GALAXY CLUSTERING IN THE COMPLETED SDSS REDSHIFT SURVEY: THE DEPENDENCE ON COLOR AND LUMINOSITY

The SDSS (York et al. 2000; Stoughton et al. 2002) was an ambitious project to map most of the high-latitude sky in the northern Galactic cap, using a dedicated 2.5 m telescope (Gunn et al. 2006). The survey started regular operations in 2000 April and completed observations (for SDSS-II) in 2008 July. A drift-scanning mosaic CCD camera (Gunn et al. 1998) imaged the sky in five photometric bandpasses (Fukugita et al. 1996; Smith et al. 2002) to a limiting magnitude of r ∼ 22.5. The imaging data were processed through a series of pipelines that perform astrometric calibration (Pier et al. 2003), photometric reduction (Lupton et al. 1999, 2001), and photometric calibration (Hogg et al. 2001; Ivezić et al. 2004; Tucker et al. 2006; Padmanabhan et al. 2008). Objects were selected for spectroscopic follow-up using specific algorithms for the main galaxy sample (Strauss et al. 2002), luminous red galaxies (Eisenstein et al. 2001), and quasars (Richards et al. 2002). Targets were assigned to spectroscopic plates using an adaptive tiling algorithm (Blanton et al. 2003a) and observed with a pair of fiber-fed spectrographs. Spectroscopic data reduction and redshift determination were performed by automated pipelines. Galaxy redshifts were measured with a success rate greater than 99% and typical accuracy of 30 km s⁻¹. To a good approximation, the main galaxy sample consists of all galaxies with Petrosian magnitude r < 17.77, with a median redshift of ∼0.1. The LRG redshift sample uses color–magnitude cuts to select galaxies with r < 19.5 that are likely to be luminous early-type galaxies, extending up to redshift ∼0.5.

Galaxy samples suitable for large-scale structure studies have been carefully constructed from the SDSS redshift data (Blanton et al. 2005b). All magnitudes are K-corrected (Blanton et al. 2003b) and evolved to rest-frame magnitudes at z = 0.1 (which is near the median redshift of the sample and thus minimizes corrections) using an updated version of the evolving luminosity function model of Blanton et al. (2003c).¹⁵ The radial selection function is derived from the sample selection criteria. When creating volume-limited samples below, we include a galaxy if its evolved, redshifted spectral energy distribution places it within the main galaxy sample's apparent magnitude and surface brightness limits at the limiting redshift of the sample. The angular completeness is characterized carefully for each sector (a unique region of overlapping spectroscopic plates) on the sky.

Due to the placement of fibers to obtain spectra, no two targets on the same plate can be closer than 55''. This results in ∼7% of targeted galaxies not having a measured redshift. We assign these galaxies the redshift of their nearest neighboring galaxy; roughly speaking, this method double-weights the galaxy that was observed, but it retains the additional information present in the angular position of the "collided" galaxy. As shown in Z05, this treatment works remarkably well for projected statistics such as w_p(r_p), above the physical scale corresponding to 55'' (r_p ≈ 0.13 h⁻¹ Mpc at the outer edge of our sample). We thus limit the measurements in this paper to scales larger than that. The median deviation of w_p(r_p) for the range of separations we utilize is 0.2%, much less than the statistical errors on the measurements (Figure 3 in Z05). It is, in fact, possible to correct for fiber collisions down to scales as small as 0.01 h⁻¹ Mpc using the ratio of small-angle pairs in the spectroscopic and photometric catalogs (Masjedi et al. 2006; Li et al. 2006; Li & White 2009), but we have not implemented this technique here.

The clustering measurements in this paper are based on SDSS DR7 (Abazajian et al. 2009), which marks the completion of the original goals of the SDSS and the end of the phase known as SDSS-II. The associated NYU Value-Added Galaxy Catalog (NYU-VAGC)¹⁵ includes approximately 700,000 main sample galaxies over about 8000 deg² on the sky. This data set can be compared to the much smaller sky coverage of the samples in previous correlation function analyses of the SDSS main galaxy sample: Zehavi et al. (2002) used an early sample of ∼700 deg², and Z05 analyzed a sample of about 2500 deg². The contiguous northern footprint of DR7 offers further advantage over earlier data sets by reducing boundary effects. Figures 1–3 show the distribution of the main sample galaxies in right ascension and redshift for slices near the celestial equator. These plots nicely illustrate the large-scale structure we aim to study using the two-point correlation function as well as the potential dependencies on galaxy properties. Diagrams that show contiguity of structure over multiple SDSS slices appear in Choi et al. (2010), who analyze the topology of large-scale structure in the DR7 main galaxy sample.

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Throughout this paper, we refer to distances in comoving units, and for all distance calculations and absolute magnitude definitions we adopt a flat ΛCDM model with Ω_m = 0.3. We quote distances in  h⁻¹ Mpc (where h ≡ H₀/100 km s⁻¹ Mpc⁻¹) and absolute magnitudes for h = 1. Our correlation function measurements are strictly independent of H₀, except that the absolute magnitudes we list as M_r are really values of M_r + 5log h. Changing the assumed Ω_m or $\Omega _\Lambda$ would have a small impact on our measurements by changing the distance–redshift relation and thus shifting galaxies among luminosity bins and galaxy pairs among radial separation bins. However, even at our outer redshift limit of z = 0.25, the effect of lowering Ω_m from 0.3 to 0.25 is only 1% in distance, so our measurements are effectively independent of cosmological parameters within their observational uncertainties.

In order to work with well-defined classes of galaxies, we study volume-limited samples constructed for varying luminosity bins and luminosity thresholds. While volume-limited subsamples include fewer galaxies than the full flux-limited sample, they are much easier to interpret. For a given luminosity bin, we discard the galaxies that are too faint to be included at the far redshift limit or too bright to be included at the near limit. We include galaxies with 14.5 < r < 17.6, with the conservative bright limit imposed to avoid small incompletenesses associated with galaxy deblending (the NYU-VAGC safe samples). We further cut these samples by color, using the K-corrected g − r color as a separator into different populations. We also study a set of luminosity-threshold samples, namely, volume-limited samples of all galaxies brighter than a given threshold, as these yield higher precision measurements than luminosity-bin samples and are somewhat more straightforward for HOD modeling. For these samples we relax the bright flux limit to r > 10.0, in order to be able to define a viable volume-limited redshift range (the NYU-VAGC bright samples). The distribution in magnitude and redshift and the cuts used to define the samples are shown in Figure 4. Details of the samples are given in Tables 1 and 2. For luminosity-threshold samples, one could improve statistics by using the flux-limited galaxy catalog and weighting galaxy pairs by the inverse volume over which they can be observed, as done by Li & White (2009, 2010) for samples weighted by stellar mass and luminosity. This procedure would extend the outer redshift limit for the more luminous galaxies above the threshold, thus reducing sample variance, but it has the arguable disadvantage of using different measurement volumes for different subsets of galaxies within the sample, and we have not implemented it here.

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Table 1. Volume-limited Correlation Function Samples Corresponding to Luminosity Bins

Mr	czmin	czmax	Ngal	Nblue	Nred	r0	γ	$\frac{\chi ^2}{{\rm dof}}$	r0d	γd	${\bar{n}}$	〈Mr〉	πmax
−23 to −22	30,900	73,500	10,251	1,797	8,452	10.47 ± 0.25	1.92 ± 0.03	2.4	10.40 ± 0.18	1.94 ± 0.02	0.004	−22.22	60
−22 to −21	19,900	47,650	73,746	27,496	46,249	5.98 ± 0.11	1.92 ± 0.02	5.0	6.30 ± 0.06	1.88 ± 0.01	0.111	−21.32	60
−21 to −20	12,600	31,900	108,629	50,879	57,749	5.46 ± 0.15	1.77 ± 0.02	3.8	5.80 ± 0.09	1.75 ± 0.01	0.530	−20.42	60
−21 to −20	12,600	19,250*	17,853	8,103	9,749	4.82 ± 0.23	1.87 ± 0.03	2.5	5.33 ± 0.13	1.81 ± 0.03	0.530	−20.42	40
−20 to −19	8,050	19,250	44,348	25,455	18,892	4.89 ± 0.26	1.78 ± 0.02	3.8	5.19 ± 0.13	1.80 ± 0.02	1.004	−19.47	60
−19 to −18	5,200	12,500	18,200	13,035	5,165	4.14 ± 0.30	1.81 ± 0.03	2.3	4.59 ± 0.18	1.93 ± 0.04	1.300	−18.48	40
−18 to −17	3,200	7,850	5,965	4,970	995	2.09 ± 0.38	1.99 ± 0.14	2.0	4.37 ± 0.37	1.91 ± 0.08	1.972	−17.46	40

Notes. All samples use 14.5 < m_r < 17.6. r₀ and γ are obtained from fitting a power law to w_p(r_p) using the full error covariance matrices, while r₀^d and γ^d are obtained when using just the diagonal elements. For all samples, the number of degrees of freedom (dof) is 9 (11 measured w_p values minus the 2 fitted parameters). ${\bar{n}}$ is measured in units of 10⁻² h³ Mpc⁻³. A handful of galaxies do not have well-measured colors, so N_blue and N_red do not sum to N_gal. The smaller −21 < M_r < −20 sample, indicated with an asterisk, is limited to a smaller redshift range to avoid the effects of the Sloan Great Wall (see the text).

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Table 2. Volume-limited Correlation Function Samples Corresponding to Luminosity Thresholds

Mmaxr	czmax	Ngal	Nblue	Nred	r0	γ	$\frac{\chi ^2}{{\rm dof}}$	r0d	γd	${\bar{n}}$	πmax
−22.0	73,500	11,385	2,145	9,237	10.71 ± 0.24	1.91 ± 0.03	3.2	10.56 ± 0.17	1.92 ± 0.02	0.005	60
−21.5	59,600	39,456	10,576	28,876	7.27 ± 0.14	2.00 ± 0.01	8.8	7.68 ± 0.08	1.94 ± 0.01	0.028	60
−21.0	47,650	83,238	30,159	53,075	5.98 ± 0.12	1.96 ± 0.02	6.1	6.46 ± 0.06	1.90 ± 0.01	0.116	60
−20.5	39,700	132,225	54,827	77,395	5.60 ± 0.12	1.90 ± 0.01	3.2	6.01 ± 0.06	1.85 ± 0.01	0.318	60
−20.0	31,900	141,733	62,862	78,868	5.54 ± 0.14	1.83 ± 0.01	3.8	6.00 ± 0.09	1.79 ± 0.01	0.656	60
−20.0	19,250*	30,245	12,733	17,510	5.24 ± 0.28	1.87 ± 0.03	1.2	5.53 ± 0.13	1.85 ± 0.02	0.656	60
−19.5	25,450	132,664	62,892	69,770	5.11 ± 0.17	1.81 ± 0.02	1.8	5.37 ± 0.08	1.81 ± 0.01	1.120	60
−19.5	19,250*	51,498	24,005	27,491	5.17 ± 0.27	1.84 ± 0.03	2.3	5.36 ± 0.13	1.85 ± 0.02	1.120	60
−19.0	19,250	77,142	39,554	37,585	4.86 ± 0.27	1.85 ± 0.03	3.2	5.23 ± 0.12	1.85 ± 0.02	1.676	60
−18.5	15,750	58,909	32,554	26,355	4.48 ± 0.33	1.86 ± 0.04	2.1	5.33 ± 0.18	1.83 ± 0.03	2.311	40
−18.0	12,500	39,027	23,159	15,868	4.10 ± 0.34	1.85 ± 0.04	1.8	4.75 ± 0.17	1.91 ± 0.04	3.030	40

Notes. All samples use 10.0 < m_r < 17.6. z_min for the samples is 0.02. r₀ and γ are obtained from fitting a power law to w_p(r_p) using the full error covariance matrices, while r₀^d and γ^d are obtained when using just the diagonal elements. For all samples, the number of degrees of freedom (dof) is 9 (11 measured w_p values minus the two fitted parameters). ${\bar{n}}$ is measured in units of 10⁻² h³ Mpc⁻³. The samples indicated with an asterisk are limited to a smaller redshift range to avoid the effects of the large supercluster (see the text).

