Issue 
A&A
Volume 579, July 2015



Article Number  A70  
Number of page(s)  18  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/201525727  
Published online  30 June 2015 
The VIMOS Public Extragalactic Redshift Survey (VIPERS)
Hierarchical scaling and biasing^{⋆}
^{1}
INAF–Osservatorio Astronomico di Bologna, via Ranzani 1,
40127
Bologna,
Italy
email:
alberto.cappi@oabo.inaf.it
^{2}
Laboratoire Lagrange, UMR 7293, Université de NiceSophia
Antipolis, CNRS, Observatoire de la Côte d’Azur, 06300
Nice,
France
^{3}
Dipartimento di Fisica e Astronomia – Alma Mater Studiorum
Università di Bologna, viale Berti
Pichat 6/2, 40127
Bologna,
Italy
^{4}
INFN, Sezione di Bologna, viale Berti Pichat 6/2,
40127
Bologna,
Italy
^{5}
Centre de Physique Théorique, UMR 6207 CNRSUniversité de
Provence, Case 907,
13288
Marseille,
France
^{6}
Dipartimento di Matematica e Fisica, Università degli Studi Roma
Tre, via della Vasca Navale
84, 00146
Roma,
Italy
^{7}
INFN, Sezione di Roma Tre, via della Vasca Navale
84, 00146
Roma,
Italy
^{8}
INAF–Osservatorio Astronomico di Roma,
via Frascati 33, 00040 Monte
Porzio Catone ( RM), Italy
^{9}
AixMarseille Université, CNRS, LAM (Laboratoire d’Astrophysique
de Marseille) UMR 7326, 13388
Marseille,
France
^{10}
INAF–Osservatorio Astronomico di Brera, via Brera 28, 20122
Milano, via E. Bianchi 46, 23807
Merate,
Italy
^{11}
Dipartimento di Fisica, Università di
MilanoBicocca, P.zza della Scienza
3, 20126
Milano,
Italy
^{12}
INAF – Osservatorio Astronomico di Torino,
10025
Pino Torinese,
Italy
^{13}
CanadaFranceHawaii Telescope, 65–1238 Mamalahoa
Highway, Kamuela,
HI
96743,
USA
^{14}
INAF–Istituto di Astrofisica Spaziale e Fisica Cosmica Milano, via
Bassini 15, 20133
Milano,
Italy
^{15}
Astronomical Observatory of the University of
Geneva, Ch. d’Écogia
16, 1290
Versoix,
Switzerland
^{16}
INAF–Osservatorio Astronomico di Trieste, via G. B. Tiepolo
11, 34143
Trieste,
Italy
^{17}
Institute ofPhysics, Jan Kochanowski University,
ul. Swietokrzyska 15,
25406
Kielce,
Poland
^{18}
Department of Particle and Astrophysical Science, Nagoya
University, Furocho, Chikusaku,
4648602
Nagoya,
Japan
^{19}
National Centre for Nuclear Research, ul. Hoza 69,
00681
Warszawa,
Poland
^{20}
Institut d’Astrophysique de Paris, UMR 7095 CNRS, Université
Pierre et Marie Curie, 98bis
boulevard Arago, 75014
Paris,
France
^{21}
Astronomical Observatory of the Jagiellonian
University, Orla
171, 30001
Cracow,
Poland
^{22}
Institute of Cosmology and Gravitation, Dennis Sciama Building,
University of Portsmouth, Burnaby
Road, Portsmouth,
PO1 3FX,
UK
^{23}
INAF–Istituto di Astrofisica Spaziale e Fisica Cosmica Bologna,
via Gobetti 101, 40129
Bologna,
Italy
^{24}
INAF–Istituto di Radioastronomia, via Gobetti 101,
40129
Bologna,
Italy
^{25}
Università degli Studi di Milano, via G. Celoria 16, 20130
Milano,
Italy
^{26}
SUPA, Institute for Astronomy, University of Edinburgh, Royal
Observatory, Blackford
Hill, Edinburgh
EH9 3HJ,
UK
^{27}
MaxPlanckInstitut für Extraterrestrische Physik,
84571
Garching b. München,
Germany
^{28}
Universitätssternwarte München, LudwigMaximillians
Universität, Scheinerstr.
1, 81679
München,
Germany
Received: 23 January 2015
Accepted: 5 May 2015
Aims. Building on the twopoint correlation function analyses of the VIMOS Public Extragalactic Redshift Survey (VIPERS), we investigate the higherorder correlation properties of the same galaxy samples to test the hierarchical scaling hypothesis at z ~ 1 and the dependence on galaxy luminosity, stellar mass, and redshift. With this work we also aim to assess possible deviations from the linearity of galaxy bias independently from a previously performed analysis of our survey.
Methods. We have measured the count probability distribution function in spherical cells of varying radii (3 ≤ R ≤ 10 h^{1} Mpc), deriving σ_{8g} (the galaxy rms at 8 h^{1} Mpc), the volumeaveraged two, three, and fourpoint correlation functions and the normalized skewness S_{3g} and kurtosis S_{4g} for different volumelimited subsamples, covering the following ranges: −19.5 ≤ M_{B}(z = 1.1) − 5log (h) ≤ −21.0 in absolute magnitude, 9.0 ≤ log (M_{∗}/M_{⊙}h^{2}) ≤ 11.0 in stellar mass, and 0.5 ≤ z< 1.1 in redshift.
Results. We have performed the first measurement of highorder correlation functions at z ~ 1 in a spectroscopic redshift survey. Our main results are the following. 1) The hierarchical scaling between the volumeaveraged two and threepoint and two and fourpoint correlation functions holds throughout the whole range of scale and redshift we could test. 2) We do not find a significant dependence of S_{3g} on luminosity (below z = 0.9 the value of S_{3g} decreases with luminosity, but only at 1σlevel). 3) We do not detect a significant dependence of S_{3g} and S_{4g} on scale, except beyond z ~ 0.9, where S_{3g} and S_{4g} have higher values on large scales (R ≥ 10 h^{1} Mpc): this increase is mainly due to one of the two CFHTLS Wide Fields observed by VIPERS and can be explained as a consequence of sample variance, consistently with our analysis of mock catalogs. 4) We do not detect a significant evolution of S_{3g} and S_{4g} with redshift (apart from the increase of their values with scale in the last redshift bin). 5) σ_{8g} increases with luminosity, but does not show significant evolution with redshift. As a consequence, the linear bias factor b = σ_{8g}/σ_{8m}, where σ_{8m} is the rms of matter at a scale of 8 h^{1} Mpc, increases with redshift, in agreement with the independent analysis of VIPERS and of other surveys such as the VIMOSVLT Deep Survey (VVDS). We measure the lowest bias b = 1.47 ± 0.18 for galaxies with M_{B}(z = 1.1) − 5log (h) ≤ −19.5 in the first redshift bin (0.5 ≤ z< 0.7) and the highest bias b = 2.12 ± 0.28 for galaxies with M_{B}(z = 1.1) − 5log (h) ≤ −21.0 in the last redshift bin (0.9 ≤ z< 1.1). 6) We quantify deviations from the linear bias by means of the Taylor expansion parameter b_{2}. We obtain b_{2} = −0.20 ± 0.49 for 0.5 ≤ z< 0.7 and b_{2} = −0.24 ± 0.35 for 0.7 ≤ z< 0.9, while for the redshift range 0.9 ≤ z< 1.1 we find b_{2} = + 0.78 ± 0.82. These results are compatible with a null nonlinear bias term, but taking into account another analysis for VIPERS and the analysis of other surveys, we argue that there is evidence for a small but nonzero nonlinear bias term.
Key words: largescale structure of Universe / cosmology: observations / dark matter / galaxies: statistics
Based on observations collected at the European Southern Observatory, Cerro Paranal, Chile, using the Very Large Telescope under programs 182.A0886 and partly 070.A9007. Also based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the CanadaFranceHawaii Telescope (CFHT), which is operated by the National Research Council (NRC) of Canada, the Institut National des Sciences de l’Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii. This work is based in part on data products produced at TERAPIX and the Canadian Astronomy Data Centre as part of the CanadaFranceHawaii Telescope Legacy Survey, a collaborative project of NRC and CNRS. The VIPERS web site is http://www.vipers.inaf.it
© ESO, 2015
1. Introduction
In the standard model of structure formation, the growth of density fluctuations from a primordial Gaussian density field is driven by gravity; it is possible to follow the evolution of these fluctuations through analytical and numerical approaches and predict the statistical properties for the dark matter field and dark matter haloes. Galaxies form in a complex process following the baryonic infall into dark matter halos: this means that the comparison between theory and observations is not straightforward, but it also implies that the spatial distribution of galaxies contains a wealth of information relevant for both cosmology and the physics of galaxy formation.
Extracting and exploiting this information from the data requires a number of different and complementary statistical approaches. For example, while the twopoint correlation function ξ_{2}(r) is the simplest and most widely used statistical indicator of galaxy clustering, a complete description of a distribution is only given by the full Jpoint correlation functions ξ_{J}, or equivalently, by the volumeaveraged correlation functions , which are related to the Jorder moments of the count probability distribution function (PDF)^{1}. The count PDF gives the probability of counting N objects as a function of volume V. Highorder correlations are particularly interesting because perturbation theory and numerical simulations can describe their behaviour for the gravitational evolution of matter density fluctuations.
The first estimates of the two and three point galaxy correlations functions on angular catalogues of galaxies were made by Groth & Peebles (1977), who found that these estimates were well described by the hierarchical relation ξ_{3}(r_{12},r_{13},r_{23}) = Q [ ξ_{2}(r_{12})ξ_{2}(r_{13}) + ξ_{2}(r_{13})ξ_{2}(r_{23}) + ξ_{2}(r_{12})ξ_{2}(r_{23}) ]. The threepoint correlation function has subsequently become a standard statistical tool for the analysis of clustering and has been applied to simulations and recent surveys of galaxies (see e.g. Marín et al. 2008; Moresco et al. 2014), while its Fourier transform, the bispectrum, has also been applied to the analysis of the Ly_{α} forest (Mandelbaum et al. 2003; Viel et al. 2004) and of the cosmic microwave background (CMB; Planck Collaboration XXIV 2014).
The scaling relation between the two and three point correlation functions was soon generalized to higher orders (Fry & Peebles 1978 up to J = 4; Sharp et al. 1984 up to J = 5) and was mathematically described by the socalled hierarchical models, where the Jpoint correlation functions are expressed as a function of products of the twopoint correlation function. Different versions of these models were suggested, but Balian & Schaeffer (1989) showed that all of them belong to the general class of scaleinvariant models, which are defined by the scaling property: (1)From a physical point of view, the hierarchical scaling of the correlation functions is expected in the highly nonlinear regime (the BBGKY hierarchy, see Davis & Peebles 1977; Fry 1984a; Hamilton 1988) and in the quasilinear regime (from perturbation theory, see Peebles 1980; Fry 1984b; Bernardeau 1992; Bernardeau et al. 2002 and references therein).
Another prediction of the hierarchical models is that the normalized highorder reduced moments should be constant. In the present paper we focus on the normalized skewness S_{3} and kurtosis S_{4}. Peebles (1980) showed that in secondorder perturbation theory, assuming Gaussian primordial density fluctuations and an Einsteinde Sitter model, S_{3m}, the normalized skewness of matter fluctuations assumes the value 34/7. Subsequent works have shown that the smoothed S_{3m} depends on the slope of the power spectrum and has a very weak dependence on the cosmological model (see Bernardeau et al. 2002).
While in standard models with Gaussian primordial fluctuations the skewness and higherorder moments assume nonzero values as a consequence of gravitational clustering, scenarios with nonGaussian primordial perturbations also predict a primordial nonzero skewness, particularly at large scales (≥10 h^{1} Mpc; Luo & Schramm 1993; Fry & Scherrer 1994; Gaztañaga & Maehoenen 1996; Gaztañaga & Fosalba 1998; Durrer et al. 2000); therefore these scenarios can in principle be constrained by measuring the highorder moments (Mao et al. 2014).
Moreover, it has been shown that the hierarchy of the Jpoint functions and the measurement of S_{3} and S_{4} can be used as a cosmological test to distinguish between the standard Λ cold dark matter (ΛCDM) and models including longrange scalar interaction between dark matter particles (“fifth force” dark matter models), as shown by Hellwing et al. (2010), who found the largest deviations in the redshift range 0.5 <z< 2.
However, the comparison between the theoretical predictions for the matter distribution and the observed galaxy distribution is not trivial, as a consequence of bias. One of the first results derived from the analysis of the first redshift surveys was that the amplitude of the twopoint correlation function depends on galaxy luminosity and galaxy colour (see Marulli et al. 2013, and references therein); therefore, the galaxy distribution must generally differ from the underlying matter distribution. A common assumption is that the galaxy and matter density fields are related by a linear relation, δ_{g} = bδ_{m}, where δ_{g} ≡ Δρ_{g}/ρ_{g} and δ_{m} ≡ Δρ_{m}/ρ_{m} are the galaxy and matter density contrast, respectively. This relation is a consequence of the scenario of biased galaxy formation, where galaxies form above a given threshold of the linear density field, in the limit of high threshold and low variance. Of course, this relation cannot have general validity: when b> 1 and δ_{m}< 0, the linear relation gives an unphysical value δ_{g}< −1.
A simple prediction of linear biasing is that the twopoint correlation function is amplified by a factor b^{2}, while S_{3} is inversely proportional to b. The analysis of the first redshift surveys revealed instead that different classes of galaxies selected in the optical and infrared bands, while differing in the amplitude of the twopoint correlation function, have similar values of S_{3} (Gaztañaga 1992; Bouchet et al. 1993; Benoist et al. 1999); the same also holds for galaxy clusters (Cappi & Maurogordato 1995). In particular, Benoist et al. (1999) analysed volumelimited samples of the Southern Sky Redshift Survey 2 (SSRS2, da Costa et al. 1994) and found that, while the twopoint correlation amplitude increases significantly with galaxy luminosity when L>L_{∗} (Benoist et al. 1996), the value of S_{3} does not scale with the inverse of the bias parameter b and is independent of luminosity and scale within the errors: this implies that the bias is nonlinear. Similar results were obtained in the Durham/UKST and StromloAPM redshift surveys (Hoyle et al. 2000) and in the larger and deeper 2dF Galaxy Redshift Survey (2dFGRS, Baugh et al. 2004; Croton et al. 2004a), which enabled a more detailed analysis: for example, Croton et al. (2004b) found evidence for a weak dependence of S_{3} on luminosity, while according to Croton et al. (2007) the S_{J} of red galaxies depends on luminosity, while blue galaxies do not show any dependence. In an analysis of the Sloan Digital Sky Survey (SDSS) Ross et al. (2006) found that the values of S_{J} are lower for latetype than for earlytypes galaxies.
