# The UV-optical colour dependence of galaxy clustering in the local universe

Yeong-Shang Loh, Robert Michael Rich, Sebastien Heinis, Ryan Scranton, Ryan P. Mallery, Samir Salim, D. Christopher Martin, Ted Wyder, Stephane Arnouts, Tom Barlow, et al.

► To cite this version:

Yeong-Shang Loh, Robert Michael Rich, Sebastien Heinis, Ryan Scranton, Ryan P. Mallery, et al. The UV-optical colour dependence of galaxy clustering in the local universe. Monthly Notices of the Royal Astronomical Society, 2010, 407 (1), pp.55-70. 10.1111/j.1365-2966.2010.16908.x. hal-01438941

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# The UV-optical colour dependence of galaxy clustering in the local universe

Yeong-Shang Loh,<fnref n="1" />* R. Michael Rich,<fnref n="1" /> Sébastien Heinis,<fnref n="2" /> Ryan Scranton,<fnref n="3" />

Ryan P. Mallery,<fnref n="1" /> Samir Salim,<fnref n="4" /> D. Christopher Martin,<fnref n="5" /> Ted Wyder,<fnref n="5" />

Stéphane Arnouts,<fnref n="2" /> Tom A. Barlow,<fnref n="5" /> Karl Forster,<fnref n="5" /> Peter G. Friedman,<fnref n="5" />

Patrick Morrissey,<fnref n="5" /> Susan G. Neff,<fnref n="6" /> David Schiminovich,<fnref n="7" /> Mark Seibert,<fnref n="5" />

Luciana Bianchi,<fnref n="8" /> Jose Donas,<fnref n="9" /> Timothy M. Heckman,<fnref n="2" /> Young-Wook Lee,<fnref n="10" />

Barry F. Madore,<fnref n="11" />,<fnref n="12" /> Bruno Milliard,<fnref n="9" /> Alex S. Szalay<fnref n="2" /> and Barry Y. Welsh<fnref n="13" />

Accepted 2010 April 22. Received 2010 April 20; in original form 2009 April 8

## ABSTRACT

We measure the UV-optical colour dependence of galaxy clustering in the local Universe. Using the clean separation of the red and blue sequences made possible by the NUV – r colour–magnitude diagram, we segregate the galaxies into red, blue and intermediate ‘green’ classes. We explore the clustering as a function of this segregation by removing the dependence on luminosity and by excluding edge-on galaxies as a means of a non-model dependent veto of highly extincted galaxies. We find that $\xi(r_p, \pi)$ for both red and green galaxies shows strong redshift-space distortion on small scales – the ‘finger-of-God’ effect, with green galaxies having a lower amplitude than is seen for the red sequence, and the blue sequence showing almost no distortion. On large scales, $\xi(r_p, \pi)$ for all three samples show the effect of large-scale streaming from coherent infall. On scales of $1 h^{-1}$ Mpc $< r_p < 10 h^{-1}$ Mpc, the projected auto-correlation function $w_p(r_p)$ for red and green galaxies fits a power law with slope $\gamma \sim 1.93$ and amplitude $r_0 \sim 7.5$ and $5.3$, compared with $\gamma \sim 1.75$ and $r_0 \sim 3.9 h^{-1}$ Mpc for blue sequence galaxies. Compared to the clustering of a fiducial $L^*$ galaxy, the red, green and blue have a relative bias of 1.5, 1.1 and 0.9, respectively. The $w_p(r_p)$ for blue galaxies display an increase in convexity at $\sim 1 h^{-1}$ Mpc, with an excess of large-scale clustering. Our results suggest that the majority of blue galaxies are likely central galaxies in less massive haloes, while red and green galaxies have larger satellite fractions, and preferentially reside in virialized structures. If blue sequence galaxies migrate to the red sequence via processes like mergers or quenching

<footnote n="1">¹Department of Physics and Astronomy, University of California, Los Angeles, CA 90095-1562, USA</footnote>

<footnote n="2">²Department of Physics and Astronomy, The Johns Hopkins University, Homewood Campus, Baltimore, MD 21218, USA</footnote>

<footnote n="3">³Department of Physics and Astronomy, University of Pittsburgh, 3941 O’Hara St., Pittsburgh, PA 15260, USA</footnote>

<footnote n="4">⁴National Optical Astronomy Observatory, 950 North Cherry Avenue, Tucson, AZ 85719, USA</footnote>

<footnote n="5">⁵California Institute of Technology, MC 405-47, 1200 East California Boulevard, Pasadena, CA 91125, USA</footnote>

<footnote n="6">⁶Laboratory for Astronomy and Solar Physics, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA</footnote>

<footnote n="7">⁷Department of Astronomy, Columbia University, New York, NY 10027, USA</footnote>

<footnote n="8">⁸Center for Astrophysical Sciences, The Johns Hopkins University, 3400 N. Charles St., Baltimore, MD 21218, USA</footnote>

<footnote n="9">⁹Laboratoire d'Astrophysique de Marseille, BP 8, Traverse du Siphon, 13376 Marseille Cedex 12, France</footnote>

<footnote n="10">¹⁰Center for Space Astrophysics, Yonsei University, Seoul 120-749, Korea</footnote>

<footnote n="11">¹¹Space Sciences Laboratory, University of California at Berkeley, 601 Campbell Hall, Berkeley, CA 94720, USA</footnote>

<footnote n="12">¹²Observatories of the Carnegie Institution of Washington, 813 Santa Barbara St., Pasadena, CA 91101, USA</footnote>

<footnote n="13">¹³Space Sciences Laboratory, University of California at Berkeley, 601 Campbell Hall, Berkeley, CA 94720, USA</footnote>

<footnote>*E-mail: yeongloh@astro.ucla.edu</footnote>
that take them through the green valley, such a transformation may be accompanied by a
change in environment in addition to any change in luminosity and colour.

**Key words:** methods: statistical – galaxies: clusters: general – galaxies: elliptical and lenticular – galaxies: evolution.

# 1 INTRODUCTION

With the advent of the Sloan Digital Sky Survey (SDSS; York et al. 2000) and its value-added galaxy catalogues, it has been possible to study the subject of galaxy bimodality and its relationship to fundamental properties, such as stellar mass and star formation history (e.g. Kauffmann et al. 2003; Schiminovich et al. 2007; Salim et al. 2007). The broad division of galaxies into star-forming discs and quiescent early-type galaxies is the fundamental principle of Hubble's tuning fork system of classification and is well established. In a plot of optical *g* − *r* colour versus *M*<sub>r</sub>, red galaxies define a clear sequence, while the locus of blue galaxies is broadened into the so-called blue cloud. The red sequence has been shown to maintain its integrity with look-back time (Bower et al. 1992) and has grown in mass since redshift ~1. Studies by Bell et al. (2004), Blanton (2006), Faber et al. (2007) and Brown et al. (2008) argue that the stellar mass contained within the red population has increased by roughly a factor of 2 in half the Hubble time.

A significant breakthrough in expressing this blue/red dichotomy occurred when photometry from the *Galaxy Evolution Explorer* (GALEX), notably the near-ultraviolet (NUV) band, was matched with SDSS photometry (Martin et al. 2007; Wyder et al. 2007; Schiminovich et al. 2007; Salim et al. 2007). When the diagram is plotted using NUV − *r* as the colour, two clear sequences emerge: the familiar red sequence and a new blue sequence in place of the 'blue cloud' of optical studies. Between the blue and red sequences there are galaxies present in a so-called green valley. Many of these are spectroscopically classified Type II active galactic nuclei (AGN) (Rich et al. 2005; Martin et al. 2007; Salim et al. 2007).

Faber et al. (2007), Martin et al. (2007) and Schiminovich et al. (2007) propose several paths by which galaxies might transition from the blue to the red sequence. The presence of AGN in the green valley suggests that AGN activity is associated with a quenching of star formation (Hopkins et al. 2006, 2007; Silk & Rees 1998). Other paths from the blue to the red sequence might, hypothetically, involve gas-rich mergers of blue galaxies (Toomre & Toomre 1972), or virial shock heating of cold gas streams (Dekel & Birnboim 2006). The red sequence might consolidate in luminosity via dissipationless mergers of red galaxies, or low-luminosity blue galaxies might acquire bulges through mergers with starbursts, retaining sufficient mass to land the evolved galaxy on the red sequence. However, the green valley might also be populated by casual visitors – red galaxies that acquire gas and form stars or feed a central AGN. Following this brief burst of star formation, these galaxies might ultimately return to the red sequence from which they started.

