The zCOSMOS Survey. The dependence of clustering on luminosity and stellar mass at z=0.2-1 Based on observation undertaken at the European Southern Observatory (ESO) Very Large Telescope (VLT) under Large Program 175.A-0839. Also based on observations with the NASA/ESA Hubble space Telescope, obtained at the Space Telescope Science Institute, operated by AURA Inc., under NASA contract NAS 5-26555, with the Subaru Telescope, operated by the National Astronomical Observatory of Japan, with the telescopes of the National Optical Astronomy Observatory, operated by the Association of Universities for Research in Astronomy, Inc. (AURA) under cooperative agreement with the National Science Foundation, and with the Canada-France-Hawaii Telescope, operated by the National Research Council of Canada, the Centre Nationla de la Recherche Scientifique de France and the University of Hawaii.

The zCOSMOS Survey. The dependence of clustering on luminosity and stellar mass at thanks: Based on observation undertaken at the European Southern Observatory (ESO) Very Large Telescope (VLT) under Large Program 175.A-0839. Also based on observations with the NASA/ESA Hubble space Telescope, obtained at the Space Telescope Science Institute, operated by AURA Inc., under NASA contract NAS 5-26555, with the Subaru Telescope, operated by the National Astronomical Observatory of Japan, with the telescopes of the National Optical Astronomy Observatory, operated by the Association of Universities for Research in Astronomy, Inc. (AURA) under cooperative agreement with the National Science Foundation, and with the Canada-France-Hawaii Telescope, operated by the National Research Council of Canada, the Centre Nationla de la Recherche Scientifique de France and the University of Hawaii.

B. Meneux Max-Planck-Institut für Extraterrestrische Physik, Giessenbachstrasse, 85748 Garching-bei-München, Germany Universitäts-Sternwarte, Scheinerstrasse 1, Munich D-81679, Germany    L. Guzzo INAF-Osservatorio Astronomico di Brera, Via Bianchi 46, I-23807 Merate (LC), Italy    S. de la Torre INAF-Osservatorio Astronomico di Brera, Via Bianchi 46, I-23807 Merate (LC), Italy Laboratoire d’Astrophysique de Marseille, UMR 6110 CNRS Université de Provence, BP8, F-13376 Marseille Cedex 12, France INAF – Istituto di Astrofisica Spaziale e Fisica Cosmica, Via Bassini 15, I-20133 Milano, Italy    C. Porciani Argelander Institute for Astronomy, Auf dem Hügel 71, D-53121, Germany Institute of Astronomy, ETH Zurich, Zurich, Switzerland    G. Zamorani INAF – Osservatorio Astronomico di Bologna, Bologna, Italy    U. Abbas INAF – Osservatorio Astronomico di Torino, Strada Osservatorio 20, I-10025 Pino Torinese (TO), Italy    M. Bolzonella INAF – Osservatorio Astronomico di Bologna, Bologna, Italy    B. Garilli INAF – Istituto di Astrofisica Spaziale e Fisica Cosmica, Via Bassini 15, I-20133 Milano, Italy    A. Iovino INAF – Osservatorio Astronomico di Brera, via Brera 28, Milano, Italy    L. Pozzetti INAF – Osservatorio Astronomico di Bologna, Bologna, Italy    E. Zucca INAF – Osservatorio Astronomico di Bologna, Bologna, Italy    S. Lilly Institute of Astronomy, ETH Zurich, Zurich, Switzerland    O. Le Fèvre Laboratoire d’Astrophysique de Marseille, UMR 6110 CNRS Université de Provence, BP8, F-13376 Marseille Cedex 12, France    J.-P. Kneib Laboratoire d’Astrophysique de Marseille, UMR 6110 CNRS Université de Provence, BP8, F-13376 Marseille Cedex 12, France    C. M. Carollo Institute of Astronomy, ETH Zurich, Zurich, Switzerland    T. Contini Laboratoire d’Astrophysique de Toulouse-Tarbes, Université de Toulouse, CNRS Toulouse, F-31400, France    V. Mainieri European Southern Observatory, Garching, Germany    A. Renzini Dipartimento di Astronomia, Università di Padova, Padova, Italy    M. Scodeggio INAF – Istituto di Astrofisica Spaziale e Fisica Cosmica, Via Bassini 15, I-20133 Milano, Italy    S. Bardelli INAF – Osservatorio Astronomico di Bologna, Bologna, Italy    A. Bongiorno Max-Planck-Institut für Extraterrestrische Physik, Giessenbachstrasse, 85748 Garching-bei-München, Germany    K. Caputi Institute of Astronomy, ETH Zurich, Zurich, Switzerland    G. Coppa INAF – Osservatorio Astronomico di Bologna, Bologna, Italy Dipartimento di Astronomia, Università degli Studi di Bologna, Bologna, Italy    O. Cucciati Laboratoire d’Astrophysique de Marseille, UMR 6110 CNRS Université de Provence, BP8, F-13376 Marseille Cedex 12, France    L. de Ravel Laboratoire d’Astrophysique de Marseille, UMR 6110 CNRS Université de Provence, BP8, F-13376 Marseille Cedex 12, France    P. Franzetti INAF – Istituto di Astrofisica Spaziale e Fisica Cosmica, Via Bassini 15, I-20133 Milano, Italy    P. Kampczyk Institute of Astronomy, ETH Zurich, Zurich, Switzerland    C. Knobel Institute of Astronomy, ETH Zurich, Zurich, Switzerland    K. Kovač Institute of Astronomy, ETH Zurich, Zurich, Switzerland    F. Lamareille Laboratoire d’Astrophysique de Toulouse-Tarbes, Université de Toulouse, CNRS Toulouse, F-31400, France    J.-F. Le Borgne Laboratoire d’Astrophysique de Toulouse-Tarbes, Université de Toulouse, CNRS Toulouse, F-31400, France    V. Le Brun Laboratoire d’Astrophysique de Marseille, UMR 6110 CNRS Université de Provence, BP8, F-13376 Marseille Cedex 12, France    C. Maier Institute of Astronomy, ETH Zurich, Zurich, Switzerland    R. Pellò Laboratoire d’Astrophysique de Toulouse-Tarbes, Université de Toulouse, CNRS Toulouse, F-31400, France    Y. Peng Institute of Astronomy, ETH Zurich, Zurich, Switzerland    E. Perez Montero Laboratoire d’Astrophysique de Toulouse-Tarbes, Université de Toulouse, CNRS Toulouse, F-31400, France    E. Ricciardelli Dipartimento di Astronomia, Università di Padova, Padova, Italy    J.D. Silverman Institute of Astronomy, ETH Zurich, Zurich, Switzerland    M. Tanaka European Southern Observatory, Garching, Germany    L. Tasca Laboratoire d’Astrophysique de Marseille, UMR 6110 CNRS Université de Provence, BP8, F-13376 Marseille Cedex 12, France INAF – Istituto di Astrofisica Spaziale e Fisica Cosmica, Via Bassini 15, I-20133 Milano, Italy    L. Tresse Laboratoire d’Astrophysique de Marseille, UMR 6110 CNRS Université de Provence, BP8, F-13376 Marseille Cedex 12, France    D. Vergani INAF – Osservatorio Astronomico di Bologna, Bologna, Italy    D. Bottini INAF – Istituto di Astrofisica Spaziale e Fisica Cosmica, Via Bassini 15, I-20133 Milano, Italy    A. Cappi INAF – Osservatorio Astronomico di Bologna, Bologna, Italy    A. Cimatti Dipartimento di Astronomia, Università degli Studi di Bologna, Bologna, Italy    P. Cassata Laboratoire d’Astrophysique de Marseille, UMR 6110 CNRS Université de Provence, BP8, F-13376 Marseille Cedex 12, France    M. Fumana INAF – Istituto di Astrofisica Spaziale e Fisica Cosmica, Via Bassini 15, I-20133 Milano, Italy    A. M. Koekemoer Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA    A. Leauthaud Berkeley Lab & Berkeley Center for Cosmological Physics, University of California, Berkeley, CA 94720, USA    D. Maccagni INAF – Istituto di Astrofisica Spaziale e Fisica Cosmica, Via Bassini 15, I-20133 Milano, Italy    C. Marinoni Centre de Physique Theorique, UMR 6207 CNRS Université de Provence, F-13288 Marseille, France    H.J. McCracken Institut d’Astrophysique de Paris, Université Pierre & Marie Curie, Paris, France    P. Memeo INAF – Istituto di Astrofisica Spaziale e Fisica Cosmica, Via Bassini 15, I-20133 Milano, Italy    P. Oesch Institute of Astronomy, ETH Zurich, Zurich, Switzerland    R. Scaramella INAF – Osservatorio Astronomico di Roma, via di Frascatti 33, 00040 Monte Porzio Catone, Italy
Received –; accepted –
Key Words.:
Cosmology: observations – Cosmology: large-scale structure of Universe – Surveys – Galaxies: evolution
offprints: B. Meneux bmeneux@mpe.mpg.de
Abstract

Context:

Aims:We study the dependence of galaxy clustering on luminosity and stellar mass at redshifts , using the first 10K redshifts from the zCOSMOS spectroscopic survey of the COSMOS field.