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The full spectroscopic survey of the SDSS DR7 Legacy survey contains 900,000 unique, survey-quality galaxy spectra over 8000 deg². Of these objects, the main galaxy sample target criteria selected 700,000. SDSS targeted the remainder as LRG candidates (around 100,000) or in other categories (e.g., as quasar candidates or in special programs on the Equator). We use a reduced footprint of 7700 deg², which excludes areas of suspect photometric calibration (Padmanabhan et al. 2008) and incomplete regions near bright stars. This reduction leaves 670,000 main sample galaxies. Because we are using an updated photometric reduction, a substantial fraction of targets are assigned fluxes fainter than the original flux limit, which further reduces the sample to about 640,000 galaxies. For uniformity we have imposed an even stricter faint limit of r = 17.6 in this paper, which yields 540,000 galaxies. About 30,000 of the original targets at that flux limit were not assigned fibers because of fiber collisions; we assign these objects the redshift of their nearest neighbor as discussed above. The resulting sample of 570,000 galaxies constitutes the parent sample for all of the volume-limited samples in this paper. When we apply a bright magnitude cut of r = 14.5, it eliminates about 6000 galaxies. Further details and the samples themselves are available as part of the public NYU-VAGC data sets.

The autocorrelation function is a powerful way to characterize galaxy clustering, measuring the excess probability over random of finding pairs of galaxies as a function of separation (e.g., Peebles 1980). To separate effects of redshift distortions from spatial correlations, it is customary to estimate the galaxy correlation function on a two-dimensional grid of pair separations parallel (π) and perpendicular (r_p) to the line of sight. Following the notation of Fisher et al. (1994), for a pair of galaxies with redshift positions v₁ and v₂, we define the redshift separation vector s ≡ v₁ − v₂ and the line-of-sight vector ${\bf l} \equiv {{1}\over {2}}({\bf v}_1+{\bf v}_2)$. The parallel and perpendicular separations are then

$$\begin{equation} \pi \equiv |{\bf s}\,{\bm \cdot}\, {\bf l}|/|{\bf l}|, \qquad {r_p}^2 \equiv {\bf s}\,{\bm \cdot}\, {\bf s} - \pi ^2. \end{equation} \tag{ 1 }$$

To estimate the pair counts expected for unclustered objects while accounting for the complex survey geometry, we generate volume-limited random catalogs with the detailed angular selection function of the samples. For the different galaxy samples, we use random catalogs with 25–300 times as many galaxies, depending on the varying number density and size of the samples. We have verified that increasing the number of random galaxies or replacing the random catalog with another one makes a negligible difference to the measurements. We estimate ξ(r_p, π) using the Landy & Szalay (1993) estimator

$$\begin{equation} \xi (r_p,\pi)=\frac{{\rm DD}-2{\rm DR}+{\rm RR}}{{\rm RR}}, \end{equation} \tag{ 2 }$$

where DD, DR, and RR are the suitably normalized numbers of weighted data–data, data–random, and random–random pairs in each separation bin. We weight the galaxies (real and random) according to the angular selection function; because we are using volume-limited samples, we do not weight by a radial selection function. We also tested the alternative ξ estimators of Hamilton (1993) and Davis & Peebles (1983) and found only small differences in the measurements. See Appendix B for these and other tests of our standard analysis procedures.

To examine the real-space correlation function, we follow standard practice and compute the projected correlation function

$$\begin{equation} w_p(r_p) = 2 \int _0^{\infty } d\pi \, \xi (r_p,\pi). \end{equation} \tag{ 3 }$$

In practice, for most samples we integrate up to π_max = 60 h⁻¹ Mpc, which is large enough to include most correlated pairs and gives a stable result by suppressing noise from distant, uncorrelated pairs. For samples with low outer redshift limits we use π_max = 40 h⁻¹ Mpc (see Tables 1 and 2). We use these π_max values consistently when performing HOD modeling of the clustering results (not including the small residual effects of redshift-space distortions). We use linearly spaced bins in π with widths of 2 h⁻¹ Mpc. Our bins in separation r_p are logarithmically spaced with widths of 0.2 dex. We checked the robustness to binning in r_p and π and found our results to be insensitive to either. The measurements are quoted at the pair-weighted average separation in the bin. We estimate that this separation varies by at most 1% from the r_p for which w_p(r_p) equals the pair-weighted average of w_p in the bin. This corresponds to a change of the same magnitude in w_p, significantly smaller than the statistical errors on the measurements, and an up to 0.5% shift in the best-fit correlation length.

The projected correlation function can be related to the real-space correlation function, ξ(r), by

$$\begin{equation} w_p(r_p) = 2 \int _{r_p}^{\infty } r\, dr\, {\xi (r)}\big(r^2-{r_p}^2\big)^{-1/2} \end{equation} \tag{ 4 }$$

(Davis & Peebles 1983). In particular, for a power law ξ(r) = (r/r₀)^−γ, one obtains

$$\begin{equation} w_p(r_p) = r_p \left(\frac{r_p}{r_0}\right)^{-\gamma } \Gamma \left(\frac{1}{2}\right) \Gamma \left(\frac{\gamma -1}{2}\right)\, \Bigr / \,\Gamma \left(\frac{\gamma }{2}\right), \end{equation} \tag{ 5 }$$

allowing one to infer the best-fit power law for ξ(r) from w_p. Alternatively, one can invert w_p to get ξ(r) independent of the power-law assumption. Here, however, we focus on w_p itself, as this is the statistic measured directly from the data that is determined by the real-space correlation function. We note that Equation (5) strictly holds only in the limit of integrating to infinity to obtain w_p. For most of our measurements used in the fits, however, r_p ≲ π_max/4, and this has a minimal effect. In this paper, we focus on HOD modeling of the measurements (using the finite π_max values consistently) and provide power-law fits only as qualitative guidelines, but see Coil et al. (2008) for a possible way to modify the power-law fitting.

We estimate statistical errors on our different measurements using jackknife resampling, as in Z05. We define 144 spatially contiguous subsamples of the full data set, each covering approximately 55 deg² on the sky. Our jackknife samples are then created by omitting each of these subsamples in turn. The error covariance matrix is estimated from the total dispersion among the jackknife samples,

$$\begin{equation} {\rm Covar}(\xi _i,\xi _j) = \frac{N-1}{N} \sum _{l=1}^{N} \big({\xi _i}^l - {\bar{\xi }_i}\big)\big({\xi _j}^l - {\bar{\xi }_j}\big), \end{equation} \tag{ 6 }$$

where N = 144 in our case and $\bar{\xi }_i$ is the mean value of the statistic ξ measured in radial bin i in all the samples (ξ denotes here the statistic at hand, whether it is ξ or w_p). In Z05 we used N = 104 for the smaller sample. The larger value here is chosen to enable better estimation of the full covariance matrix, while still allowing each excluded subvolume to be sufficiently large.

Norberg et al. (2009) have recently studied a variety of error estimators for dark matter correlation functions in N-body simulations, comparing internal methods such as jackknife and bootstrap to external estimates derived from multiple independent catalogs, each comparable in size to our L_* galaxy samples. They find good agreement between jackknife and external estimates for the variance in w_p(r_p) on large scales, with jackknife errors somewhat overestimating the externally derived errors on small scales (r_p ≲ 2 h⁻¹ Mpc). Our own tests of the jackknife method on PTHalos mock catalogs (Scoccimarro & Sheth 2002), described by Zehavi et al. (2002, 2004) and Z05, show that it yields error and covariance estimates similar to those derived from multiple independent catalogs (see Figure 2 in Z05). In principle, covariance matrices derived from large numbers of realistic mock catalogs are preferable because they use larger total volumes and include cosmic variance on scales of the full survey (while jackknife or bootstrap estimates only include cosmic variance on the scale of individual subsamples). However, the tests mentioned above suggest that jackknife estimates are sufficient for our purposes, and they are a far more practical tool when working with many subsamples of different clustering properties, as the mock catalog approach would require a new set of realizations mimicking the clustering signal of each one. The Norberg et al. (2009) tests suggest that parameter uncertainties derived using the jackknife errors should, if anything, be conservative. Because of potential noise or systematics in jackknife estimates of the full covariance matrix, we also present some model fits below that use only diagonal elements.

We interpret the clustering measurements in the HOD framework, which describes the bias between galaxies and mass in terms of the probability distribution P(N|M_h) that a halo of virial mass M_h contains N galaxies of a given type. Our modeling effectively translates galaxy clustering measurements for each class of galaxies into halo occupation functions 〈N(M_h)〉, the mean number of galaxies as a function of halo mass. Other aspects of the HOD—the form of P(N|〈N〉) and the galaxy distribution within halos—are specified by theoretical expectations.

We adopt a spatially flat "concordance" ΛCDM cosmological model with matter density parameter Ω_m = 0.25, and baryon density parameter Ω_b = 0.045, consistent with recent determinations from the cosmic microwave background (CMB; WMAP5; Hinshaw et al. 2009; Dunkley et al. 2009; Komatsu et al. 2009), supernova Ia (Kowalski et al. 2008; Kessler et al. 2009), and baryon acoustic oscillations (Percival et al. 2010). Accordingly, we assume a primordial density power spectrum with fluctuations at 8  h⁻¹ Mpc scale of σ₈ = 0.8. The Hubble constant we use is H₀ = 70 km s⁻¹ Mpc⁻¹, and we assume an inflationary spectral index n_s = 0.95. Lowering Ω_m to 0.25 has only a small effect on our clustering measurements (see Appendix B).