In Salim et al. (2007), a plot of mass against specific star formation rate (SFR) reveals a clear division between lower mass, star-forming, blue sequence galaxies and more massive AGN, which are not detected in large numbers until stellar mass $M \sim 3 \times 10^{10} M_{\odot}$. The process responsible for populating the green valley and for potentially contributing to evolution from blue to red must bear

some relationship to environment and to the dark matter haloes in
which the galaxies reside. In this study, we investigate the cluster-
ing environment of the blue and red sequences, and for the green
valley.

The current paradigm of structure formation assumes that galaxies are assembled in dark matter haloes. The dependence of the clustering on galaxy properties may provide clues to the baryonic processes that are important to galaxy formation and evolution. The dependence of galaxy clustering on galaxy type has been known since the earliest studies of extragalactic astronomy (Hubble 1936; Zwicky et al. 1968). In the modern era of large-scale galaxy surveys, Davis & Geller (1976) showed that the angular auto-correlation of ellipticals has a steeper power-law slope than those of spirals. Recent redshift surveys using the Two-Degree Field (2dF) Galaxy Survey and SDSS confirms these earlier results and the apparent bimodal nature of galaxy clustering (Madgwick et al. 2003; Budavari et al. 2003; Zehavi et al. 2005; Li et al. 2006a; Wang et al. 2007).

Studies using SDSS have further revealed that galaxy colour is the property most predictive of local environment. Blanton et al. (2005b) found that at fixed luminosity and colour, density does not correlate with surface brightness nor the Sersic index, and argue that morphological properties of galaxies are less closely related to environment than their star formation history, and are traced by broad-band optical colours. [See Park et al. (2007) for an alternative analysis and point of view.] Li et al. (2006a) found that the dependence of clustering on optical *g* − *r* colour and *D*<sub>4000</sub> is much stronger than structural parameters like concentration and surface brightness, and extend to 5 *h*<sup>−1</sup> Mpc, beyond what is expected from the localized halo paradigm of structure formation. They concluded that at fixed stellar mass, the clustering properties of the surrounding dark matter haloes are correlated with the colour of the selected galaxies. They further argued that different physical processes may be required to explain environmental trends in star formation, distinct from those established by galaxy structure.

In this paper, we will consider the colour dependence of the
two-point physical correlation function of galaxies using samples
constructed from *GALEX*, augmented with redshift and optical data
from SDSS. In particular, we use the natural separation from the
*NUV* − *r* colour to assign galaxies into three subsamples of red,
green and blue galaxies. Our study complements recent work by
Heinis et al. (2009) who investigate the physical clustering of galax-
ies as a function of star formation history in the local universe, as
well as earlier studies by Milliard et al. (2007), Heinis et al. (2007)
and Basu-Zych et al. (2008) who investigate the angular-correlation
function of rest-frame *UV*-selected galaxies and their evolution. We
measure the auto-correlation function (ACF) of each of the differ-
ent subsamples of galaxies, as well as the CCF (CCF) between the
subsamples. In Section 2, we describe in detail the data used in this
analysis. In Section 3 we describe the method for estimating cor-
relation functions. We present our results in Section 4, and discuss
their implications for the nature of green valley galaxies and the
formation of red sequence galaxies. We summarize our findings in Section 6.

# 2 DATA

## 2.1 GALEX and SDSS data

The ultraviolet imaging portion of the data set is from the GALEX that was launched in 2003 April (Martin et al. 2005; Morrissey et al. 2005, 2007). GALEX obtains wide field imaging in both the far-UV (FUV; centred at 1540 Å) and the near-UV (NUV, 2300 Å) over a 1.22 diameter field of view, with 5 arcsec images. Here, we use data from the Medium Imaging Survey (MIS); these images are ≈1500 s in duration reaching NUV ≈ 23 mag, covering an orbital shadow crossing. The MIS pointing that defines our sample targets the North Galactic Cap, which overlaps the SDSS spectroscopic footprint; this part of the program was designed from studies cross-matching SDSS and GALEX data. The data set used for our current analysis is from the Galaxy Release 3 (GR3) which is available from the Multi-mission Archive at STScI (MAST). The GALEX pipeline uses SExTRATOR (Bertin & Arnouts 1996) to detect sources and measure fluxes. We use the ‘MAG_AUTO’ output from SExTRATOR as our default flux measurement; it is essentially a Kron (1980) magnitude with an elliptical aperture.

Because our analysis requires each galaxy to have a spectroscopic redshift, we start our cross-matching with a galaxy from the SDSS Main spectroscopic survey. For each SDSS galaxy, we search for the closest GALEX detection within 4 arcsec radius from the location of the SDSS spectroscopic fibre. Only GALEX sources within 0.55 of the tile centre field-of-view (FOV) are retained, since astrometry degrades towards the periphery of the FOV (a problem which will be resolved in later releases) and the incidence of artefacts increases as well. After the matching of GALEX and SDSS sources, we further trim the sample to create a statistically complete data set following the procedure of Wyder et al. (2007). We will refer the reader to their table 1 for the full details. Here, we list a few essential parameters and the minor modifications we employed: 14.0 < r < 17.6, 0.03 < z < 0.25, σr < 0.2, zconf > 0.67, GALEX exposure time t > 750 s and 16.0 < NUV < 23.0.<fnref n="1" /> All magnitudes are AB magnitudes and corrected for Galactic foreground extinction.

## 2.2 SDSS large-scale structure sample

Considerable effort has been invested by the SDSS team to prepare the redshift data for large-scale structure (LSS) studies. This secondary data set known as the New York University Value Added Catalogue (NYU-VAGC) is documented in Blanton et al. (2005a) and available from the NYU web site. Proprietary versions of this catalogue have been used by many groups within the SDSS collaboration for various investigations of clustering and luminosity function of galaxies. We match the GALEX–SDSS catalogue constructed above with the SDSS LSS DR5 sample. The version used for our analysis includes all of the detailed radial and angular selection functions for the various statistical subsamples used in the previous analyses (e.g. Zehavi et al. 2005). This sample is ideal because we can compare our result with that of Zehavi et al. (2005) which was based solely on selection from optical criteria.

## 2.3 GALEX–SDSS overlapping footprint

In order to statistically define our combined GALEX–SDSS survey, we need to have the understanding of the angular sampling function of the two surveys, which varies across different regions of the sky. GALEX’s MIS survey consists of overlapping circular tiles (radius 0.55), while the SDSS spectroscopic survey is a combination of circular spectroscopic plates, but with fibre placement based on a rectangular imaging survey that runs along great circles. To combine the footprints of both surveys, we use the GESTALT footprint server. GESTALT<fnref n="2" /> uses a hierarchical pixelization system, enabling one to encode observations of arbitrary geometry while tagging information about completeness and the masking of artefacts. We first encode the GALEX MIS survey using GESTALT. We then obtain the detailed observational footprint of the SDSS LSS sample from the NYU-VAGC web site. The SDSS footprint is expressed as a set of disjoint polygons using the software MANGLE (Hamilton & Tegmark 2002) which takes into account the complex angular mask and geometry of the SDSS survey. We convert these polygons into the pixelization scheme of GESTALT and consider the intersection of the two surveys. There are approximately 490 deg<fnref n="2" /> in the GALEX–SDSS overlapping footprints after masking for holes, bright stars and satellite trails, and excluding defects.

In order to measure the correlation function, a random sample needs to be constructed to normalize the galaxy pair counts. Angular sampling completeness as a function of position in the sky encoded in the footprint server is used to generate random samples of density roughly 50 times the galaxy density. We adopt the method proposed by Li et al. (2006a) where we assign each galaxy in our sample to a random position on the sky but keep all other attributes the same (e.g. redshift, colour, magnitude). This random sample by construction has the same redshift distribution as the original sample, and thus does not smooth out the redshift structure like those generated via the luminosity function. As noted in Li et al. (2006a), this approach works well in surveys with a wide-angular sky coverage (e.g. much larger than the typical LSS), and with small variation in survey depth. We note here that our random sample would inherit the redshift correlation function of the colour–magnitude distribution of the parent galaxy sample; only spatial distribution has been randomized, hence any excess in clustering must be due to positional differences.

In the SDSS spectroscopic survey, no two galaxies with separation θ less than 55 arcsec can both be assigned spectroscopic fibres for observation on any given observing plate. Hence, a large fraction of galaxy pairs with θ < 55 arcsec are missing. We correct for this ‘fibre-collision’ problem by using the observed angular correlation function, a method first suggested by Li et al. (2006a), and described in detail in the companion paper by Heinis et al. (2009). In brief, the observed projected two-point angular correlation function for both the photometric sample wph(θ) and the spectroscopic sample wsp(θ) is measured and used to construct the pair weighting ratio:

$$F(\theta) = \frac{w_{ph}(\theta) + 1}{w_{sp}(\theta) + 1} \qquad (1)$$

as a function of separation. Empirically, Heinis et al. (2009) find F(θ) ~ 3 for θ < 55 arcsec and zero otherwise for the all GALEX–SDSS cross-matched galaxies. We apply this correction to each of the red, green and blue subsamples equally.