Methods:We measure the redshift-space correlation functions and and the projected function, for sub-samples covering different luminosity, mass and redshift ranges. We explore and quantify in detail the observational selection biases due to the flux-limited nature of the survey, using ensembles of realistic semi-analytic mock samples built from the Millennium simulation. We use the same mock data sets to carefully check our covariance and error estimate techniques, comparing the performances of methods based on the scatter in the mocks and on bootstrapping schemes. We finally compare our measurements to the cosmological model predictions from the mock surveys.

Results:At odds with other measurements at similar redshift and in the local Universe, we find a weak dependence of galaxy clustering on luminosity in all three redshift bins explored. A mild dependence on stellar mass is instead observed, in particular on small scales, which becomes particularly evident in the central redshift bin (), where  shows strong excess power on scales  h Mpc. This is reflected in the shape of the full that we interpret as produced by large-scale structure dominating the survey volume and extending preferentially in direction perpendicular to the line-of-sight. Comparing to measurements, we do not see any significant evolution with redshift of the amplitude of clustering for bright and/or massive galaxies.

Conclusions:This is consistent with previous results and the standard picture in which the bias evolves more rapidly for the most massive halos, which in turn host the highest-stellar-mass galaxies. At the same time, however, the clustering measured in the zCOSMOS 10K data at for galaxies with is only marginally consistent with the predictions from the mock surveys. On scales larger than h Mpc, the observed clustering amplitude is compatible only with  1% of the mocks. Thus, if the power spectrum of matter is CDM with standard normalization and the bias has no “unnatural” scale-dependence, this result indicates that COSMOS has picked up a particularly rare, positive fluctuation in a volume of  h Mpc. These findings underline the need for larger surveys of the Universe to appropriately characterize the level of structure at this epoch.

1 Introduction

In the canonical scenario of galaxy formation, galaxies are thought to form through the cooling of baryonic gas within extended dark matter halos (White & Rees, 1978). The mass of the hosting halo is expected to play a significant role in the definition of the visible properties of the galaxy, as the total mass in gas and stars, its luminosity, color, star formation rate and, possibly, morphology.

Since it is the baryons that form the visible fabric of the Universe, a major challenge in testing the galaxy formation paradigm is to build clear connections between these observed properties and those of the hosting dark-matter halos. This is a difficult task, as any direct connection existing initially between the dark-matter mass and the baryonic component cooling within the halo is modified by all subsequent dynamical processes affecting the halo-galaxy system, as merging or dynamical friction. This is confirmed by simulations, that also show however that galaxy luminosity and stellar mass do retain in fact memory of the ”original” (not actual) halo mass, i.e. before it experiences a major merger or is accreted by a larger halo (Conroy et al., 2006; Wang et al., 2006, 2007). This gives some hope that measuring the dependence of the galaxy distribution on galaxy properties one is actually constraining the relationship between the dark and luminous components of galaxies.

Measurements of first moments, as the luminosity function or the stellar mass function, provide a way to understand how these are related to the total halo mass functions, which can be obtained from analytic predictions (e.g. Press & Schechter, 1974) or n-body simulations (e.g. Warren et al., 2006). Similar investigations can be made on the second moment, i.e. the two-point correlation function (e.g. Springel et al., 2006). Studies of galaxy clustering in large local surveys have shown how clustering at does depend significantly on several specific properties. These include luminosity (Hamilton, 1988; Iovino et al., 1993; Maurogordato & Lachieze-Rey, 1991; Benoist et al., 1996; Guzzo et al., 2000; Norberg et al., 2001, 2002; Zehavi et al., 2005), color or spectral type (Willmer et al., 1998; Norberg et al., 2002; Zehavi et al., 2002), morphology (Davis & Geller, 1976; Giovanelli et al., 1986; Guzzo et al., 1997), stellar mass (Li et al., 2006) and environment (Abbas & Sheth, 2006).

In recent years it has become possible to extend these investigations to high redshift, obtaining first indicative results on how these dependences evolve with time (Coil et al., 2006; Pollo et al., 2006; Phleps et al., 2006; Meneux et al., 2006; Daddi et al., 2003; Meneux et al., 2008). The VIMOS-VLT Deep Survey (VVDS) (Pollo et al., 2006) and the DEEP2 survey (Coil et al., 2006) in particular, have provided new insights on the way galaxies of different luminosity cluster at . More specifically, Pollo et al. (2006) have shown that at these epochs galaxies already show a luminosity segregation, with more luminous galaxies being more clustered than faint objects. At the same time, however, a significant steepening with luminosity of the shape of their two-point correlation function for separations  h Mpc, is observed. This behaviour is at variance with that at . A similar trend has been observed at the same redshift by the DEEP2 survey (Coil et al., 2006). Complementarily, Meneux et al. (2008) have shown a positive trend of clustering with stellar mass also at , with a clear evidence for a stronger evolution of the bias factor for the most massive galaxies (see also Brown et al., 2008; Wake et al., 2008).

The interpretation of the evolution in shape and amplitude of  with respect to luminosity and redshift is particularly interesting in the context of the halo model for galaxy formation. In this framework, the observed shape of (or ) is interpreted as composed by the sum of two components: (a) the 1-halo term, which dominates on small scales ( h Mpc at the current epoch), where correlations are dominated by pairs of galaxies living within the same dark-matter halo (i.e. in a group or cluster); (b) the 2-halo term on large scales, which is characterized by pairs of galaxies occupying different dark-matter halos (see Cooray & Sheth (2002) for a review). Zheng et al. (2007) have modelled the luminosity-dependent  from both the DEEP2 (at ) and SDSS (at ) surveys, within such Halo Occupation Distribution (HOD) framework. In this way they establish evolutionary connections between galaxies and dark-matter halos at these two epochs, providing a self-consistent scenario in which the growth of the stellar mass depends on the halo mass. Similar results are obtained more recently in a combined analysis of the VVDS-Deep and SDSS data (Abbas et al., 2009).

In this paper we use the first 10,000 redshifts from the zCOSMOS redshift survey (the “10K sample”) to further explore these high-redshift trends of clustering with luminosity and mass based on a new, independent sample. Although shallower than VVDS-Deep and DEEP2 ( vs. 24 and 23.5, respectively), zCOSMOS covers a significantly larger area and samples a volume of  h Mpc to redshift z=1.2. This should hopefully help reducing the effect of cosmic variance (still strong for samples this size (Garilli et al., 2008; Stringer et al., 2009)), while providing a better sampling of the high-end tail of the luminosity and mass functions. However, one main result from this analysis will be the explicit demonstration of how strong cosmic variance still is within volumes of the Universe this size. The clustering properties of the zCOSMOS sample in the volume contained within the redshift range seem to lie at the extreme high end of the distribution of fluctuations on these scales, as it was in fact already suggested by the angular clustering of the COSMOS data (McCracken et al., 2007). As we shall see, these results and those presented in the zCOSMOS series of clustering papers (Porciani et al., 2009; de la Torre et al., 2009; Abbas et al., 2009) indicate how cautious one should be in drawing far-reaching conclusions from the modelling of current clustering results from deep galaxy surveys.

A significant part of this paper is dedicated to discussing in detail these cosmic-variance effects, together with the impact of incompleteness on the derived results. This is particularly important when constructing mass-limited sub-samples from a magnitude-limited survey, which introduces a mass incompleteness that depends on redshift and stellar mass. The intrinsic scatter in the galaxy mass-luminosity relation determines a progressive loss of faint galaxies with high mass-to-light ratio. We study in detail the effect of this incompleteness on the measured clustering both using the data themselves and mock samples built from the Millennium simulation. At the same time, we explore in quite some detail our ability to characterize measurement errors and the covariance matrix of our data, comparing estimates from the mock samples to those from bootstrap resamplings of the data themselves.

The paper is organised as follows. In Sect. 2 and 3 we describe the zCOSMOS survey and the simulated mock samples used in the analysis, while in Sect. 4 we describe the selection of luminosity- and mass-limited subsamples, discussing extensively the incompleteness related to this operation; in Sect. 5 we describe our clustering estimators, while in Sect. 6 we discuss in detail the observational biases and selection effects, how we account for them and what is their effect on the measured quantities; in Sect. 7 we explore and discuss the error budget and how to estimate the covariance properties of our measurements; in Sect. 8 and 9 we present our measurements of clustering as a function of luminosity and mass, respectively, while in Sect. 10 we compare these results with those from other surveys and with simple model predictions; finally, in Sect. 11 we place these findings in a broader context and discuss future developments.

