In the scheme outlined above, the function
*b*() contains the
information about the mean biasing (via the parameter
) and its
non-linear features (e.g., via
/
).
The next quantity of interest in the case of stochastic biasing is
the conditional standard deviation, the function
_{b}(),
and its variance over ,
_{b}^{2}.
In order to evaluate the actual effects of non-linear and stochastic
biasing on the various measurements of
, one should try
to evaluate these functions or parameters
from simulations, theoretical approximations and observations.

In an ongoing study that generalizes earlier investigations
[10,
46],
we are investigating the biasing in high-resolution
*N*-body simulations of several cosmological scenarios, both for galactic
halos and for galaxies as identified using semi-analytic models
[54].
We refer here to a representative cosmological model:
= 1 with a
CDM power spectrum which roughly obeys
the constraints from large-scale structure.
The simulation mass resolution is 2 x 10^{10}
*M*_{} inside a box
of comoving side 85 *h*^{-1} Mpc. The present epoch is
identified with
_{8} = 0.6.
Figure 1
demonstrates the qualitative features of the biasing scheme.
The non-linear behavior at < 1
is characteristic
of all masses, times, and smoothing scales:
*b*() << 1 near
= -1 and it
steepens to *b*() > 1 towards
= 0.
At > 1
the behavior strongly depends on the mass, time
and smoothing scale.
The scatter in the figure includes both shot noise and physical scatter
which are hard to separate properly.
In the case shown at *z* = 0, the non-linear parameter is
^{2} /
^{2} = 1.08,
and the scatter parameter is
_{b}^{2} /
^{2} = 0.15.
The effects of stochasticity and non-linearity in this specific
case thus lead to moderate differences in the various measures of
, on the order of 20-30%.
Gas-dynamics and other non-gravitational processes
may extend the range of estimates even further.

Given the distribution
*P*() of the matter fluctuations,
the biasing function *b*()
should obey by definition at least the following two constrains.
First, *g* -1 everywhere,
because the galaxy density _{g} cannot be negative, with
*g* = -1 at = -1, because
there are no galaxies where there is no matter. Second,
<*g*> = 0 because *g* describes fluctuations about the mean
galaxy density.
An example for a simple functional form that obeys the constraint at
= -1 and reduces to the linear
relation near = 0 is
[17]

The constraint <*g*> = 0 is to be enforced by a specific choice of
the factor *c* for a given *b*.
With *b* > 1, this functional form indeed provides a reasonable fit
to the simulated halo biasing relation in the
< 0 regime.
However, the same value of *b* does not
necessarily fit the biasing relation in the
> 0 regime.
A better approximation could thus be provided by a combination of two
functions like Eq. (18)
with two different parameters *b*_{n} and
*b*_{p} in the regimes
0 and
> 0 respectively.
The parameter *b*_{n} is always larger than unity
while *b*_{p} ranges from slightly below unity to much
above unity.
The best fit to Fig. 1 at *z* = 0
has *b*_{n} ~ 2 and *b*_{p} ~ 1.
At high redshift both *b*_{n} and *b*_{p}
become significantly larger.

The non-linear biasing function can alternatively be parameterized by

Since *g* must average to zero, this general power series can be
written as

where ^{2}
<^{2}>,
*S*
<^{3}>, etc.
This determines the constant term *b*_{0}.
The constraint at -1 provides another relation between the parameters.
Therefore, the expansion to third order contains only two free parameters
out of four.

In order to evaluate the parameters
and
for these non-linear
toy models, we approximate the distribution
*P*() as
log-normal in /
= 1 +
[12,
39],
where is the single free parameter.
With Eq. (20), Assuming *b*_{2} << *b*_{1} and
<< 1, one obtains
^{2} /
^{2}
1 + (1/2)
(*b*_{2} / *b*_{1})^{2}
^{2}.
This is always larger than unity, but the deviation is small.
Alternatively, using the functional form of Eq. (18), with
*b*_{n} ranging from 1 to 5 and
*b*_{p} ranging from 0.5 to 3,
and with = 0.7, we find
numerically that /
is in the range 1.0 to 1.15.
These two toy models, calibrated by
the *N*-body simulations, indicate that despite the obvious
non-linearity, especially in the negative regime, the non-linear parameter
/
is typically only slightly
larger than unity.
This means that the effects of non-linear biasing on measurements of
are likely to be relatively small.