<footnote n="1">¹Our faint-end limit of 23.0 is about 0.5 mag fainter than those employed by Wyder et al. (2007).</footnote>

<footnote n="2">²http://nvogre.phyast.pitt.edu:8080/gestalt_tutorial/</footnote>
# 3 METHODOLOGY

## 3.1 The two-point correlation function

In brief, the two-point ACF $\xi(r)$ measures the excess probability of finding a galaxy pair with separation $r$ from a random galaxy distribution,

$$dP = n[1 + \xi(r)]dV, \qquad (2)$$

where $n$ is the mean number density of galaxy sample. For the last 40 years, the correlation function has served as the primary method for cosmologists to quantify the clustering properties of galaxies from large-scale surveys (Totsuji & Kihara 1969; Peebles 1980). If the underlying density distribution is Gaussian, then the correlation function fully describes all statistical properties of a given distribution. Since early cosmological models are often based on primordial Gaussian dark matter density fields, the use of the correlation function leads to a straightforward comparison between empirical studies on the statistical distribution of galaxies with such models. Recently, with the widespread adoption of a standard $\Lambda$-dominated cosmology (Ostriker & Steinhardt 1995; Riess et al. 1998; Perlmutter et al. 1999; Spergel et al. 2007), the correlation function is used instead to probe the growth of structure in the universe and the range of formation scenarios for varying kinds of galaxies.<fnref n="3" /> The correlation length – the amplitude from a power-law correlation function – provides information on the mass of dark matter haloes in which the various galaxies reside, linking observation with theoretical description of structure formation (Bower et al. 2006; Croton et al. 2006)

For a galaxy survey with a well-defined angular selection function, $\xi$ can be estimated using an optimal estimator like the Landy & Szalay (1993) estimator:

$$\hat{\xi}_{\text{LS}} = \frac{DD - 2DR + RR}{RR} \qquad (3)$$

where $DD$, $DR$ and $RR$ are normalized counts of galaxy pairs in the data-data, data-random and random-random catalogues.<fnref n="4" /> The Landy & Szalay (1993) estimator is preferred because it is relatively insensitive to the size of the random catalogue and to edge corrections (Kerscher et al. 2000).

Because we observe galaxies in redshift and projected space and not in physical space, in practice, the correlation function is first measured on a two-dimensional grid of separations: $\pi$ along the line of sight (redshift space) and $r_p$ for angular separation on the sky. In addition to providing information about the underlying mass distribution through the amplitude of the clustering signals, the two-dimensional correlation function $\xi(\pi, r_p)$ contains additional information about the dynamics of the galaxies (Peebles 1980). At small projected separations $r_p$ random motions within virialized overdensity (e.g. clusters of galaxies) cause an elongation along the line-of-sight ($\pi$ direction) known as the 'finger-of-God' effect. On large scales, coherent streaming of galaxies into potential wells causes an apparent compression of structure along the line-of-sight (Sargent & Turner 1977; Kaiser 1987; Hamilton 1992). Various studies have used $\xi(\pi, r_p)$ to extract cosmological and dynamical information (e.g. Tinker 2007).

### 3.1.1 Real-space correlation

Because redshift-space distortion only affects the line-of-sight component of $\xi(\pi, r_p)$, we can recover the true space correlation function $\xi(r)$ by following the standard procedure of computing the projected correlation function:

$$w_p(r_p) = 2 \int_0^\infty d\pi \xi(r_p, \pi). \qquad (4)$$

In practice, we integrate along the line-of-sight direction out to $\pi = 30 h^{-1} \text{Mpc}$. This is large enough to include almost all correlated pairs but also stable enough to suppress noise from distant uncorrelated pairs. The projected correlation function can in turn be related to the real-space correlation function $\xi(r)$,

$$w_p(r_p) = 2 \int_0^\infty r dr \xi(r) (r^2 - r_p)^{-1/2}. \qquad (5)$$

If the real-space correlation function follows a power law $\xi(r) = (r/r_0)^\gamma$, we can infer its parameters: the correlation length $r_0$ and the power-law slope $\gamma$ from the best-fitting power law to $w_p(r_p)$ using the following deprojection:

$$w_p(r_p) = r_p \left(\frac{r_p}{r_0}\right)^{-\gamma} \Gamma\left(\frac{1}{2}\right) \Gamma\left(\frac{\gamma - 1}{2}\right) / \Gamma\left(\frac{\gamma}{2}\right). \qquad (6)$$

### 3.1.2 Cross-correlation

Related to the ACF is the CCF between two classes of galaxies. The CCF $\xi_{1,2}(r)$, measures the clustering of one type of galaxy around another. $\xi_{1,2}(r)$ is essentially the probability of finding a galaxies of type 1 around a galaxy of type 2 as a function of separation $r$. For our analysis, we use the cross-correlation version of the classical Davis & Peebles (1983) estimator:

$$\hat{\xi}_{1,2} = \frac{D_1 D_2}{D_1 R_2} - 1 \qquad (7)$$

where $D_1 D_2$ are the normalized counts of cross-pairs, while $D_1 R_2$ are cross-pairs of type 1 with random galaxies having the same redshift distributions as galaxies of type 2.

Recently, the CCF has been used extensively in clustering studies of galaxy properties (Zehavi et al. 2005; Wang et al. 2007; Coil et al. 2008; Chen 2009; Padmanabhan et al. 2009) since the ACF alone tells us little about how galaxies of various types relate to one another. Two populations of galaxies may both have comparable correlation strength, and yet be physically unrelated if they are spatially segregated. Specifically, the CCF may be used in conjunction with the ACFs of the galaxies to glean information as to how they mix statistically. For example, if the two populations are well mixed, i.e. they are consistent with being drawn stochastically from their respective ACF equally, their projected CCF $w_p^{1,2}(r_p)$ would follow that of the geometric mean of their ACFs

$$w_p^{1,2}(r_p) = \sqrt{w_p^{1,1}(r_p)w_p^{2,2}(r_p)}. \qquad (8)$$

On the other hand, if the populations were segregation between the populations and do not distribute evenly in all space, e.g. they sit on different haloes, the amplitude of $w_p^{1,2}(r_p)$ would be lower than that of $w_p^{1,2}(r_p)$ known in the literature as stochastic anti-bias (e.g. Blanton 2000; Swanson et al. 2008).

### 3.1.3 Bootstrap errors

We estimate the errors for the correlation measurements using a modified bootstrap (Efron 1981) method for spatial statistics known

<footnote n="3">³The use of correlation function to probe the growth of structure predates the advent of $\Lambda$ cold dark matter, especially in the study of faint blue galaxies (e.g. Efstathiou et al. 1991).</footnote>

<footnote n="4">⁴Normalized counts of galaxy pairs are weighted by the selection function of the galaxies involved.</footnote>
as marked-point bootstrap (Loh 2008). We first estimate the contribution of each galaxy to the correlation function – the marks:

$$\xi_i(r) = \sum_{j=1, j \neq i}^{n} \phi(|x_i - x_j| \in (r - dr, r + dr)} \quad (9)$$

such that

$$\xi(r) = \sum_{i=1}^{n} \xi_i \quad (10)$$

where *n* is the number of galaxies and *φ* is the chosen correlation estimator (e.g. $\hat{\xi}_{LS}$). Hence, the mark $\hat{\xi}_i$ associated with galaxy $x_i$ is the number of excess pairs at a distant *r* from $x_i$. These $\hat{\xi}_i$ are (roughly) independent, and identically distributed, and can be used to replicate *B* bootstrap samples by the sampling with replacement. An estimate of the correlation function $\hat{\xi}_j^*$ can be computed from each of the *j*th bootstrap samples. We use *B* = 999 bootstrap samples for our analysis. The distribution of $\hat{\xi}^*(r_i)$ can be used to estimate the error bars of the correlation function at each separation $r_i$. For an equivalent of $1\sigma$ errors, we ranked $\hat{\xi}^*(r_i)$ and take the 159th and 840th for the lower and upper error bounds. Because the $\hat{\xi}$ errors for the $r_i$ are correlated, when we estimate parameters for the two-parameter power-law model (equation 6), we refitted each bootstrap sample to obtain a two-dimensional distribution of the parameters.

We chose this method over the conventional jackknife methods because of the fragmented nature of the GALEX–SDSS footprint. Our bootstrap errors are consistent with the jackknife errors estimated by excluding one GALEX tile at a time. We note here that jackknife errors are known to overestimate the variance on small scale, and often bias the overall correlation estimates (Norberg et al. 2009). Since bootstrap is a form of internally estimated errors, and internally estimated errors are known not to reproduce externally estimated errors accurately (Norberg et al. 2009), our clustering results should be treated with caution.