Throughout the paper we adopt a cosmology with , . When needed, we also adopt a value for the normalization of the matter power spectrum; this is chosen for consistency with the Millennium simulation, also used for comparison to model predictions. The Hubble constant is parameterized via to ease comparison with previous works. Stellar masses are quoted in unit of . All length values are quoted in comoving coordinates.

Figure 1: Distribution on the sky of the galaxies with measured redshift (crosses) forming the zCOSMOS “10K” sample. The large blue dots mark the centres of independent VIMOS pointings, each including four quadrants on the sky (as described by the red solid lines).

2 The zCOSMOS survey data

The zCOSMOS survey (Lilly et al., 2007) is performed with the VIMOS multi-object spectrograph at the ESO Very Large Telescope (Le Fèvre et al., 2003). 600 hours of observation have been allocated to this program. These are invested to measure spectra for galaxies in the COSMOS field (Scoville et al., 2007a), targeting: (a) 20000 galaxies brighter than (zCOSMOS Bright); 10000 sources at redshift pre-selected using color-color criteria (Lilly et al., 2007) (zCOSMOS Faint). So far, the survey has observed about half of the total “Bright” sample, This is the so called “10K” sample used for the analysis presented in this paper, and is based on the observations of 83 VIMOS pointings over 44 distinct telescope positions on the sky (Lilly et al., 2009). These are shown in Fig. 1, where the footprint of VIMOS (4 quadrants of  arcmin separated by a cross about 2 arcmin wide) is evident. About every galaxy has been observed in the field. The final “20K” zCOSMOS sample will be twice larger, reaching a sampling around 60-70%. The correction of the complex angular selection function will be discussed later in the context of our galaxy clustering measurement.

Observations are performed using the medium resolution RED grism, corresponding to and covering the spectral range Å. The average error on the redshift measurements has been estimated from the repeated observations of 632 galaxies, and is found to be 100 km s (Lilly et al., 2009). This roughly corresponds to a radial distance error of 1 h Mpc. The reduction of the data to the redshift assignment was carried out independently at two institutes before a reconciliation process to solve discrepancies. The quality of each measured redshift was then quantified via a quality flag that provides us with a confidence level (see Lilly et al. (2007, 2009) for definition). For the present work, we only use redshifts with flags 1.5 to 4.5 and 9.3 to 9.5, corresponding to confidence levels greater than 98%.

The zCOSMOS survey benefits of the large multi-wavelength coverage of the COSMOS field (Capak et al., 2007), that with the latest additions now comprises 30 photometric bands (Ilbert et al., 2009) extending well into the infrared. These include in particular accurate -band and Spitzer-IRAC photometry over the whole area, which have allowed us to derive relevant physical properties as rest-frame luminosity and stellar mass with unprecedented accuracy (Zucca et al., 2009; Bolzonella et al., 2009; Pozzetti et al., 2009).

Figure 2: Selection boundaries of the different sub-samples of the zCOSMOS 10K survey used in this paper. Left: luminosity-redshift selection, which accounts for the average luminosity evolution of galaxies; Right: mass-redshift selection.

3 Mock survey catalogues

In this paper we shall make intense use of mock surveys constructed from the Millennium simulation (Springel et al., 2005). This is done (a) to understand the effect of our selection criteria on the measured quantities (Sec. 6.3); (b) to estimate the measurement errors and covariance of the data (Sec. 7).

We use two sets of light cones, constructed as explained in Kitzbichler & White (2007) and Blaizot et al. (2005) by combining dark-matter halo trees from the Millennium run to the Munich semi-analytical model of galaxy formation (De Lucia & Blaizot, 2007). The two sets contain 24 deg mocks built by Kitzbichler & White (2007) and 40 deg mocks built by De Lucia & Blaizot (2007), that we shall name KW24 and DLB40, respectively. The main difference between the two sets, in addition to the different survey area, is that the DLB40 set contains all galaxies irrespective to any criteria down to the simulation limit that corresponds roughly to , up to redshift , whereas the KW24 set contains only galaxies brighter than . This implies that the DLB40 set allows us to select in stellar mass down to very low masses and test selection effects. The observing strategy of the zCOSMOS 10K sample was only applyied to the KW24 set, allowing us to do a carefull error analysis of our measurements.

The Millennium run contains particles of mass  h M in a cubic box of size  h Mpc. The simulation was built with a CDM cosmological model with , , and  km s Mpc.

4 Luminosity- and Mass-selected subsamples

4.1 Luminosity selection

Absolute magnitudes have been derived for the 10K galaxies using the code ALF (Ilbert et al., 2005; Zucca et al., 2009), which is based on fitting a Spectral Energy Distribution to the observed multi-band photometry. There are various sources of uncertainties to take into account (errors on apparent magnitudes, number of available photometric bands, method used, …). A direct comparison with absolute magnitudes derived with the independent code ZEBRA (Feldmann et al., 2006) shows consistent estimates with a small dispersion of magnitudes, in particular in the B band. This can resonably be considered as the typical error on our absolute magnitudes.

For our analysis, the goal is to define luminosity-limited samples that are as close as possible to truly volume-limited samples, i.e. with a constant number density. This should be possibly done within a few independent redshift ranges. The size of the redshift slices in which to split the sample has to be chosen as a compromise between two aspects: (a) we want it to be large enough as to have sufficient statistics and provide a good measurement of clustering; however, (b) we do not want it to be too large, as to avoid significant evolution within each redshift bin.

However, we know that all through the overall redshift range covered by the zCOSMOS survey () the luminosity of galaxies evolves, with a clear change in the characteristic parameters of the luminosity function (Ilbert et al., 2005). This evolution does depend on the morphological/spectral type of the galaxy considered. To be able to select a nearly volume-limited sample within a given redshift interval, we need to take the corresponding evolution into account. This can be realistically done only in a statistical way, looking at the population-averaged evolution of the global luminosity function.

We have therefore considered the observed luminosity function measured from the same data (Zucca et al., 2009) and modelled its change with redshift as a pure luminosity evolution (i.e. keeping a constant slope and normalisation factor ), which is a fair description of the observed behaviour. We find that the characteristic absolute magnitude evolves with redshift as

(1)

where . In the companion paper, de la Torre et al. (2009) split the zCOSMOS galaxy samples in 3 morphological classes. They observe different luminosity evolutions for elliptical, spiral and irregular galaxies, with varying from for to but with large uncertainties making compatible for all classes. Porciani et al. (2009) reach similar conclusions when dividing the zCOSMOS 10K sample into 3 color classes.

We therefore define our luminosity-limited samples by an effective absolute magnitude cut at , and including all galaxies with . The resulting selection loci for different values of are plotted over the data in the luminosity-redshift plane in the left panel of Fig. 2. As evident from the figure, the faintest allowed threshold depends on the redshift range considered, i.e. z=[0.2-0.5], z=[0.5-0.8] and z=[0.8-1.0]. The details of the resulting samples are described in Table 1.

Sample Redshift Mean Number of
range redshift (z=0) galaxies
L1.1 0.2-0.5 0.37 -18.00 1892
L1.2 0.2-0.5 0.37 -18.50 1311
L1.3 0.2-0.5 0.37 -19.00 811
L1.4 0.2-0.5 0.37 -19.50 469
L2.1 0.5-0.8 0.67 -19.00 1848
L2.2 0.5-0.8 0.67 -19.50 1025
L2.3 0.5-0.8 0.67 -20.00 441
L3.1 0.8-1.0 0.91 -19.50 971
L3.2 0.8-1.0 0.91 -20.00 447
Table 1: Properties of the luminosity-selected samples
Figure 3: The observed relationship between stellar mass and luminosity for galaxies in the 10K sample, within the three redshift ranges studied in this paper. The left panel shows an aspect of the galaxy bimodality, with red galaxies more massive and brighter than blue ones.

4.2 Mass selection

Sample Redshift Mean Number of
range redshift range median galaxies
M1.1 0.2-0.5 0.36 9.80 2159
M1.2 0.2-0.5 0.37 10.09 1445
M1.3 0.2-0.5 0.36 10.36 827
M1.4 0.2-0.5 0.37 10.66 275
M2.1 0.5-0.8 0.66 9.97 2831
M2.2 0.5-0.8 0.66 10.12 2276
M2.3 0.5-0.8 0.67 10.38 1366
M2.4 0.5-0.8 0.67 10.68 477
M3.1 0.8-1.0 0.90 10.46 755
M3.2 0.8-1.0 0.90 10.73 344
Table 2: Properties of the mass-selected samples

Stellar mass has become a quantity routinely measured in recent years, thanks to surveys with multi-wavelength photometry, extending to the near-infrared (e.g. Rettura et al., 2006), although some uncertainties related to the detailed modelling of stellar evolution remain (Pozzetti et al., 2007). This has made studies of clustering as a function of stellar mass possible for large statistical samples. We used stellar masses estimated by fitting the Spectral Energy Distribution (SED), as sampled by the large multi-band photometry, with a library of stellar population models based on Bruzual & Charlot (2003). We used the code Hyperzmass, a modified version of the photometric redshift code Hyperz (Bolzonella et al., 2000). The typical error on stellar masses is 0.2 dex. The method and accuracy of these measurements are fully described in Bolzonella et al. (2009) and Pozzetti et al. (2009).