In more recent years, deeper surveys enabled exploring the effects of the evolution of gravitational clustering and bias, thus placing stronger constraints on models of galaxy formation and evolution.
Wolk et al. (2013) measured the hierarchical clustering of the CFHTLSWide from photometric redshifts. They found an indication that at small scales the hierarchical moments increase with redshift, while at large scales their results are still consistent with perturbation theory for ΛCDM cosmology with a linear bias, but suggest the presence of a small nonlinear term.
From the analysis of the VIMOSVLT Deep Survey, based on spectroscopic redshifts, Marinoni et al. (2005; see also Marinoni et al. 2008) found that the value of S_{3} for luminous (M_{B}< −21) galaxies is consistent with the local value at z< 1 while decreasing beyond z ~ 1, and that the bias is nonlinear.
In this paper we analyse the highorder correlations and moments of the first release of the VIMOS Public Extragalactic Redshift Survey (VIPERS^{2}) in the redshift range 0.5 <z ≤ 1.1 as a function of luminosity and stellar mass. We also derive an estimate of the nonlinear bias. Our analysis extends those presented in a number of recent works that have investigated various aspects of galaxy clustering in the VIPERS sample. Some works have focused on twopoint statistics, like the standard galaxygalaxy twopoint correlation function to estimate redshift space distortions (de la Torre et al. 2013) and its evolution and dependence on galaxy properties (Marulli et al. 2013). A different type of twopoint statistics, the clustering ratio, has been introduced by Bel & Marinoni (2014) and applied to VIPERS galaxies (Bel et al. 2014) to estimate the mass density parameter Ω_{M}. Micheletti et al. (2014) have searched the VIPERS survey for galaxy voids and characterized their properties by means of the galaxyvoid crosscorrelation. Bel et al. (in prep.) have proposed a method to infer the onepoint galaxy probability function from counts in cells that Di Porto et al. (2014) have exploited to search for and detect deviations from linear bias; a result that we directly compare our results with. Finally, Cucciati et al. (2014) studied different methods for accounting for gaps in the VIPERS survey and assessing their impact on galaxy counts.
As cosmological parameters we have adopted H_{0} = 70 km s^{1} Mpc^{1}, Ω_{M} = 0.25, Ω_{Λ} = 0.75, but all cosmologydependent quantities are given in H_{0} = 100 km s^{1} Mpc^{1} units associated with the corresponding power of h = H_{0}/ 100.
2. Highorder statistics
In this section we resume the formalism and define the statistical quantities measured in our work.
The volumeaveraged Jpoint correlation functions are given by (2)where for spherical cells (used in this work) is a function of the cell radius R and V = 4πR^{3}/ 3.
The volumeaveraged twopoint correlation function gives the variance of the density contrast: (3)The volumeaveraged Jpoint correlation functions can be easily derived from the moments of the count PDF P(N,R), that is, the probability of counting N objects in a randomly chosen spherical volume of radius R (see Peebles 1980). For simplicity, in the following we omit the dependence on R. At a fixed scale R, the centred moments of order J are (4)where is the mean number of objects in a cell of radius R.
The volumeaveraged correlation functions correspond to the reduced moments and up to the fourth order are given by the following relations: (5)An alternative way to estimate the highorder correlations is through the factorial moments m_{k}: (6)where (7)is the falling factorial power of order k (see e.g. Graham et al. 1994).
In fact, for a local Poisson process the moments about the origin of a stochastic field are given by the factorial moments of N; as our variable is the number density contrast , we have to convert the factorial moments m_{k} into the moments about the mean (central moments) through the standard relations (8)We can finally derive the volumeaveraged Jpoint correlation functions (9)and the normalized moments S_{J}(10)The normalized moments can also be obtained through a recursive formula (Szapudi & Szalay 1993; Colombi et al. 2000): (11)where (12)The values given in this paper were calculated using factorial moments.
At a fixed scale R, the deterministic bias parameter b can be directly measured through the square root of the ratio of the galaxy variance to the matter variance : (13)In the case of linear biasing, the galaxy density contrast δ_{g} is proportional to the matter density contrast δ_{m} by a constant factor b, δ_{g} = bδ_{m}: there is no dependence on scale, and b is the only parameter that completely defines the relation between the galaxy and matter distribution.
As we have noted in the introduction, the linear biasing cannot have a general validity. It is more general and realistic to assume a local, deterministic nonlinear bias b(z,δ_{m},R), which can be written as a Taylor expansion (Fry & Gaztanaga 1993): (14)where b_{1} ≡ b. Fry & Gaztanaga (1993) have shown that such a local bias transformation preserves the hierarchical properties of the underlying matter distribution in the limit of small fluctuations (large scales).
In the case of linear bias, b_{k} = 0 for k> 1, and the galaxy and matter normalized moments are then related by the following equation: (15)In general, the deviation from linear biasing is measured by taking the second order of the expansion. In this case, the galaxy normalized skewness is given by the following relation: (16)
3. VIPERS survey
Definition of the samples.
The VIMOS Public Extragalactic Redshift Survey (VIPERS) is an ongoing ESO Large Programme aimed at determining redshifts for ~10^{5} galaxies in the redshift range 0.5 <z< 1.2, to accurately and robustly measure clustering, the growth of structure (through redshiftspace distortions) and galaxy properties at an epoch when the Universe was about half its current age (Guzzo & The Vipers Team 2013; Guzzo et al. 2014). The survey is divided into two separate areas and will cover ~24 deg^{2} when completed. The two areas are the socalled W1 and W4 fields of the CanadaFranceHawaii Telescope Legacy Survey Wide (CFHTLSWide); the CFHTLS optical photometric catalogues^{3} constitute the parent catalogue from which VIPERS spectroscopic targets were selected. The VIPERS survey strategy is optimized to achieve a good completeness in the largest possible area (Scodeggio et al. 2009). Galaxies are selected to a limit of i_{AB}< 22.5, further applying a simple and robust gri colour preselection to effectively remove galaxies at z< 0.5. In this way, only one pass per field is required, allowing us to double the galaxy sampling rate in the redshift range of interest with respect to a pure magnitudelimited sample (~40%). The final volume of the survey will be 5 × 10^{7}h^{3} Mpc^{3}, comparable to that of the 2dFGRS at z ~ 0.1.
VIPERS spectra are obtained using the VLT Visible MultiObject Spectrograph (VIMOS, Le Fèvre et al. 2002, 2003) at moderate resolution (R = 210), with the LR Red grism at R = 210 and a wavelength coverage of 5500–9500 Å. The typical radial velocity error is 140(1 + z) km s^{1}. A discussion of the survey data reduction and the first management infrastructure were presented in Garilli et al. (2012) and the detailed description of the survey was given by Guzzo et al. (2014).
The data set used in this and the other published papers is the VIPERS Public data release 1 (PDR1) catalogue, made available to the public in 2013 (Garilli et al. 2014). It includes about 47 000 reliable spectroscopic redshifts of galaxies and active galactic nuclei (AGNs). We here only selected galaxies with reliable redshift, that is, with spectroscopic quality flags 2,3,4, or 9 (see Garilli et al. 2014 for the definition).
To avoid regions dominated by large gaps, we here selected a subset of the total area covered by VIPERS: our limits are 02^{h}01^{m}00^{s} ≤ RA ≤ 02^{h}34^{m}50^{s}, −5.08° ≤ Dec ≤ −4.17° (7.67 square degrees) in W1 and 22^{h}01^{m}12^{s} ≤ RA ≤ 22^{h}18^{m}00^{s}, 0.865° ≤ Dec ≤ 2.20° (5.60 square degrees) in W4.
We defined volumelimited subsamples with different absolute magnitude and stellar mass limits, following the same criteria as in Marulli et al. (2013). The choice of these particular samples is discussed in detail in that paper; here we recall their main properties.
The restframe Bband absolute magnitude and the stellar mass were estimated through the HYPERZMASS program (Bolzonella et al. 2000, 2010), which applies a spectral energy distribution (SED) fitting technique. To take into account luminosity evolution, we fixed as a reference limit the luminosity at our maximum redshift (z = 1.1) and assumed an evolution M(z) = M(0) − z (see Meneux et al. 2009 and also Ilbert et al. 2005; Zucca et al. 2009).
We did not correct the mass limit of the stellarmasslimited subsamples; this limit was therefore kept fixed within each redshift bin because the evolution of M_{∗} is negligible in our redshift range (Pozzetti et al. 2007, 2010; Davidzon et al. 2013).
The respective numbers of galaxies for the different subsamples are given in Table 1. We note that these numbers are slightly different from those in Marulli et al. (2013) because we applied more stringent angular limits to avoid regions nearby prominent gaps that might affect the counts in spherical cells (while the direct estimate of the twopoint correlation function through counts of galaxy pairs can be easily corrected for by using a random catalogue with the same survey geometry).
4. Analysis of mock catalogues
We used mock catalogues derived from cosmological simulations to estimate not only the statistical errors and the uncertainty related to cosmic variance, but also the systematic errors that are due to the inhomogeneous spectroscopic completeness and the specific geometry of the two fields. A detailed description of the way these mocks were built was given by de la Torre et al. (2013).
We analysed a set of 26 independent mock catalogues based on the dark matter halo catalogue of the MultiDark simulation (Prada et al. 2012), which assumes a flat ΛCDM cosmology with (Ω_{M}, Ω_{Λ}, Ω_{b}, h, n, σ_{8m}) = (0.27, 0.73, 0.0469, 0.7, 0.95, 0.82). This catalogue was populated with galaxies using halo occupation distribution prescriptions, as described in de la Torre et al. (2013). In particular, the original halo catalogue was repopulated with halos below the resolution limit with the new technique of de la Torre & Peacock (2013), which enables reproducing the range in stellar mass and luminosity probed by VIPERS data. For luminositylimited subsamples, galaxy luminosities were calibrated using VIPERS data, while for stellar masslimited subsamples masses were assigned to galaxies using the stellartohalo mass relation (SHMR) of Moster et al. (2013). From the parent mock catalogues, a set of spectroscopic catalogues was derived by applying the same angular, photometric, and spectroscopic selection functions as were applied to the real data. For a more detailed and complete description of the mock catalogues see de la Torre et al. (2013).
From the mock spectroscopic catalogues we derived volumelimited subsamples with cuts in blue absolute magnitude and stellar mass corresponding to the observed ones. First of all, these mocks were used to test the effect of the gaps in the survey. As VIMOS is made of four quadrants 7 × 8 separated by 2 arcmin, characteristic crossshaped gaps are left in the survey; a further gap is present between the rows of pointings at different declination; finally, there are a few missing quadrants due to failed pointings. Cells whose projection on the sky includes a gap can potentially miss some galaxies, which affects final counts.
These gaps might be avoided by conservatively only counting galaxies in the cells that are completely included in one single quadrant, but in this way, only small scales would be sampled (the exact value obviously depends on the cell distance but it is generally lower than R ~ 5 h^{1} Mpc). Alternatively, the counts in each cell might be associated with the effective volume of the cell, subtracting the volume falling into the gaps; but this less drastic choice, which would slightly alter the shape of the cells, would still limit the range of the sampled scales.
Another option would be filling the gaps. Cucciati et al. (2014) applied two algorithms that use the photometric redshift information and assign redshifts to galaxies based upon the spectroscopic redshifts of the nearest neighbours. In this way, it is also possible to take into account the varying completeness from field to field. Tests on mocks have shown that these algorithms are successful in reconstructing the lowest and highest density environments at a scale of 5 h^{1} Mpc, but not in recovering the count PDF and its moments due to systematic biases.
Fig. 1 Comparison between mock catalogues with a sampling rate of 100% and without gaps (red triangles), and with a sampling rate and gaps as in VIPERS (blue triangles). The subsamples are limited at M_{B}(z = 1.1) − 5log (h) ≤ −20.5. From top to bottom: volumeaveraged two, three, and fourpoint correlation functions, normalized skewness S_{3} and kurtosis S_{4} in redshift space. First column: 0.5 ≤ z< 0.7; second column: 0.7 ≤ z< 0.9; third column: 0.9 ≤ z< 1.1. 

