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

Equation 12 (12)

where the averaging is over the ensembles at points 1 and 2 separated by r. We define the biasing as local if xiem(r) = 0 for any r and xiepsilonepsilon(r) = 0 for r > rb, where rb 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

Equation 13 (13)

Note that the biasing parameter that appears here is b1, not bvar.

To see how the power spectra are affected by the biasing scatter, we approximate the local biasing by a step function: xiepsilonepsilon(r) = sigmab2 sigma2 for r < rb and zero otherwise. Recalling that the power spectra are the Fourier transforms of the corresponding correlation functions, we get for k << rb-1, from Eq. (13),

Equation 14 (14)

where Vb is the volume associated with rb. We see that the local biasing scatter adds a constant to Pgg(k).

We can now proceed to estimating beta via redshift distortions [36, 28, 29, 30, 25, 31, 11, 24, 40]. To first order, the local galaxy density fluctuations in redshift space (gs) and real space (g) are related by gs = g - ðu / ðr, where u is the radial component of the peculiar velocity v. Assuming no velocity biasing, linear GI predicts ðu / ðr = - µ2 f(Omega) delta, where µ2 is a geometrical factor depending on the angle between v and x. Thus, the basic linear relation for redshift distortions is gs = g + fµ2delta. The general expression for redshift distortions is obtained from this basic relation by averaging < g1s g2s > over the distributions of delta at a pair of points separated by r:

Equation 15 (15)

Recalling that the power spectra are the Fourier transforms of the corresponding correlation functions, one can equivalently write an expression involving Pggs(k), Pgg(k), Pgm(k) and Pmm(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 Pgg = b1Pgm = b12Pmm, so the distortion relation reduces to Kaiser's formula [36], Pggs = Pgg(1 + µ2beta1)2, where beta1 ident f(Omega) / b1.

In the more realistic case of linear, local, and stochastic biasing, first at zero lag, xiepsilonepsilon(0) = sigmab2 sigma2 and xigg(0) = bvar2 ximm(0). Then, via Eq. (5) and Eq. (7), the general distortion relation, Eq. (15), reduces to

Equation 16 (16)

In this local equation both bvar and r are involved in a non-trivial way; the distortions depend on the scatter, reflecting the sigmab2 term in Eq. (5). On the other hand, at large separations r > rb, where xiepsilonepsilon vanishes, one obtains instead, from Eq. (13),

Equation 17 (17)

This is simply the Kaiser formula again, which, unlike Eq. (16), is independent of the biasing scatter! It involves only the mean biasing parameter b1, 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 xigg 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 b1-1 and b1-2 in Eq. (17) for xi are multiplied by [1 - sigmab2 sigma2 Vb / Pgg(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 bvar 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 Pgg(k) = bvar(k)2Pmm(k) and Pgm(k) = bvar(k)r(k) Pmm(k). (Pen's beta refers to his b1, which is our bvar, except that he allows it to vary with k). In the case of local biasing, a comparison to our Eq. (14) yields bvar(k)2 = b12 + sigmab2 sigma2Vb / Pmm(k) and bvar(k) r(k) = b1. In the k range near the peak of Pmm(k), where the constant term in Eq. (14) may be negligible, one has bvar(k) = b1 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 del . v = -fdelta, 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(delta) and the one- and two-point probability distribution functions of delta.

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