For the analysis of redshift-distortion we need to deal with spatial correlations. Given the random biasing field , we define the two-point biasing-matter cross-correlation function and the biasing auto-correlation function by

where the averaging is over the ensembles at points 1 and 2 separated by
*r*. We define the biasing as *local* if
_{em}(*r*)
= 0 for any *r* and
_{}(*r*) = 0 for *r* > *r*_{b},
where *r*_{b} is on the order of the basic smoothing scale.
Using lemmas that are two-point equivalents of Eq. (4),
one obtains analogous relations to Eqs. (7) and (5).
In the case of *linear* and *local* biasing, these become

Note that the biasing parameter that appears here is
*b*_{1}, not *b*_{var}.

To see how the power spectra are affected by the biasing scatter,
we approximate the local biasing by a step function:
_{}(*r*) =
_{b}^{2}
^{2} for *r* <
*r*_{b} and zero otherwise.
Recalling that the power spectra are the Fourier transforms of
the corresponding correlation functions,
we get for *k* << *r*_{b}^{-1}, from Eq. (13),

where *V*_{b} is the volume associated with *r*_{b}.
We see that the *local* biasing scatter adds a constant to
*P*_{gg}(*k*).

We can now proceed to estimating
via redshift distortions
[36,
28,
29,
30,
25,
31,
11,
24,
40].
To first order, the local galaxy density fluctuations in redshift space
(*g*_{s}) and real space (*g*) are related by
*g*_{s} = *g* - ð*u* / ð*r*,
where *u* is the radial component of the peculiar velocity **v**.
Assuming no velocity biasing, linear GI predicts
ð*u* / ð*r* = - *µ*^{2}
*f*()
,
where *µ*^{2} is a geometrical
factor depending on the angle between **v** and **x**.
Thus, the basic linear relation for redshift distortions is
*g*_{s} = *g* +
*f**µ*^{2}.
The general expression for redshift distortions
is obtained from this basic relation by averaging
< *g*_{1}^{s} *g*_{2}^{s} >
over the distributions of
at a pair of
points separated by *r*:

Recalling that the power spectra are the Fourier transforms of the
corresponding correlation functions, one can equivalently write an
expression involving *P*_{gg}^{s}(*k*),
*P*_{gg}(*k*), *P*_{gm}(*k*) and
*P*_{mm}(*k*),
or the analogous spherical harmonics.

Next we tie in the biasing scheme.
In the simplified case of linear and deterministic biasing, one simply has
*P*_{gg} = *b*_{1}*P*_{gm} =
*b*_{1}^{2}*P*_{mm}, so the distortion
relation reduces to Kaiser's
formula [36],
*P*_{gg}^{s} =
*P*_{gg}(1 + *µ*^{2}_{1})^{2},
where _{1}
*f*() / *b*_{1}.

In the more realistic case of *linear, local*, and *stochastic*
biasing, first at zero lag, _{}(0) =
_{b}^{2}
^{2} and
_{gg}(0) =
*b*_{var}^{2} _{mm}(0). Then, via Eq. (5) and Eq. (7), the general
distortion relation, Eq. (15), reduces to

In this local equation both *b*_{var} and `r` are
involved in a
non-trivial way; the distortions depend on the scatter, reflecting
the _{b}^{2}
term in Eq. (5).
On the other hand, at large separations *r* > *r*_{b},
where _{} vanishes,
one obtains instead, from Eq. (13),

This is simply the Kaiser formula again, which, unlike Eq. (16), is
*independent* of the biasing scatter!
It involves only the mean biasing parameter *b*_{1}, in an
expression that is indistinguishable from the deterministic case.
This is a straightforward result of the assumed locality of the biasing scheme:
the biasing scatter at two distant points is uncorrelated and therefore its
contribution to
_{gg} cancels out.

The distortion relation for *P(k)* becomes more complicated because
of the additive term in Eq. (14). For linear biasing, when substituting
Eq. (14) in the linear distortion relation, the terms analogous to the ones
involving *b*_{1}^{-1} and
*b*_{1}^{-2} in Eq. (17) for
are multiplied by [1 -
_{b}^{2}
^{2} *V*_{b} /
*P*_{gg}(*k*)], a function of *k*.
The distortion relation for *P(k)* is thus affected by the biasing
scatter in a complicated way.
However, there may be a significant *k* range around the peak
of *P(k)* in which the additive scatter term is small compared
to the rest. In this range the relation reduces to an expression
similar to Eq. (17) for the corresponding power spectra.
Still, the scatter term always dominates Eq. (14) at small
and at large *k*'s.

Equation (7) of Pen [50],
which involves *b*_{var} and `r` like our
Eq. (16), may leave the impression that the redshift-distortion
expression depends on the scatter. In order to obtain his relation
from the general distortion relation, one has to define
*k*-dependent biasing parameters by *P*_{gg}(*k*) =
*b*_{var}(k)^{2}*P*_{mm}(*k*)
and *P*_{gm}(*k*) =
*b*_{var}(*k*)`r`(*k*)
*P*_{mm}(*k*).
(Pen's refers to
his *b*_{1}, which is
our *b*_{var}, except that he allows it to vary with *k*).
In the case of local biasing, a comparison to our Eq. (14)
yields *b*_{var}(*k*)^{2} =
*b*_{1}^{2} +
_{b}^{2}
^{2}*V*_{b} / *P*_{m}*m(k)*
and *b*_{var}(*k*) `r`(*k*) =
*b*_{1}.
In the *k* range near the peak of *P*_{mm}(*k*),
where the constant term
in Eq. (14) may be negligible, one has *b*_{var}(*k*)
= *b*_{1} and `r`(*k*) = 1,
and there is indeed no sign of the stochasticity in the distortion relation.

While its sensitivity to stochasticity is indirect,
the redshift distortion analysis *is* sensitive to
the *non-linearity*, of the biasing. A proper analysis
would require a non-linear treatment including
a non-linear generalization of the GI relation
**.** **v** =
-*f*,
because the non-linear effects of biasing and gravity enter at the same
order.
The result is more complicated than Eq. (15), but is calculable
in principle once one knows the function
*b*() and the one-
and two-point probability distribution functions of
.