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We therefore here adopted another solution. The tests on mocks have shown that when cells are not allowed to cross the gaps by more than 40% of their volume, the nonobserved regions and the varying sampling rate can be approximated by a random Poisson sampling, and the original count PDF can be recovered with good precision (Bel et al., in prep.). This means that to obtain good estimates of the quantities we discuss here (Jpoint correlations and normalized moments), which depend on the density contrast Δρ/ρ, it is sufficient to implement the restriction on the volume of the cells falling into the gaps.
In our analysis, we conservatively only considered spherical cells for which no more than 30% of the volume falls in a gap. Moreover, to improve the statistics, we combined the counts of the W1 and W4 fields.
In Fig. 1 we show the results obtained from the analysis of mock subsamples limited at M_{B}(z = 1.1) − 5log (h) ≤ −20.5 in the three redshift bins [0.5, 0.7], [0.7, 0.9], [0.9, 1.1]. We compare the ideal case with 100% completeness and no gaps to the more realistic case with gaps and the same spectroscopic incompleteness as in our observed catalogue, that is, including the effects of the target sampling rate, TSR(Q), and the spectroscopic sampling rate, SSR(Q), where Q indicates the quadrant dependence.
Two other selection effects were not taken into account: the colour sampling rate, CSR(z), and the smallscale bias due to the constraints in the spectroscopic target selection (slits cannot overlap). The first effect depends on redshift but it is weak in our redshift range (see Fig. 5 of Guzzo et al. 2014), while the second effect is negligible because the angular radii of our cells are generally larger than the size of one quadrant.
We note that other sources of systematic errors, as discussed by Hui & Gaztañaga (1999), are the integral constraint bias, affecting the Jpoint correlation functions, and the ratio bias, affecting the estimate of S_{J}. Given the large size of our volumes, such systematic effects are weaker than the other errors, however, and can be neglected.
Figure 1 shows that the original values are recovered with good precision (within 1σ error), particularly in the scale range between 4 and 10 h^{1} Mpc.
Fig. 2 Fractional difference of the average , , , S_{3} and S_{4} (from top to bottom) for the same set of mock catalogues as defined in Fig. 1, i.e. with 100% sampling rate and without gaps, and with sampling rate and gaps as in VIPERS. The subsamples are limited at M_{B}(z = 1.1) − 5log (h) ≤ −20.5. Red triangles: 0.5 ≤ z< 0.7; blue squares: 0.7 ≤ z< 0.9; green hexagons: 0.9 ≤ z< 1.1. 