## 4 DEPENDENCE OF CLUSTERING ON UV-OPTICAL COLOUR

Fig. 1 shows the NUV–*r* colour–magnitude diagram (CMD) in the local universe. All absolute magnitudes are *k*-corrected using the *k CORRECT* program by Blanton & Roweis (2007) and are quoted

in units with Hubble constant $H_0 = 100 h \text{ km s}^{-1} \text{ Mpc}^{-1}$, i.e. $M_r \equiv M_r - 5 \log(h = 1)$. Compared with the optical CMD (e.g. $g - r$ versus $M_r$), the NUV–*r* colour axis has a higher dispersive power. The scatter plot shows two clear sequences: red and blue, and an intermediate ‘valley’ of ‘green’ population. Conventional optical diagrams only display a single red sequence with an extended blue cloud. Hence, diagnostic diagrams like Fig. 1 separate galaxies into three natural groupings: (1) a red sequence of bulge-dominated galaxies with old stellar systems, (2) a blue sequence of star-forming systems consisting of mainly late-type galaxies and (3) galaxies in a ‘green valley’ that in principle might exhibit transitional properties (Martin et al. 2007; Schiminovich et al. 2007). Detailed luminosity functions and physical properties analysis of these galaxies derived from this UV-optical CMD has been reported elsewhere (Wyder et al. 2007; Martin et al. 2007; Salim et al. 2007; Schiminovich et al. 2007). Here, we investigate the clustering properties of these three subpopulations of galaxies, separated by the two tilted horizontal lines, using the two-point ACF, as well as their relation to each other using the CCF. Our results are not sensitive to 0.25 mag shift in the oblique equation used to classify the galaxies.

### 4.1 Dust

Dust content within each galaxy can modify their broad-band colours, hence the resultant CMD, as galaxies move from one group to another (Martin et al. 2007; Schiminovich et al. 2007). This is a concern, as it might affect our estimate of the correlation function by either increasing the scatter between groups or biasing the result in a systematic but unknown manner. Indeed, a fraction of the green valley galaxies are dusty star-forming galaxies whose intrinsic colour would have placed them on the blue sequence in the absence of dust (Wyder et al. 2007). However, the available procedures for dust correction are highly uncertain. They give inconsistent results depending on whether one uses a primarily photometric approach (Salim et al. 2007; Johnson et al. 2006) or one based on spectroscopic indices (Kauffmann et al. 2003). The former, for example, has the side effect of reducing the red sequence number counts substantially (Schiminovich et al. 2007; Heinis et al. 2009), while the latter approach requires spectroscopy of modest signal-to-noise ratio (S/N), which is not available for part of our sample.

<figure><img src="image_3.png" /><figcaption>**Figure 1.** NUV–*r* versus $M_r$ CMD of galaxies in the local Universe. The left-hand panel shows the full distribution. The middle panel shows the distribution drawn from face-on galaxies with $b/a > 0.6$, used to correct for the effect of dust on the CMD. The right-hand panel shows the distribution of edge-on/face-on ratio overlay on to the CMD contours. The green valley is dominated by the region of the CMD with high relative fraction of edge-on galaxies. By limiting ourselves to galaxies with $M_r < -20.3$ (dashed vertical lines) for the volume-limited sample, we also eliminate region of the CMD with the highest fraction of edge-on galaxies.</figcaption></figure>
A comprehensive analysis of the clustering as a function of star formation history with a dust-corrected CMD using the method of Johnson et al. (2006) is done in Heinis et al. (2009).

Here, we adopt a geometric approach. Because dust lanes in galaxies usually appear when viewed edge-on, we can reduce the effect of dust on galaxy colour by restricting our analysis to galaxies that are primarily face-on. The middle panel of Fig. 1 shows the ‘dust-corrected’ CMD obtained by including only galaxies with isophotal (minor-over-major) axis ratio $b/a > 0.6$. To the extent that each of the subpopulations has the same intrinsic distribution of $b/a$, removing edge-on galaxies will give a correct mixture of galaxies that mimic the intrinsic colour–magnitude distribution. Comparing the dust-corrected CMD with the CMD on the left-hand panel (hereafter the full distribution), the density of green galaxies around $M_r \sim -21$ is reduced substantially, with the contours showing a more pronounced ‘valley’ separating the sequence of galaxies.

The right-hand panel of Fig. 1 shows the median ratio of edge-on ($b/a < 0.4$) to face-on ($b/a > 0.8$) as a function of colour and magnitude, with the same density contours from the middle panel. The region with the highest edge-on/face-on ratio falls in the green valley and towards the faint end of the luminosity distribution (Choi et al. 2007; Martin et al. 2007; Schiminovich et al. 2007). For the galaxies with the magnitude range with $-23.5 < M_r < -19.0$ used to construct our flux-limited sample, the median $b/a$ for red and blue sequence galaxies are 0.73 and 0.72, respectively, while for green valley galaxies it is 0.66. Choi et al. (2007) find that edge-on galaxies are also statistically fainter due to the internal extinction, and it affects morphologically late-type galaxies (classified by eye) more than early-type galaxies. Hence, by limiting our analysis to face-on galaxies, we would reduce biases associated with differential dimming in addition to the biases from colour shifts. Following Choi et al. (2007), we restrict our analysis to galaxies with $b/a > 0.6$ as a proxy for a galaxy distribution derived from a dust-corrected CMD.

## 4.2 Luminosity bias

The clustering of galaxies is luminosity dependent. Norberg et al. (2002), Zehavi et al. (2005) and Li et al. (2006a) show that the amplitude of the projected correlation function $w_p$ increases monotonically as a function of luminosity (or stellar mass) on scales ranging from $0.2 h^{-1}$ Mpc to $10 h^{-1}$ Mpc. In order to study the colour dependence on clustering, or to compare the clustering intrinsic to the membership of discrete subpopulations, it is vital to remove this known luminosity dependence. The top two panels of Fig. 2 show the luminosity distribution of the red, green and blue population of galaxies with $M_r < -19$ from our flux-limited sample. The series of dashed histograms (in red, green and blue) show the original luminosity distributions. Blue galaxies are more numerous and less luminous, on average, compared to red and green galaxies, reflecting a steeper blue luminosity function at the faint-end (Wyder et al. 2007). To remove this luminosity dependence, we resample the luminosity histograms to match the number counts of the smallest of the three distributions at each magnitude bin. The result of this resampling is shown by the solid histograms. The left-hand panel shows histograms from the full distribution, while the right shows those from a dust-corrected distribution. There are 4177 (full) and 3148 (dust-corrected) galaxies in each of the resampled luminosity distribution of red, green and blue galaxies, with a common median luminosity of $M_r \sim -21.0$, about half a magnitude brighter than $M^*$, the typical luminosity (Blanton et al. 2003). Note that if the fundamental attribute that drive clustering is stellar mass, removing

the luminosity dependence like what we have done here would still leave residual clustering due to the difference in mass-to-light ratio of the respective colour-selected sample.

## 4.3 Volume-limited sample

While the resampled red, green and blue galaxies are matched in luminosity, they are not matched in volume and have different redshift distributions. This makes the interpretation of the physical correlation function problematic. To this end, we select the largest possible volume within our catalogue, using a redshift cut of $0.03 < z < 0.12$ and an additional luminosity cut at $M_r < -20.3$ to construct a volume-limited sample.<fnref n="5" /> The luminosity distributions of this sample are shown on the bottom row of Fig. 2. As before, the left-hand panel is for the full CMD, while the right-hand panel is for the dust-corrected (restricted) CMD. Similar to the flux-limited case, the (original) blue dashed histograms are substantially different from the red and the green, and are weighted more heavily towards the faint-end. We resample the histograms to match in luminosity. Note that in the case the procedure outlined in Section 4.2 essentially amounts to looking for an optimal function $W(L)$ such that $W(L)\Phi(L)$ gives the minimum luminosity function of the three subsamples. The new red, green and blue volume-limited subsamples each consists of 1971 (full) and 1226 (dust-corrected) galaxies with a common median luminosity of $M_r \sim 20.9$, almost identical to the flux-limited case.