We have thus constructed a set of mass-selected samples, containing galaxies more massive than a given threshold. We choose the same redshifts ranges as used for the luminosity selected samples. The properties of the selected subsamples are summarized in Table 2 and represented in Fig. 2.

4.3 Mass completeness

Figure 4: Estimate of how the completeness in stellar mass changes as a function of redshift, due to the survey flux limit (). The shaded grey area and green contours describe the loci of constant completeness. They are derived from the DLB40 mock samples of deg and defined as the fraction of observed () galaxies over the total number in a given cell with size and . The red points superimposed correspond to the actual data of the 10K sample. The yellow points and dotted line show the 95% M/L ratio completeness level derived independently by Pozzetti et al. (2009) (see text), directly from the observed data. The agreement between the two estimates, from the data and from the Millennium mocks, is remarkable and adds confidence in the use of the simulated samples.

Due to the flux-limited nature of surveys like zCOSMOS () the lowest-mass samples are affected to varying degree by incompleteness related to the scatter in the mass-luminosity relation (Fig. 3). This introduces a bias against objects which would be massive enough to enter the mass selected samples, but too faint to fulfill the apparent magnitude limit of the survey. These missed high mass-to-light ratio galaxies will be those dominated by low-luminosity stars, i.e. the red and faint objects. Clearly, if this is not accounted for in some way, it would inevitably affect the estimated clustering properties, with respect to a truly complete, mass-selected sample (Meneux et al., 2008). It is therefore necessary to understand in detail the effective completeness level in stellar mass of the samples that we have defined for our analysis.

Meneux et al. (2008) have used 2 different methods to explore and quantify the completeness limit in stellar mass as a function of redshift. The first is based on the observed scatter in the mass-luminosity relation, obtained from the data themselves and extrapolated to fainter fluxes. The second instead makes use of mock survey samples, under the hypothesis that they provide a realistic description of the mass-luminosity relation and its scatter: the DLB40 set of mock survey catalogues that are complete in stellar mass are “observed” under the same conditions as the real data, i.e., selected at . The completeness is then simply defined, for a given redshift range and mass threshold, as the ratio of the number of galaxies brighter than the zCOSMOS flux limit over those at any flux. Interestingly, even if this method is model-dependent (in particular, on the prescription of galaxy formation used in the semi-analytical models), this approach leads to similar completeness limits than the first one. The results of this second exercise are shown, as a function of redshift and mass threshold and for a flux limit , in Fig. 4. Completeness is estimated in narrow redshift ranges () for different mass thresholds increasing from to with a step of . A large fraction of low-mass objects is clearly missed at high redshift.

The yellow points and dashed line in Fig. 4 have been estimated from the observed scatter in the M/L relation of the data, and are defined at each redshift as the lower boundary, , including above it 95% of the mass distribution (Pozzetti et al., 2009).

It is very encouraging to notice the very good agreement between this independent estimation from the data and that based on the DLB40 set of mock catalogues. Table 3 summarises the completeness estimates derived from these mock catalogues for each of the 10 zCOSMOS galaxy samples defined in Table 2. The sample M2.1 shows the strongest incompleteness: 65.1% of the galaxies more massive than are fainter than at z=[0.5-0.8] and then, not included in our sample. In Sec. 6.3 we shall discuss the effects of this incompleteness on the galaxy clustering measurement.

Sample Redshift Stellar mass () Completeness
range range
M1.1 0.2-0.5 9.0 0.783
M1.2 0.2-0.5 9.5 0.972
M1.3 0.2-0.5 10.0 1.000
M1.4 0.2-0.5 10.5 1.000
M2.1 0.5-0.8 9.0 0.349
M2.2 0.5-0.8 9.5 0.652
M2.3 0.5-0.8 10.0 0.919
M2.4 0.5-0.8 10.5 0.996
M3.1 0.8-1.0 10.0 0.571
M3.2 0.8-1.0 10.5 0.882
Table 3: The completeness in stellar mass of mass-selected mock sub-samples reproducing the properties and selection criteria of our 10K data samples. Completeness is defined as the percentage of all galaxies above the mass limit which are actually included in the sample.

5 Estimating the two-point correlation function

The two-point correlation function is the simplest estimator used to quantify galaxy clustering, being related to the second moment of the galaxy distribution, i.e. its variance. In practice, it describes the excess probability to observe a pair of galaxies at a given separation , with respect to that of a random distribution (Peebles, 1980). Here we shall estimate the redshift-space correlation function , which allows one to account and correct for the effect of peculiar motions on the pure Hubble recession velocity. In this case, galaxy separations are split into the tangential and radial components, and (Davis & Peebles, 1983; Fisher et al., 1994).

The real-space correlation function can be recovered by projecting  along the line-of-sight, as

(2)
Figure 5: Overall radial distribution of the zCOSMOS 10K sample, compared to three different smoothed distributions. These are obtained by filtering the observed data with a Gaussian kernel of increasing =150, 250 and 450 h Mpc. The first two smoothed curves retain information of the two large structures located at 1000 and 1800 h Mpc along the line-of-sight, while the third one overestimates the number density of galaxies at low and high redshift.
Figure 6: The radial distribution of the luminosity selected sample L2.1 is compared to a smoothed curved (with a kernel of =450 h Mpc - dashed curve) and a radial distribution generated from the integration of the luminosity function (solid curve). The latter is consistent with that expected for such galaxy sample.

For a power-law correlation function, , this integral can be solved analytically and fitted to the observed  to find the best-fitting values of the correlation length  and slope  (e.g., Davis & Peebles, 1983). In computing , a finite upper integration limit has to be chosen in practice. Its value has to be large enough as to include most of the clustering signal dispersed along the line of sight by peculiar motion. However, it must not be too large, to avoid adding just noise, which is dominant above a certain . Previous works (Pollo et al., 2005) have shown that, for similar data, the best results are obtained with an integration limit between 20 and 40 h Mpc. Our tests show that the scatter in the recovered  is obtained using the lowest value in this range. This can introduce a 5-10% under-estimate in the recovered large-scale amplitude, which can be accounted for when fitting a model to . In the following, we shall in general use  hMpc and show examples of how the amplitude is biased by this choice for the real data.

To estimate  from each galaxy sample, we use the standard estimator of Landy & Szalay (1993):

(3)

where is the mean galaxy density (or, equivalently, the total number of objects) in the sample; is the mean density of a catalogue of random points distributed within the same survey volume and with the same selection function as the data; is the number of independent galaxy-galaxy pairs with separation between and and between and ; is the number of independent random-random pairs within the same interval of separations and represents the number of galaxy-random cross pairs.

6 Observational biases and selection effects

6.1 Correction of VIMOS angular foot-print and varying sampling

To properly estimate the correlation function from the 10K zCOSMOS data, we need to correct for its spatial sampling rate, which is on average 30%, but varies with the position on the sky due to the VIMOS foot-print and the superposition of multiple passes (see Fig. 1). The correction scheme used here is an evolution of that discussed in Pollo et al. (2005), but with a simplified weighting scheme. The main differences of this sample with respect to the VVDS-Deep data used by Pollo et al. (2005), are that: (a) this sample is 1.5 magnitudes brighter, (b) the spectra are taken with higher resolution, which produces longer spectra and thus less objects along the dispersion direction and, (c) there are as many as 8 repeated observations (“passes”) covering each point on the sky in the central area of the COSMOS field. The net result of these differences can be appreciated in Fig. 1: the sample is characterized by a well-sampled central region, but also by rather sparsely sampled VIMOS pointings in the outskirts of the field. In particular, these external pointings clearly show target galaxies concentrated along rows. This effect is produced by the significant length of the spectra on the CCD in medium-resolution mode: not more than 2 spectra can be aligned on top of each other on the detector in each quadrant, which results into the observed two “stripes”. This is significantly different from what happens in the low-resolution observing mode (as e.g. in VVDS-Deep, Le Fèvre et al., 2005a), where spectra are shorter and up to four of them can be packed along the same column on the CCD.

We have tested three different algorithms to correct for the angular selection function of the survey, obtaining comparable results. Other weighting schemes use in particular the angular correlation functions of the 10K sample and the photometric catalogue to correct for the non uniform spatial sampling rate. These methods are discussed in the parallel clustering analyses by de la Torre et al. (2009) and Porciani et al. (2009). In the latter paper in particular, comparative tests of the three algorithms are presented.