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A more detailed analysis of the differences is possible with Fig. 2, which gives the fractional difference for , , , S_{3}, and S_{4} as a function of scale for the same mock subsamples as in Fig. 1: it shows that in most cases we can retrieve the Jpoint correlation functions and S_{J} with only a small systematic difference. In the first redshift bin (0.5 ≤ z< 0.7) at a radius R = 8 h^{1} Mpc, is overestimated by 8%, while is underestimated by 3% and by 6%: this translates into an underestimate of S_{3} by 16% and of S_{4} by 26%. We have similar values in the second redshift bin (0.7 ≤ z< 0.9). In the last redshift bin (0.9 ≤ z< 1.1) the Jpoint correlation functions show the largest difference, increasing with order J: but these deviations at different orders are correlated, so that finally the value of S_{3} at 8 h^{1} Mpc is underestimated by only 10% and of S_{4} by 20%, which is comparable to what is found for the other two redshift bins. The cause of the larger deviations in the last redshift bin is the lower density of the subsample; we take these systematics into account in the discussion of our results.
It is interesting to point out that we find values between 1.8 and 2.1 for S_{3} and between 8 and 10 for S_{4} for mocks; as an example, the analysis of the mock subsamples limited at M_{B}(z = 1.1) − 5log (h) ≤ −20.5 in the redshift bin [0.7, 0.9] gives S_{3} ~ 2.13 ± 0.16 and S_{4} ~ 9.8 ± 1.6 at R = 8 h^{1} Mpc. S_{3} and S_{4} show no significant redshift evolution, and their values are also comparable within the errors to the value measured in local redshift surveys for galaxies in a similar luminosity range.
Because we know both the cosmological and the “observed” redshift for galaxies in the mock samples, including the peculiar velocity and measurement error, we can estimate the conversion factor from redshift to real space from the mock samples. We need this factor to compare our results with secondorder perturbation theory predictions. Figure 3 shows the difference between the estimates in real and redshift space for the subsamples limited at M_{B}(z = 1.1) − 5log (h) ≤ −20.5 in the three redshift ranges. The redshift space correlation functions show the expected loss of power at small scales and the reverse trend at large scales. The estimate of the volumeaveraged twopoint correlation function in redshift space is flatter than the corresponding estimate in real space; the difference becomes significant on scales smaller than ~4 h^{1} Mpc. While the real space values of S_{3} and S_{4} increase at smaller scales, the increase is suppressed in redshift space; the difference becomes small beyond ~4 h^{1} Mpc. However, at small scales we have large errors due to the small number of objects in the cells. For these reasons we focus our analysis on the 4–10 h^{1} Mpc range, and particularly at 8 h^{1} Mpc, where we expect to be in the quasilinear regime and predictions of secondorder perturbation theory should hold.
We recall here another bias affecting massselected galaxy samples, which has been discussed and tested with mock catalogues by Marulli et al. (2013). The lowest stellar mass subsamples suffer from incompleteness because VIPERS is magnitude limited (i_{AB}< 22.5); as a consequence, we can miss high masstolight ratio galaxies. From the analysis of mocks, Marulli et al. (2013) found that these galaxies are faint and red and that the clustering amplitude can be suppressed up to 50% on scales below 1 h^{1} Mpc. However, as discussed by Marulli et al. (2013), the abundance of red and faint galaxies is overpredicted by the semianalytic model used for the tests, and the clustering of red galaxies appears to be overestimated with respect to real data (de la Torre et al. 2011; Cucciati et al. 2012), so that the amplitude of the effect might be overestimated. As we have previously noted, we did not analyse small scales and did not correct for stellar mass incompleteness.
Fig. 3 Comparison between mock catalogues with 100% sampling rate and without gaps in real space (red squares) and redshift space (blue triangles). The subsamples are limited at M_{B}(z = 1.1) − 5log (h) ≤ −20.5. First column: 0.5 ≤ z< 0.7; second column: 0.7 ≤ z< 0.9; third column: 0.9 ≤ z< 1.1. 