Hydrodynamic simulations show that the most massive galaxy in a halo typically resides at or near the halo center (e.g., Berlind et al. 2003; Simha et al. 2009), in accord with the expectations of semianalytic models (e.g., White & Frenk 1991; Kauffmann et al. 1993; Cole et al. 1994). For HOD parameterization, it is useful to separate the contributions of these central galaxies from those of the additional, satellite galaxies in each halo (Kravtsov et al. 2004; Zheng et al. 2005). For samples of galaxies brighter than a given luminosity, the mean occupation function can be well characterized by a smoothed step function for the central galaxy and a power-law number of satellites increasing with halo mass. We model it in this work using the following form:

$$\begin{eqnarray} \langle N(M_h)\rangle &= & \frac{1}{2} \left[1+ {\rm erf}\left(\frac{\log M_h-\log M_{\rm min}}{\sigma _{\log M}}\right)\right]\nonumber\\ &&\times \left[1+\left(\frac{M_h-M_0}{M_1^\prime }\right)^\alpha \right], \end{eqnarray} \tag{ 7 }$$

where erf is the error function ${\rm erf}(x)=\frac{2}{\sqrt{\pi }} \int _0^{\rm {x}} e^{-t^2} dt$. The mean occupation function of the central galaxies (the left square brackets times the 1/2 factor) is a step-like function with a cutoff profile softened to account for the scatter between galaxy luminosity and halo mass (see also More et al. 2009). The mean occupation of the satellite galaxies (the second term in the right square brackets multiplied by the left square brackets times the 1/2 factor) is a power law modified by a similar cutoff profile. The five free parameters are the mass scale M_min and width σ_log M of the central galaxy mean occupation, and the cutoff mass scale M₀, normalization M'₁, and high-mass slope α of the satellite galaxy mean occupation function.

This specific form is motivated by the theoretical study presented in Zheng et al. (2005) and is identical to the five-parameter model adopted in Zheng et al. (2007; see also Appendix B of Zheng et al. 2009). It is more flexible than the three-parameter model used in Z05, which has the same basic shape. The five-parameter model introduces two additional parameters to characterize the cutoff profiles of central and satellite galaxies, allowing excellent descriptions of the 〈N(M_h)〉 functions predicted by hydrodynamic simulations and semianalytic models (Zheng et al. 2005).

Two characteristic halo masses come into play in the modeling, which set the mass scales of halos that host central galaxies and satellites. M_min characterizes the minimum mass of a halo hosting a central galaxy above the luminosity threshold. The exact definition of M_min can vary between different HOD parameterizations; in the form we adopt (Equation (7)), it is the mass for which half of such halos host galaxies above the luminosity threshold, i.e., 〈N_cen(M_min)〉 = 0.5. It can also be interpreted as the mass of halos in which the median luminosity of central galaxies is equal to the luminosity threshold (see Zheng et al. 2007 for details, except that it was incorrectly labeled as mean rather than median luminosity there). The second characteristic mass scale is M₁, the mass of halos that on average have one additional satellite galaxy above the luminosity threshold, defined by 〈N_sat(M₁)〉 = 1. Note that M₁ is different from M'₁ in Equation (7), though it is obviously related to the values of M'₁ and M₀.

As is common practice, the distributions of the occupation number of central galaxies and satellite galaxies are assumed to follow the nearest-integer and Poisson distributions, respectively, consistent with theoretical predictions (Kravtsov et al. 2004; Zheng et al. 2005). Boylan-Kolchin et al. (2010) have recently argued that the distributions of subhalo counts in high-mass halos become super-Poisson at high 〈N_sat〉, but we expect such a distribution to have minimal quantitative impact on our clustering predictions. The spatial distribution of satellite galaxies within halos is assumed to be the same as that of the dark matter, which follows to a good approximation an Navarro–Frenk–White profile (Navarro et al. 1996). For the halo concentration parameter c(M_h), we adopt the relation given by Bullock et al. (2001), modified to be consistent with our halo definition that the mean density of halos is 200 times the background density: c(M_h) = c₀(M_h/M_nl)^β(1 + z)⁻¹, where c₀ = 11, β = −0.13, and M_nl = 2.26 × 10¹²h⁻¹ M_☉ is the nonlinear mass scale at z = 0.

The two-point correlation function of galaxies in our model is calculated using the method described by Tinker et al. (2005), specifically their "$\bar{n}_g^\prime$–matched" method, which improves the algorithm in Zheng (2004) by incorporating a more accurate treatment of the halo exclusion effect. The method, calibrated and tested using mock catalogs, is accurate to 10% or better. We use the measured values of w_p(r_p) with the full error covariance matrices. We also incorporate the observed number density of galaxies in each subsample as an additional constraint on the HOD model, with an assumed 5% uncertainty. That is, we form the χ² as

$$\begin{equation} \chi ^2= \mathbf {(w_p-w_p^*)^T C^{-1} (w_p-w_p^*)} +(n_g-n_g^*)^2/\sigma _{n_g}^2, \end{equation} \tag{ 8 }$$

where $\mathbf {w_p}$ and n_g are the vectors of the two-point correlation function and the number density of the sample, and $\mathbf {C}$ is the full covariance matrix. The measured values are denoted with a superscript "*".

We implement a Markov Chain Monte Carlo (MCMC) code to explore the HOD parameter space. At each point of the chain, a random walk is taken in the parameter space to generate a new set of HOD parameters. The step-size of the random walk for each parameter is drawn from a Gaussian distribution. The probability to accept the new set of HOD parameters is taken to be 1 if χ²_new ⩽ χ²_old and exp [ − (χ²_new − χ²_old)/2] if χ²_new > χ²_old, where χ²_old and χ²_new are the values of χ² for the old and new models. Flat priors in logarithmic space are adopted for the three parameters related to mass scales, and flat priors in linear space are used for the other two HOD parameters. The length of the chain for each galaxy sample is typically 10,000 and we find convergence by comparing multiple realizations of chains.

Before turning to our observational results, it is worth noting the similarities and differences between HOD modeling and two closely related methods, conditional luminosity functions (CLFs) and subhalo abundance matching (SHAM). Each well-defined class of galaxies, e.g., a luminosity-bin or luminosity-threshold sample, has its own HOD. A CLF (Yang et al. 2003) provides a global model of the full galaxy population, specifying the luminosity function at each halo mass. An HOD can be calculated from a CLF by integrating the latter over luminosity, and a CLF can be calculated from a series of HODs by smoothed differentiation. The virtue of the CLF is its completeness, but when fitting data it typically requires stronger prior assumptions, such as a functional form for the luminosity function itself and functional forms for the dependence of luminosity function parameters on halo mass. By contrast, the five-parameter HOD model used here is already flexible enough to provide a near-perfect description of theoretical model predictions for luminosity-threshold samples (Zheng et al. 2005). The SHAM method (Conroy et al. 2006; Vale & Ostriker 2006) assumes a monotonic relation between the luminosity or stellar mass of a galaxy and the mass or circular velocity of its parent halo or subhalo; the method can be generalized to allow scatter in this relation. While an HOD model takes the space density and clustering of a galaxy population as input for parameter fits, a SHAM model takes only the space density as input and predicts the clustering, effectively using a theoretical prior to specify the satellite occupation function. SHAM models are remarkably successful at matching the Z05 correlation functions of luminosity-threshold samples (Conroy et al. 2006).

The correlation functions of sub-L_* galaxies at low redshift are typically well described by power laws on scales r_p ≲ 10 h⁻¹ Mpc (Totsuji & Kihara 1969; Peebles 1974; Gott & Turner 1979), but deviations from a power law become progressively stronger for brighter galaxies (Zehavi et al. 2004; Z05; this paper) and at higher redshifts (Conroy et al. 2006 and references therein). In the context of HOD models, Watson et al. (2011) discuss the physical processes that lead both to an approximate power-law correlation function and to deviations from a power law (see also Benson et al. 2000; Berlind & Weinberg 2002; and Appendix A of Zheng et al. 2009).

We study the clustering dependence on luminosity using sets of volume-limited samples constructed from the full SDSS sample, corresponding to different luminosity bins and thresholds. Details of the individual samples are given in Tables 1 and 2 and illustrated in Figure 4.

As a representative case, we show in Figure 5 the two-dimensional correlation function, ξ(r_p, π), as a function of separations perpendicular, r_p, and parallel, π, to the line of sight, calculated for the −20 < M_r < −19 sample. In the absence of redshift-space distortions the contours would have been isotropic, a function only of total separation ($\sqrt{{r_p}^2+{\pi }^2}$). Redshift-space distortions enter in the line-of-sight direction and are clearly evident in the plot. For small projected separations, the contours are elongated along the line-of-sight direction, reflecting the "fingers-of-God" effect of small-scale virial motions in collapsed objects. On larger scales, we see the compression caused by coherent large-scale streaming into overdense regions and out of underdense regions (Sargent & Turner 1977; Kaiser 1987).

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We isolate real-space correlations by calculating the projected correlation function, w_p(r_p), according to Equation (3). Figure 6 shows the projected correlation functions obtained for the volume-limited samples defined by luminosity bins and by luminosity thresholds. For the luminosity bins, we find a pronounced dependence of clustering on luminosity for the bright samples, with the more luminous galaxies exhibiting higher clustering amplitudes. The dependence on luminosity is more subtle for the fainter luminosity-bin samples, with little change for scales r_p < 2 h⁻¹ Mpc. Behavior for the luminosity thresholds is similar, with nearly identical correlation functions for the M_r < −18.5 and < − 19.5 samples, a slow but significant increase in clustering strength moving to M_r < −20.5 and M_r < −21.0, then a rapid increase going to M_r < −21.5 and M_r < −22.0. Measurements for the luminosity-threshold samples are less noisy, and one can see that the shapes of w_p(r_p) are similar for all samples at r_p ≳ 3 h⁻¹ Mpc, while the brighter samples exhibit a stronger inflection in w_p(r_p) at r_p ≈ 2 h⁻¹ Mpc and a steeper correlation function at smaller scales. The error covariance matrices exhibit significant correlation between the measurements on different scales, particularly for the relatively faint, smaller volume, galaxy samples. Similar behavior is found by McBride et al. (2011). Power-law fits for these clustering measurements, using the measured data points for r_p < 20 h⁻¹ Mpc, are presented in Tables 1 and 2. We include fits computed both with and without the off-diagonal terms in the covariance matrix (i.e., setting the off-diagonal terms to zero). We caution that when the power-law fit is an inadequate description of the data, as indicated by large χ²/dof values, the r₀ and γ uncertainties have limited meaning—fitting over different ranges of the data could produce different values of the fit parameters. Tests in Appendix B on the M_r < −21 sample show that changes to our analysis procedures cause r₀ and γ changes that are smaller than (or at most comparable to) the statistical error bars.