Since the sub-samples analyzed in this work are essentially volume-limited (above the luminosity/mass completeness limits), we do not need to apply any further minimum-variance weighting scheme (as e.g. the weighting, Fisher et al., 1994). This is normally necessary for purely flux-limited surveys in which the selection function varies significantly as a function of redshift, such that different parts of the volume are sampled by galaxies with different luminosities and number densities (e.g., Li et al., 2006).

6.2 Construction of reference random samples

A significant source of uncertainty that we encountered in estimating two-point functions from our 10K sub-samples is related to the construction of the random sample and in particular to its redshift distribution. We soon realized that the strongly clustered nature of the COSMOS field along the line of sight, with several dominating structures at different redshifts, required some particular care as not to generate systematic biases in the random sample. These superclusters are already evident as vertical stripes in Fig. 2 and even more clearly in the redshift histogram of Fig. 5. Note the big “walls” at , 0.75 and 0.9, which are also clearly identified by the density field reconstruction of Kovač et al. (2009).

A standard way to generate a random redshift coordinate accounting for the radial selection function of the data uses a Gaussian-filtered version of the data themselves. This is normally obtained using smoothing kernels with a dispersion (in co-moving coordinates) in the range 150250 h Mpc. The results of applying this technique to the current 10K data are shown in Fig. 5. One notes how for smoothing scales of 150 and 250 h Mpc the curves still retain memory of the two largest galaxy fluctuations. These are erased only when a very strong smoothing filter (450 h Mpc) is adopted. However, in this case the smoothed curve is unable to follow correctly the global shape of the distribution, over-estimating the number density in the lowest and highest redshift ranges. The situation for our specific analysis, however, is somewhat simpler than this general case. Our luminosity-limited or mass-limited samples are in principle “volume-limited”, i.e. samples that – if properly selected – should have a constant density within the specific redshift bin. One such case is shown in the zoom of Fig. 6, where the redshift distribution in the range is plotted.

An alternative way to generate the radial distribution of the random sample is to integrate the galaxy luminosity function (LF) in steps along the redshift direction, computing at each step a value for the density of galaxies expected at that redshifts. Ideally, the LF can be measured from the sample itself and would include any detected evolution of its parameters. This is what we did here, using the evolving LF parameters presented in the companion dedicated paper (Zucca et al. 2009). The red dashed line in Fig. 6 shows the result one obtains if smoothing with a kernel of 450 h Mpc (dashed), compared to that obtained from the integration of the LF (solid). The latter is fully consistent with that expected from a truly volume-limited sample with the given selection criteria, with the number of objects increasing as the square of the radial comoving distance.

6.3 Effect of mass incompleteness on

As discussed in Sec. 4.3 when we constructed our mass-limited samples, a fraction of galaxies more massive than the formal mass threshold are in fact lost due to the limiting flux cut of the survey. This becomes more and more important with increasing redshift. As we said this population of missing galaxies is inevitably dominated by red objects with high mass-to-light ratio (Meneux et al., 2008), which are known to cluster more strongly than the average population (Meneux et al., 2006; Coil et al., 2008; McCracken et al., 2008). Having defined our clustering tools, we can now further extend the analysis of Sec. 4.3 and use the DLB40 mock samples to quantify directly the effect this has on the measured . We thus computed the statistics for each of the mocks, which are complete down to very small masses (), with and without applying the apparent-magnitude cut. Clearly, we are making here a very strong hypothesis, i.e. that the simulated samples have intrinsic clustering properties (and their relation to the galaxy’s ratio), that are similar to those of real data.

The ratio of these two estimates (“true” over “observed”) averaged over the 40 mock catalogues is shown on Fig. 7. For a mass selection which is 100% complete within the given redshift bin we would measure at all separations. We can see that the only mass range for which this is strictly happening at any redshift is that with 10.5. For smaller mass samples we see a clear reduction of the clustering amplitude. However, we can also see that for most samples the shape of  is distorted mainly only below  h Mpc. Above this scale, the mass incompleteness introduces an amplitude reduction up to in the worst cases. This will have to be considered when comparing our measurements with models (although keeping in mind that these estimates come from simulated data, not from real observations). For general comparisons, however, the amount of amplitude reduction of  is typically negligible on scales larger than 1 h Mpc, given the statistical errors of the data measurements.

Figure 7: The effect of stellar mass incompleteness on the measured , estimated from the deg Millennium mock samples. The figure shows the average of the quantity over 40 mock samples as a function of . is defined as the ratio of the estimates of  with and without the flux cut at , i.e. for a sample mimicking the 10K selection and a sample 100% complete in stellar mass.

7 Systematic and statistical errors on correlation estimates

Figure 8: Ratio of the diagonal errors on , obtained through the bootstrap resampling method to the “true” ones obtained from the variance of 24 mock catalogues. Filled and open symboles correspond to two different bootstrapping techniques, resampling respectively sub-volumes of the survey or single galaxies. The former technique clearly provides a standard deviation which is closer to the “true” one obtained from repeated measurements.

The derivation of realistic errors on the galaxy correlation function has been the subject of debate since its early measurements (see e.g., Bernstein, 1994). In particular, it is well-known that the measured values of the two-point correlation function on different scales are not independent. This means that, e.g., the bins of  have a degree of correlation among them, that needs to be taken into account when fitting a model to the observed values. This can be done if we are able to reconstruct the covariance (or correlation) matrix of the bins (Fisher et al., 1994).

In a recent paper, Norberg et al. (2008) compare in detail three different methods for estimating the covariance matrix of a given set of measurements. These use: (a) the ensemble variance from a set of mock catalogues reproducing as accurately as possible the clustering properties and selection function of the real data; (b) a set of bootstrap resamplings of the volume containing the data, and (c) a so-called jack-knife sub-set of volumes of the survey. In this latter case, the survey volume is divided into sub-volumes and the statistics under study is re-computed each time excluding one of the sub-parts. In the “block-wise” incarnation of the bootstrap technique (Porciani & Giavalisco, 2002, method “b”), instead, sub-volumes are selected each time with repetition, i.e. excluding some of them, but counting two or more times some others as to always get a global sample with the same total volume. We note however that historically there are two possible ways to resample internally the data set. The classical “old” bootstrap (Ling et al., 1986) entailed boot-strapping the sample “galaxy-by-galaxy”. This means picking randomly each time a sample of galaxies among our data set of galaxies, allowing repetitions. In this way, within one bootstrap realization a galaxy can be selected more than once, while some others are never selected. This technique has been shown to lead in general to some under-estimation of the diagonal errors (Fisher et al., 1994). Here we shall test directly also this aspect.

The advantage of using mock samples is that, under the assumption that these are a realistic realization of the real data, they allow us to obtain a true ensemble average and standard deviation from samples with the same size as the data sample, including both Poissonian noise and cosmic variance. Unfortunately, the covariance properties derived from mock samples are not necessarily a good description of those of the real data, thus making the use of the derived covariance matrix (e.g. in model fitting) doubtful. Conversely, depending on the sample size, jack-knife or volume-bootstrap covariance matrices can exacerbate peculiarities of some sub-regions, again not representing adequately the true covariance properties of the data.

For the present investigation, we spent considerable effort to understand how to best estimate a sensible covariance matrix for our measurements. The available mock samples were crucial as to allow us to perform direct comparisons of the performances of the different techniques. After some initial attempts, we excluded the jack-knife method because of the limited size of the survey volume. We then performed a direct comparison of the covariance matrices derived through the bootstrap technique and from the KW24 mock catalogues. For the bootstrap method, we decided to test directly how galaxy- and volume-bootstrap were performing. We concentrated on the redshift range z=[0.5-0.8] selecting simulated galaxies brighter than . After computing the correlation function for all 24 mock samples, we constructed for each of them: (a) 100 galaxy-galaxy bootstrap samples and (b) 100 volume-volume bootstrap samples. In the latter case, we considered 8 equal sub-volumes, defined as redshift slices within the redshift range considered. The number of sub-volumes was choosen as the best compromise between having enough of them and not having too small volumes. With this choice, their volume is  h Mpc for the samples with z=[0.5-0.8] and z=[0.8-1.0] and  h Mpc for z=[0.2-0.5]. The two bootstrap techniques lead to a total of 4800 samples and corresponding estimates of . We then calculated the covariance (and correlation) matrices for each of these two cases along with the one derived from the correlation function of the 24 mocks themselves.

