### 4.1 Concepts and definitions of biasing

As discussed above, luminous objects, such as galaxies and quasars, are not direct tracers of
the mass in the Universe. In fact, a difference of the spatial distribution between luminous
objects and dark matter, or a bias, has been indicated from a variety of observations. Galaxy
biasing clearly exists. The fact that galaxies of different types cluster differently (see, e.g., [16])
implies that not all of them are exact tracers of the underlying mass distribution (see also
Section 6).
In order to confront theoretical model predictions for the mass distribution against observational data,
one needs a relation of density fields of mass and luminous objects. The biasing of density peaks in a
Gaussian random field is well formulated [37, 4], and it provides the first theoretical framework for the
origin of galaxy density biasing. In this scheme, the galaxy-galaxy and mass-mass correlation functions are
related in the linear regime via

where the biasing parameter is a constant independent of scale . However, a much more
specific linear biasing model is often assumed in common applications, in which the local density
fluctuation fields of galaxies and mass are assumed to be deterministically related via the relation
Note that Equation (116) follows from Equation (117), but the reverse is not true.
The above deterministic linear biasing is not based on a reasonable physical motivation. If , it
must break down in deep voids because values of below are forbidden by definition. Even in the
simple case of no evolution in comoving galaxy number density, the linear biasing relation is not preserved
during the course of fluctuation growth. Non-linear biasing, where varies with , is
inevitable.

Indeed, an analytical model for biasing of halos on the basis of the extended Press–Schechter
approximation [59] predicts that the biasing is nonlinear and provides a useful approximation for its
behavior as a function of scale, time, and mass threshold. -body simulations provide a more accurate
description of the nonlinearity of the halo biasing confirming the validity of the Mo and White
model [35, 103].