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At large scales, we expect the real-space galaxy correlation function to be a scale-independent multiple of the dark matter correlation function ξ_gg(r) = b²_gξ_mm(r), where the bias factor b_g will differ from one class of galaxies to another. For each sample, we calculate the best-fitting bias factor of the measured w_p(r_p) with respect to the z = 0.1 dark matter w_p(r_p) over the separation range 4 h⁻¹ Mpc < r_p < 30 h⁻¹ Mpc, using the full error covariance matrix of the measurements. (The average of r_p weighted by the inverse of the uncertainty in w_p corresponds to a separation of ∼8 h⁻¹ Mpc, which can be regarded as a rough estimate of the effective radius for this fit.) The w_p(r_p) predicted for the nonlinear matter distribution of our ΛCDM cosmological model is computed from the nonlinear power spectrum with the method of Smith et al. 2003. Filled circles in Figure 7 show these bias factors b_g(L) and b_g(> L) for our luminosity-bin and luminosity-threshold samples. We refer to these below as "DM-ratio" bias factors. In Z05, we defined bias factors via the value of w_p(r_p) at one representative separation, r_p = 2.67 h⁻¹ Mpc. Open triangles in Figure 7 show this measurement of "single-r_p" bias factors for comparison. For the −21 < M_r < −20 bin and the M_r < −20 and M_r < −19.5 thresholds, we use the samples with cz_max = 19, 250 km s⁻¹ (see Tables 1 and 2), for the reasons discussed in Section 3.2 below. Because b_g(L) changes so rapidly between M_r = −21 and M_r = −22, we have also divided the −22 < M_r < −21 bin into two half-magnitude bins and computed bias factors separately for each. The open circles, discussed further in Section 3.3, show large-scale bias factors derived from HOD model fits to the full projected correlation functions ("HOD bias factors"; computed at the mean redshift of each sample).

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In agreement with previous studies (Norberg et al. 2001; Tegmark et al. 2004; Z05), b_g(L) is nearly flat for luminosities L ⩽ L_*, then rises sharply at brighter luminosities.¹⁶ Dotted and dashed curves in the left panel show the empirical fits to b_g(L)/b_g(L_*) proposed by Norberg et al. (2001) and Tegmark et al. (2004), respectively, where we take as b_g(L_*) the "DM-ratio" bias factor estimated for the −21 < M_r < −20 luminosity bin using the large-scale w_p(r_p) ratio. The Norberg et al. (2001) form appears to fit our measurements better, but the differences between the curves only become large for the −18.0 < M_r < −19.0 sample, where the single-r_p and DM-ratio bias factors differ noticeably, and where the tests discussed in Section 3.2 below suggest that cosmic variance fluctuations are still significant. The HOD bias factors are in good agreement with the "DM-ratio" ones.

The luminosity-threshold samples allow more precise bias measurements, and they avoid binning effects that can influence the estimates of b(L) when it changes rapidly across a bin. The HOD and DM-ratio values of b_g(> L) agree well for all luminosity-threshold samples except M_r < −18.0, where the HOD fit overpredicts the large-scale w_p(r_p) measurements (see Figure 10 below). The HOD bias points are fit to 3% or better by the functional form

$$\begin{equation} b_g({>}L) \times (\sigma _8/0.8) = 1.06+0.21(L/L_*)^{1.12}, \end{equation} \tag{ 9 }$$

where L is the r-band luminosity corrected to z = 0.1 and L_* corresponds to M_r = −20.44 (Blanton et al. 2003c). Except for the M_r < −18 point, this formula also accurately describes the DM-ratio bias factors. The HOD and DM-ratio bias factors scale as σ⁻¹₈ to a near-perfect approximation, since at large scales ξ_gg = b²_gξ_mm∝b²_gσ²₈. We consider Equation (9) to be our most robust estimate of the dependence of large-scale bias on galaxy luminosity, applicable over the range 0.16 L_* < L < 6.3 L_* (−22.5 < M_r < −18.5). Fitting the DM-ratio bias values for the luminosity-bin samples yields

$$\begin{equation} b_g(L) \times (\sigma _8/0.8) = 0.97+0.17(L/L_*)^{1.04}, \end{equation} \tag{ 10 }$$

which is close to the formula derived by Norberg et al. (2001) for b_J-selected galaxies, but has a slightly steeper rise at high luminosities.

Our volume-limited, luminosity-bin samples span different ranges in redshift (specified in Table 1), with intrinsically brighter galaxies observed over larger volumes. It is thus important to test for the robustness of the detected luminosity dependence to "cosmic variance" of the structure in these different volumes. (We follow common practice in referring to these finite-volume effects as cosmic variance, though a more precise term would be "sample variance"; Scott et al. 1994.) Figure 8 compares projected correlation functions of adjacent luminosity bins when using their respective full volume-limited redshift range (points with error bars) and when restricting both to their common overlap range (lines). The overlap volume is similar to the full volume of the fainter sample (differing only because of the r > 14.5 bright limit), so the filled points and solid lines are usually in close agreement.

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The most significant cosmic variance effect on the measurements appears to be due to the Sloan Great Wall (SGW), a huge supercluster at z ∼ 0.08, which is the largest coherent structure detected in the SDSS (Figure 2; see also Gott et al. 2005). Its distance places it right at the edge of the −21 < M_r < −20 sample (see Table 1). Its exclusion from this sample when limiting to the overlap range with the −20 < M_r < −19 sample causes the decrease in clustering amplitude on large scales seen in the bottom left panel. Hence, this structure also causes the flattening in the projected correlation function of this sample at large separations seen in Figure 6. Conversely, restricting the brighter −22 < M_r < −21 sample to the smaller overlap range accentuates the supercluster's dominance and gives rise to the increased clustering seen in the top right panel (dashed line). Taken together, these results strongly suggest that the most reliable estimate of w_p(r_p) for the −21 < M_r < −20 luminosity bin comes from the SGW-excluded sample rather than the full sample. Brighter, larger volume samples are much less affected by the SGW, while fainter samples do not extend as far and are thus not affected. These results are very similar to those of an identical test performed with the smaller samples of Z05.

We have performed similar tests with the luminosity-threshold samples, and we find an analogous effect of the SGW, mostly for the M_r < −20 sample and, to a lesser degree, for the M_r < −19.5 sample. We have also done tests where we have excluded specifically the SGW region with angular and redshift cuts, confirming its significant impact on the large-scale clustering measurement for these samples. It is striking that even with the full SDSS sample, the effect of the SGW is still significant, and one should use caution in interpreting clustering measurements for relatively large separations (r_p > 5 h⁻¹ Mpc) for the few specific samples whose redshift range extends just up to (and including) this structure. For this reason, in Tables 1 and 2, we also provide power-law fits of these samples when restricted to a redshift limit that excludes the SGW (cz_max = 19, 250 km s⁻¹, the same limiting redshift as for the −20 < M_r < −19 and the M_r < −19 samples). The HOD parameters derived for these samples are relatively insensitive to the choice of sample volume, and the correlation functions corresponding to these HOD models differ much less than the power-law fits (see, e.g., Figure 15 in Z05). This insensitivity reflects the constrained nature of HOD fits for a specified cosmological model: there is little freedom within these fits to adjust the large-scale correlation amplitude relative to the more robust measurements at r_p < 2 h⁻¹ Mpc, so the HOD modeling just accepts the χ² penalty of missing the large-scale data points.

Other large structures in the survey volume may affect the clustering measurements in more subtle ways. McBride et al. (2011) investigated the impact of such structures on the three-point correlation function by analyzing the residuals of individual jackknife samples. The two-point correlation function is much less sensitive to such effects, with individual deviations at the few percent level at most scales (C. K. McBride 2011, private communication). Of course, it is the cumulative impact of these residuals that determines the jackknife error bars, so the fact that they exist does not in itself imply that errors are larger than our estimates. The tests of Z05 and Norberg et al. (2009) suggest that jackknife error estimates for w_p(r_p) are typically accurate or conservative (see Section 2.2). An investigation of the individual residuals also suggests that the jackknife errors follow a Gaussian distribution to a fair approximation, but with some noticeable skewness (C. K. McBride 2011, private communication; see also Norberg et al. 2011). However, we find only a small difference between measuring w_p(r_p) from the full data sample and taking the median of the jackknife w_p(r_p) values in each separation bin, which suggests that any non-Gaussianity will have minimal impact on our clustering results.

Additional cosmic variance concerns have to do with the relatively small volumes associated with the lowest luminosity samples we consider. Figure 8 can shed some light on this issue as well. Specifically, the bottom-right panel checks the sensitivity to the volume probed in the correlation functions measured for the −19 < M_r < −18 and −20 < M_r < −19 luminosity bins. The general agreement of the curves (calculated in the overlap volume) to the respective sets of points (calculated for the full volume-limited sample), within the measured uncertainties, is reassuring. We perform similar tests with the magnitude-threshold samples, looking at the robustness of the clustering measurements of different samples when limiting to the inner small volumes associated with the fainter thresholds. Here, we find substantial volume effects when considering the volume of the faintest threshold sample M_r < −18, but very weakened effects for the M_r < −18.5 and brighter thresholds (see Figure 9). We are unsure why the effects for the M_r < −18.0 sample appear larger than those for the −19.0 < M_r < −18.0 luminosity-bin sample, as they have the same outer redshift limit; however, the galaxy populations are different in the two cases, with the luminosity-threshold sample including also brighter galaxies. We find these finite-volume effects to be smaller than those found in the earlier samples of Z05, due to the larger sky coverage of SDSS DR7.

The comparisons of solid and dashed curves in Figure 8 provide the fairest test of luminosity dependence between the samples, as the effects of cosmic variance are essentially removed by matching volumes. For the fainter samples shown in the lower panels, the evidence for luminosity dependence is marginal relative to the error bars. The detection is stronger in the upper right panel and overwhelming for the brightest galaxies in the upper left. The difference between the dashed line and the open points in this panel is plausibly explained by the small sample (∼2600 galaxies) of −23 < M_r < −22 galaxies in the overlap volume: the larger volume of the full sample is required to give a robust measurement of large-scale clustering for these rare galaxies. These conclusions—evidence for increased clustering at M_r ≈ −21.5 and dramatically increased clustering at M_r ≈ −22.5—are consistent with the b(L) data points in Figure 7.

