Modeling biasing

4.2 Modeling biasing

Biasing is likely to be stochastic, not deterministic [15

]. An obvious part of this stochasticity can be attributed to the discrete sampling of the density field by galaxies, i.e., the shot noise. In addition, a statistical, physical scatter in the efficiency of galaxy formation as a function of $δ m$ is inevitable in any realistic scenario. For example, the random variations in the density on smaller scales is likely to be reflected in the efficiency of galaxy formation. As another example, the local geometry of the background structure, via the deformation tensor, must play a role too. Such ‘hidden variables’ would show up as physical scatter in the density-density relation [87

Consider the density contrasts of visible objects and mass, $δ (x,z|R ) obj$ and $δ (x,z |R ) m$ , at a position $x$ and a redshift $z$ smoothed over a scale $R$ [86 ]. In general, the former should depend on various other auxiliary variables $⃗𝒜$ defined at different locations $x′$ and redshifts $z ′$ smoothed over different scales $R ′$ in addition to the mass density contrast at the same position, $δm(x,z|R )$ . While this relation can be schematically expressed as

it is impossible even to specify the list of the astrophysical variables $𝒜⃗$ , and thus hopeless to predict the functional form in a rigorous manner. Therefore if one simply focuses on the relation between $δobj(x, z|R)$ and $δ (x,z |R ) m$ , the relation becomes inevitably stochastic and nonlinear due to the dependence on unspecified auxiliary variables $⃗ 𝒜$ .

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 of $δ m$ may be approximated by

a local stochastic nonlinear bias,
a local deterministic nonlinear bias,

and
a local deterministic linear bias,

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 $bobj(z,R )$ 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 ].