**6.6. The Halo model - II: biased galaxy populations**

In relating the distribution of galaxies to that of the mass, there are two distinct ways in which a degree of bias is inevitable:

- Halo occupation numbers. For low-mass haloes, the
probability of obtaining an
*L*^{*}galaxy must fall to zero. For haloes with mass above this lower limit, the number of galaxies will in general not scale linearly with halo mass. - Nonlocality. Galaxies can orbit within their
host haloes, so the probability of forming a galaxy depends
on the overall halo properties, not just the density at a point.
Also, the galaxies can occupy special places within
the haloes: for a halo containing only one galaxy, the
galaxy will clearly mark the halo centre. In general,
we will
*assume*one central galaxy and a number of satellites.

The first mechanism leads to large-scale bias, because
large-scale halo correlations depend on mass, and are some
biased multiple of the mass power spectrum:
_{h}^{2} = *b*^{2}(*M*)
^{2}.
As discussed earlier,
the linear bias parameter for a given class of
haloes, *b*(*M*), depends on the rareness of
the fluctuation and the rms of the underlying field:

(129) |

(Kaiser 1984;
Cole & Kaiser 1989;
Mo & White 1996),
where =
_{c} /
, and
^{2} is the
fractional mass variance at the redshift of interest.
This formula is not perfectly accurate, but the
deviations may be traced to the fact that the Press-Schechter
formula for the number density of haloes (which is assumed
in deriving the bias) is itself systematically in error; see
Sheth & Tormen
(1999).

If we do not wish to assume that the number of galaxies in
a halo of mass *M* is strictly proportional to *M*, we
are in effect giving haloes a mass-dependent weight, as
was first considered by
Jing, Mo & Börner
(1998).
A simple but instructive model for this is

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A model in which mass traces light would have *M*_{c}
0 and
= 1. We will show
below that, empirically, we should choose
< 1.

The bias formula applies to haloes of a given , i.e. of a given mass, so the effect of mass-dependent weights is

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Where *F*( > ) is the
fraction of the mass in haloes exceeding a given
;
*dF* / *d*
exp(- ^{2} / 2)
according to Press-Schechter theory.
The total model for the galaxy power spectrum is then

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where

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The key ingredient needed to make this machinery work is
the occupation number, which in principle needs to be
calculated via a detailed numerical model of galaxy formation.
However, for a given assumed background cosmology, the
answer may be determined empirically.
Galaxy redshift surveys have been analyzed via grouping
algorithms similar to the `friends-of-friends' method
widely employed to find virialized clumps in *N*-body
simulations. With an appropriate correction for the
survey limiting magnitude, the observed number of galaxies in
a group can be converted to an estimate of the total
stellar luminosity in a group. This allows a determination of the All
Galaxy System (AGS) luminosity function: the distribution of virialized
clumps of galaxies as a function of their total
luminosity, from small systems like the Local Group to
rich Abell clusters.

The AGS function for the CfA survey was investigated by Moore, Frenk & White (1993), who found that the result in blue light was well described by

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where ^{*} = 0.00126 *h*^{3}
Mpc^{-3},
= 1.34,
= 2.89;
the characteristic luminosity is
*L*^{*} = 7.6 × 10^{10}
*h*^{-2}
*L*_{}.
One notable feature of this function is that it is
rather flat at low luminosities, in contrast to the
mass function of dark-matter haloes (see Sheth & Tormen 1999).
It is therefore clear that any fictitious galaxy catalogue
generated by randomly sampling the mass is unlikely to be a
good match to observation.
The simplest cure for this deficiency is to assume that the
stellar luminosity per virialized halo is a monotonic, but nonlinear,
function of halo mass. The required luminosity-mass
relation is then easily deduced by finding the luminosity
at which the integrated AGS density
( > *L*) matches the
integrated number density of haloes with mass > *M*.
The result is shown in figure 14.

We can now calculate the halo-based galaxy power spectrum
and use semi-realistic occupation numbers, *N*, as a function of
mass. This is needs a little care at small numbers,
however, since the number of haloes with occupation number unity
affects the correlation properties. These
haloes contribute no correlated pairs, so they simply
dilute the signal from the haloes with *N*
2.
This means that we need in principle to use different
weights for the large-scale bias and the halo term:

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respectively
(Seljak 2000).
In practice, this correction has
a rather small effect, provided the relation between *N* and *M*
has no scatter. If, in contrast, the distribution of *N*
for given *M* is assumed to obey a Poisson distribution,
the small-scale clustering properties are strongly affected, and do
not match the data well
(Benson et al. 2000a).
Finally, we need to put the
galaxies in the correct location, as discussed above.
If one galaxy always occupies the halo centre, with others
acting as satellites, the small-scale correlations automatically
follow the slope of the halo density profile, which keeps them
steep. The results of this exercise are shown in
figure 15.
This shows that, depending on the range of halo masses chosen,
the galaxies can be positively or negatively biased with
respect to the mass, as expected. What is particularly
interesting is that the shape of the galaxy spectrum is
expected to differ from that of the mass. For an appropriate
mass range, the galaxy power spectrum can be very close to
a power law, which has been a long-standing puzzle to explain.
Interestingly, the power-law should not be perfect; small
deviations have long been suspected, and were confirmed by
Hawkins et al. (2002) and
Zehavi et al. (2003).
The inflection is at a scale of
~ 0.5 *h* Mpc^{-1}, as expected from the halo model.
Figure 15 also shows that
the results of this simple model are encouragingly similar to
the scale-dependent bias found in the detailed calculations of
Benson et al. (2000a),
shown in figure 11.
There are thus grounds for optimism that we may
be starting to attain a physical understanding of the
origin of galaxy bias.