4.5 Halo occupation function
approach for galaxy biasing
Since the clustering of dark matter halos is well understood now,
one can describe the galaxy biasing if the halo model is combined
with the relation between the halos and luminous objects. This is
another approach to galaxy biasing, halo
occupation function (HOF), which has become very popular
recently. Indeed the basic idea behind HOF has a long history, but
the model predictions have been significantly improved with the
recent accurate models for the mass function, the biasing and the
density profile of dark matter halos. We refer the readers to an
extensive review on the HOF by Cooray and Sheth [13]. Here we briefly
outline this approach.
We adopt a simple parametric form for the average
number of a given galaxy population as a function of the hosting
halo mass:
The above statistical and empirical relation is the essential
ingredient in the current modeling characterized by the minimum
mass
of halos which host the population of galaxies, a
normalization parameter which can be interpreted as the critical
mass
above which halos typically host more than one
galaxy (note that
may exceed
since the above
relation represents the statistical expected value of number of
galaxies), and the power-law index
of the mass
dependence of the efficiency of galaxy formation. We will put
constraints on the three parameters from the observed number
density and clustering amplitude for each galaxy population. In
short, the number density of galaxies is most sensitive to
which changes the average number of galaxies per
halo. The clustering amplitude on large scales is determined by the
hosting halos and thus very sensitive to the mass of those halos,
. The clustering on smaller scales, on the other
hand, depends on those three parameters in a fairly complicated
fashion; roughly speaking,
changes the amplitude, while
, and to a lesser extent
as well, change the
slope.
With the above relation, the number density of
the corresponding galaxy population at redshift
is given by
where
denotes the halo mass function.
The galaxy two-point correlation function on
small scales is dominated by contributions of galaxy pairs located
in the same halo. For instance, Bullock et al. [8] adopted the
mean number of galaxy pairs
within a halo of mass
of the form:
In the framework of the halo model, the galaxy
power spectrum consists of two contributions, one from galaxy pairs
located in the same halo (1-halo term) and the other from galaxy
pairs located in two different halos (2-halo term):
The 1-halo term is written as
Seljak [77
] chose
for
and
for
. The 2-halo term on the assumption of
the linear halo bias model [59
] reduces to
where
is the linear dark matter power
spectrum,
is the halo bias factor, and
is the Fourier transform of the halo dark matter
profile normalized by its mass,
[77].
The halo occupation formalism, although simple,
provides a useful framework in deriving constraints on galaxy
formation models from large data sets of the upcoming galaxy
redshift surveys. For example, Zehavi et al. [105] used the halo
occupation formalism to model departures from a power law in the
SDSS galaxy correlation function. They demonstrated that this is
due to the transition from a large-scale regime dominated by galaxy
pairs in different halos to a small-scale regime dominated by those
in the same halo. Magliocchetti and Porciani [47] applied the halo
occupation formalism to the 2dFGRS clustering results per spectral
type of Madgwick et al. [45
]. This provides
constraints on the distribution of late-type and early-type
galaxies within the dark matter halos of different mass.