connection 73
sive haloes are more strongly biased. Consequently, the
clustering strength of galaxies of a give luminosity is a
direct measure of the mass of the haloes in which these
galaxies reside: since more luminous galaxies are more
strongly clustered than less luminous galaxies, and more
massive haloes are more strongly clustered than less
massive haloes, more luminous galaxies have to reside
in more massive haloes.
At the MPIA, the research group of Frank van den
Bosch, in collaboration with Xiaohu Yang (Shanghai
Observatory) and Houjun Mo (University of Massachusetts),
has developed a novel statistical technique based
on this principle. The technique aims at describing the
halo occupation statistics via the so-called conditional
luminosity function (hereafter CLF), which expresses the
average number of galaxies of luminosity L that reside
in a halo of mass M. This CLF completely specifies the
bias of galaxies as a function of their luminosity and it
allows one to compute the total, average luminosity of
all galaxies that reside in a halo of a given mass. In other
words, the CLF completely specifies the average relation
between light and mass in the Universe.
Using the luminosity function, which expresses the
abundance of galaxies as function of their luminosity,
and the luminosity dependence of the clustering strength
obtained from the 2dFGRS, one of the largest galaxy
r 0 [h –1 Mpc]
log((M / L) / (h M � / L � ))
10
8
6
4
–16 –18 –20 –22
M –5 logh
bJ
4
3.5
3
2.5
2
1.5
2dFGRS
10 12
log(M / (h –1 M � ))
14