In Fig. 8 we show a comparison of the standard deviations derived from the two bootstrap techniques, to that derived from the 24 mocks. In each case, these values correspond by definition to the square root of the diagonal elements of the covariance matrix. In the plot we show the mean (over the 24 mocks) of the ratio of from the bootstrap to the “true” one from the ensemble of mock surveys. This shows clearly how the r.m.s. values obtained with the single-galaxy bootstrap grossly underestimate the true variance, up to one order of magnitude on large scales. Bootstrapping by volumes produces a better result, providing a realistic estimate of between 0.1 and 1 h Mpc, and a 20-25% under-estimate on larger scales.

eigenvalue bootstrap mocks
galaxies volumes catalogues
3.87784 8.28193 11.82062
1.95834 2.17866  0.15708
1.36841 0.71635  0.01794
1.11316 0.38252  0.00436
0.91683 0.20340  0.00000
Table 4: The 5 main eigenvalules of the correlation matrix derived with the bootstrap resampling, respectively of galaxies (first column) and sub-volumes (col. 2), and from the ensemble variance of the 24 mocks (col. 3). For the two latter cases, each mock sample is used in turn as “data” and the reported eigenvalues are the obtained as the average over the 24 mocks.

Each element of the the correlation matrix is obtained from the corresponding element of the covariance matrix as . By definition, the off-diagonal terms of the correlation matrix will then range between -1 and 1, indicating the degree of correlation between different scales of the function . Considering the redshift range z=[0.5-0.8], we show in Fig. 9 the mean of the 24 correlation matrices derived resampling the galaxies (left panel), or resampling 8 equal sub-volumes (center), 100 times each. These are compared to the correlation matrix directly derived from the 24 mocks (right panel). The first case shows a mainly diagonal correlation matrix where off-diagonal terms are mostly noise. In the second case they instead decrease smoothly from 1 to 0 as a function of bin separation. The matrix derived from the 24 mocks shows high correlation at all scales.

Figure 9: Mean of the 24 correlation matrix derived resampling the galaxies of eack KW24 mocks (left), or resampling 8 equal sub-volumes (center). These are compared to the correlation matrix derived directly from the KW24 mocks (right). The redshift range considered here is z=[0.5-0.8]. The averaging over the 24 realisations of the 2 left matrices suppress the negative off-diagonal terms which are sometimes present for a given mock catalogues. Correlation coefficients are then color-coded from 0 to 1.

In order to compare directly the properties of the correlation matrices derived with the 3 methods, we diagonalize all of the matrices by computing their principal components and the amplitudes of the corresponding eigenvalues (). Note that the sum of the eigenvalues of a correlation matrix is always equal to its dimension, i.e. 12 in our case. We report in Table 4 the values of the five main eigenvalues obtained with the 24 mocks (first column) compared to the averages over the 24 mocks of those obtained with the 2 resampling methods. The numbers show that the correlation matrix derived from the 24 mocks contains essentially four principal components and is mostly dominated by one of them. This indicates a strong correlation in the data. The bootstrap matrices, instead, show more than 5 non-negligible components, with the fifth one being of the same order of magnitude of the second in the mock matrix. This implies a lower correlation. We note, however, that volume resampling tends to produce a matrix whose structure is closer to that of the mocks, with 1-2 dominant components. This is another indication of how volume bootstrapping, although also not reproducing perfectly the intrinsic covariance properties of the sample, provides a better estimate of the variance and correlation in the data with respect to galaxy-galaxy bootstrap.

These experiments are extended and further discussed in our parallel accompanying papers, in particular by Porciani et al. (2009). The bottom-line result of our extensive investigations is that volume bootstrap, if enough resamplings are used, provides a sufficiently good reconstruction of the intrinsic covariance matrix of the data set. This is obtained at the expenses of a slightly less accurate account of cosmic variance on large scale, with respect to what can be obtained from the scatter among mock samples, where wavelengths longer than the survey size can be sampled. However, we have shown (Fig. 8) that this effect on scales  h Mpc is limited to .

8 Dependence of galaxy clustering on luminosity

8.1 Luminosity dependence at fixed redshift

Figure 10: Projected correlation function  measured as a function of galaxy luminosity within three redshift ranges. No significant dependence on luminosity is observed within the explored ranges. Note the very flat slope of  in the central redshift bin, compared to the two other slices.
Figure 11: Iso-correlation contours of (here smoothed with a gaussian kernel) for galaxies brighter than , computed in 3 redshift ranges. The amplitude is color-coded according to the side bar, while the black contour corresponds to . White values correspond to

Figure 10 shows the projected correlation function  estimated for our 9 luminosity selected sub-samples at different redshifts. Error bars correspond to the dispersion provided by 200 volume bootstrap resamplings, as extensively discussed in Sect. 7.

We note that no clear dependence on luminosity is observed within any of the three redshift ranges. Also, in the shape of  there is some hint for the usual “shoulder”, i.e. a change of slope around 1 h Mpc, but no clear separation between the expected 1-halo term on small scales and the 2-halo component above this scale (see the Introduction for definitions). In particular in the intermediate redshift bin, all sub-samples show a rather flat large-scale slope, with no evidence for the usual breakdown above  h Mpc.

To try and understand the origin of the observed flat shape, it is interesting to look directly at the contour plots of the bi-dimensional redshift-space correlation function . These are shown in Fig. 11 for the three luminosity-selected sub-samples L1.4, L2.2 and L3.1 (see Table 1 for definitions), that include galaxies brighter than . The three contour plots show some interesting features. First, one clearly notices the much stronger distortion along the line of sight , in the central panel. At the same time, in the same redshift range a much more extended signal is also observed along the perpendicular direction . It is tempting to interpret both these effects as produced in some way by the two dominating structures that we evidenced in Fig. 6. The excess signal along the line of sight is very plausibly due to the distortions by “Fingers of God”, due to an anomalous number of virialized systems (groups and clusters) within these structures. At the same time, the extension along is indicating that there is also an excess of pairs perpendicularly to the line of sight, with respect to an isotropic distribution. In fact, we know (Scoville et al., 2007b; Guzzo et al., 2007) that the large-scale structure at extends over a large part of the COSMOS area. This evidently biases the observed number of pairs along , for simple geometrical reasons. We cannot exclude that part of the large-scale compression observed in is also generated by an excess of galaxy infall onto this structure, thus producing what is known as the Kaiser effect (Kaiser, 1987). This effect is proportional to the growth of structure (see e.g., Guzzo et al., 2008, for a recent direct estimate at similar redshift) and can be extracted when the underlying clustering can be assumed to be isotropic. In this case it is in practice impossible to disentangle this dynamical distortion from the geometrical anisotropy generated by having one dominating structure elongated perpendicularly to the line of sight.

The flatter shape in  in Fig. 10 is also consistent with the overdense samples of Abbas & Sheth (2007), who noticed not only a higher amplitude for the most overdense (10% and 30%) samples of mock and SDSS galaxies, but also a flattening in the correlation function with respect to the full sample. This is another line of evidence in favour of the hypothesis that the zCosmos field is centered on an overdensity.

The plots of Fig. 11 also show explicitly the reasons for our choice of  h Mpc as the upper integration limit in the computation of : this value provides a reasonable compromise between including most of the signal and excluding the noisiest regions in the upper part the diagrams. In the central redshift bin, however, some real clustering power may still be present above this scale, for small ’s. In Fig. 12 we show directly how  changes, when is extended from 20 to 30 h Mpc. We see that, somewhat counter-intuitively, below 1 h Mpc no extra amplitude is gained, while – as indicated by the mock experiments (see Sect. 5) – the scatter is increased. Conversely, one can see the slight scale-dependent bias on the amplitude at larger separations, which gets up to at 15 h Mpc when increasing .

Figure 12: Sensitivity of the projected function  to the upper integration limit , for one luminosity-selected sample in the central redshift bin.

8.2 Redshift evolution at fixed (evolving) luminosity

In section 4.1 we have discussed how our luminosity selection has been devised as to account for the average evolution in the luminosity of galaxies, assuming this to be the dominant effect in modifying the mean density of objects at a given luminosity. Under this assumption, it is then interesting to test how  for galaxies within the same (de-evolved) luminosity interval changes with redshift. This is shown in Fig. 13 for , i.e. for the same three samples for which is plotted in Fig. 11. No coherent evolution of the amplitude and shape of the projected correlation function is observed among the three samples, characterized by mean redshifts 0.37, 0.61 and 0.91. The three curves are consistent with each other within the error bars, with – as expected – the intermediate-redshift bin (triangles) showing a systematically higher amplitude than the other two. Again, this is easily intepreted as a local effect, resulting from the extreme large-scale clustering observed in this redshift range. On the other hand, the overall lack of apparent amplitude evolution of luminous galaxies with redshift is consistent with previous results from the VVDS-Deep (Pollo et al., 2006), DEEP2 (Coil et al., 2006) and SDSS (Zehavi et al., 2005) surveys for galaxies brighter than . The only evolutionary effect evidenced in particular by the VVDS data is a steepening of the small-scale slope of  (i.e. the 1-halo term) for high-redshift luminous galaxies (Pollo et al., 2006). McCracken et al. (2008) also show a lack of clustering amplitude evolution for a large sample of luminous () galaxies based on accurate photometric redshifts in the CFHTLS Deep fields; interestingly, they show that such invariance is maintained also when splitting the sample into early- and late-type galaxies.