To investigate further the implications of the luminosity-dependent clustering, we turn to HOD modeling. We find the best-fit HOD models for our set of volume-limited luminosity-threshold samples, using the five-parameter model described in Section 2.3. Figure 10 shows the HOD best fits to the projected correlation functions (staggered by 0.25 dex for clarity). Here, we use the full volume-limited samples, with no attempt to remove the SGW. The values of the fitted parameters, inferred using the full error covariance matrix, are given in Table 3. We also list f_sat, the fraction of sample galaxies that are satellites from the HOD modeling results. We see that the HOD models provide reasonable fits to the projected correlation functions, with deviations from a power law more apparent for the brighter samples. The characteristic inflections in w_p(r_p) at r_p = 1–2 h⁻¹ Mpc arise at the transition from the small-scale, one-halo regime, where most correlated pairs come from galaxies in the same halo, to the large-scale, two-halo regime, where the shape of ξ(r) approximately traces the shape of the matter correlation function (Berlind & Weinberg 2002; Zehavi et al. 2004). The χ² values for these fits are also specified in the table and can be compared to the corresponding values for the power-law fits (Table 2). In all cases the HOD model has a better goodness of fit than the best-fit power-law model. We note, however, that the χ² values still tend to be somewhat large, particularly for the bright samples. These might reflect uncertainties in the jackknife error covariance estimate, residual systematics or a limitation of the restricted HOD model. For the M_r < −18.0 sample, the HOD model overpredicts the amplitude of w_p(r_p) at large scales, but the tests in Figure 9 suggest that the large-scale clustering of this sample is significantly affected by the small sample volume.

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Table 3. HOD and Derived Parameters for Luminosity-threshold Samples

Mmaxr	log Mmin	σlog M	log M0	log M'1	α	log M1	fsat	bg	$\frac{\chi ^2}{{\rm dof}}$
−22.0	14.06 ± 0.06	0.71 ± 0.07	13.72 ± 0.53	14.80 ± 0.08	1.35 ± 0.49	14.85 ± 0.04	0.04 ± 0.01	2.16 ± 0.05	1.8
−21.5	13.38 ± 0.07	0.69 ± 0.08	13.35 ± 0.21	14.20 ± 0.07	1.09 ± 0.17	14.29 ± 0.04	0.09 ± 0.01	1.67 ± 0.03	2.3
−21.0	12.78 ± 0.10	0.68 ± 0.15	12.71 ± 0.26	13.76 ± 0.05	1.15 ± 0.06	13.80 ± 0.03	0.15 ± 0.01	1.40 ± 0.03	3.1
−20.5	12.14 ± 0.03	0.17 ± 0.15	11.62 ± 0.72	13.43 ± 0.04	1.15 ± 0.03	13.44 ± 0.03	0.20 ± 0.01	1.29 ± 0.01	2.7
−20.0	11.83 ± 0.03	0.25 ± 0.11	12.35 ± 0.24	12.98 ± 0.07	1.00 ± 0.05	13.08 ± 0.03	0.22 ± 0.01	1.20 ± 0.01	2.1
−19.5	11.57 ± 0.04	0.17 ± 0.13	12.23 ± 0.17	12.75 ± 0.07	0.99 ± 0.04	12.87 ± 0.03	0.23 ± 0.01	1.14 ± 0.01	1.0
−19.0	11.45 ± 0.04	0.19 ± 0.13	9.77 ± 1.41	12.63 ± 0.04	1.02 ± 0.02	12.64 ± 0.04	0.33 ± 0.01	1.12 ± 0.01	1.8
−18.5	11.33 ± 0.07	0.26 ± 0.21	8.99 ± 1.33	12.50 ± 0.04	1.02 ± 0.03	12.51 ± 0.04	0.34 ± 0.02	1.09 ± 0.01	0.9
−18.0	11.18 ± 0.04	0.19 ± 0.17	9.81 ± 0.62	12.42 ± 0.05	1.04 ± 0.04	12.43 ± 0.05	0.32 ± 0.02	1.07 ± 0.01	1.4

Notes. See Equation (7) for the HOD parameterization. Halo mass is in units of h⁻¹ M_☉. Error bars on the HOD parameters correspond to 1σ, derived from the marginalized distributions. M₁, f_sat, and b_g are derived parameters from the fits. M₁ is the mass scale of a halo that can on average host one satellite galaxy above the luminosity threshold and f_sat is the fraction of satellite galaxies in the sample. b_g is the large-scale galaxy bias factor and is degenerated with the amplitude of matter clustering σ₈, so that this is in fact b_g × (σ₈/0.8). A 2% systematic shift in the w_p values would correspond to a 1% change in b_g, effectively doubling the tiny error bars on it. For all samples, the number of degrees of freedom (dof) is 9 (13 measured w_p values plus the number density minus the five fitted parameters). The parameters of the sharp-cutoff models plotted in Figure 10 for the six fainter samples (see the text) are specified hereby as (M^max_r, log M_min, σ_log M, log M₀, log M'₁, α): (−18.0, 11.14, 0.02, 9.84, 12.40, 1.04); (−18.5, 11.29, 0.03, 9.64, 12.48, 1.01); (−19.0, 11.44, 0.01, 10.31, 12.64, 1.03); (−19.5, 11.56, 0.003, 12.15, 12.79, 1.01); (−20.0, 11.78, 0.02, 12.32, 12.98, 1.01); (−20.5, 12.11, 0.01, 11.86, 13.41, 1.13).

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The high-mass slope α of the satellite mean occupation function is around unity for most samples. For the brightest sample (M_r < −22.0), α is noticeably higher than unity, but with large error bars. The right panel of Figure 10 presents the halo occupation functions themselves. When going toward brighter samples, the main effect is a shift of the halo occupation function toward higher halo masses, a shift that affects both the central galaxy cutoff and the satellite occupation. More luminous galaxies occupy more massive halos, which leads to their stronger clustering. For the six fainter samples, there are models with sharp central galaxy cutoffs (σ_log M = 0) that have Δχ² < 1 compared to the best-fit model; we have chosen to plot these sharp-cutoff 〈N(M_h)〉 curves in Figure 10. For the M_r < −21.0, M_r < −21.5, and M_r < −22.0 samples, however, a non-zero value of σ_log M, indicating scatter between halo mass and central galaxy luminosity, is required to simultaneously fit the galaxy number density and projected correlation function. A sharper cutoff would predict an excessive clustering amplitude for the measured number density because of the rising b(M_h) relation. Figure 11 illustrates the level of statistical uncertainty in the HOD fits, plotting 〈N(M_h)〉 for ten models randomly chosen from the MCMC chain that have Δχ² < 1 relative to the best-fit model for each of three luminosity thresholds. The cutoff profile is generally better constrained for brighter samples because of the steeper form of b(M_h) at high M_h. The satellite occupations are tightly constrained in all cases (other than a relatively large scatter in M₀ for the M_r < −20.5 sample).

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Figure 12(a) shows the two characteristic halo mass parameters M_min and M₁ (see Section 2.3) as a function of the threshold luminosity. Both halo mass scales increase with the sample's threshold luminosity, with a steeper dependence for brighter galaxies. Because central galaxies dominate the total number density for any luminosity threshold (Zheng et al. 2005), the approximate form of the M_min curve follows simply from matching the space densities of galaxies and halos (e.g., Conroy et al. 2006; Vale & Ostriker 2006). In our HOD parameterization, M_min can be interpreted as the mass of halos in which the median luminosity of central galaxies is equal to the threshold luminosity. We propose the following form for the relation between median central galaxy luminosity L_cen and halo mass M_h (see also Kim et al. 2008),

$$\begin{equation} L_{\rm cen}/L_* =A\left(\frac{M_h}{M_t}\right)^{\alpha _M}\exp \left(-\frac{M_t}{M_h}+1\right), \end{equation} \tag{ 11 }$$

where A, M_t, and α_M are three free parameters. That is, the median central galaxy luminosity has a power-law dependence on halo mass at the high-mass end (with a power-law index of α_M) and drops exponentially at the low-mass end. The transition halo mass is characterized by M_t, and the normalization factor A is the median luminosity of central galaxies (in units of L_* = 1.20 × 10¹⁰h⁻² L_☉ in the r band; Blanton et al. 2003c) in halos of transition mass. The fit (solid curve) shown in Figure 12(a) has A = 0.32, M_t = 3.08 × 10¹¹h⁻¹ M_☉, and α_M = 0.264. Figure 12(b) shows the M_h/L_cen ratio as a function of halo mass. The solid curve is derived from the fit in Figure 12(a):

$$\begin{equation} \frac{M_h}{L_{\rm cen}}= \left(\frac{M_h}{L_{\rm cen}}\right)_{M_t} \left(\frac{M_h}{M_t}\right)^{1-\alpha _M} \exp \left(\frac{M_t}{M_h}-1\right), \end{equation} \tag{ 12 }$$

where $(M_h/L_{\rm cen})_{M_t}=80h\,M_\odot /L_\odot$ is the mass-to-light ratio in halos of transition mass. The transition mass (times 1/(1 − α_M), to be exact) also marks the scale at which M_h/L_cen reaches a minimum. Halos of M_h ≈ 4.2 × 10¹¹ h⁻¹ M_☉ are maximally efficient at converting their available baryons into r-band light of their central galaxy. Other authors have reached a similar conclusion using HOD, CLF, or SHAM methods (e.g., Yang et al. 2003; Tinker et al. 2005; Vale & Ostriker 2006; Zheng et al. 2007; Kim et al. 2008; Guo et al. 2010; Moster et al. 2010).

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In Figure 12(a), the sharp upturn in M_h (M_min) arises because the galaxy luminosity function drops exponentially in a regime where the halo mass function remains close to a power law. The sharp rise in b(L) (Figure 7) is driven both by this upturn in M_h (M_min) and by the steepening of the b(M_h) relation itself (Mo & White 1996; Jing 1998; Sheth et al. 2001; Tinker et al. 2010). As discussed by Zheng et al. (2009, Appendix A), the greater departures from a power law w_p(r_p) evident for brighter galaxies arise mainly because M_min and M₁ are larger than the characteristic halo mass M*_h where the halo mass function begins to drop exponentially; this change in the halo mass function shape leads to a sharper transition between the one-halo and two-halo regimes of the correlation function.

There is a considerable gap between the values of M_min and M₁ at all luminosities. As in earlier works, we find an approximate scaling relation of M₁ ≈ 17M_min, implying that a halo hosting two galaxies (one central galaxy and one satellite) above the luminosity threshold has to be about 17 times more massive on average than a halo hosting only one (central) galaxy above the luminosity threshold. Halos in this "hosting gap" mass range tend to host more luminous (higher mass) central galaxies rather than multiple galaxies, consistent with the predictions of Berlind et al. (2003) based on hydrodynamic simulations and semianalytic models. As can be seen in Figure 12(a), this scaling factor is somewhat smaller at the high-luminosity end, corresponding to massive halos that host rich groups or clusters. This latter trend likely reflects the relatively late formation of these massive halos, which leaves less time for satellites to merge onto central galaxies and thus lowers the satellite threshold M₁. Physical effects that shape the M₁/M_min relation are discussed by Zentner et al. (2005) using analytic descriptions of halo and galaxy merger rates.