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5. Results
5.1. Volumeaveraged correlation functions
In this section we present the results of our statistical analysis on the combined W1 and W4 samples.
Fig. 4 Volumeaveraged twopoint correlation functions as a function of the B absolute luminosity (upper panels) and stellar mass (lower panels). The limits of the subsamples in absolute magnitude and stellar mass are shown in the left upper and lower panels. Black triangles: M_{B}(z = 1.1) − 5log (h) ≤ −19.5 (log (M_{∗}/M_{⊙}h^{2} ≥ 9.0 M_{⊙}); red squares: M_{B}(z = 1.1) − 5log (h) ≤ −20.0 (log (M_{∗}/M_{⊙}h^{2} ≥ 9.5 M_{⊙}); blue pentagons: M_{B}(z = 1.1) − 5log (h) ≤ −20.5 (log (M_{∗}/M_{⊙}h^{2} ≥ 10.0 M_{⊙}); green hexagons: M_{B}(z = 1.1) − 5log (h) ≤ −21.0 (log (M_{∗}/M_{⊙}h^{2} ≥ 10.5 M_{⊙}); magenta heptagons: M_{B}(z = 1.1) − 5log (h) ≤ −21.5 (log (M_{∗}/M_{⊙}h^{2} ≥ 11.0 M_{⊙}). Dotted lines: realspace for the M_{B}(z = 1.1) − 5log (h) ≤ −20.5 (log (M_{∗}/M_{⊙}h^{2} ≥ 10.0 M_{⊙}) subsamples in the redshift bin [0.5, 0.7], predicted from the powerlaw fit of ξ_{2} in Marulli et al. (2013). 

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Figure 4 shows the volumeaveraged twopoint correlation function obtained from counts in cells for luminosity and stellar masslimited subsamples in the three different redshift bins.
In the same figure, as a reference for comparing the results in the different redshift bins, we plot the expected real space powerlaw in the redshift bin [0.5, 0.7] for the M_{B}(z = 1.1) − 5log (h) ≤ −20.5 subsamples (top panels) and M_{∗} ≥ 10.0 M_{⊙} (bottom panels), derived from the ξ_{2} estimate of Marulli et al. (2013); we converted their twopoint correlation function to the volumeaveraged correlation function through the formula (Peebles & Groth 1976): (17)The line shows the effects of redshift space distortions, which lower the value of on small scales and increase it on large scales.
It is clear that the amplitude of increases with both luminosity and stellar mass at all redshifts. appears to have a stronger dependence on stellar mass than on luminosity, in agreement with the results of Marulli et al. (2013): see their Fig. 3 for the redshift space twopoint correlation functions.
There are some fluctuations: for example, the dependence on luminosity appears to be sligthly weaker in the intermediate and distant redshift bins. However, these variations are consistent when taking into account statistical errors and sample variance, which are included in error bars. We conclude that the dependence of the twopoint correlation function on luminosity and stellar mass does not evolve significantly up to z ~ 1.
Fig. 5 Volumeaveraged threepoint correlation functions as a function of the B absolute luminosity (upper panels) and stellar mass (lower panels). The limits of the subsamples in absolute magnitude and stellar mass are shown in the left upper and lower panels (symbols and colours are the same as in Fig. 4). 

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Fig. 6 Volumeaveraged fourpoint correlation functions as a function of the B absolute luminosity (upper panels) and stellar mass (lower panels). The limits of the subsamples in absolute magnitude and stellar mass are shown in the left upper and lower panels (symbols and colours are the same as in Fig. 4). 

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In Figs. 5 and 6 we show the volumeaveraged three and fourpoint correlation functions. Their behaviour reflects the twopoint correlation functions, showing a stronger dependence of the correlation amplitude on stellar mass than on luminosity.
Fig. 7 Scaling relation of the volumeaveraged two and threepoint correlations as a function of the B absolute luminosity (upper panels) and stellar mass (lower panels). The limits of the subsamples in absolute magnitude and stellar mass are shown in the left upper and lower panels (symbols and colours are the same as in Fig. 4). The dashed line represents the scaling relation . 

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Fig. 8 Scaling relation of the volumeaveraged two and fourpoint correlations as a function of the B absolute luminosity (upper panels) and stellar mass (lower panels). The limits of the subsamples in absolute magnitude and stellar mass are shown in the left upper and lower panels (symbols and colours are the same as in Fig. 4). The dashed line represents the scaling relation . 

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The specific signature of the hierarchical scaling is the powerlaw relation between highorder correlation functions (Eq. (10)). In Figs. 7 and 8 we show the three and fourpoint volumeaveraged correlation functions as a function of the twopoint volumeaveraged correlation functions. The data clearly follow the hierarchical scaling relations and . These relations appear to hold at all luminosities and masses in the the first two redshift bins, but some systematic differences appear in the last redshift bin, particularly for the stellarmass limited subsamples, where points are systematically higher than the reference scaling law, but in this case the values are also consistent with the same scaling relation observed at lower redshifts.
As we have previously discussed, the existence of these scaling relations has been verified in the local Universe: they are expected for the matter distribution in the quasilinear regime, as a consequence of gravitational clustering. In this case, it is natural that they do not evolve with redshift: however, it is not an obvious result to observe the same hierarchical behaviour for the galaxy distribution at all redshifts, given the evolution of bias.
5.2. Skewness and kurtosis
From the counts in cells we derived the rms σ (Eq. (3)), the normalized skewness S_{3} and kurtosis S_{4} (Eq. (11)) for the different VIPERS subsamples. Their values at R = 8 h^{1} Mpc are given in Cols. 4–6 of Table 1. The R = 8 h^{1} Mpc reference radius is nearly optimal because it is large enough to enter into the quasilinear regime, and at the same time it is in the scale range for which we have a good sampling.
In Figs. 9 and 10 we show S_{3} and S_{4} as a function of luminosity and stellar mass in the three redshift bins. We also show the predictions of secondorder perturbation theory in real space for the matter distribution and the corresponding predictions for galaxies, derived from the matter value assuming the linear bias estimated from , and corrected for redshift space distortion using the factors obtained from mocks. This derivation is described in the next subsection. The theoretical curves for S_{3} and S_{4} are shown for radii larger than ~6 h^{1} Mpc, as they are calculated in the quasilinear regime.
Fig. 9 Normalized skewness S_{3} as a function of the B absolute luminosity (upper panels) and stellar mass (lower panels). The limits of the subsamples in absolute magnitude and stellar mass are shown in the left upper and lower panels (symbols and colours are the same as in Fig. 4). We note that black points corresponding to galaxies with M< −19.50 or M_{∗} ≥ 9.0 M_{⊙} are only plotted for the first redshift bin ([0.5, 0.7]), but most of them are not visible as they lie below the points of the other samples. The dashed line corresponds to S_{3} = 2. In the top panels, the dotted line is the prediction from secondorder perturbation theory for the matter distribution in real space; the solid line is the prediction of S_{3} for galaxies with M_{B}(z = 1.1) − 5log (h) ≤ −20.5 (to be compared to blue pentagons): it was obtained from the matter values, converted to redshift space and divided by the corresponding linear bias factor. 