Figure 13: Evolution of the projected function  of galaxies with between redshift and .

9 Dependence of galaxy clustering on stellar mass

The relation of clustering properties to galaxy stellar masses is in principle more informative and straightforward to interpret as stellar mass is a more fundamental physical parameter than luminosity.

9.1 Mass dependence at fixed redshift

Also in the case of stellar mass dependence, it is interesting to look at the shape of the full correlation function in the three redshift ranges. We show in Fig. 14, the result obtained for the 3 samples M1.3, M2.3, M3.2 (see Table 2), that include galaxies more massive than . Also in this case the central panel is significantly different from the other two, with remaining positive out to much larger scales in both and directions. We note that the small-scale stretching along seems to be less extended than that obtained for the corresponding luminosity-selected sample, although it is hard to say whether this difference is significative.

Figure 14: Example of full redshift-space correlation function (here smoothed with a gaussian kernel) for the sub-sample containing galaxies more massive than computed in the same 3 redshift ranges as in Fig. 11. is computed in cells of 1 h Mpc side in both and and the iso-correlation levels are color coded according to the side bar. The thick black contour corresponds to and the white values to . Note how the central panel () shows extra power in both directions, perpendicular and parallel to the line-of-sight. The small-scale “Finger-of-God” effect is however less pronounced than in the case of the luminosity-selected sample of Fig. 11
Figure 15: The projected correlation function  as a function of galaxy stellar masses in the zCOSMOS 10K sample, within three redshift ranges.

Figure 15 shows the projected correlation function of the 10 mass-selected samples. The plotted points are not corrected for the residual stellar mass incompleteness (see Sec. 6.3). Errors are estimated as in the luminosity case using 200 bootstrap resamplings of 8 equal sub-volumes of each dataset. The figure shows a weak mass dependence of clustering in the low- and high-redshift bins, in particular at small separations. At the same time, a strong dependence at all separations is evident in the intermediate redshift slice. In this range the slope of  remains extremely flat out to the largest explored scales, even more strongly than in the luminosity-selected cases. Finally, we note also that in the low- and high-redshift bins there is evidence for a steeper “1-halo term” contribution at  h Mpc (with no clear indication for an evolution in redshift of the transition scale to the 2-halo term). Conversely, the central redshift range seems to be characterized by the same, flat power-law shape down to 0.2 h Mpc, where a sudden steepening is then observed. The slope below 0.2 h Mpc seems to depend directly on the limiting mass, with more massive galaxies showing a steeper correlation function. In summary, no clear overall trend can be evidenced among the three redshift ranges, with the central volume again displaying peculiar clustering properties that apparently dominate over any possible cosmological effect.

9.2 Clustering evolution at fixed stellar mass

It is also interesting to compare directly the evolution of  with redshift, when a specific class of stellar mass is selected. As mentioned earlier in this section, this is particularly interesting, as in principle it does not require accounting for strong galaxy evolutionary trends as in the case of luminosity. We are assuming here that stellar mass does not significantly grow between and , which we know is only partially true. A factor of growth in stellar mass is in fact expected on average between and , which however would have little effect on the estimated correlation function, if taken into account. In Fig. 16 we show  computed for the same three samples with of Fig. 14. Similarly to the luminosity samples, we do not see any clear evolution with redshift of the amplitude and shape of the projected correlation function. The three curves are consistent with each other within .

Figure 16: Evolution of the projected function  of galaxies with between redshift and .

10 Comparison with independent measurements and models

10.1 Redshift evolution of

An accurate reference measurement of as a function of stellar mass has been obtained by the SDSS (Li et al., 2006). Meneux et al. (2008) do find evidence for evolution of the amplitude of  for galaxies less massive than , when comparing this to the measurements from VVDS-Deep at . The SDSS and zCOSMOS stellar masses were derived with the same initial mass function (Chabrier, 2003) and normalised to h=1. They are directly comparable. The SDSS clustering measurements were obtained within differential stellar mass ranges (Li et al., 2006), while ours correspond to galaxies more massive than a given threshold. However, from Fig. 2 we see that the zCOSMOS sample includes a very small number of galaxies more massives than  h, due to the much smaller volume when compared to the SDSS. Any of our mass-selected samples has therefore, in practice, an upper bound at this value of mass. This implies that we can coherently compare two of the SDSS measurements of (for their galaxy samples with stellar masses in the ranges and ) to those from our samples M3.1 and M3.2, that include galaxies more massive than and  h respectively, within the redshift range z=[0.8-1.0]. This comparison (Fig. 17) does not show a clear evolution with redshift. For both samples, the large-scale amplitude of  is virtually the same as in the local SDSS samples. Considering simple evolution of structures, this implies that the bias for galaxies more massive than  h, has evolved significantly between and today, as to keep their apparent clustering amplitude substantially unchanged. This implies in practice that the bias must evolve in a way such that , where is the linear growth rate of density fluctuations. In the standard model, this implies that at the approximate mean redshifts of our redshift bins, , the bias of massive galaxies must have been respectively 1.2, 1.44 and 1.53 times its value at the current epoch. Meneux et al. (2008) observed the same effect at in the VVDS data but only for galaxies more massive than  h, with lower-mass objects showing a weaker bias evolution. A non evolution of the clustering of the most massive galaxies was also noticed in the NDWFS (Brown et al., 2008) and 2SLAQ surveys (Wake et al., 2008).

Figure 17: The measured projected function  for galaxies at z=[0.8-1.0] with (left) and (right) from the zCOSMOS survey (filled diamonds), compared to the estimate by the SDSS (Li et al., 2006) (blue curve).

10.2 Observed and predicted shape of at

The only available measurement of clustering as a function of stellar mass at redshifts comparable to those explored by our sample is that from the VVDS-Deep survey (Meneux et al., 2008) at . VVDS-Deep goes 1.5 magnitude deeper (although over a smaller area of deg), which allows the analysis to be extended beyond . To provide a qualitative, yet meaningful comparison of these two data sets, we can re-compute the correlation function for the 10K data within the largest usable redshift range overlapping with the VVDS interval, i.e. [0.5-1.0]. We applied the same stellar mass selection limits, keeping in mind the residual incompleteness that will affect the highest redshift part of the sample. The result is shown in Fig. 18, where the VVDS and zCOSMOS mass-selected samples are directly compared. The difference in shape and amplitude in the  derived from the two data sets is rather striking. The zCOSMOS points show in general a much flatter relation than those from the VVDS. The amplitudes for a given mass selection also seem to be incompatible at several standard deviations between the two samples, especially above 1 h Mpc.

Figure 18: Direct comparison of the dependence of clustering on stellar mass in the VVDS-Deep and zCOSMOS samples, over a similar redshift range. This plot evidences the significant intrinsic difference between the two surveys, with the zCOSMOS sample showing a significantly larger clustering amplitude for all stellar masses. In addition, the much flatter shape of indicates the predominance of coherent structure perpendicularly to the line of sight, due to the “walls” discussed in the text.
Figure 19: The zCOSMOS (solid circles) and VVDS-Deep (open circles)  of galaxies with mass larger than and , compared to model predictions. These include the non-linear mass projected correlation function computed using HALOFIT (lower solid line, Smith et al., 2003), and, for reference, the corresponding  for a population of halos with bias (dashed curve). The latter curve is a very good description of the full non-linear  (light solid blue line), obtained averaging the forty DLB40 mocks from the Millennium run after applying the same sampling, magnitude and mass selections of the 10K sample. The shaded areas (thick and thin shaded) give, respectively, the and confidence corridors around the mean. The large-scale zCOSMOS  in this redshift range is thus marginally compatible with a rare, positive fluctuation in a standard CDM Universe.

