Consider the density contrasts of visible objects
and mass, and
, at a
position
and a redshift
smoothed over a scale
[86
]. In general, the
former should depend on various other auxiliary variables
defined at different locations
and redshifts
smoothed over different
scales
in addition to the mass density contrast at the same
position,
. While this relation can be
schematically expressed as
For illustrative purposes, we define the biasing factor as the ratio of the density contrasts of luminous objects and mass:
Only in very idealized situations, the above nonlocal stochastic nonlinear factor in terms ofFrom the above point of view, the local
deterministic linear bias is obviously unrealistic, but is still a
widely used conventional model for biasing. In fact, the time- and
scale-dependence of the linear bias factor was neglected in many previous studies of biased
galaxy formation until very recently. Currently, however, various
models beyond the deterministic linear biasing have been seriously
considered with particular emphasis on the nonlinear and stochastic
aspects of the biasing [71, 15
, 87
, 86
].