## 4.4 Red, green, blue auto-correlation

Figs 3 and 4 show the two-dimensional ACF $\xi(\pi, r_p)$ as a function of line-of-sight ($\pi$) and projected ($r_p$) separation for the flux- and volume-limited samples. The panels on the first row in each figure are from the full distribution, while the second row is from the dust-corrected distribution. The panels are for red sequence (left), green valley (middle) and blue sequence (right) galaxies. We have binned $\pi$ and $r_p$ linearly at $1 h^{-1}$ Mpc. For all panels, the grey-scale has the same range, and the contours are boxcar smoothed at $2 h^{-1}$ Mpc. The $\xi(\pi, r_p)$ contours indicate the constant probability of finding pairs at a given $\pi$ and $r_p$. The heavy solid line marks $\xi(\pi, r_p) = 1$, and contours are spaced with increment (inner) and decrement (outer) by a factor of 2. The effect of redshift-space distortion is clearly seen in all panels, manifested by their departure from isotropy (concentric circles). At small $r_p$, the contours are elongated along the line-of-sight ($\pi$) due to virial motions of galaxies in clusters. At large $r_p$, the contours are compressed in the $\pi$ direction due to the coherent large-scale streaming as galaxy infall into potential well. We recover the results of Zehavi et al. (2005): red sequence galaxies show the strongest finger-of-God effect and the larger correlation amplitude; and all three subpopulations (of all samples and distribution) show clear signs of large-scale compression.

We see a clear finger-of-God effect for green valley galaxies, but not for blue sequence galaxies. In Fig. 3, if one compares the first contour inwards from $\xi = 1$ (heavy line) for red galaxies to the $\xi = 1$ contour of green galaxies, green and red galaxies have identical kinematics, differing only by a scaling in the amplitude. The blue sequence appears to have a dynamical structure dominated by large-scale streaming distinct from both red and green galaxies, similar

<footnote n="5">⁵This is not strictly volume-limited since NUV is not complete at these redshifts. An additional weighting is applied using the luminosity function of Wyder et al. (2007).</footnote>
<figure><img src="image_4.png" /><figcaption>**Figure 2.** The luminosity distribution of our flux-limited (top row) and volume-limited (bottom row) samples. The two panels on the left are for the full distribution, the right is for dust-corrected CMD, using only galaxies with $b/a > 0.6$. The dashed histograms show the original luminosity distribution for each of the subsamples of red, green and blue galaxies. We resampled the luminosity to match the number counts of the minimal of the three distribution at each magnitude bin, shown as solid histograms. Even for the volume-limited samples, the variations in the original luminosity distribution among the three subsamples are substantial. Blue galaxies are heavily weighted towards the less luminous end compared to red or green galaxies.</figcaption></figure>

to the optical blue cloud sample of Zehavi et al. (2005). We will consider the implications of this in the discussion below. The same conclusion can be drawn from the volume-limited sample shown in Fig. 4. The dust-corrected samples (bottom row) are substantially noisier. We also note that the contours of the blue sequence of the volume-limited full sample (top right panel) are roughly circular for $\sim 4 < \pi$, $r_p < 10 h^{-1}$ Mpc where the finger-of-God elongation is balanced by compression due in infall.

Figs 5 and 6 show the projected correlation function $w_p(r_p)$ for the flux-limited and volume-limited samples. The coloured (red, green, blue) points are the respective measured correlation function. The panel on the left (right) is for the full (dust-corrected) distribution. As expected, in all four panels, the red sample has the largest amplitude, the blue the lowest and green in between.

On large scales ($r_p \gtrsim 1 h^{-1}$ Mpc), the ratio of projected ACF of the respective samples is constant as a function of scales, as expected from linear theory. All correlation functions appear to have a form that is well fit by a broken power law with the dust-corrected distribution having a more pronounce convexity. It is noteworthy that the flux-limited analysis (Fig. 5) is more noisy than the volume-limited analysis (Fig. 6), despite having a factor of 2 greater numbers. We believe that this is due to the inclusion of uncorrelated (in redshift-space) galaxies, which dilute the clustering signal, especially for blue galaxies on small scales. From here onwards, we will restrict our analysis to the volume-limited sample.

When we fit the projected correlation function of the full distribution with the standard two-parameter power law for scales from $1 < r_p < 10 h^{-1}$ Mpc, red and green galaxies appear to have similar
<figure><img src="image_5.png" /><figcaption>**Figure 3.** Contours and normalized counts of the two-dimensional correlation function $\xi(\pi, r_p)$ for red (left), green (middle) and blue (left) populations for the flux-limited sample. The panels on the top row are for the full distribution, bottom row are restricted to face-on galaxies ($b/a > 0.6$). The contours obtained after $2 \times 2 h^{-1}$ Mpc boxcar smoothing. The levels are 0.0 (dotted lines), 0.25, 0.5, 1.0 (heavy lines), 2.0, 4.0, 8.0 and 16.0. Red galaxies have the strongest finger of God effect (extension in the $\pi$ direction). The finger-of-God effect is seen in the green sample as well, while it is not present in the blue. Galaxies in the green valley appear to have clustering characteristics of the red sequence.</figcaption></figure>

slope with $\gamma \sim 1.93$, while blue galaxies have a substantially shallower slope, with $\gamma \sim 1.75$. The best-fitting correlation lengths $r_0$ are $7.5 h^{-1}$ Mpc (red), $5.3 h^{-1}$ Mpc (green) and $3.9 h^{-1}$ Mpc (blue). The result of the covariance analysis using 999 bootstrap samples is shown as confidence interval contours – 68 per cent (inner) and 95 per cent (outer) in Fig. 7 (left-hand panel). These results are in excellent agreement with Heinis et al. (2009). The correlation function from the dust-corrected distribution is substantially noisier, in part due to the smaller number of galaxies. Applying the same power-law fit, we obtain a larger uncertainty in $\gamma$, but comparable uncertainty in $r_0$. This is shown on the right-hand panel of Fig. 7 and is tabulated in Table 1.

On the left-hand panel, the projected correlation function of green galaxies is evidently intermediate between the red and the blue on a range of scales ($0.2 < r_p < 6 h^{-1}$ Mpc), and runs faithfully parallel with the red correlation function, but with a lower amplitude. The picture for the dust-corrected distribution is slightly different. While the green still runs in between the red and blue for $r_p < 1 h^{-1}$ Mpc and mostly parallel to the red with a lower amplitude for broad range of scales, for larger scales ($r_p \gtrsim 2 h^{-1}$ Mpc), it coincides with the blue. This is a consequence of the more prominent two-halo excess on large scales seen in the blue correlation function.

One way to compare the clustering between subpopulations of galaxies is through their relative bias (e.g. Norberg et al. 2002),

defined as

$$ \frac{b_i}{b_*}(r) = \sqrt{\frac{\xi_i(r)}{\xi_*(r)}} \approx \frac{r_{0,i}^{\gamma_i/2}}{r_0^{\gamma_i/2}} r^{(\gamma-\gamma_i)/2} \quad (11) $$

where $\xi_i$ is the correlation function of interest and $\xi_*$ is the fiducial correlation function that all correlation function is compared with. The approximate equality holds when the correlation function takes a power-law form. Zehavi et al. (2005) found that a typical $L^*$ galaxies in the local universe with $-20 < M_r < -21$ from the SDSS main spectroscopic sample have a fiducial power-law correlation function with $\xi_*(r) = (r/5 h^{-1} \text{ Mpc})^{-1.8}$ on scale $1 < r_p < 10 h^{-1} \text{ Mpc}$. By fitting the correlation function using a constant $\gamma = 1.8$ on scales $1 < r_p < 10 h^{-1} \text{ Mpc}$, we obtained the bias relative at $r_p = 5 h^{-1} \text{ Mpc}$ to a fiducial $L^*$ galaxies for each of our subpopulations: $b_{\text{red}}/b_* = 1.53 \pm 0.08$, $b_{\text{green}}/b_* = 1.08 \pm 0.06$ and $b_{\text{blue}}/b_* = 0.81 \pm 0.06$ for the full-sample, and $b_{\text{red}}/b_* = 1.47 \pm 0.10$, $b_{\text{green}}/b_* = 1.13 \pm 0.09$ and $b_{\text{blue}}/b_* = 0.92 \pm 0.09$ for the dust-corrected samples (Table 1). In Fig. 8, we plot $b_i/b_*(r = 5 h^{-1} \text{ Mpc}) = (r_0/5 h^{-1} \text{ Mpc})^\gamma$ as a function of the comoving number density and compare our values with those obtained by Zehavi et al. (2005). Comoving number density of our sample is estimated using the $1/V_{\text{max}}$ method and further corrected to match those obtained by Wyder et al. (2007). While all the samples by construction have almost identical comoving density, there is a range of relative bias, with the red having
<figure><img src="image_6.png" /><figcaption>**Figure 4.** Contours and normalized counts of the two-dimensional correlation function $\xi(\pi, r_p)$ for red (left), green (middle) and blue (left) populations for the volume-limited sample. The panels on the top row are for the full distribution, bottom panels are restricted to face-on galaxies ($b/a > 0.6$). The contours obtained after $2 \times 2 \text{ h}^{-1} \text{ Mpc}$ boxcar smoothing. The levels are 0.0 (dotted lines), 0.25, 0.5, 1.0 (heavy lines), 2.0, 4.0, 8.0 and 16.0. Red galaxies have the strongest finger of God effect (extension in the $\pi$ direction). Similar to Fig. 3, but with lower S/N, the finger-of-God effect is seen in the green sample as well, while it is not present in the blue.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Figure 5.** The projected correlation function, $w_p(r_p)$, of red sequence, green valley and blue sequence galaxies for the flux-limited sample. The left (right) panel is from the full (dust-corrected) distribution. From red to blue, there is an increase in convexity, with the progressive shallowing of the slope on large scales, indicating an excess of two-halo signals at scale ($r_p \gtrsim 1 \text{ h}^{-1} \text{ Mpc}$).</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>**Figure 6.** The projected correlation function, $w_p(r_p)$, of red sequence, green valley and blue sequence galaxies for the volume-limited sample. Similar to Fig. 5, from red to blue, there is an increase in convexity, with the progressive shallowing of the slope on large scale, indicating an excess of two-halo signals, at scale ($r_p \gtrsim 1 h^{-1} Mpc$). For the full distribution (left panel), the green $w_p(r_p)$ is parallel to the red with a lower amplitude for scales $0.2 < r_p < 10 h^{-1} Mpc$. For the dust-corrected distribution (right panel), the blue $w_p(r_p)$ approaches the green on large scales.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Figure 7.** Confidence contours of the power-law fits to $w_p(r_p)$ on scales $1 h^{-1} Mpc < r_p < 10 h^{-1} Mpc$ for the full (left panel) and dust-corrected (right) distributions from the volume-limited sample. The three colours are for red, green and blue subsamples, with the inner (outer) contours encircling the 68 (95) per cent confidence region in the $r_0-\gamma$ space.</figcaption></figure>

