Madgwick et al.  applied the Principal Component Analysis to compress each galaxy spectrum into one quantity, . Qualitatively, is an indicator of the ratio of the present to the past star formation activity of each galaxy. This allows one to divide the 2dFGRS into -types, and to study, e.g., luminosity functions and clustering per type. Norberg et al.  showed that, at all luminosities, early-type galaxies have a higher bias than late-type galaxies, and that the biasing parameter, defined here as the ratio of the galaxy to matter correlation function varies as . Figure 25 indicates that for galaxies, the real space correlation function amplitude of early-type galaxies is higher than that of late-type galaxies.
Figure 26 shows the redshift-space correlation function in terms of the line-of-sight and perpendicular to the line-of-sight separation . The correlation function calculated from the most passively (‘red’, for which the present rate of star formation is less than 10% of its past averaged value) and actively (‘blue’) star-forming galaxies. The clustering properties of the two samples are clearly distinct on scales . The ‘red’ galaxies display a prominent finger-of-God effect and also have a higher overall normalization than the ‘blue’ galaxies. This is a manifestation of the well-known morphology-density relation. By fitting over the separation range for each class, it was found that , and corresponding pairwise velocity dispersions of and . At small separations, the real space clustering of passive galaxies is stronger than that of active galaxies: The slopes are respectively 1.93 and 1.50 (see Figure 27) and the relative bias between the two classes is a declining function of separation. On scales larger than the biasing ratio is approaching unity.
Another statistic was applied recently by Wild et al.  and Conway et al. , of a joint counts-in-cells on 2dFGRS galaxies, classified by both color and spectral type. Exact linear bias is ruled out on all scales. The counts are better fitted to a bivariate log-normal distribution. On small scales there is evidence for stochasticity. Further investigation of galaxy formation models is required to understand the origin of the stochasticity.
Zehavi et al.  analyzed the Early Data Release (EDR) sample of the SDSS 30,000 galaxies to explore the clustering of per luminosity and color. The inferred real-space correlation function is well described by a single power-law: for . The galaxy pairwise velocity dispersion is for projected separations . When divided by color, the red galaxies exhibit a stronger and steeper real-space correlation function and a higher pairwise velocity dispersion than do the blue galaxies. In agreement with 2dFGRS there is clear evidence for a scale-independent luminosity bias at . Subsamples with absolute magnitude ranges centered on ,
, and have real-space correlation functions that are parallel power laws of slope with correlation lengths of approximately , , and , respectively.
Figures 27 and 28 pose an interesting challenge to the theory of galaxy formation, to explain why the correlation functions per luminosity bins have similar slope, while the slope for early type galaxies is steeper than for late type.
Let us move next to the three-point correlation functions (3PCF) of galaxies, which are the lowest-order unambiguous statistic to characterize non-Gaussianities due to nonlinear gravitational evolution of dark matter density fields, formation of luminous galaxies, and their subsequent evolution. The determination of the 3PCF of galaxies was pioneered by Peebles and Groth  and Groth and Peebles  using the Lick and Zwicky angular catalogs of galaxies. They found that the 3PCF obeys the hierarchical relation:
As we have seen in Section 6.3.2, galaxy clustering is sensitive to the intrinsic properties of the galaxy samples under consideration, including their morphological types, colors, and luminosities. Nevertheless the previous analyses were not able to examine those dependences of 3PCFs because of the limited number of galaxies. Indeed Kayo et al.  were the first to perform the detailed analysis of 3PCFs explicitly taking account of the morphology, color, and luminosity dependence. They constructed volume-limited samples from a subset of the SDSS galaxy redshift data, ‘Large-scale Structure Sample 12’. Specifically they divided each volume limited sample into color subsamples of red (blue) galaxies, which consist of 7949 (8329), 8930 (8155), and 3706 (3829) galaxies for , , and , respectively.
Figure 29 indicates the dimensionless amplitude of the 3PCFs of SDSS galaxies in redshift space,[82, 53, 50, 85] indicate that decreases with scale in both real and redshift spaces. This trend is not seen in the observational results.
In order to demonstrate the expected dependence in the current samples, they compute the biasing parameters estimated from the 2PCFs,.
As an illustrative example, consider a simple bias model in which the galaxy density field for the -th population of galaxies is given by
Such behavior is unlikely to be explained by any simple model inspired by the perturbative expansion like Equation (176). Rather it indeed points to a kind of regularity or universality of the clustering hierarchy behind galaxy formation and evolution processes. Thus the galaxy biasing seems much more complex than the simple deterministic and linear model. More precise measurements of 3PCFs and even higher-order statistics with future SDSS datasets would be indeed valuable to gain more specific insights into the empirical biasing model.
© Max Planck Society and the author(s)