Our results are consistent with previous measurements of these trends (Z05; Zheng et al. 2007). Z05 found a slightly larger scale factor of ≈23, likely because of slight differences in the HOD parameterizations and the corresponding definitions of the halo mass scales. Zheng et al. (2007) found, for that same early SDSS sample but using the current HOD model, M₁ ≈ 18M_min, in excellent agreement with our results for the final SDSS sample. These results are also in agreement with predictions of galaxy formation models. In particular, the scale factor in the M_min − M₁ scaling relation is in good agreement with the predictions presented by Zheng et al. (2005).

Returning to Figure 7, open circles in the right-hand panel show the values of b_g(> L) corresponding to our best-fit HOD models, i.e., the asymptotic bias on large scales where the bias factor is scale independent. These bias estimates necessarily depend on the assumptions associated with our HOD modeling, principally that 〈N(M_h)〉 has the form defined by Equation (7) and that 〈N(M_h)〉 is independent of a halo's large-scale environment (no "assembly bias"). These assumptions allow us to use constraints from smaller scale clustering and the galaxy number density, greatly reducing the error bars on b_g(> L) and reducing the sensitivity to cosmic variance in the large-scale clustering. Given the significant finite-volume variations that remain even in SDSS DR7 (Section 3.2), we consider these HOD-based b_g(> L) values to be our most robust estimates of the luminosity dependence of galaxy bias, despite their dependence on an assumed model.

To obtain HOD-based bias factors for luminosity-bin samples, we have taken the central and satellite occupation functions for each bin to be simply the difference of the occupation functions for the bracketing threshold samples, yielding the open circles in the left panel of Figure 7. The right panel of Figure 13 shows the resulting mean occupation functions, and the left panel compares the w_p(r_p) predicted by these threshold-difference HODs to the observed w_p(r_p) from Figure 6. After fitting the luminosity-threshold correlation functions, there are no parameter adjustments made to fit the luminosity-bin data. The agreement is generally good. For the faintest luminosity bin −19 < M_r < −18 on large scales, the model slightly overpredicts the observed w_p(r_p) (with χ² = 20 for 13 data points). This tension could indicate that our HOD model does not allow a good description of galaxies in this luminosity range, or it could be that our jackknife method underestimates the cosmic variance uncertainties for this small-volume sample. We have already noted that the HOD model of the bracketing M_r < −18.0 sample overpredicts its observed large-scale clustering (Figure 10), and that this sample appears to have significant finite-volume effects (Figure 9). We revisit this overprediction in Section 4.5 below, where we separately examine the clustering of red and blue galaxies in this luminosity bin.

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The clustering of galaxies is known to depend on other galaxy properties in addition to luminosity, such as color, surface brightness, and profile shapes. These latter quantities are correlated with each other and produce similar trends in w_p(r_p) (e.g., Zehavi et al. 2002). Moreover, galaxy luminosity and color have been shown to be the two properties most predictive of galaxy environment (Blanton et al. 2005a), such that any residual dependence on morphology or surface brightness at fixed luminosity and color is weak. We focus in this section on the color dependence of galaxy clustering using our luminosity-bin samples. Figure 2 qualitatively illustrated the clustering differences between blue and red galaxies. To study this difference quantitatively, we divide our sample into "blue" and "red" galaxies according to the well-known color bimodality in the color–magnitude plane (e.g., Strateva et al. 2001; Baldry et al. 2004). Following Z05, we use a magnitude-dependent color cut defined by

$$\begin{equation} (g-r)_{\rm cut} = 0.21 - 0.03 M_r. \end{equation} \tag{ 13 }$$

This tilted cut, shown below in Figure 18, appropriately separates the red E/S0 ridgeline from the blue cloud, following the division into two populations as a function of luminosity. An identical color cut is used by Swanson et al. (2008) and McBride et al. (2011), while other works (e.g., Blanton & Berlind 2007; Skibba & Sheth 2009) use a very slightly modified division. Our results are not sensitive to the exact choice of the cut.

While color most directly measures star formation history, it can also be viewed as a proxy of morphology, where blue galaxies are mostly spirals and red galaxies tend to be spheroid dominated. (The two classification schemes are certainly not identical, however; see, e.g., Choi et al. 2007; Bamford et al. 2009; Blanton & Moustakas 2009; Skibba et al. 2009.) Figure 14 shows ξ(r_p, π) separately for blue and red galaxies, for a representative case of the −20 < M_r < −19 volume-limited sample. The difference between the two populations is striking. The red galaxies exhibit a substantially higher clustering amplitude and much stronger finger-of-God distortions on small scales, as seen in the elongation along the π direction for small r_p separations. These differences reflect the expected color–density relation, with red galaxies residing in more massive halos that have a stronger bias and higher velocity dispersions. The large-scale coherent distortion is more apparent in the blue sample. In linear theory, the coherent distortion depends on the parameter β ≈ Ω^0.6_m/b (Kaiser 1987; Hamilton 1998), as a lower bias implies a larger gravitational perturbation for a given galaxy overdensity. The blue galaxies are better tracers of the "field" and are less biased and thus exhibit a stronger large-scale compression and only a weak finger-of-God distortion. The ξ(r_p, π) diagram of the full −20 < M_r < −19 sample (Figure 5) is, of course, intermediate between these two. These results are in qualitative agreement with previous SDSS measurements (Z05) and with DEEP2 measurements at z ∼ 1 (Coil et al. 2008).

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Figure 15 shows the corresponding projected correlation functions, w_p(r_p). The correlation function for the red sample has a higher amplitude and steeper slope than the blue sample. Fitting power laws results in a correlation length of r₀ = 6.63 ± 0.41 h⁻¹ Mpc and slope γ = 1.94 ± 0.03 for the red galaxies versus r₀ = 3.62 ± 0.15 h⁻¹ Mpc and γ = 1.66 ± 0.03 for the blue (see Table 4). These trends are similar for all the luminosity samples, but the differences in clustering are weaker with increasing luminosity, as is shown in Section 4.3.

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The autocorrelation functions of red and blue galaxies separately do not include blue–red galaxy pairs. Another useful measurement is then the cross-correlation between blue and red galaxies. In the large-scale, linear bias approximation, where δ_red = b_redδ_m and δ_blue = b_blueδ_m, the cross-correlation must be the geometric mean of the autocorrelations. To the extent that halos have correlation coefficient $r \equiv \xi _{hm}/\sqrt{\xi _{mm}\xi _{hh}}=1$ with the matter distribution, the geometric mean result should hold throughout the two-halo regime, even if the halo bias is scale dependent and the matter field is nonlinear. On small scales, in the one-halo regime, the cross-correlation encodes information on the mixing of galaxy populations within the halos. Any tendency of red or blue galaxies to segregate from one another will be reflected as a deviation of the cross-correlation function from the geometric mean. For example, if some halos contained only red galaxies while other halos contained only blue galaxies, this would depress the number of one-halo pairs and push the cross-correlation function below the geometric mean. However, the prediction of the cross-correlation function has a number of subtleties; we discuss these issues and provide some simplified estimates in Appendix A. We also show in this appendix that the cross-correlation function of two galaxy populations is mathematically determined if one knows the autocorrelation of the individual populations and of the combined population; nonetheless, the cross-correlation presents this implicit information in a more intuitive form.

We measure the cross-correlation function of the blue and red galaxy samples in an analogous way to the autocorrelations, using the Landy–Szalay estimator. Specifically, we use Equation (2) with D₁D₂ replacing DD, R₁R₂ replacing RR, and D₁R₂ + D₂R₁ replacing 2DR, with the subscripts denoting the two cross-correlated subsamples. Error bars are obtained similarly via jackknife resampling. Filled green circles in Figure 15 show the resulting cross-correlation function for the −20 < M_r < −19 sample. On large scales, as expected, we find that the cross-correlation result follows the geometric mean of the blue and red autocorrelations. On small scales (for r_p ≲ 2 h⁻¹ Mpc) we find that the cross-correlation falls below the geometric mean, possibly indicating a slight segregation of blue and red galaxies within the halos. This deviation is significant given the small error bars on these scales. Note, however, that this is very far from suggesting a full segregation into "red halos" and "blue halos." That extreme case would lead to no one-halo contribution at all, making the projected cross-correlation approximately flat for r_p < 2 h⁻¹ Mpc.

We find similar behavior for the cross-correlation of red and blue galaxies in all of our luminosity subsamples. However, the depression of the cross-correlation below the geometric mean is stronger for the relatively faint samples and smaller for brighter galaxies (consistent with Z05, who showed the cross-correlation function for an M_r < −21 galaxy sample). Our results are also in agreement with Wang et al. (2007), who investigated in detail the cross-correlation between galaxies of different luminosities and color using an earlier SDSS sample, and with Ross & Brunner (2009), who measured angular clustering of an SDSS photometric sample. A similar depression of the cross-correlation below the geometric mean is observed by Coil et al. (2008).

Using an SDSS group catalog, Weinmann et al. (2006) find that the colors of satellite galaxies are correlated with those of their central galaxy. However, this trend, which they termed "galactic conformity," is found to have roughly the same strength independent of luminosity, so its connection to our findings is unclear. It is known that the fraction of red galaxies that are satellites becomes larger with decreasing luminosity (e.g., Z05; see also related discussions in the following subsections). Thus, the luminosity-dependent suppression of the cross-correlation function in the one-halo regime may be simply related to the relative paucity of blue galaxies compared to red ones within large halos (see also van den Bosch et al. 2008b; Hansen et al. 2009). In future work, we will model the cross-correlation results with HOD in detail, and study the implication of these measurements for the distribution of red and blue galaxies within dark matter halos.

We now turn to the luminosity dependence of clustering within the red and blue galaxy populations individually, using the luminosity-dependent color division of Equation (13). Figure 16 shows projected correlation functions for the volume-limited luminosity-bin samples, separately for the red (left panel) and blue (right panel) galaxies. Figure 17 shows the correlation length r₀ and slope γ of power-law fits to these samples. Because some of the samples are quite small, making jackknife estimates of the covariance matrix noisy, we fit using the diagonal error bars only, which is enough to capture the trends visible in the w_p(r_p) plots. Figure 17 also shows r₀ and γ from diagonal fits to the full luminosity-bin samples. The differences between the different color samples are particularly distinct for the fainter samples, and they decrease with increasing luminosity.