<table><caption>**Table 1.** Power-law fit to the projected ACF of volume-limited samples ($1 h^{-1} Mpc < r_p < 10 h^{-1} Mpc$).</caption><thead><tr><th>Full-sample</th><th>N</th><th>z</th><th>Median M<sub>r</sub></th><th>r<sub>0</sub></th><th>γ</th><th>Relative bias</th><th>χ<sup>2</sup>/d.o.f.</th></tr></thead><tbody><tr><td>Red.........</td><td>1971</td><td>(0.03, 0.12)</td><td>-20.9</td><td>7.5 ± 0.27</td><td>1.94 ± 0.07</td><td>1.53 ± 0.08</td><td>1.36</td></tr><tr><td>Green........</td><td>1971</td><td>(0.03, 0.12)</td><td>-20.9</td><td>5.3 ± 0.19</td><td>1.93 ± 0.08</td><td>1.08 ± 0.06</td><td>0.40</td></tr><tr><td>Blue.........</td><td>1971</td><td>(0.03, 0.12)</td><td>-20.9</td><td>3.9 ± 0.25</td><td>1.73 ± 0.10</td><td>0.81 ± 0.06</td><td>1.54</td></tr><tr><td>Face-On</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Red.........</td><td>1226</td><td>(0.03, 0.12)</td><td>-20.9</td><td>7.6 ± 0.38</td><td>1.81 ± 0.11</td><td>1.47 ± 0.10</td><td>2.47</td></tr><tr><td>Green........</td><td>1226</td><td>(0.03, 0.12)</td><td>-20.9</td><td>5.5 ± 0.24</td><td>1.92 ± 0.15</td><td>1.13 ± 0.09</td><td>0.48</td></tr><tr><td>Blue.........</td><td>1226</td><td>(0.03, 0.12)</td><td>-20.9</td><td>4.6 ± 0.31</td><td>1.74 ± 0.17</td><td>0.92 ± 0.09</td><td>1.17</td></tr></tbody></table>
<figure><img src="image_10.png" /><figcaption>**Figure 8.** Bias relative to the typical $L^*$ galaxy with a $\xi^{fid}(r) = (r/r_0)^{-1.8}$ at $r_p = 5 h^{-1} \text{ Mpc}$ versus comoving number density. We also plotted points from SDSS (Zehavi et al. 2005). The solid circles are from their luminosity range analysis, i.e. galaxy with absolute magnitude between $M_1$ and $M_2$, while the open circles are from their luminosity threshold analysis, i.e. $M_r < M_{lim}$. While far from unique, the higher red bias suggests that on average red galaxies are more likely to be satellite galaxies than a typical galaxy with those number densities. If we assume the lower blue and green galaxies are all central galaxies in the halo they reside in, the lower observed number densities suggest that only a fraction of these haloes, as described by their bias, host a blue or green galaxy, as their expected number density is much higher.</figcaption></figure>

the highest bias and blue the lowest. The spread among the bias is much smaller among the dust-corrected samples. Red galaxies have clustering strength above the nominal strength (inferred from a typical SDSS galaxy) for a given number density. One plausible scenario suggests that red galaxies have a larger than average satellite fraction. On the other hand, blue and green galaxies have lower observed number density compared to their clustering. If we assume all blue and green galaxies are central galaxies in the halo they reside in, the lower observed number densities ($\sim 0.001 h^3 \text{ Mpc}^{-3}$) suggest that only a small fraction of these haloes, as described by their lower bias, host a blue or green galaxy, as their expected number density (inferred from SDSS to be $\sim 0.01 h^3 \text{ Mpc}^{-3}$) is much higher. These fractions are increased if the average phases of green and blue are shorter than the lifetime of the haloes (Haiman & Hui 2001; Martini & Weinberg 2001). In the case of the green galaxies, the transitional nature of these galaxies may be closely related to the AGN they host, or to minor mergers and starbursts, with triggering cycles corresponding to the duty cycles for these respective phenomena. We defer a more detail analysis of halo occupation and the life cycle of transition galaxies using SFR tracers of different lags to a future paper (Heinis et al. in preparation). Using a theoretical bias function of dark matter haloes (Seljak & Warren 2004) and normalizing to $M_* = 10^{12} h^{-1} \text{ M}_\odot$,<fnref n="6" /> red galaxies cluster similar to haloes with mass $\sim 10^{12.7} h^{-1} \text{ M}_\odot$, green galaxies $10^{12.2} h^{-1} \text{ M}_\odot$ and blue galaxies $10^{11.6} h^{-1} \text{ M}_\odot$.

## 4.5 Cross-correlation functions

Fig. 9 shows the two-dimensional $\xi(r_p, \pi)$ CCF for the three cross-pairs of galaxies in our volume-limited sample, with the panels on the top for the full distribution, and the bottom panels for the dust-corrected distribution. The large-scale infall effect is seen in all three panels suggesting that the galaxies, on average, trace a similar matter distribution as expected from linear theory. On small scales, the finger-of-God effect is strongest for the red–green CCF, intermediate for red–blue and green–blue. In Fig. 10, we plot the relative amplitude of $\xi$ at the projected distance $r_p = 1 h^{-1} \text{ Mpc}$ as a function of line-of-sight distance $\pi$. The circles are for green–red cross-correlation, squares for red–blue and triangles for green–blue. Green–red have largest amplitude for a wide range of $\pi$, while red–blue and green–blue have smaller amplitudes. The stronger finger-of-God component suggests that the dynamics of red and green are more strongly coupled compared with red–blue or green–blue pairs. Note that red–blue CCF have a very different $\xi(\pi, r_p)$ compared with the green ACF, the former has a weaker finger-of-God and stronger infall compression.

As discussed in Section 3.1.2, one way to understand the relationship between two populations is to compare the projected CCF between the two with their geometric mean. If the two populations are mixed evenly, the CCFs should trace the geometric mean. If they are spatial segregated (partially) beyond what was expected from their respective ACF, the CCF should be systematically below the geometric mean. The projected CCFs for the red and blue galaxies $w_p^{r,b}(r_p)$ are plotted as open circles in Fig. 11. Also plotted for comparison are the red and blue ACFs (solid lines), and the red–blue geometric mean $w_p^{r,b}(r_p)$ (dashed lines). On scales larger than $\sim 3 h^{-1} \text{ Mpc}$, the $w_p^{r,g}(r_p)$ approaches the geometric mean. For scales $0.2 < r_p < 2 h^{-1} \text{ Mpc}$, $w_p^{r,g}(r_p)$ is systematically below the geometric mean for the full distribution (left-hand panel). For the dust-corrected distribution (right-hand panel), $w_p^{r,b}(r_p)$ starts to inch below the geometric mean only from $r_p \lesssim 0.7 h^{-1} \text{ Mpc}$ onwards. Our results are consistent with the partial morphology segregation within galaxy clusters (Dressler 1980). At the one-halo regime ($r_p \lesssim 1–3 h^{-1} \text{ Mpc}$), the relevant scales for galaxy clusters, red galaxies tend to occupy the cores of the clusters, while blue galaxies tend to lie towards the periphery. The lower level of spatial mixing on these scales suppresses amplitude of the CCF. Note that the green ACF (green solid line) does trace the red–blue geometric mean on large scales, suggesting that blue and red galaxies do mix evenly on these scales.

