Spatial and velocity biases 439procedure is quite complicated. First, the density field is constructed. Second,
the density (with a negative sign) is treated as a potential in which particles start
to move as in a viscous fluid. Eventually, particles sink to the bottom of the
potentials (which are also maxima density). Third, only particles with negative
energy (relative to their group) are retained. Just as in the case of FOF, we can
easily imagine situations when (this time) DENMAX should fail; for example,
two colliding galaxies in a cluster of galaxies. They should just pass each other
because of large relative velocity. In the moment of collision DENMAX ceases to
‘see’ both galaxies because all particles have positive energies. This is probably
a quite unlikely situation. The method is definitely one of the best at present.
The only problem is that it seems to be too complicated for the present state of
simulations. DENMAX has two siblings—SKID (Stadelet al) and BDM (Klypin
and Holtzman 1997)—which are frequently used.
‘Overdensity 200’. There is no name for this method, but it is often used.
Find the density maximum, place a sphere and find the radius, within which the
sphere has the mean overdensity 200 (or 177 if you really want to follow the
top-hat model of nonlinear collapse).
15.3 Spatial and velocity biases
15.3.1 Introduction
The distribution of galaxies is probably biased with respect to the dark matter.
Therefore, galaxies can be used to probe the matter distribution only if we
understand the bias. Although the problem of bias has been studied extensively
in the past (e.g. Kaiser 1984, Daviset al1985, Dekel and Silk 1986), new data
on high redshift clustering and the anticipation of coming measurements have
recently generated substantial theoretical progress in the field. The breakthrough
in an analytical treatment of the bias was the paper by Mo and White (1996),
who showed how bias can be predicted in the framework of the extended
Press–Schechter approximation. A more elaborate analytical treatment has been
developed by Catelanet al(1998a, b), Porcianiet al(1998) and Sheth and Lemson
(1999). The effects of nonlinearity and stochasticity were considered in Dekel and
Lahav (1999) (see also Taruya and Suto 2000).
Valuable results are produced by ‘hybrid’ numerical methods in which low-
resolutionN-body simulations (typical resolution∼20 kpc) are combined with
semi-analytical models of galaxy formation (e.g. Diaferioet al1999, Bensonet al
2000, Somervilleet al2001). Typically, the results of these studies are very close
to those obtained with brute-force approach of high-resolution (.2 kpc)N-body
simulations (e.g. Col ́ınet al1999, Ghignaet al1998). This agreement is quite
remarkable because the methods are very different. It may indicate that the biases
of galaxy-size objects are controlled by the random nature of the clustering and
merging of galaxies and by dynamical effects, which cause the merging, because
those are the only common effects in those two approaches.