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Fig. 10 Normalized skewness S_{4} as a function of the B absolute luminosity (upper panels) and stellar mass (lower panels). The limits of the subsamples in absolute magnitude and stellar mass are shown in the left upper and lower panels (symbols and colours are the same as in Fig. 4). We note that black points corresponding to galaxies with M< −19.50 or M_{∗} ≥ 9.0 M_{⊙} are only plotted for the first redshift bin ([0.5, 0.7]), but most of them are not visible as they lie below the points of the other samples. The dashed line corresponds to S_{4} = 8. In the top panels, the dotted line is the prediction from secondorder perturbation theory for the matter distribution in real space; the solid line is the prediction of S_{4} for galaxies with M_{B}(z = 1.1) − 5log (h) ≤ −20.5 (to be compared to blue pentagons): it was obtained from the matter values, converted to redshift space and divided by the corresponding linear bias factor. 

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In the first redshift bin, both for luminosity and stellar mass limited samples, the value of S_{3} is constant and around 2 at small and intermediate scales, but it starts decreasing beyond R ~ 8 h^{1} Mpc. In principle, variations of S_{3} with scale can be due to changes in the slope of the power spectrum or to a scaledependent bias. However, such a systematic effect can be ascribed to the small number of independent cells at large scales, as shown by mocks and reflected in the large error bars. In the same redshift bin, S_{4} shows a small decrease at large scales and is consistent with a constant value of ~7.3 between 4 and 10 h^{1} Mpc. In the range 6–8 h^{1} Mpc, the best scales to compare with perturbation theory (on larger scales the errors increase significantly), the theoretical predictions for S_{3} and S_{4} are slightly higher than the observed values corresponding to the M_{B}(z = 1.1) − 5log (h) ≤ −20.5 subsample, but only at 1σ level.
In the second redshift bin the value of S_{3} for luminositylimited subsamples is around 1.8, sligthly lower than in the first bin, but still consistent within the errors; moreover, it is consistent with a constant value in the whole range of scales. The value of S_{3} for masslimited subsamples is also constant in the whole range of scales and is consistent with the value in the first redshift bin. S_{4} has an analogous behaviour: while showing a systematic decrease, particularly in luminositylimited subsamples, it is still consistent with a constant value in the range 4–16 h^{1} Mpc. As in the case of the first redshift bin, in the range 6–8 h^{1} Mpc the theoretical predictions for S_{3} and S_{4} are slightly higher than the corresponding observed values.
In the third redshift bin the values of S_{3} and S_{4} for luminosity and stellarmasslimited subsamples increase systematically with scale. Moreover, in contrast with the two previous redshift bins, in the range 6–8 h^{1} Mpc, the theoretical predictions for S_{3} and S_{4} are lower than the observed values.
To better appreciate the significance of these deviations, we note that of 26 mocks, 3 show an increase of the values of S_{3} and S_{4} similar to what we find in the last redshift bin.
In fact, higherorder statistics are very sensitive to largescale structure, and the correlated variations in the measured values of S_{3} and S_{4} probably indicate genuine fluctuations in the galaxy distribution (see e.g. the discussion in Croton et al. 2004b).
In our case, this interpretation is suggested by checking the W1 and W4 fields separately: we find that in the outermost redshift shell, both S_{3} and S_{4} are larger in W1 than in W4. For example, for the M_{B}(z = 1.1) − 5log (h) ≤ −21.0 subsample, at R = 8 h^{1} Mpc, we find S_{3} = 2.7 ± 0.5 in W1 and S_{3} = 1.6 ± 0.3 in W4. Analogously, for the log (M/M_{⊙}h^{2}) ≥ 10.5 subsample at R = 8 h^{1} Mpc, we find S_{3} = 3.4 ± 0.5 in the W1 field and S_{3} = 2.0 ± 0.3 in the W4 field. This difference might be regarded as the imprint of spatially coherent structures more prominent in W1.
In conclusion, the values of S_{3} and S_{4} do not show any significant dependence on luminosity or on stellar mass: the points corresponding to different subsamples are consistent within the error bars (we discuss a possible weak dependence on luminosity in the next subsection). There is no evidence of evolution in redshift either, apart from the systematic increase of S_{3} and S_{4} with scale in the last redshift bin.
Taking into account the behaviour of mocks, the observed systematic variations in the values of highorder moments are consistent with the fluctuations expected for comparable volumes randomly extracted from a ΛCDM universe.
It is possible to compare our results on S_{3} and S_{4} with those obtained by Wolk et al. (2013) for the four CFHTLSWide fields. They have divided the galaxies in the photometric catalogue into four redshift bins through the estimated photometric redshifts; for galaxies with M_{g}< −20.7, they have estimated S_{J} as a function of angular scale and the corresponding 3D values through deprojection, which, as they discussed, rely on some approximations. Their work is therefore complementary to ours: they have a larger area and number of objects, but we can directly estimate the 3D (redshift space) S_{J}; they can sample smaller, highly nonlinear scales where we do not have enough statistics, but we can better sample the quasilinear scales; finally, we can also test the dependence of S_{3} on luminosity and stellar mass.
A comparison with their Fig. 12 shows that, as expected (see our Fig. 3), their deprojected values for S_{3} and S_{4} on small scales (R< 5 h^{1} Mpc) are higher than our redshift space values. On larger scales, the redshift space effect on S_{3} and S_{4} becomes negligible, and their estimate is consistent with ours.
We note that Wolk et al. (2013) found significant deviations in the results for the W3 field, while we have found differences between W1 and W4 in our last redshift bin: this shows that sample variance is still significant for highorder statistics on the scale of CFHTLS Wide Fields.
5.3. Implications for biasing
We now discuss the implications of our analysis for biasing. We concentrate on the reference scale R = 8 h^{1} Mpc, where secondorder perturbation theory predictions can be applied and results are still reliable (errors and systematic deviations increase on larger scales). Because we aim to compare our results with the matter density field, statistical quantities referring to galaxies are indicated with a subscript g and those relative to matter with a subscript m.
Figure 11 shows the values of σ_{8g} (top panel) and S_{3g} (bottom panel) at R = 8 h^{1} Mpc for the VIPERS volumelimited subsamples with different limiting absolute magnitudes and in the different redshift bins. In the same figure we also show the corresponding VVDS estimates (Marinoni et al. 2005) and the 2dFGRS estimates for the local Universe (Croton et al. 2004b) for galaxies with a similar luminosity as ours.
At a given redshift, VIPERS subsamples with a brighter absolute magnitude limit have higher values of σ_{8g}, but there is no significant evolution of σ_{8g} with redshift. The same holds when combining our results with those of the 2dFGRS in the local Universe and those of the VVDS at higher redshift: σ_{8g} shows no significant evolution from z = 0 to z = 1.4 (VVDS points are systematically lower but at the 1σ level). This implies (see e.g. the discussion in Marinoni et al. 2005) a strong evolution of the linear bias b with redshift because σ_{8m} increases with time (see Eq. (13)). There are various models that describe the evolution of b(z) and explain its decrease with time (see e.g. Blanton et al. 2000); from an empirical point of view, we note that the available data can be fitted by the simple relation b(z) ∝ 1/σ_{8m}.
The skewness S_{3g} of the VIPERS subsamples measured at 8 h^{1} Mpc and plotted as a function of redshift has more fluctuations than σ_{8g}, with a minimum value in the redshift bin [0.7,0.9], but it does not show a significant dependence on luminosity and is still consistent with a constant value independent of redshift. The values of S_{3g} in the VVDS below z = 1.2 are lower than VIPERS values, but are consistent within the errors, while they start to decrease beyond z ~ 1.1.
The absence of a significant evolution of S_{3g} with redshift is not limited to our redshift range: the values of S_{3g} measured in VIPERS are similar to those measured in the 2dFGRS, that is, S_{3} ~ 2.0 ± 0.2, where depending on the subsample S_{3g} varies from 1.95 to 2.58 (while not shown in the figure, the values of S_{4} are also consistent with the 2dFGRS ones). Therefore, taking into account all data points, starting from the local value for the 2dFGRS up to z = 1.1 (VIPERS and VVDS data), S_{3g} is consistent with a constant value ~2: in VIPERS the strongest but marginal deviations of the S_{3g} value are for M_{B}(z = 1.1) − 5log (h) ≤ −20.0 galaxies in the nearest redshift range [0.5, 0.7] and for M_{B}(z = 1.1) − 5log (h) ≤ −21.0 galaxies in the most distant redshift interval [0.9, 1.1], both giving a value of S_{3g} that is 15% higher.
Fig. 11 σ_{8g}(upper panel) and S_{3g} (lower panel) as a function of redshift. Colours represent the different absolute magnitude limits as in previous figures. The open hexagon represents the corresponding value for 2dFGRS galaxies in the absolute magnitude M_{bJ} range [–21, –20] (Croton et al. 2004b). Black open squares are the values for VVDS galaxies brighter than M_{B} = −21 (Marinoni et al. 2005). For better visibility, points corresponding to different redshifts are slightly shifted in magnitude. 

