6.3 Luminosity and
spectral-type dependence of galaxy clustering
Although biasing was commonly neglected until the early 1980s, it
has become evident observationally that on scales
different galaxy populations exhibit
different clustering amplitudes, the so-called morphology-density
relation [16]. As
discussed in Section 4, galaxy biasing is naturally
predicted from a variety of theoretical considerations as well as
direct numerical simulations [37, 59, 15, 87, 86, 103]. Thus, in this
Section we summarize the extent to which the galaxy clustering is
dependent on the luminosity, spectral-type, and color of the galaxy
sample from the 2dFGRS and SDSS.
6.3.1
2dFGRS: Clustering
per luminosity and spectral type
Madgwick et al. [45
] applied the
Principal Component Analysis to compress each galaxy spectrum into
one quantity,
. Qualitatively,
is an indicator of the ratio of the present to the past star
formation activity of each galaxy. This allows one to divide the
2dFGRS into
-types, and to study, e.g., luminosity
functions and clustering per type. Norberg et al. [61
] showed that, at all
luminosities, early-type galaxies have a higher bias than late-type
galaxies, and that the biasing parameter, defined here as the ratio
of the galaxy to matter correlation function
varies as
. Figure 25 indicates that for
galaxies, the real space correlation function
amplitude of
early-type galaxies is
higher than that of late-type galaxies.
Figure 26 shows the
redshift-space correlation function in terms of the line-of-sight
and perpendicular to the line-of-sight separation
. The correlation function calculated from the most
passively (‘red’, for which the present rate of star formation is
less than 10 % of its past averaged value) and actively (‘blue’)
star-forming galaxies. The clustering properties of the two samples
are clearly distinct on scales
. The
‘red’ galaxies display a prominent finger-of-God effect and also
have a higher overall normalization than the ‘blue’ galaxies. This
is a manifestation of the well-known morphology-density relation.
By fitting
over the separation range
for each class, it was found that
,
and corresponding pairwise velocity dispersions
of
and
[45
]. At small
separations, the real space clustering of passive galaxies is
stronger than that of active galaxies: The slopes
are respectively 1.93 and 1.50 (see Figure 27) and the relative
bias between the two classes is a declining function of separation.
On scales larger than
the biasing ratio is
approaching unity.
Another statistic was applied recently by Wild et
al. [98] and Conway et
al. [12], of a joint
counts-in-cells on 2dFGRS galaxies, classified by both color and
spectral type. Exact linear bias is ruled out on all scales. The
counts are better fitted to a bivariate log-normal distribution. On
small scales there is evidence for stochasticity. Further
investigation of galaxy formation models is required to understand
the origin of the stochasticity.
6.3.2
SDSS: Two-point
correlation functions per luminosity and color
Zehavi et al. [104
] analyzed the Early
Data Release (EDR) sample of the SDSS 30,000 galaxies to explore
the clustering of per luminosity and color. The inferred real-space
correlation function is well described by a single power-law:
for
. The galaxy pairwise
velocity dispersion is
for projected separations
. When divided by color, the red galaxies exhibit a
stronger and steeper real-space correlation function and a higher
pairwise velocity dispersion than do the blue galaxies. In
agreement with 2dFGRS there is clear evidence for a
scale-independent luminosity bias at
.
Subsamples with absolute magnitude ranges centered on
,
, and
have
real-space correlation functions that are parallel power laws of
slope
with correlation lengths of
approximately
,
, and
, respectively.
Figures 27 and 28 pose an interesting
challenge to the theory of galaxy formation, to explain why the
correlation functions per luminosity bins have similar slope, while
the slope for early type galaxies is steeper than for late
type.
6.3.3
SDSS: Three-point
correlation functions and the nonlinear biasing of galaxies per
luminosity and color
Let us move next to the three-point correlation
functions (3PCF) of galaxies, which are the lowest-order
unambiguous statistic to characterize non-Gaussianities due to
nonlinear gravitational evolution of dark matter density fields,
formation of luminous galaxies, and their subsequent evolution. The
determination of the 3PCF of galaxies was pioneered by Peebles and
Groth [70] and Groth and
Peebles [27] using the Lick and
Zwicky angular catalogs of galaxies. They found that the 3PCF
obeys the hierarchical relation:
with
being a constant. The value of
in real space deprojected from these angular
catalogues is
for
. Subsequent analyses of redshift
catalogs confirmed the hierarchical relation, at least
approximately, but the value of
(in redshift space) appears
to be smaller,
.
As we have seen in Section 6.3.2, galaxy clustering is
sensitive to the intrinsic properties of the galaxy samples under
consideration, including their morphological types, colors, and
luminosities. Nevertheless the previous analyses were not able to
examine those dependences of 3PCFs because of the limited number of
galaxies. Indeed Kayo et al. [39
] were the first to
perform the detailed analysis of 3PCFs explicitly taking account of
the morphology, color, and luminosity dependence. They constructed
volume-limited samples from a subset of the SDSS galaxy redshift
data, ‘Large-scale Structure Sample 12’. Specifically they divided
each volume limited sample into color subsamples of red (blue)
galaxies, which consist of 7949 (8329), 8930 (8155), and 3706
(3829) galaxies for
,
, and
, respectively.
Figure 29 indicates the
dimensionless amplitude of the 3PCFs of SDSS galaxies in redshift
space,
for the equilateral triplets of galaxies. The overall conclusion is
that
is almost scale-independent and ranges between 0.5
and 1.0, and that no systematic dependence is noticeable on
luminosity and color. This implies that the 3PCF itself does depend
on the galaxy properties since two-point correlation functions
(2PCFs) exhibit clear dependence on luminosity and color. Previous
simulations and theoretical models [82, 53, 50, 85] indicate that
decreases with scale in both real and redshift
spaces. This trend is not seen in the observational results.
In order to demonstrate the expected dependence in
the current samples, they compute the biasing parameters estimated
from the 2PCFs,
where the index
runs over each sample of galaxies with
different colors and luminosities. The predictions of the mass
2PCFs in redshift space,
, in the
cold dark matter model are computed
following [28].
As an illustrative example, consider a simple
bias model in which the galaxy density field
for the
-th population of galaxies is given
by
If both
and
are constant
and the mass density field
,
Equation (174) implies that
Thus the linear bias model (
) simply
implies that
is inversely proportional to
, which is plotted in Figure 30. A comparison of
Figures 29 and 30 indicates that the
biasing in the 3PCFs seems to compensate the difference of
purely due to that in the 2PCFs.
Such behavior is unlikely to be explained by any
simple model inspired by the perturbative expansion like
Equation (176). Rather it indeed
points to a kind of regularity or universality of the clustering
hierarchy behind galaxy formation and evolution processes. Thus the
galaxy biasing seems much more complex than the simple
deterministic and linear model. More precise measurements of 3PCFs
and even higher-order statistics with future SDSS datasets would be
indeed valuable to gain more specific insights into the empirical
biasing model.