Large-Scale Structure of the Universe and Cosmological ...

Table 19
Some measurements of S3 and S4 in redshift catalogs. In most cases, only the mean
values over a range of scales were published. In cases where measurements of the
individual values for each scale are reported in the literature, we quote the actual
range of estimates over the corresponding range of scales. In most cases error bars
should be considered only as rough estimates, see text for discussion.
S3 S4 Scales Sample Year Ref
2 ± 1 − 6 ± 4 — 5 − 20 QDOT 1991 [549]
1.5 ± 0.5 4.4 ± 3.7 0.1-50 IRAS 1.2Jy 1992 [88],[92]
1.9 ± 0.1 4.1 ± 0.6 2-22 CfA 1992 [246]
2.0 ± 0.1 5.0 ± 0.9 2-22 SSRS 1992 [246]
2.1 ± 0.3 7.5 ± 2.1 3-10 IRAS 1.9Jy 1994 [224]
2.4 ± 0.3 13 ± 2 2-10 PPS 1996 [267]
2.8 ± 0.1 6.9 ± 0.7 8-32 IRAS 1.2Jy 1998 [377]
1.8 ± 0.1 5.5 ± 1 1-10 SSRS2 1999 [35]
1.9 ± 0.6 7 ± 4 1-30 PSCz 2000 [632]
1.82 ± 0.21 ∼ 3 12.6 Durham/UKST 2000 [319]
2.24 ± 0.29 ∼ 8 18.2 Stromlo/APM 2000 [319]
8.3.4 Skewness, Kurtosis and Higher-Order Cumulants
Table 19 shows different estimates for S3 = ξ3/ξ 2
2 and S4 = ξ4/ξ 3
2 , the ratios
of the cumulants ξN obtained by counts-in-cells. The shape of the cells
correspond to top-hat spheres, unless stated otherwise.
The QDOT results by [549] were obtained from counts-in-cells with a Gaussian
window. The errors, from a minimum variance scheme, are quite large but they
suggest a hierarchical scaling ξ3 ≃ ξ 2
2 , with a value of S3 consistent with gravity
from Gaussian initial conditions, as argued in [137].
Figure 59 displays the 1.2 Jy IRAS results ([88], [92], left panel) and CfA-SSRS
results ([246], right panel). There is a convincing evidence for the hierarchical
scaling in ξ3 and ξ4 (denoted by straight lines) but the resulting S3 and S4
amplitudes are probably affected by sampling biases (see discussion below).
Note that the scaling is preserved well into the non-linear regime, this is in
agreement with expectations from N-body simulations which show that in
redshift space the growth of Sp parameters towards the non-linear regime is
suppressed by velocity dispersion from virialized regions ([391,437], see e.g.
Fig. 49).
In their analysis of higher-order moments in the CfA, SSRS, and IRAS 1.9
Jy catalogs, [224] studied the sensitivity of Sp to redshift distortions, by calculating
moments in spherical cells and conical cells. The latter were argued
to be less sensitive to the redshift space mapping that acts along the line of
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