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Figure 12 shows σ_{8g} (top panel) and S_{3g} at 8 h^{1} Mpc (bottom panel) as a function of absolute magnitude for the three redshift bins. σ_{8g} shows a systematic increase with luminosity (reflecting the dependence of the correlation amplitude on luminosity), but at a given absolute luminosity its value is similar in the three redshift bins.
S_{3g} appears to be independent of absolute magnitude, with fluctuations from sample to sample. However, if we exclude the points relative to the last redshift bin, where S_{3g} has a higher value, the data might suggest a small decrease of S_{3g} with increasing luminosity, reminiscent of the results of Croton et al. (2004b) for the 2dFGRS.
A trend of S_{3g} with luminosity is interesting because in the hypothesis of linear biasing, S_{3g} is inversely proportional to the bias factor b: knowing from the twopoint correlation function of our samples that b increases with luminosity, we expect a corresponding decrease of S_{3g}.
Fig. 12 σ_{8g} (upper panel) and S_{3g} at R = 8 h^{1} Mpc (lower panel) as a function of galaxy luminosity for the three redshift bins. Red triangles: 0.5 ≤ z< 0.7; blue squares: 0.7 ≤ z< 0.9; green hexagons: 0.9 ≤ z< 1.1. For better visibility, points corresponding to different redshifts are slightly shifted in magnitude. 

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To test whether our results are consistent with the linearity of bias, we therefore estimated the bias of galaxies with respect to the underlying matter density field at R = 8 h^{1} Mpc, using the observed σ_{8g} and S_{3g} of the galaxy distribution and estimating σ_{m} and S_{3m} of the matter distribution through perturbation theory.
Juszkiewicz et al. (1993) and Bernardeau (1994a, 1994b; see also Bernardeau et al. 2002, and references therein) have shown that for a smoothed density field with primordial Gaussian fluctuations, Peebles’ unsmoothed value of S_{3m} = 34/7 (Peebles 1980) has to be corrected according to the expression (18)where d is the logarithmic slope of the linear variance of the matter density field smoothed with a spherical tophat function of radius R, (19)For a powerlaw spectrum P(k) ∝ k^{n}, Eq. (18) becomes S_{3m} = 34/7 − (n + 3).
Similar relations hold for higher orders, involving higherorder derivatives.
The values obtained from perturbation theory have been tested with numerical simulations, and it has been shown that in the range we are studying, that is, at R = 8 h^{1} Mpc and for σ_{8m} ~ 1, they are very accurate: for example, the difference in the S_{3} values is smaller than a few percent (Baugh et al. 1995; Fosalba & Gaztanaga 1998; Bernardeau et al. 2002).
Applying Eqs. (18) and (19) and using the software CAMB (Lewis & Bridle 2002), we have computed the values of σ_{8m} and S_{3m} for a power spectrum with the new cosmological parameters derived from the Planck mission (Planck Collaboration XVI 2014) and with the old Millennium parameters (first year WMAP data and 2dFGRS, with Ω_{M} = 0,25, Ω_{Λ} = 075, n = 1 and σ_{8m} = 0.9).
We here assumed that the standard ΛCDM model is correct. With other assumptions, such as a dark energy component with an evolving equation of state or modified gravity, the clustering and bias evolution would be affected (see e.g. Munshi et al. 2004), as would the redshift distortions (Hellwing et al. 2013). This dependence on cosmology will be studied in a future work.
We also converted the observed σ_{8g} and S_{3g} to real space values by applying correction factors directly derived from the mocks.
For the subsample limited at M_{B}(z = 1.1) − 5log (h) ≤ −20.50, we give in Table 2 the redshift range (Col. 1), the values of σ_{8g} (Col. 2), σ_{8m} (Col. 3), b = σ_{8g}/σ_{8M} (Col. 4), S_{3g} (Col. 5), S_{3m} (Col. 6), all measured at a scale of R = 8 h^{1} Mpc.
In Fig. 13 we plot our estimates for the linear bias term b as a function of redshift, with the correponding estimates for VIPERS of Marulli et al. (2013) and Di Porto et al. (2014). As expected, the estimates are fully consistent, with b increasing with luminosity and redshift. As discussed by Di Porto et al. (2014), there is only a difference in the last redshift bin where the estimate of Marulli et al. is lower than that of Di Porto et al. (2014). The difference is probably due to the way b is estimated (counts in cells in our case and in Di Porto et al. 2014, pair counts in Marulli et al. 2013). Our estimate is consistent with both the other two estimates at the 1σ level, however.
In Fig. 14 we compare the linear bias directly measured from the ratio of the galaxy and matter rms, b = σ_{8g}/σ_{8m}, with the ratio of the galaxy and matter skewness, S_{3m}/S_{3g}. Under the hypothesis of linear biasing, the two ratios should have the same value. For the first two redshift bins we find slightly different values: the skewness ratio is systematically higher than the bias directly computed from the variance. The third redshift bin shows the largest discrepancy, but with the opposite behaviour, that is, the skewness ratio is lower than the bias directly computed from the variance. This different behaviour is a consequence of the fact that the value of S_{3g} in the last redshift bin increases with scale and becomes higher than at lower redshifts.
We can quantify the degree of nonlinearity by directly estimating the secondorder term b_{2} from Eq. (16): (20)where we used the real space values S_{3g} and b obtained from the redshift space values by using the conversion factor calculated from the mocks. We note that this correction is small (a few percent) at our scale of R = 8 h^{1} Mpc, because this scale is at the transition from the regime of smallscale velocity dispersion (where redshift space correlation functions are lower than real space ones) to the regime of infall where redshift space correlations are higher than real space ones (see Fig. 3).
In this formalism, if b> 0, b_{2} is negative when σ_{8g}/σ_{8m}<S_{3m}/S_{3g}. This is what happens in the first two redshift bins, where at nearly all magnitudes b_{2} is negative: for example, for the subsample limited at M ≤ −20.5(z = 1.1) − 5log (h), we find b_{2} = −0.20 ± 0.49 in the first redshift bin and b_{2} = −0.24 ± 0.35, in the second redshift bin. In contrast, we find a positive b_{2} in the third bin, with b_{2} = + 0.78 ± 0.82.
As we have noted above when discussing the results of our tests on mocks, the assumption that masked regions and inhomogeneities can be described as a Poissonian random sampling gives a small bias with an overestimate of b of a few percent and an underestimate of S_{3} around 10–15%. Using the correction factors derived from the average of the mocks, we find for the subsample limited at M ≤ −20.5(z = 1.1) − 5log (h)b_{2} = −0.03 ± 0.49 in the first redshift, b_{2} = −0.25 ± 0.35 in the second redshift bin, and b_{2} = + 0.72 ± 0.82 in the third bin. The differences are well within 1σ error.
It would be tempting to interpret these results as suggesting a possible evolution of the nonlinear bias b_{2} with redshift, with a similar trend, for example, as for the model of Sefusatti & Komatsu (2007). Unfortunately, the problem is the extreme sensitivity of b_{2} to the errors on b and S_{3g}, amplified by a factor b^{2}, and we have seen that subsamples in the last redshift bin are affected by larger errors and systematic trends.
Real space values of σ_{8} and S_{3} of galaxies with M_{B}(z = 1.1) − 5log (h) ≤ −20.50 vs. those expected from secondorder perturbation theory (with Millennium and Planck cosmological parameters).
Fig. 13 Linear bias b as a function of redshift. Top panel: M_{B} ≤ −20.0(z = 1.1) + 5log (h); middle panel: M_{B} ≤ −20.5(z = 1.1) + 5log (h); bottom panel: M_{B} ≤ −21.0(z = 1.1) + 5log (h). Red hexagons: our estimates of b = σ_{8g}/σ_{8m}. Blue squares: estimates of Marulli et al. (2013). Magenta triangles: Di Porto et al. (2014). 

