3. TWO-POINT CORRELATIONS: REDSHIFT DISTORTIONS

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

(12)

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

(13)

Note that the biasing parameter that appears here is b₁, 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),

(14)

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 - ðu / ðr, where u is the radial component of the peculiar velocity v. Assuming no velocity biasing, linear GI predicts ðu / ðr = - µ² f() , where µ² is a geometrical factor depending on the angle between v and x. Thus, the basic linear relation for redshift distortions is g_s = g + fµ². The general expression for redshift distortions is obtained from this basic relation by averaging < g₁^s g₂^s > over the distributions of at a pair of points separated by r:

(15)

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 = b₁P_gm = b₁²P_mm, so the distortion relation reduces to Kaiser's formula [36], P_gg^s = P_gg(1 + µ²₁)², where ₁ f() / b₁.

In the more realistic case of linear, local, and stochastic biasing, first at zero lag, (0) = _b^{2 2} and _gg(0) = b_var² _mm(0). Then, via Eq. (5) and Eq. (7), the general distortion relation, Eq. (15), reduces to

(16)

In this local equation both b_var and r are involved in a non-trivial way; the distortions depend on the scatter, reflecting the _b² term in Eq. (5). On the other hand, at large separations r > r_b, where vanishes, one obtains instead, from Eq. (13),

(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 _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 b₁^-1 and b₁^-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)²P_mm(k) and P_gm(k) = b_var(k)r(k) P_mm(k). (Pen's refers to his b₁, 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 = b₁² + _b^{2 2}V_b / P_mm(k) and b_var(k) r(k) = b₁. 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) = 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 . 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 .