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These plots display the same general trends seen in previous sections: the large-scale clustering amplitude increases with luminosity for both red and blue populations, and red galaxies generically have higher clustering amplitude and a steeper correlation function. Within the individual populations, however, the luminosity trends are remarkably different. The projected correlation functions of the blue galaxies are all roughly parallel, with slopes 1.6 ⩽ γ ⩽ 1.8, and the amplitude (or correlation length) increases steadily with luminosity. For the red galaxies, on the other hand, the shape of w_p(r_p) is radically different for the two faintest samples, −18 < M_r < −17 and −19 < M_r < −18, with a strong inflection at r_p ≈ 3 h⁻¹ Mpc indicating a high-amplitude one-halo term. These two samples have the strongest small-scale clustering, matched only by the ultraluminous, −23 < M_r < −22 galaxies. The large-scale clustering (at r_p ≈ 5–10 h⁻¹ Mpc) shows no clear luminosity dependence until the sharp jump at the −23 < M_r < −22 bin, though it is consistent with a weak but continuous trend at lower luminosities. Power-law fits yield significantly steeper power-law slopes for the correlation functions of the two faintest samples, together with a mild increase in the correlation lengths (see Figure 17). We caution that the −18 < M_r < −17 sample is very small, containing only about 5000 blue galaxies and 1000 red galaxies, and might be sensitive to cosmic variance; we have not used it in earlier sections but include it here to show the extension of the luminosity trends to the faintest galaxies we can effectively study.

The strong clustering of intrinsically faint red galaxies has been previously observed (Norberg et al. 2002; Hogg et al. 2003; Z05; Swanson et al. 2008; Cresswell & Percival 2009). We build on these studies, confirming this intriguing clustering signal and presenting its most significant measurement obtained with the largest redshift sample available. The red galaxy samples analyzed here include ∼25,000 galaxies below L_*, about 6000 of them in the two faintest bins, more than triple the size of the samples studied in Z05. The strong clustering is an indication that most of the faint red galaxies are satellites in fairly massive halos (Berlind et al. 2005; Z05; Wang et al. 2009). We present HOD models of a few of these samples in Section 4.5 below but defer a detailed examination of this population to future work.

The large size of the DR7 main galaxy sample allows us to measure w_p(r_p) for narrow bins of color in addition to the broad "blue" and "red" classifications used in Sections 4.1–4.3 and in most earlier work. Figure 18 shows the cuts we adopt to divide galaxies into "bluest," "bluer," "redder," and "reddest" populations. We also define an intermediate population of "green" galaxies, located near the minimum of the observed color bimodality along the red/blue dividing line, associated with the so-called green valley galaxies (e.g., Wyder et al. 2007; Loh et al. 2010). We include all galaxies within Δ(g − r) = 0.05 of the tilted dividing line of Equation (13) (analogous to the "green" galaxy population studied by Coil et al. 2008). In addition, we add a sample of galaxies along the cusp of the red sequence galaxies, denoted as "redseq," defined as all galaxies within Δ(g − r) = 0.03 of the redder/reddest dividing line. Note that the last two classes are not distinct populations: the "green" sample contains a subset of the redder and bluer samples, while the "redseq" sample contains a subset of the redder and reddest samples. Details of the individual samples are given in Table 4.

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Table 4. Color Subsets of the Volume-limited −20 < M_r < −19 Sample

Sample	Ngal	${\bar{n}}$	r0	γ	$\frac{\chi ^2}{{\rm dof}}$
All	44,348	1.004	4.89 ± 0.26	1.78 ± 0.02	3.79
Red	18,892	0.428	6.63 ± 0.41	1.94 ± 0.03	5.07
Blue	25,455	0.576	3.62 ± 0.15	1.66 ± 0.03	1.66
Reddest	10,278	0.233	7.62 ± 0.42	2.07 ± 0.03	1.87
Redseq	7,542	0.171	7.23 ± 0.28	1.95 ± 0.03	1.06
Redder	8,614	0.195	5.48 ± 0.43	1.91 ± 0.04	1.84
Green	5,543	0.126	5.06 ± 0.42	1.79 ± 0.05	1.35
Bluer	11,156	0.253	4.14 ± 0.21	1.69 ± 0.04	0.89
Bluest	14,299	0.324	3.15 ± 0.15	1.71 ± 0.05	1.10

Notes. All samples use 14.5 < m_r < 17.6. cz_min = 8050 km s⁻¹ and cz_max = 19, 250 km s⁻¹. ${\bar{n}}$ is measured in units of 10⁻² h³ Mpc⁻³. The number of degrees of freedom (dof) is 9 (11 measured w_p values minus the two fitted parameters). The subsamples are defined using tilted color cuts as described in the text.

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Figure 19 shows the projected correlation functions of all these color samples, for the representative luminosity bin −20 < M_r < −19. We find a continuous trend with color, in both amplitude and slope: the redder the color of the sample, the higher and steeper the correlation function. We find the same trends in the other luminosity bins, although the dependence on color is weaker at higher luminosities, as already seen for the red/blue division in Figure 17.

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Differences in clustering strengths should be reflective of the different environments of the galaxies. The steady trend of w_p(r_p) with color at fixed luminosity is consistent with the findings of Hogg et al. (2003), who investigated the density of galaxy environments as a function of luminosity and color. The trend across our three red samples indicates that redder galaxies within the red sequence populate denser regions, again consistent with Hogg et al. (2003). Hogg et al. (2004) examined the color–magnitude diagram as a function of environment and did not find a significant shift of the red sequence location with density, but examining their results in detail does reveal mild changes in the locus for bulge-dominated galaxies. The trends observed in Hogg et al. (2003, 2004) are subtle, but they appear consistent with our results.

Coil et al. (2008) have carried out an analysis similar to ours at z ∼ 1, using projected correlation functions of fine color bins in the DEEP2 galaxy survey. They find qualitatively similar results for blue galaxies and for the difference between blue and red galaxy clustering, but they find no significant change in the amplitude or slope of w_p(r_p) among their red samples (see their Figure 12). The difference from our results could be a consequence of details of sample definition, or possibly a consequence of color-dependent incompleteness in DEEP2 (e.g., Gerke et al. 2007), though Coil et al. (2008) account for this in their analysis. The difference could also be an evolutionary effect reflecting the buildup of galaxies on the red sequence. One plausible explanation is that variations in star formation history and dust content contaminate and scatter galaxies within the red sequence, and from the "blue cloud" into the red sequence, when the universe is younger (e.g., Brammer et al. 2009), while evolution to z = 0 allows galaxy populations to separate more cleanly, yielding a tighter correlation between color, stellar population age, and environment.

The clustering of the green galaxies falls between that of the blue and red galaxy samples and clearly follows the continuous trend with color in both amplitude and slope. We do not find the apparent break in the green galaxies' projected correlation function seen in DEEP2 (Coil et al. 2008), where the clustering amplitude is similar to that of blue galaxies on small scales and to that of the red galaxies on large scales. Loh et al. (2010) investigate the clustering properties of "green valley" galaxies using UV imaging from Galaxy Evolution Explorer matched to SDSS spectroscopy, and find that the clustering of green galaxies is intermediate between that of the blue and red galaxies, in qualitative agreement with our results. However, they find that the green galaxies have a large-scale clustering amplitude similar to that of the blue galaxies (in contrast with Coil et al. 2008). When fitting an overall power law to the projected correlation function, they find the green galaxies' clustering amplitude to be between that of the blue and red samples, with a similar slope to that of the red galaxies, while we find the green galaxy population to be intermediate in both amplitude and slope. These differences may have to do with the different sample definition and selection in each of these and warrant further investigation.

To model the color dependence of w_p(r_p) presented in Section 4.4, we adopt a simplified HOD model based on the parameterized form of the mean occupation function specified in Equation (7) for luminosity-threshold samples. For the −20 < M_r < −19 luminosity bin, we set the central galaxy occupation function to the difference of the M_r < −19 and M_r < −20 modeling results shown in Section 3.3. For simplicity, we also fix the slope of the satellite occupation function, α, to 1. We also assume that the occupation number of satellites at fixed halo mass follows a Poisson distribution and is independent of the central galaxy occupation number. The modeling is thus a one-parameter family, in which only M'₁ is varied to fit w_p(r_p), changing the relative normalization of the central and satellite occupation functions with color. The overall normalization is determined by matching the observed number density of galaxies in the color bin. In this simple model, the relative fraction of blue and red satellites has no dependence on halo mass. Different modeling approaches and more detailed parameterizations are possible, of course (e.g., Scranton 2002; Cooray 2005; Z05; Ross & Brunner 2009; Simon et al. 2009; Skibba & Sheth 2009), but this form is sufficient to explain the main trends of the color dependence. We note that our model guarantees that the sum of central galaxy occupation functions of independent color samples equals that of the full −20 < M_r < −19 bin sample. By construction, the sum of the satellite mean occupation functions, each of which follows a power law with soft cutoff, differs in shape slightly from the bin-sample satellite occupation, which is the difference of two power-law curves with soft cutoffs. We have verified, however, that the sum in our fits is close to the satellite mean occupation function of the overall bin sample, especially in the range where the occupation number is close to unity and the contribution to the small-scale clustering signal is dominant.

Figure 20 presents the results of this modeling. Points with error bars in the upper left panel are the w_p(r_p) measurements for fine color bins repeated from Figure 19, with 0.25 dex offsets added between bins for visual clarity. The curves show the model predictions corresponding to the best-fit HODs, exhibited in the upper-right panel. Going from bluer galaxies to redder galaxies, the number of central galaxies steadily decreases and the number of satellite galaxies steadily increases. Although the central-to-satellite ratio is the only tunable parameter in our simplified HOD model, this is sufficient to explain the main trends observed in Figure 19: going from bluer to redder galaxies, the large-scale amplitude of w_p(r_p) increases, the correlation function steepens, and the inflection at the one-to-two-halo transition becomes stronger. Table 5 lists the best-fit HOD parameters and χ² values. We find χ²/dof of 0.5–1.3 for most of the color samples, the exception being the bluest sample, which has χ²/dof ∼2. The fits can be improved by adding flexibility to the HOD model; for example, the fit for the bluest galaxies can be improved by allowing the slope of the satellite occupation function and the halo concentration to change.

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Table 5. HOD and Derived Parameters for −20 < M_r < −19 Fine-color Subsamples

Sample	log M'1	fnorm	fsat	log Mmed	$\frac{\chi ^2}{{\rm dof}}$
Reddest	12.11 ± 0.06	0.10	0.75 ± 0.03	12.61 ± 0.07	1.3
Redseq	12.39 ± 0.05	0.11	0.62 ± 0.03	12.15 ± 0.09	1.2
Redder	12.67 ± 0.04	0.18	0.46 ± 0.02	11.80 ± 0.02	1.0
Green	12.87 ± 0.05	0.14	0.34 ± 0.02	11.70 ± 0.01	0.9
Bluer	13.11 ± 0.03	0.33	0.24 ± 0.01	11.65 ± 0.01	0.5
Bluest	13.36 ± 0.05	0.47	0.15 ± 0.01	11.62 ± 0.01	2.0

Notes. The shape of the mean occupation function for central galaxies is assumed to be the difference of those of M_r < −19 and M_r < −20 samples. The mean occupation function for satellites follows a modified power law. The relative normalization of the mean occupation functions for central and satellite galaxies is determined by M'₁. The overall normalization f_norm is obtained from matching the observed number density (see the text). Halo mass is in units of h⁻¹ M_☉. The satellite fraction, f_sat, and the median mass of host halos, M_med, are derived parameters. For all samples, the number of degrees of freedom (dof) is 12 (13 measured w_p values minus one fitted parameter).

