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The two-point correlation function has long been known to depend on galaxy properties and can vary as a function of galaxy luminosity, morphological or spectral type, color, stellar mass, and redshift. The general trend is that galaxies that are more luminous, early-type, bulge-dominated, optically red, and/or higher stellar mass are more clustered than galaxies that are less luminous, late-type, disk-dominated, optically blue, and/or lower stellar mass. Presented below are relatively recent results indicating how clustering properties depend on galaxy properties from the largest redshifts surveys currently available. The physical interpretation behind these trends is presented in Section 8 below.

6.1. Luminosity Dependence

Fig. 3 shows the large scale structure reflected in the galaxy distribution at low redshift. What is plotted is the spatial distribution of galaxies in a flux-limited sample, meaning that all galaxies down to a given apparent magnitude limit are included. This results in the apparent lack of galaxies or structure at higher redshift in the figure, as at large distances only the most luminous galaxies will be included in a flux-limited sample. In order to robustly determine the underlying clustering, one should, if possible, create volume-limited subsamples in which galaxies of the same luminosity can be detected at all redshifts. In this way the mean luminosity of the sample does not change with redshift and galaxies at all redshifts are weighted equally.

The left panel of Fig. 6 shows the projected correlation function, wp(rp), for galaxies in SDSS in volume-limited subsamples corresponding to different absolute magnitude ranges. The more luminous galaxies are more strongly clustered across a wide range in absolute optical magnitude, from -17 < Mr < -23. Power law fits on scales from ~ 0.1 h-1 Mpc to ~ 10 h-1 Mpc show that while the clustering amplitude depends sensitively on luminosity, the slope does not. Only in the brightest and faintest magnitude bins does the slope deviate from gamma ~ 1.8 and have a steeper value of gamma ~ 2.0. Across this magnitude range r0 varies from ~ 2.8 h-1 Mpc at the faint end to ~ 10 h-1 Mpc at the bright end. This same general trend is found in the 2dFGRS and other redshift surveys (e.g., Norberg et al. 2001).

The right panel of Fig. 6 shows the relative bias of SDSS galaxies as a function of luminosity, relative to the clustering of L* galaxies, measured at the scale of rp = 2.7 h-1 Mpc, which is in the non-linear regime where delta > 1 (Zehavi et al. 2005). L* is the characteristic galaxy luminosity, defined as the luminosity of the break in the galaxy luminosity function. The relative bias is seen to steadily increase at higher luminosity and rise sharply above L*. This is in good agreement with the results from Tegmark et al. (2004), using the power spectrum of SDSS galaxies measured in the linear regime on a scale of ~ 100 h-1 Mpc. The data also agree with the clustering results of galaxies in the 2dFGRS from Norberg et al. (2001). The overall shape of the relative bias with luminosity indicates a slow rise up to the value at L*, above which the rise is much steeper. As discussed in Section 8.2 below, this trend shows that brighter galaxies reside in more massive dark matter halos than fainter galaxies.

Figure 6a Figure 6b

Figure 6. Luminosity-dependence of galaxy clustering. On the left is shown the projected correlation function, wp(rp), for SDSS galaxies in different absolute magnitude ranges, where brighter galaxies are seen to be more clustered. On the right is the relative bias of galaxies as a function of luminosity. Both figures are from Zehavi et al. (2005).

6.2. Color and Spectral Type Dependence

The clustering strength of galaxies also depends on restframe color and spectral type, with a stronger dependence than on luminosity. Fig. 7 shows the spatial distribution of galaxies in SDSS, color coded as a function of restframe color. Red galaxies are seen to preferentially populate the most overdense regions, while blue galaxies are more smoothly distributed in space. This is reflected in the correlation function of galaxies split by restframe color. Red galaxies have a larger correlation length and steeper slope than blue galaxies: r0 ~ 5-6 h-1 Mpc and gamma ~ 2.0 for red L* galaxies, while r0 ~ 3-4 h-1 Mpc and gamma ~ 1.7 for blue L* galaxies in SDSS Zehavi et al. (2005). Clustering studies from the 2dFGRS split the galaxy sample at low redshift by spectral type into galaxies with emission line spectra versus absorption line spectra, corresponding to star forming and quiescent galaxies, and find similar results: that quiescent galaxies have larger correlation lengths and steeper clustering slopes than star forming galaxies (Madgwick et al. 2003).

Figure 7

Figure 7. The spatial distribution of galaxies in the SDSS main galaxy sample as a function of redshift and right ascension, projected through 8° in declination, color coded by restframe optical color. Red galaxies are seen to be more clustered than blue galaxies and generally trace the centers of groups and clusters, while blue galaxies populate further into the galaxy voids. Taken from Zehavi et al. (2011).