10.3 Comparison to analytic and semi-analytic models

It is at this point relevant to compare the available observations from zCOSMOS and VVDS with model predictions in a standard CDM scenario. We can do this in two ways. We first used the HALOFIT public code (Smith et al., 2003), that uses the halo model to compute the expected non-linearly evolved power spectrum at , that we take as a reasonable mean redshift for the two samples (our conclusions would not differ at all if predictions for or 0.9 were used). The corresponding projected function  is then computed by Fourier-transforming the power spectrum and projecting the resulting real-space correlation function. The result gives the expected  of the mass density field at . Secondly, we can compute the expectation value and the scatter expected in the same redshift range () for  using the available semi-analytic mock surveys built from the Millennium run. To this end, we use the DLB40 mocks for which we have full control over stellar masses, selecting simulated galaxies with and , reproducing the sampling rate of the 10K data. In Fig. 19 we plot both the HALOFIT prediction for the dark matter (lower solid line) and for an arbitrarily biased population of halos with (dashed line), together with the BDL40 mean  (blue, lighter solid line) and the corresponding and scatter corridor from the 40 mock surveys (solid- and dotted-dashed areas, respectively). We note as a consistency check the rather good agreement between the analytic HALOFIT result and the expectation value from the full n-body plus semi-analytic simulation. On these model predictions, we overplot the corresponding zCOSMOS and VVDS estimates. The zCOSMOS points agree rather well with the models (at better than 68% confidence) in both shape and amplitude on scales smaller than 1 h Mpc. On larger scales, however, the observed  would require a strongly scale-dependent bias to be compatible with the model predictions. Such a scale-dependence would also behave oppositely to what models and very general considerations suggest, implying a bias which grows with scale, rather than declines. From the plot we see in fact that the 10K data are compatible with on small scales, but would require on 10 h Mpc scales. The shaded area shows that this large-scale excess is marginally compatible with the model predictions, representing a very strong positive fluctuation. A few percent of volumes this size would show such a high clustering amplitude (on these scales and for this kind of galaxies), in a CDM Universe.

It is interesting to note, at the same time, how the VVDS measurements lie on the opposite side of the distribution, at about from the mean, but with a shape which is compatible with the model prediction over the whole range (corresponding to a linear bias ). Comparison with Abbas & Sheth (2007) also suggests that the zCOSMOS field is centered on an overdensity, whereas the VVDS field is a less significant underdensity.

These results show how a full HOD model fitting to the  measured from the 10K data – that we originally planned to include in this paper – would add no meaningful information to the current analysis. Our first experiments with HOD models based on the universal halo mass function indicate that rather unrealistic sets of parameters are required to reproduce the observed function. An interesting possibility would be to use in such modeling a halo mass function that depends on local environment (e.g. Abbas & Sheth, 2005, 2006), to take into account the evidence that a large part of this sample is dominated by an overdensity. We plan to explore this possibility using the larger 20K zCOSMOS sample which is now nearly complete.

11 Discussion

Together with previous analyses (McCracken et al., 2007; Kovač et al., 2009), these results suggest that a significant fraction of the volume of Universe bounded by the COSMOS field is indeed characterized by particularly extreme density fluctuations. We have seen how, in statistical terms, these seem to lie at the limit of the distribution of amplitudes expected in volumes of a few  h Mpc. We should consider, however, that these conclusions are drawn from measurements that are strongly affected by the angular distribution of structure. The McCracken et al. (2007) result is based on the angular correlation function, while here we have studied the projected function . The latter, although making use of the redshift information is in practice a clustering measure dominated by galaxy pairs lying almost perpendicularly to the line of sight. The underlying assumption when measuring  is that the geometrical distribution of structures within the sample being analyzed is completely isotropic. In other words, that there are superclusters aligned along several directions, such that the only remaining radial signal is produced by galaxy peculiar velocities. The very reason of using  is indeed to get rid of the distortions introduced in the shape of (the redshift-space, angle-averaged correlation function) by galaxy motions. If this is true, and only in this case, then  is fully equivalent to an integral over the real-space correlation function and therefore carries the same cosmological information. However, if, as in the case we have encountered here, there is one or more dominating structures extending preferentially along one direction, then the use of  to infer cosmological information is inappropriate.

One may thus wonder whether more robust cosmological information could instead be inferred looking directly at the simplest, angle-averaged redshift-space correlation function . The expectation is that the average over all directions reduces the weight of the excess pair counts produced by just a few structures oriented along one preferential direction. In such case any analytic modelling (e.g. with HOD models) should also include an appropriate model for the linear and non-linear redshift distortions (Scoccimarro, 2004; Tinker et al., 2007). More simply, we can use the available mock samples in redshift space to compute the non-linear redshift-space and its variance and compare it to the data, as we did for . In Fig. 20 we first plot for the 10K sample, computed for the usual four mass ranges in the broad redshift range . The four data sets show a smooth power-law behaviour, with some evidence for a mass dependence of the clustering amplitude, in particular at the upper mass limit. The overall shape is well described by a rather flat power-law , with slope and a correlation length between 6 and  h Mpc. These values for the shape and amplitude of are similar to those measured for luminous red galaxies at in the 2SLAQ survey (see Fig. 7 in Ross et al. (2007)) and for luminous early-type galaxies in the 2dFGRS (Norberg et al., 2002). This is consistent with the most massive objects in the 10K sample being predominantly red, early-type galaxies which show moderate or no evolution in the overall clustering amplitude with redshift.

In Fig. 21, instead, we compare of our “reference” sample with , with the mean and scatter (at and confidence, respectively) of the similarly-selected set of BDL40 mocks. Despite the angular average, we note a behaviour which is similar to that observed in   although now the agreement extends to slightly larger scales. The observed clustering is compatible with the predictions of the standard model (to better than the 68% level) on scales smaller than  h Mpc. On larger scales, also shows excess power with respect to the models, which places the zCOSMOS volume at the upper limit of the statistical distribution obtained from the mocks. This exercise shows that even after angle-averaging our clustering estimator, the amount of structure present in this specific volume of the Universe remains outstanding in comparison to the model expectations. The conclusion can only be that either we have been very unlucky in the selection of the COSMOS field and picked up a fluctuation which has a probability of  1% to be found in such a volume, or fluctuations with this amplitude are in reality more common than what the standard cosmology predicts.

Figure 20: The redshift-space, angle-averaged correlation function of the mass-selected samples in the redshift bin . A mild systematic mass dependence is visible.
Figure 21: Comparison of the redshift-space correlation function for galaxies with at in the zCOSMOS sample (filled circles), with the model predictions from the BDL40 mock samples. The solid line gives the average of the 40 mocks, with the dashed areas corresponding to the and confidence error corridor. Similarly to what we found for , the agreement between the models and the zCOSMOS measurement at 1 h Mpc is remarkable. Nevertheless, the large-scale shape of the zCOSMOS is at the mean amplitude of the mock catalogues.

12 Summary

We have used the 10K zCOSMOS spectroscopic sample to study galaxy clustering as a function of galaxy luminosity and stellar mass, in the range of redshift [0.2,1].

To this end, we built luminosity and mass-selected samples from the 10K catalogue sampling three separate redshift ranges. We used mock catalogues to quantify the effect of stellar mass incompleteness on the measured clustering, as a function of redshift. We carefully checked our covariance and error estimate techniques, comparing the performances of methods based on the scatter in the mocks and on bootstrapping schemes. We adopted the latter, based on 200 resamplings of 8 sub-volumes of the survey, as the most appropriate description of of the covariance properties of the data.

By measuring the redshift-space correlation functions and and the projected function for these sub-samples, we found the following results.

  • Surprisingly, we do not see any clear dependence on luminosity of the correlation function at all redshifts. This is at odd with results in the local Universe by the 2dFGRS and with mesurements at similar redshift by the VVDS and DEEP2 surveys, that found a significant steepening of  with luminosity.

  • We find a mildly more evident (although not striking) dependence of  on stellar mass, especially on small scales. The central redshift bin (), displays in general a more evident effect, with a very flat shape of  on scales  h Mpc. The overall shape of the corresponding map of shows strong distortions that we interpret as the effect of dominant structure extending preferentially perpendicularly to the line of sight.

  • From comparison to the SDSS measurements at , we do not see any significant evolution with redshift of the amplitude of clustering for bright and/or massive galaxies. Together with previous results from VVDS, this is consistent with a more rapid evolution of the linear bias for the most massive objects, with respect to the general population. In the zCOSMOS sample this invariance in the clustering amplitude between and seems to remain valid down to smaller masses than in the VVDS, an effect easily explained by the overall larger clustering amplitude observed in general for in this sample. This is evidenced by a much flatter shape (higher amplitude) of  of zCOSMOS galaxies with respect to VVDS galaxies, when selected with the same criteria.

  • This particularly high level of structure is confirmed by comparison of the measured  and at with model predictions, concentrating on the sample with . On scales smaller than h Mpc, the observations agree very well with the model expectation values in the standard CDM scenario for a linar bias . On these scales, the measured values are compatible to better than 68% with the BDL40 mocks. On larger scales, however, the observed clustering amplitude is reproduced in only a few percent of the mocks. In other words, if the shape of the power spectrum is that of CDM and the bias has no “innatural” scale-dependence, COSMOS has picked up a volume of the Universe which is rare, positive fluctuation. This conclusion is corroborated also by comparison with the VVDS measurements, which on the other hand lie on the lower side of the distribution, at about .

Acknowledgements.
We thank the anonymous referee for a detailed review of the manuscript that helped to improve the paper. We thank G. De Lucia, J. Blaizot, S. Phleps and A. Sànchez for their thorough comments on an early version of the manuscript. This work was supported by grant ASI/COFIS/WP3110 I/026/07/0.

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