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The lower panels of Figure 20 display the trends of satellite fraction more clearly. In the lower right panel, we scale each occupation function by a constant factor so that the central galaxy components have the same normalization. The amplitude of the satellite occupation function increases steadily going from the bluest galaxies to the reddest galaxies. The lower-left panel plots the satellite fraction f_sat of each color bin against the median halo mass of galaxies in that bin. The satellite fraction rises from ∼15% for the bluest bin to ∼75% for the reddest bin, and the median halo mass increases as the fraction of galaxies that are satellites in massive halos grows. Green valley galaxies have occupation functions intermediate between the red and blue galaxies, consistent with the idea that they are a transitional population (e.g., Coil et al. 2008; Martin et al. 2007).

As discussed in Section 3.3, the trend of clustering strength with luminosity is explained principally by a rise in the central galaxy halo mass, and the satellite fraction drops with increasing luminosity because the halo mass function steepens at higher masses. In contrast, the trend with color at fixed luminosity can be explained with a constant halo mass for central galaxies and a steady increase of satellite fraction with redder color. The increase in typical host halo mass leads to an increase in the large-scale bias factor and thus a higher clustering amplitude at large scales. However, increasing f_sat drives the one-halo term up more rapidly than the bias factor, so the correlation function steepens for redder galaxies as well. The success of our simple HOD model does not rule out a shift in central galaxy halo mass for redder galaxies, but explaining the strong observed color trend solely through the central galaxy occupation would require placing moderate luminosity red galaxies at the centers of very massive halos, and it might well be impossible to match the clustering and number density constraints simultaneously.

Returning to the joint dependence on color and luminosity (Section 4.3), Figure 21 presents HOD model fits to the blue and red galaxy populations for three of the luminosity bins shown in Figure 16. We use the same modeling approach adopted above for the fine color bins: we difference the central galaxy occupation functions of two luminosity-threshold samples to get the central galaxy occupation function of the luminosity bin, fix the satellite slope to α = 1, and vary only the relative central and satellite normalizations within each population to fit the red and blue w_p(r_p) measurements. With the other HOD parameters fixed previously by fitting the correlation functions of the full luminosity-threshold samples, this modeling has just one adjustable parameter within each luminosity bin. The model explains the rather complex color–luminosity trends from Figure 16 fairly well. In particular, it is able to reproduce the small-scale clustering of red galaxies increasing toward low luminosities, both in absolute terms and relative to the large-scale clustering, while the shape of w_p(r_p) for the blue galaxies stays roughly constant. The fraction of red galaxies that are satellites increases sharply with decreasing luminosity, from 33% to 60% to 90% in the three luminosity bins, while the fraction of blue satellites (13%, 19%, 19%) is smaller and only weakly dependent on luminosity. The precise values of the satellite fractions depend on the HOD parameterization used to fit w_p(r_p), but the general trend is robust: most blue galaxies at these luminosities are central, and the satellite fraction for red galaxies is higher and increases toward faint luminosities.

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The largest quantitative failure of this model is its overprediction of the large-scale w_p(r_p) for the faintest red galaxies (and, to a smaller extent, for the faintest blue galaxies). It could be that our jackknife method underestimates the errors for this small-volume sample, and we have already noted (Figure 13) that our HOD model overpredicts the total (red+blue) galaxy correlation function in this luminosity bin. However, this discrepancy could indicate a limitation of our restricted HOD parameterization. To investigate this possibility, we have considered models for the faint red galaxy population in which we vary the satellite slope α, the concentration parameter of red galaxies in halos, and, most notably, the satellite cutoff parameter M₀ in Equation (7).

The dot-dashed curves in the upper panels of Figure 21 show an example in which red satellites arise only in halos above 10¹³h⁻¹ M_☉, reducing the satellite fraction from 90% in our original fit to 34%, thereby lowering the large-scale bias factor. The physical motivation for such a model is that gas accretion (and subsequent star formation) by a satellite system might be shut off only if it enters a halo whose mass is much larger than the "birth" halo in which it was a central galaxy (Simha et al. 2009; see also Font et al. 2008; Kang & van den Bosch 2008; Skibba 2009). The lower satellite fraction of this model is more consistent with the results of Wang et al. (2009), who argue, based on group catalogs, that 30%–60% of faint red galaxies (significantly fainter than those modeled here) are central rather than satellite galaxies. The fit to the smallest scale data points is improved by increasing the galaxy concentration parameter (van den Bosch et al. 2008a) to twice the dark matter value, steepening the profile of the one-halo term. Visually, the w_p(r_p) prediction of this model is not much better than that of the original model, but the χ²/dof drops from 3.3 to 1.4 (since the mid-range points where the deviation is largest are the most covariant), a large statistical improvement. Despite its flexibility, this model underpredicts w_p(r_p) in the one-halo regime and overpredicts it in the two-halo regime, emphasizing how difficult it is to simultaneously reproduce the strong small-scale clustering and low large-scale bias factor of this galaxy sample. This tension could be a sign of environment-dependent effects on the HOD, but the expected form of "assembly bias" for low-mass halos, putting the redder central galaxies into older, more clustered halos, would exacerbate the discrepancies with the data further.

Overall, our inferences from HOD modeling accord well with the theoretical predictions of Berlind et al. (2005), who compared the results of cosmological smoothed particle hydrodynamic simulations to Hogg et al.'s (2003) measurements of galaxy environments as a function of luminosity and color. In particular, Berlind et al. (2005) find that the environment of satellite galaxies in the simulations is strongly correlated with stellar population age (hence color), and that for low and intermediate luminosities the environmental dependence of the overall galaxy population tracks that of satellite galaxies. Berlind et al. (2005) also find that the great majority of faint red galaxies in the simulation are satellites, though the simulation they use to study this population is small. The success of our simple HOD models in reproducing the observed color dependence of w_p(r_p) contrasts with the recent conclusions of Ross & Brunner (2009), who find that some segregation of early- and late-type galaxies into separate halos is required to reproduce their measured angular clustering of an SDSS photometric galaxy sample, which extends to smaller separations.

Zu et al. (2008) consider the general case of the relation between the two-point autocorrelation functions of a galaxy sample and the auto- and cross-correlation functions of its subsamples (see their appendix). Here, we focus on the specific case of the correlation functions of red, blue, and all galaxies. The point we make here is that only three of the four correlation functions (blue–blue, red–red, all–all autocorrelation functions, and red–blue cross-correlation functions) are independent. That is, if we measured blue–blue, red–red, and all–all autocorrelation functions, there would be no new information from the red–blue cross-correlation functions.

To demonstrate that this is the case, we recall that the two-point correlation function represents a galaxy pair count. The total number of pairs of all galaxies in the parent sample is simply the sum of the numbers of red galaxy pairs, blue galaxy pairs, and red–blue galaxy pairs. That is

$$\begin{eqnarray} \frac{1}{2}n_{\rm all}^2 (1+\xi _{\rm all}) &=& \frac{1}{2}n_{\rm blue}^2 (1+\xi _{\rm blue})+ \frac{1}{2}n_{\rm red}^2 (1+\xi _{\rm red})\nonumber\\ && + n_{\rm blue} n_{\rm red} (1+\xi _{\rm cross}), \end{eqnarray} \tag{ A1 }$$

where n_all, n_red, and n_blue are the mean number density of the parent sample and the red/blue subsamples, ξ_all, ξ_blue, and ξ_red are the two-point autocorrelation functions, and ξ_cross is the red–blue two-point cross-correlation function. The factor of 1/2 in front of the autocorrelation terms is to avoid the double count of autopairs. Since n_all = n_red + n_blue, the above identity reduces to

$$\begin{equation} n_{\rm all}^2 \xi _{\rm all} = n_{\rm blue}^2 \xi _{\rm blue} + n_{\rm red}^2 \xi _{\rm red} + 2 n_{\rm blue} n_{\rm red} \xi _{\rm cross}. \end{equation} \tag{ A2 }$$

The same relation holds for projected correlation functions w_p. Thus, the red–blue cross-correlation function can be derived from the three autocorrelation functions.

As a test of this relation, we predicted the red–blue cross-correlation function based on the measured all–all, red–red, and blue–blue autocorrelation functions for the −20 < M_r < −19 volume-limited galaxy sample. The prediction agrees essentially perfectly with the measured cross-correlation function shown in Figure 15, with deviations much smaller than the 1σ error bars.

There is thus, in theory, no new information provided by the cross-correlation function, when one has measured the three individual autocorrelation functions. In practice, the relation in Equation (A2) and that for projected two-point correlation functions can be used for a consistency check. Furthermore, for understanding the mixture among different galaxy populations, the cross-correlation function is more readily interpreted than the consistency relation itself. For example, segregation of "red" and "blue" halos would produce a distinctive suppression of the one-halo term of the cross-correlation function, while its effect on the autocorrelation functions (boosting the red and blue autocorrelations relative to the all autocorrelation) might be difficult to disentangle from changes in satellite occupation slopes, concentration parameters, and so forth.

The two-point cross-correlation function of two populations (e.g., red and blue galaxies we study here) is often compared to the geometric mean of the two-point autocorrelation functions to infer the information about the mixing of the two populations. On large scales, where the two-halo term dominates the correlation functions, the cross-correlation function is guaranteed to be the geometric mean of the autocorrelation functions.¹⁷ On small scales, where the one-halo term dominates, it is not obvious what we can infer if there are deviations of the cross-correlation function from the geometric mean. We show here that the situation becomes even less clear for projected correlation functions.

As an example, consider the two-point autocorrelation functions of red and blue galaxies, ξ_red and ξ_blue, and their cross-correlation function ξ_cross. Under the assumption that these correlation functions are positive, we have

$$\begin{eqnarray} \left(\int \sqrt{\xi _{\rm red}(r_p,\pi) \xi _{\rm blue}(r_p,\pi)} d\pi \right)^2 &\le& \int \xi _{\rm red}(r_p,\pi) d\pi \nonumber\\ &&\times\int \xi _{\rm blue}(r_p,\pi) d\pi\nonumber\\ \end{eqnarray} \tag{ A3 }$$

from the Cauchy–Schwartz inequality. Even if we had $\xi _{\rm cross}=\sqrt{\xi _{\rm red}\xi _{\rm blue}}$ on all scales, the above inequality would mean that the projected correlation functions satisfy

$$\begin{equation} w_{p,{\rm cross}} \le \sqrt{ w_{p,{\rm red}} w_{p,{\rm blue}} }. \end{equation} \tag{ A4 }$$

The equality only holds for the case where both galaxy populations trace the same dark matter distribution and ξ_red and ξ_blue are parallel to each other, which is true on large scales but not on small scales.