Fig. 12 shows the projected CCF between the green and red $w_p^{g,r}(r_p)$, and green and blue $w_p^{g,b}(r_p)$. The solid green line is the ACF of the green sample ($w_p^{gg}(r_p)$), while the dashed lines are the geometric means: $w_p^{g,r}(r_p)$ and $w_p^{g,b}(r_p)$. For the full distribution (left-hand panel), $w_p^{g,r}(r_p)$ is systematically below $w_p^{g,r}$ for a range of scales within errors. $w_p^{g,b}(r_p)$ are consistent, or slightly above $w_p^{g,b}(r_p)$ for $r_p \gtrsim 1 h^{-1} \text{ Mpc}$, and systematically below for $0.3 \lesssim r_p \lesssim 1 h^{-1} \text{ Mpc}$. These inferences are shown more clearly in Fig. 13 (left for full, right for dust-corrected), where we plot the normalized cross clustering strength – the ratio $w_p^{g,x}/w_p^{gg}$, where x is either red or blue – relative to the auto-correlation of the green galaxies. The additional solid lines are the normalized ACF of the red and blue. This suggests that red and green are consistent with being drawn from the same statistical sample on average for a large range of scales with a slight anti-bias on small scales. For the green and blue, there is a stronger anti-bias for $r \lesssim 1 h^{-1} \text{ Mpc}$ as they avoid each other on these scales. Our results suggest that on scales typical

<footnote n="6">⁶The halo mass that a typical $L^*$ galaxy resides in.</footnote>
<figure><img src="image_11.png" /><figcaption>**Figure 9.** Contours and normalized counts of the two-dimensional CCF $\xi(\pi, r_p)$ for red–blue (left), green–red (middle) and green–blue (left) for the volume-limited sample. The panels on the top row are for the full distribution, bottom panels are restricted to face-on galaxies ($h/a > 0.6$). The contours obtained after $2 \times 2 h^{-1}$ Mpc boxcar smoothing. The levels are 0.0 (dotted lines), 0.25, 0.5, 1.0 (heavy lines), 2.0, 4.0, 8.0 and 16.0. The large-scale infall effect is seen in all three panels. On small scales, the finger-of-God effect is strongest for the red–green, intermediate for red–blue and weakest for green–blue, as expected. Note that red–blue CCF $\xi(\pi, r_p)$ has a weaker finger-of-God and stronger infall compression compare to the green valley ACF (Fig. 4, middle panels).</figcaption></figure>

of dark matter haloes, galaxies drawn from the green and blue populations are less associated than would be predicted by their respective ACF.

# 5 DISCUSSION

## 5.1 Comparison with DEEP2 $z \sim 1$

The GALEX NUV-selected sample used in our analysis is directly comparable to the high-redshift ($z \approx 1$) galaxy sample of the DEEP2 survey (Coil et al. 2008) since their optical selection mimics the rest-frame NUV. In that paper, clustering analysis was done on green valley galaxies for the first time. In their analysis, the green valley $\xi(r_p, \pi)$ appears to display kinematic structure intermediate between red and blue galaxies, with an intermediate finger-of-God effect and an intermediate overall clustering amplitude. When redshift-space distortion is removed, the projected correlation function $w_p(r_p)$ shows a scale dependence convergence. At $r_p > 1 h^{-1}$ Mpc the functions converge to those of red populations, while for $r_p < 1 h^{-1}$ Mpc it tends towards the clustering of the blue population.

In contrast, we find that our green valley $\xi(r_p, \pi)$ has a strong finger-of-God effect consistent with that measured for red sequence galaxies, differing only in their amplitude. This can be seen most

clearly in the $w_p(r_p)$ analysis for the volume-limited sample (Fig. 6), where the green function has the form similar to that of the red function for scales $0.2 < r_p < 15 h^{-1}$ Mpc, but is displaced to lower amplitude. It is noteworthy that the slope of the blue correlation function begins shallowing at $r \sim 1 h^{-1}$ Mpc, displaying the kind of one- and two-halo segregation expected from a correlation function dominated by central galaxies. In contrast to Coil et al. (2008), the green ACF converges to that of the blue at $r_p > 1 h^{-1}$ Mpc. We emphasize that the green galaxy sample of Coil et al. (2008) is defined differently from ours. We divide the CMD into three disjoint parts to separate our galaxies into the subpopulations while in Coil et al., the green overlaps both the red and the blue.

## 5.2 Green valley

Many recent studies reveal that blue sequence mass has remained roughly constant since $z \sim 1$ (Blanton 2006; Faber et al. 2007) because the average ongoing star formation over $0 < z \lesssim 1$ is balanced by mass flux off the blue sequence, presumably towards the build-up of the red sequence since $z \approx 1$ (Bell et al. 2004; Martin et al. 2007). Hence, green valley galaxies occupy a position where one expects to find many transitional galaxies. Martin et al. also note that the AGN fraction peaks at the green valley. For our
<figure><img src="image_12.png" /><figcaption>**Figure 10.** This figure shows the relative amplitude of $\xi$ at projected distance $r_p = 1 h^{-1}$ Mpc as a function of the line-of-sight distance $\pi$. The circles are for green-red cross-correlation, squares for red-blue and triangles for green-blue. Green-red have largest amplitude for a wide range of $\pi$, while red-blue and green-blue have smaller amplitudes. These results suggest that the finger-of-God component of green-red CCF is stronger than those from red-blue or green-blue.</figcaption></figure>

green valley definition using the oblique colour cuts from Fig. 1, the AGN fraction is ~50 per cent.<fnref n="7" />

In their study of AGN using SDSS, Constantin & Vogeley (2006) found that the redshift-space two-point correlation function of Seyferts is less clustered than that of low-ionization nuclear emission-line regions survey (LINERS). However, Miller et al. (2003) and Li et al. (2006b) found that AGN as a whole cluster similarly as typical $L^*$ galaxies, if one takes into account luminosity bias. Wang & Kauffmann (2008) argue that almost all galaxies in the local universe with stellar mass $\gtrsim 10 h^{-1} M_{\odot}$ have active nuclei, often LINERS with lines too weak to be detected spectroscopically in SDSS. Our green galaxies, with luminosities peaked at $M_r \sim -21$, have stellar masses well above $10 h^{-1} M_{\odot}$, and could be dominated by LINERS (either detected or undetected). This would in part explain the clustering effect we see in Figs 3 and 4 (middle panels) where those bulge-dominated LINERS display kinematics similar to red sequence (primarily non-active) galaxies. The reason that AGN from $r$-band-selected survey (e.g. Miller et al. 2003) cluster on average much like typical $L^*$ galaxies may merely be coincidental.

The lack of such behaviour in the $z \sim 1$ green valley (Coil et al. 2008) may be attributable to evolution in the AGN population. There may be fewer LINERS, or the red sequence galaxies may have been experiencing more gas infall, feeding their AGN. We note that with a relative bias ~1.1, green galaxies cluster similarly to a typical galaxy with $L^* - 0.5 \approx -21$, the median luminosity of our sample.

As was discussed in Section 4.1, a substantial fraction of the galaxies in the green valley are dusty star-forming galaxies. Because dust content can modify the CMD we use to separate the galaxies into red, green and blue populations, this might potentially change

the behaviour of the correlation function. We argue here that to the extent that dust modifies the number counts of green galaxies, it is to promote the migration from the blue sequence to the green valley (Choi et al. 2007). Our results from Fig. 4 (for the volume-limited sample) suggest that their influence is modest at best, and merely acts as additional Poisson noise to the two-dimensional $\xi(\pi, r_p)$ without altering the kinematics in the sample. The projected correlation functions of Fig. 6 show similar characteristics.

## 5.3 Blue (UV) versus blue (optical)

In their studies using $g-r$ colour, Zehavi et al. (2005) found that the correlation function of blue galaxies exhibits a lower amplitude and shallower correlation functions. By fitting the $w_p(r_p)$ with halo occupation distribution (HOD) models, they found that the majority (~70–90 per cent) of blue galaxies are central galaxies in dark matter haloes, usually haloes with mass $\lesssim 10^{13} M_{\odot}$, in contrast to only the most luminous red galaxies being central objects of massive haloes, while the majority of (less luminous) red galaxies are satellites. This fits well with the notion that blue galaxies are field galaxies – the central objects of low-mass haloes; red galaxies, with the exception of the central galaxy in clusters and groups, are mostly satellites of massive haloes.

