4.4 Biasing of galaxies in
cosmological hydrodynamic simulations
Popular models of the biasing based on the peak or the dark halos
are successful in capturing some essential features of biasing.
None of the existing models of bias, however, seems to be
sophisticated enough for the coming precision cosmology era. The
development of a more detailed theoretical model of bias is needed.
A straightforward next step is to resort to numerical simulations
which take account of galaxy formation even if phenomenological at
this point. We show an example of such approaches from Yoshikawa et
al. [103
] who apply
cosmological smoothed particle hydrodynamic (SPH) simulations in
the LCDM model with particular attention to the comparison of the
biasing of dark halos and simulated galaxies (see also [78]).
Galaxies in their simulations are identified as
clumps of cold and dense gas particles which satisfy the Jeans
condition and have the SPH density more than 100 times the mean
baryon density at each redshift. Dark halos are identified with a
standard friend-of-friend algorithm; the linking length is 0.164
times the mean separation of dark matter particles, for instance,
at
. In addition, they identify the surviving
high-density substructures in dark halos, DM cores (see [103
] for further
details).
Figure 9 illustrates the
distribution of dark matter particles, gas particles, dark halos,
and galaxies at
where galaxies are more strongly
clustered than dark halos. Figure 10 depicts a close-up
snapshot of the most massive cluster at
with a mass
. The circles in the lower panels
indicate the positions of galaxies identified in our
simulation.
Figure 11 shows the joint
distribution of
and
with the mass density field
at redshift
,
, and
smoothed over
. The conditional mean
relation
computed directly from the simulation
is plotted in solid lines, while dashed lines indicate theoretical
predictions of halo biasing by Taruya and Suto [87
]. For a given
smoothing scale, the simulated halos exhibit positive biasing for
relatively small
in agreement with the predictions. On
the other hand, they tend to be underpopulated for large
, or anti-biased. This
is mainly due to the exclusion effect of dark halos due to their
finite volume size which is not taken into account in the
theoretical model. Since our simulated galaxies have smaller spatial extent than
the halos, the exclusion effect is not so serious. This is clearly
illustrated in the lower panels in Figure 11, and indeed they show
much better agreement with the theoretical model.
We turn next to a more conventional biasing parameter
defined through the two-point statistics:
where
and
are two-point
correlation functions of objects
and of dark matter,
respectively. While the above biasing parameter is ill-defined
where either
or
becomes
negative, it is not the case at clustering scales of interest (
).
Figure 12 shows two-point
correlation functions of dark matter, galaxies, dark halos, and DM
cores (upper and middle panels), and the profiles of biasing
parameters
for those objects (lower panels) at
,
, and
. In the lower panels, we also
plot the parameter
, which are defined in terms
of the one-point statistics (variance), for comparison on smoothing
scales
,
, and
at
for each kind
of objects by different symbols. In the upper panels, we show the
correlation functions of DM cores identified with two different
maximum linking lengths,
and
. Correlation functions of DM cores identified with
are similar to those of galaxies. On
the other hand, those identified with
exhibit
much weaker correlation, and are rather similar to those of dark
halos. This is due to the fact that the present algorithm of group
identification with larger
tends to pick up lower mass
halos which are poorly resolved in our numerical resolution.
The correlation functions of galaxies are almost
unchanged with redshift, and the correlation functions of dark
halos only slightly evolve between
and 2. By
contrast, the amplitude of the dark matter correlation functions
evolve rapidly by a factor of
from
to
. The biasing parameter
is larger at a higher redshift, for example,
at
. The biasing parameter
for dark halos is systematically lower than that of
galaxies and DM cores again due to the volume exclusion effect. At
, galaxies and DM cores are slightly anti-biased
relative to dark matter at
. In
lower panels, we also plot the one-point biasing parameter
at
for comparison.
In general we find that
is very close to
at
, but systematically lower than
at higher redshifts.
For each galaxy identified at
, we define its formation redshift
by the epoch when half of its cooled gas particles satisfy our criteria of
galaxy formation. Roughly speaking,
corresponds to the
median formation redshift of stars in
the present-day galaxies. We divide all simulated galaxies at
into two populations (the young population with
and the old population with
) so as to approximate the observed number ratio of
for late-type and early-type galaxies.
The difference of the clustering amplitude can be
also quantified by their two-point correlation functions at
as plotted in Figure 13. The old population
indeed clusters more strongly than the mass, and the young
population is anti-biased. The relative bias between the two
populations
ranges
and 2 for
, where
and
are the two-point correlation functions
of the young and old populations.