Consider the density contrasts of visible objects and mass, and , at a position and a redshift smoothed over a scale . 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 asstochastic and nonlinear due to the dependence on unspecified auxiliary variables .
For illustrative purposes, we define the biasing factor as the ratio of the density contrasts of luminous objects and mass:nonlocal stochastic nonlinear factor in terms of may be approximated by
From 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].
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