6.3 Luminosity and spectral-type dependence of galaxy clustering

Although biasing was commonly neglected until the early 1980s, it has become evident observationally that on scales ≲ 10h −1 Mpc different galaxy populations exhibit different clustering amplitudes, the so-called morphology-density relation [16]. As discussed in Section 4, galaxy biasing is naturally predicted from a variety of theoretical considerations as well as direct numerical simulations [3759158786103]. Thus, in this Section we summarize the extent to which the galaxy clustering is dependent on the luminosity, spectral-type, and color of the galaxy sample from the 2dFGRS and SDSS.

6.3.1 2dFGRS: Clustering per luminosity and spectral type

Madgwick et al. [45Jump To The Next Citation Point] applied the Principal Component Analysis to compress each galaxy spectrum into one quantity, η ≈ 0.5pc1 + pc2. Qualitatively, η 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 η-types, and to study, e.g., luminosity functions and clustering per type. Norberg et al. [61Jump To The Next Citation Point] 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 ∘ ------ b ≡ ξg∕ξm varies as b∕b∗ = 0.85 + 0.15L ∕L∗. Figure 25View Image indicates that for L ∗ galaxies, the real space correlation function amplitude of η early-type galaxies is ∼ 50% higher than that of late-type galaxies.

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Figure 25: The variation of the galaxy biasing parameter with luminosity, relative to an L ∗ galaxy for the full sample and for subsamples of early and late spectra types. (Figure taken from [61].)

Figure 26View Image shows the redshift-space correlation function in terms of the line-of-sight and perpendicular to the line-of-sight separation ξ(σ,π ). 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 −1 ≲ 10h Mpc. 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 ξ (π, σ) over the separation range 8– 20h− 1 Mpc for each class, it was found that βactive = 0.49 ± 0.13, βpassive = 0.48 ± 0.14 and corresponding pairwise velocity dispersions σ P of 416 ± 76 km s−1 and 612 ± 92 km s−1 [45Jump To The Next Citation Point]. At small separations, the real space clustering of passive galaxies is stronger than that of active galaxies: The slopes γ are respectively 1.93 and 1.50 (see Figure 27View Image) and the relative bias between the two classes is a declining function of separation. On scales larger than 10h− 1 Mpc the biasing ratio is approaching unity.

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Figure 26: The two point correlation function ξ(σ,π) plotted for passively (left panel) and actively (right panel) star-forming galaxies. The line contours levels show the best-fitting model. (Figure taken from [45Jump To The Next Citation Point].)

Another statistic was applied recently by Wild et al. [98] and Conway et al. [12], 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.

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Figure 27: The correlation function for early and late spectral types. The solid lines show best-fitting models, whereas the dashes lines are extrapolations of these lines. (Figure taken from [45].)

6.3.2 SDSS: Two-point correlation functions per luminosity and color

Zehavi et al. [104Jump To The Next Citation Point] 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: ξ(r) = (r∕6.1 ± 0.2h −1 Mpc )−1.75±0.03 for 0.1h −1 Mpc ≤ r ≤ 16h −1 Mpc. The galaxy pairwise velocity dispersion is σ12 ≈ 600 ± 100 km s−1 for projected separations 0.15h −1 Mpc ≤ rp ≤ 5h−1 Mpc. 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 r ∼ 10h− 1 Mpc. Subsamples with absolute magnitude ranges centered on M ∗ − 1.5,

M ∗, and M + 1.5 ∗ have real-space correlation functions that are parallel power laws of slope ≈ − 1.8 with correlation lengths of approximately −1 7.4h Mpc, −1 6.3h Mpc, and −1 4.7h Mpc, respectively.

Figures 27View Image and 28View Image 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.

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Figure 28: The SDSS (EDR) projected correlation function for blue (squares), red (triangles) and the full sample, with best-fitting models over the range 0.1 < r < 16h− 1 Mpc p (upper panel), and the SDSS (EDR) projected correlation function for three volume-limited samples, with absolute magnitude and redshift ranges as indicated and best-fitting power-law models (lower panel). (Figure taken from [104].)

6.3.3 SDSS: Three-point correlation functions and the nonlinear biasing of galaxies per luminosity and color

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 [70] and Groth and Peebles [27] using the Lick and Zwicky angular catalogs of galaxies. They found that the 3PCF ζ (r ,r ,r ) 12 23 31 obeys the hierarchical relation:

ζ(r12,r23,r31) = Qr [ξ(r12)ξ(r23) + ξ(r23)ξ(r31) + ξ(r31)ξ(r12)], (173 )
with Qr being a constant. The value of Qr in real space deprojected from these angular catalogues is 1.29 ± 0.21 for r < 3h −1 Mpc. Subsequent analyses of redshift catalogs confirmed the hierarchical relation, at least approximately, but the value of Qz (in redshift space) appears to be smaller, Qz ∼ 0.5– 1.

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. [39Jump To The Next Citation Point] 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 − 22 < Mr − 5log h < − 21, − 21 < Mr − 5 logh < − 20, and − 20 < Mr − 5 log h < − 19, respectively.

Figure 29View Image indicates the dimensionless amplitude of the 3PCFs of SDSS galaxies in redshift space,

ζ(s12,s23,s31) Qz (s12,s23,s31) ≡ ---------------------------------------, (174 ) ξ(s12)ξ (s23) + ξ(s23)ξ (s31) + ξ(s31)ξ(s12)
for the equilateral triplets of galaxies. The overall conclusion is that Qz is almost scale-independent and ranges between 0.5 and 1.0, and that no systematic dependence is noticeable on luminosity and color. This implies that the 3PCF itself does depend on the galaxy properties since two-point correlation functions (2PCFs) exhibit clear dependence on luminosity and color. Previous simulations and theoretical models [82535085] indicate that Q decreases with scale in both real and redshift spaces. This trend is not seen in the observational results.
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Figure 29: Dimensionless amplitude of the three-point correlation functions of SDSS galaxies in redshift space. The galaxies are classified according to their colors; all galaxies in open circles, red galaxies in solid triangles, and blue galaxies in crosses. (Figure taken from [39Jump To The Next Citation Point].)

In order to demonstrate the expected dependence in the current samples, they compute the biasing parameters estimated from the 2PCFs,

∘ ----------- --ξz,i(s)--- bz,i(s ) ≡ ξ (s), (175 ) z,ΛCDM
where the index i runs over each sample of galaxies with different colors and luminosities. The predictions of the mass 2PCFs in redshift space, ξ (s) z,ΛCDM, in the Λ cold dark matter model are computed following [28].

As an illustrative example, consider a simple bias model in which the galaxy density field δg,i for the i-th population of galaxies is given by

δg,i = bg,i(1)δmass + bg,i(2)δ2mass. (176 )
If both bg,i(1) and bg,i(1) are constant and the mass density field δmass ≪ 1, Equation (174View Equation) implies that
--1-- bg,i(2) Qg,i = b Qmass + b2 . (177 ) g,i(1) g,i(1)
Thus the linear bias model (bg,i(2) = 0) simply implies that Qg,i is inversely proportional to b g,i(1), which is plotted in Figure 30View Image. A comparison of Figures 29View Image and 30View Image indicates that the biasing in the 3PCFs seems to compensate the difference of Qg purely due to that in the 2PCFs.
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Figure 30: Same as Figure 29View Image, but for the inverse of the biasing parameter defined through the two-point correlation functions. (Figure taken from [39].)

Such behavior is unlikely to be explained by any simple model inspired by the perturbative expansion like Equation (176View Equation). 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.

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