4.3 Density peaks and dark
matter halos as toy models for galaxy biasing
Let us illustrate the biasing from numerical simulations by
considering two specific and popular models: primordial density
peaks and dark matter halos [86
]. We use the
-body simulation data of
again
for this purpose [36]. We select density
peaks with the threshold of the peak height
, 2.0, and 3.0. As for the dark matter halos, these
are identified using the standard friend-of-friend algorithm with a
linking length of 0.2 in units of the mean particle separation. We
select halos of mass larger than the threshold
,
, and
.
Figures 5 and 6 depict the
distribution of dark matter particles (upper panel), peaks (middle
panel), and halos (lower panel) in the LCDM model at
and
within a circular slice
(comoving radius of
and thickness of
). We locate
a fiducial observer in the center of the circle. Then the comoving position vector
for a particle with a comoving
peculiar velocity
at a redshift
is observed at the
position s in redshift space:
where
is the Hubble parameter at
.
The right panels in Figures 5 and 6 plot the observed
distribution in redshift space, where the redshift-space distortion
is quite visible: The coherent velocity field enhances the
structure perpendicular to the line-of-sight of the observer (squashing) while the virialized clump
becomes elongated along the line-of-sight (finger-of-God).
We use two-point correlation functions to quantify
stochasticity and nonlinearity in biasing of peaks and halos, and
explore the signature of the redshift-space distortion. Since we
are interested in the relation of the biased objects and the dark
matter, we introduce three different correlation functions: the
auto-correlation functions of dark matter and the objects,
and
, and their cross-correlation function
. In the present case, the subscript o refers to
either h (halos) or
(peaks). We also use the superscripts R
and S to distinguish quantities defined in real and redshift
spaces, respectively. We estimate those correlation functions using
the standard pair-count method. The correlation function
is evaluated under the distant-observer
approximation.
Those correlation functions are plotted in
Figures 7 and 8 for peaks and halos,
respectively. The correlation functions of biased objects generally
have larger amplitudes than those of mass. In nonlinear regimes
(
) the finger-of-God effect suppresses the amplitude
of
relative to
, while
is larger than
in linear regimes (
) due to the coherent velocity field.