### 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 = 1.0, 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.