One can infer a similar conclusion from the auto-correlation $w_p(r_p)$ plot for the blue galaxies (Fig. 6). If we decompose the correlation function into two parts, one due to the one-halo term and the other due to the two-halo term, blue galaxies show strong two-halo excess on scales $\gtrsim 1 h^{-1}$ Mpc, implying that on those such scales, the majority of galaxy pairs are from different haloes. This was seen in the optical analysis of Zehavi et al. (2005) but prominently in our blue sequence sample. For the dust-corrected volume-limited sample (right-hand panel of Fig. 6), the amplitude of the ACF of the blue sample beyond ~3 $h^{-1}$ Mpc actually rises to match the clustering strength of the green sample.

Li et al. (2006a) found that the dependence of $w_p(r_p)$ on optical $g-r$ colour extends beyond $5 h^{-1}$ Mpc, suggesting that the conventional wisdom that clustering should converge at large scales may not occur until at a scale larger than $5 h^{-1}$ Mpc. Here, we argue that this is an effect due to the mixture of population between blue galaxies, and green and red, and that the optical $g-r$ colour does not have sufficient power to separate the green from the blue. Blue galaxies, by themselves, have a very pronounced two-halo excess and are dominated entirely by central galaxies, flattening the correlation function substantially at large scales, compared to the red and green population. To the extent that one can eliminate or correct for the internal reddening due to dust, $NUV - r$ colour is very efficient in isolating a sequence of purely star-forming galaxies.

# 6 SUMMARY AND CONCLUSIONS

We have constructed a *GALEX* and SDSS-matched catalogue, where we have used the GR3 catalogue from *GALEX* and the SDSS DR5 main spectroscopic galaxy sample. We construct the galaxy distribution of $NUV - r$ versus $M_r$ CMD, and divide the distribution into populations of red sequence, green valley and blue sequence. Since our main goal is to study the colour dependence of clustering, we took substantial care in matching the luminosity distribution of each population. For each population, we measure the two-dimensional correlation function $\xi(\pi, r_p)$, and the one-dimensional projected correlation function $w_p$. We also perform cross-correlation analyses between each of the subpopulations.

<footnote n="7">⁷This is a lower limit since the fraction of low-luminosity AGN and composite objects is unknown.</footnote>
<figure><img src="image_13.png" /><figcaption>**Figure 11.** The projected CCFs between red and blue sequence galaxies $w_p^{r,b}(r_p)$ are shown as open circles. The solid lines are the ACF of the red and blue galaxies. The dashed line is the red-blue geometric mean $w_p^{g,b}(r_p)$. On scales larger than ~3 $h^{-1}$ Mpc, the cross-correlation approaches the geometric mean. For scales $0.2 h^{-1}$ Mpc < $r_p$ < $2 h^{-1}$ Mpc, the CCF is systematically below the geometric mean. This result suggests that on small scales, blue and red galaxies are spatially segregated beyond what was expected from their ACF. This is consistent with the morphology density relation (Dressler 1980).</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Figure 12.** The projected CCF of green sample with blue $w_p^{g,b}(r_p)$ (blue open circles) and red $w_p^{r,g}(r_p)$ (red open circles) subsamples. The solid line shows the ACF of the green sample. The dashed lines are the geometric mean of the green–red (in red) and green–blue (in blue) auto-correlation. On the left (right) we show the analysis from the full (dust-corrected) distribution from the volume-limited sample (see Fig. 13 for more details).</figcaption></figure>

Our principal finding is that the red sequence and green valley appear to show similar clustering properties, as expressed in the finger-of-God effect in the ACF. The projected correlation function is consistent with red and green galaxies residing as satellites of massive haloes, while the blue sequence shows what appears to be a clear two-halo signature, hence primarily serving as central galaxies of less massive haloes. The CCF also shows that green and blue galaxies, on small scales, are not a mere statistical mix, but are spatially segregated from each other.

The findings would appear to place the green valley population with the red sequence. The green valley would largely consist of

massive galaxies that reside in massive haloes, and which cluster like the red sequence. We note that Martin et al. (2007), Wyder et al. (2007) and Salim et al. (2007) show that a large fraction of type II AGN are found in the green valley. Salim et al. show that in the plot of specific SFR versus stellar mass, the AGN tend to be found in massive (>3 × 10$^{10}$ M$_{\odot}$) galaxies. The AGN occupy a region in these plots that strongly resembles that of the reddest class of galaxies, the ‘no-H$\alpha$’ red sequence galaxies. Significantly, the AGN are clearly offset from the locus of the blue sequence, in the plot of specific SFR versus mass. The significance is that while a minority of AGN are found with properties that coincide with those
<figure><img src="image_15.png" /><figcaption>**Figure 13.** The ratio of projected correlation function ($w_p(r_p)$) with the ACF ($w_{gg}^p(r_p)$) of the green subsample. The two ratios from the CCFs, $w_p^{g,r}(r_p)$ and $w_p^{g,b}(r_p)$, are indicated by coloured open circles. Dashed lines are for geometric means, as in Fig. 12. The additional solid lines are ratios from the ACF of red and blue, respectively. The fact that both $w_p^{g,r}(r_p)$ and $w_p^{g,b}(r_p)$ are below their respective geometric mean on small scales suggests that both red and blue galaxies are found in different environments compared to the green on these scales, albeit with large uncertainties.</figcaption></figure>

of the more massive blue sequence galaxies, green valley galaxies – the subsample with the largest AGN fraction – exhibit properties similar to those of the red sequence, but showing mildly elevated star formation.

In this study, we have shown that the green valley population clusters in ways that are characteristic of, but also less strongly than, the red sequence. One may suggest that these studies paint a picture in which both the properties of the green valley and the ‘demographics’ are different from those of the blue sequence, at the present epoch. These findings do not necessarily contradict the studies that find an increase in the total mass of red sequence galaxies since $z \sim 1$. They do suggest, however, that if blue sequence galaxies evolve by some process to the green valley, and ultimately to the red sequence, such evolution must also be accompanied by a transition from the field environment to a group/cluster environment. Such a change could conceivably occur if the blue population resides along filaments that infall into clusters over time. Our cross-correlation results show that green galaxies avoid both red and blue galaxies on small scales, which is consistent with the change in environment hypothesis. We note that models like ram-pressure stripping (Gunn & Gott 1972), starvation of (cold) gas and the virial shock heating model of Dekel & Birnboim (2006) naturally incorporate environmental factors in their mechanism for colour transformation in galaxies.

It is also possible that the downsizing (Cowie et al. 1996) effect is so strong that most star-forming galaxies are evolving rapidly with redshift (e.g. Tinsley 1968). One must recall that star-forming activity at $z \sim 1$ resides in considerably more massive galaxies, and that a colour-based population separation, as we have done, will refer to much higher masses; the rest-frame colours may be similar, but the fundamental nature of the galaxies would not.

One may speculate that the green valley is occupied by nominally red galaxies that experience the infall of a gas-rich system that either induces star formation and/or fuels the AGN, rendering it visible via its emission lines. However, the feedback of an AGN might inhibit

star formation and move a blue sequence galaxy to the green valley. Any number of environmental effects (e.g. harassment, starvation) might speed the consumption of gas in a disc, again moving a galaxy to the green valley. A small sample of optically quiescent members of the green valley that nominally have an UV excess shows clear star formation signatures (spirals) when imaged in the UV using *Hubble Space Telescope* (Salim & Rich 2009). There is still the issue of the origin of the low-mass red sequence, and the evolution of blue sequence, into the green valley and ultimately the red sequence might have an important role in the growth of the lower mass portion of the red sequence. This was partially addressed by semi-analytical work of Benson et al. (2003) and Bower et al. (2006).

In contrast to the construction of CMDs for stellar populations, the environment, for galaxies, is a critical physical variable in their evolution. This is true in the sense of both their dark matter environment and the presence of detectable companion stellar systems. In considering the major processes driving galaxy evolution, it would appear that evolution of both of these observables must be considered as a function of look-back time.

## ACKNOWLEDGMENTS

YSL would like to thank C. Hirata, S. Salim, C. Park, J. Kormendy and Z. Zheng for helpful discussions. This work has made extensive use of IDLUTILS⁸ and Goddard IDL libraries. RMR acknowledges support from grant GO-11182 from the Space Telescope Science Institute.

GALEX is a NASA Small Explorer, launched in 2003 April. We gratefully acknowledge NASA’s support for construction, operation and science analysis for the GALEX mission, developed in

<footnote>8http://spectro.princeton.edu/idlutils</footnote>
cooperation with the Centre National d'Etudes Spatiales of France
and the Korean Ministry of Science and Technology.

Facilities: GALEX, SDSS

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