Red and blue galaxies have distinct luminosity-dependent clustering properties. As shown in Fig. 8, the general trends seen in r0 and gamma with luminosity for all galaxies are well-reflected in the blue galaxy population; however, at faint luminosities (L ltapprox 0.5 L*) red galaxies have larger clustering amplitudes and slopes than L* red galaxies. This reflects the fact that faint red galaxies are often found distributed throughout galaxy clusters.

Figure 8

Figure 8. The clustering scale length, r0 (left), and slope, gamma (right), for all, red, and blue galaxies in SDSS as a function of luminosity. While all galaxies are more clustered at brighter luminosities, and red galaxies are more clustered than blue galaxies at all luminosities, below L* the red galaxy clustering length increases at fainter luminosities. The clustering slope for faint red galaxies is also much steeper than at other luminosities. Taken from Zehavi et al. (2011).

Galaxy clustering also depends on other galaxy properties such as stellar mass, concentration index, and the strength of the 4000Å break (D4000), in that galaxies that have larger stellar mass, more centrally concentrated light profiles, and/or larger D4000 measurements (indicating older stellar populations) are more clustered (Li et al. 2006). This is not surprising given the observed trends with luminosity and color and the known dependencies of other galaxy properties with luminosity and color. Clearly the galaxy bias is a complicated function of various galaxy properties.

6.3. Redshift Space Distortions

The fact that red galaxies are more clustered than blue galaxies is related to the morphology-density relation (Dressler 1980), which results from the fact that galaxies with elliptical morphologies are more likely to be found in regions of space with a higher local surface density of galaxies. The redshift space distortions seen for red and blue galaxies also show this.

As discussed in Section 4, redshift space distortions arise from two different phenomena: virialized motions of galaxies within collapsed overdensities such as groups and clusters, and the coherent streaming motion of galaxies onto larger structures that are still collapsing. The former is seen on relatively small scales (rp ltapprox 1 h-1 Mpc) while the latter is detected on larger scales (rp gtapprox 1 h-1 Mpc). While the presence of redshift space distortions complicates the measurement of the real space xi(r), these distortions can be used to uncover information about the thermal motions of galaxies in groups and clusters as well as the amplitude of the mass density of the Universe, Omegamatter.

Fig. 9 shows xi(rp, pi) for quiescent and star forming galaxies in 2dF. The quiescent galaxies on the left show larger "Fingers of God" than the star forming galaxies on the right, reflecting the fact that red, quiescent galaxies have larger motions relative to each others. This naturally arises if red, quiescent galaxies reside in more massive, virialized overdensities with larger random peculiar velocities than star forming, optically blue galaxies. The large scale coherent infall of galaxies is seen both for blue and red galaxies, though it is often easier to see for blue galaxies, due to their smaller "Fingers of God".

Figure 9

Figure 9. Two-dimensional redshift space correlation function xi(rp, pi) (as in Fig. 6 here sigma is used instead of rp) for quiescent, absorption line galaxies on the left and star forming, emission line galaxies on the right. The redshift space distortions are different for the different galaxy populations, with quiescent and/or red galaxies showing more pronounced "Fingers of God". Both galaxy types exhibit coherent infall on large scales. Contours show lines of constant xi at xi = 10, 5, 2, 1, 0.5, 0.2, 0.1. Taken from Madgwick et al. (2003).

These small scale redshift space distortions can be quantified using the sigma12 statistic, known as the pairwise velocity dispersion (Davis et al. 1978, Fisher et al. 1994). This is measured by modeling xi(rp, pi) in real space, which is then convolved with a distribution of random pairwise motions, f(v), such that

Equation 23 (23)

where the random motions are often taken to have an exponential form, which has been found to fit observed data well:

Equation 24 (24)

In the 2dFGRS Madgwick et al. (2003) find that sigma12 = 416 ± 76 km s-1 for star forming galaxies and sigma12 = 612 ± 92 km s-1 for quiescent galaxies, measured on scales of 8-20 h-1 Mpc. Using SDSS data Zehavi et al. (2002) find that sigma12 is ~ 300-450 km s-1 for blue, star forming galaxies and ~ 650-750 km s-1 for red, quiescent galaxies. It has been shown, however, that this statistic can be sensitive to large, rare overdensities, such that samples covering large volumes are needed to measures it robustly.

Madgwick et al. (2003) further measure the large scale anisotropies seen in xi(rp, pi) for galaxies split by spectral type and find that beta = 0.49 ± 0.13 for star forming galaxies and beta = 0.48 ± 0.14 for quiescent galaxies. This implies a similar bias for both galaxy types on large scales, though they find that on smaller scales integrated up to 8 h-1 Mpc, the relative bias of quiescent to star forming galaxies is brel = 1.45 ± 0.14.

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