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Fig. 14 Estimates of b = σ_{8g}/σ_{8m} (solid lines, filled symbols) and b′ ≡ S_{3m}/S_{3g} (dotted lines, open symbols); b′ = b under the assumption of linear bias. Red lines with triangles: 0.5 ≤ z< 0.7; blue lines with squares: 0.7 ≤ z< 0.9; green lines with hexagons: 0.9 ≤ z< 1.1. 

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With these caveats, we can check the consistency of our results with other works in the same redshift and luminosity ranges. In this comparison, one has to take into account the sensitivity of b_{2} to the different methods and, as pointed out by Kovač et al. (2011), to sample variance. In fact, even local measurements of the nonlinear term have given different values (see e.g. Verde et al. 2002 and Pan & Szapudi 2005, and the discussion in Gaztañaga et al. 2005 and Marinoni et al. 2008).
First of all, Di Porto et al. (2014) have analysed the VIPERS data reconstructing the bias relation from the estimate of the probability distribution function: they found a small (<3%) but significant deviation from linear bias.
In their analysis of the four CFHTLS Wide fields, Wolk et al. (2013) have found that perturbation theory predictions agree well with their measurements when taking into account the linear bias, but note that there is still a small discrepancy that can be explained by the presence of a nonlinear bias term. This is also consistent with what we found.
Marinoni et al. (2005) have analysed VVDS volumelimited samples limited at M_{B}< −20 + 5log (h) (this limit was fixed and did not take into account luminosity evolution) in the redshift bins 0.7 <z< 0.9 and 0.9 <z< 1.1, finding b_{2} = −0.20 ± 0.08 and b_{2} = −0.12 ± 0.08 (here the errors do not include sample variance): these values are consistent with ours below z = 1.
Kovač et al. (2011) analysed the zCOSMOS galaxy overdensity field and estimated the mean biasing function between the galaxy and matter density fields and its second moment, finding a small nonlinearity, with the nonlinearity parameter (defined in the formalism of Dekel & Lahav 1999) at most 2% with an uncertainty of the same order.
Gaztañaga et al. (2005) have found and from the measurement of the Q_{3} parameter in the threepoint correlation function of the 2dFGRS for the local Universe.
The nonlinear term we have measured in the redshift interval between z = 0.5 and z = 0.9 is therefore similar to what has been measured in the above surveys. We conclude that there is general evidence for a small but nonzero nonlinear b_{2} term. It is also clear that no evolution of b_{2} with redshift can be detected in the available data, in contrast to the linear bias term.
6. Conclusions
We have analysed the highorder clustering of galaxies in the first release of VIPERS, using counts in cells to derive the volumeaveraged correlation functions and normalized skewness S_{3g} and kurtosis S_{4g}. We have analysed volumelimited subsamples with different cuts in absolute magnitude and stellar mass in three redshift bins; these subsamples are the same as in Marulli et al. (2013).
Errors were estimated through a set of mock catalogues, derived from dark matter halo catalogues repopulated with the method of de la Torre & Peacock (2013). The mocks were built to reproduce the properties of VIPERS, including masks and selection effects. Our analysis has shown that the highorder statistical properties of these mocks are consistent with observations.
We also studied the dependence of the second and thirdorder statistics of galaxy counts on the bias, deriving the linear bias term b and the first nonlinear term b_{2}, and comparing our results with predictions from perturbation theory and with other works in the literature.
Here are our main conclusions.

We showed that the hierarchical scaling relations and hold in the range of scales and redshifts we could sample, that is, 3 ≤ R ≤ 10 h^{1} Mpc and 0.5 ≤ z< 1.1. These relations are consistent with predictions from gravitational clustering and with the scaling observed in local surveys.

S_{3g} and S_{4g} appear to be independent of luminosity; however, if we do not not take the last redshift bin into account, there is a slight decrease of S_{3g} with increasing luminosity, an effect previously detected locally in the 2dFGRS by Croton et al. (2004b).

The values of S_{3g} and S_{4g} are scaleindependent within the errors and do not evolve significantly at least up to z = 0.9. We detected a systematic increase with scale in the last redshift bin (beyond ~10 h^{1} Mpc), mainly due to one of the two CFHTLS fields (W1); this deviation is consistent with what can be expected from the sample variance shown by mock catalogues.

The observed values of S_{3g} ~ 2 ± 0.2 and S_{4g} ~ 8 ± 0.4 are similar to those measured in local surveys for galaxies in the same luminosity range. This confirms the substantial absence of evolution of S_{3g} in the redshift range 0 <z< 1 at the level of ~10%. This result is expected for S_{3m}, but is not trivial for S_{3g}, given the evolution of bias.

At second order, galaxies with higher luminosity or stellar mass have a larger amplitude (greater linear bias parameter) of the volumeaveraged twopoint correlation function, consistently with the direct analysis of the twopoint correlation function by Marulli et al. (2013). We showed that our estimate of the linear bias parameter b = σ_{8g}/σ_{8m} is consistent within 1σ with those of Marulli et al. (2013) and Di Porto et al. (2014). The linear bias increases both with luminosity and with redshift: in our redshift range, we measured the lowest bias b = 1.47 ± 0.18 for galaxies with M_{B}(z = 1.1) − 5log (h) ≤ −19.5 in the redshift bin 0.5 ≤ z< 0.7 and the largest bias b = 2.12 ± 0.28 for galaxies with M_{B}(z = 1.1) − 5log (h) ≤ −21.0 in the redshift bin 0.9 ≤ z< 1.1.

For a given luminosity class, σ_{8g} does not evolve with redshift. For example, comparing our values for M_{B}(z = 1.1) − 5log (h) ≤ −20.5 to the corresponding value measured in the 2dFGRS, we found that σ_{8g} is consistent with a constant value 1.0 (our 1σ error is 10%), from z = 0 to z ~ 1. Given that σ_{8m} increases with time, we have the empirical relation b(z) ∝ 1 /σ_{8m}(z).

The value of the nonlinear bias parameter b_{2} measured below z ~ 1 at the scale R = 8 h^{1} Mpc, that is, in the quasilinear regime, is negative but not statistically different from zero when taking into account the error; however, taking into account the ensemble of results coming from this and other surveys in the redshift range 0.5 ≤ z< 1 (Marinoni et al. 2005; Kovač et al. 2011; Wolk et al. 2013; Di Porto et al. 2014), there is evidence for a small but nonzero nonlinear term. Including the results from local surveys as well, no evolution of b_{2} with redshift can be detected in the available data.

The comparison with the properties of mocks and with the predictions of perturbation theory shows that our results are consistent with the general scenario of biased galaxy formation and gravitational clustering evolution in a standard ΛCDM cosmology.
In conclusion, we have provided an independent check on the secondorder statistical studies of the galaxy distribution through our analysis; we explored the galaxy bias with an independent technique; finally, we determined the higherorder statistical properties of the galaxy distribution in the redshift range between 0.5 and 1.1, thanks to the combination of volume and density of galaxies in the VIPERS survey. When VIPERS is complete, it will be possible to perform a more general analysis, which will allow us not only to decrease error bars, but also to include the dependence of highorder statistics on galaxy colour, to apply other highorder statistical tools such as the void probability function, and to give better constraints on the nonlinear bias.
However, there is the important exception of the lognormal distribution, see Coles & Jones (1991) and Carron (2011).
Mellier et al. (2008), http://terapix.iap.fr/cplt/oldSite/Descart/CFHTLST0005Release.pdf
Acknowledgments
This work is based on observations collected at the European Southern Observatory, Cerro Paranal, Chile, using the Very Large Telescope under programs 182.A0886 and partly 070.A9007. Also based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the CanadaFranceHawaii Telescope (CFHT), which is operated by the National Research Council (NRC) of Canada, the Institut National des Sciences de l’Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii. This work is based in part on data products produced at TERAPIX and the Canadian Astronomy Data Centre as part of the CanadaFranceHawaii Telescope Legacy Survey, a collaborative project of NRC and CNRS. The VIPERS web site is http://www.vipers.inaf.it/. We acknowledge the crucial contribution of the ESO staff for the management of service observations. In particular, we are deeply grateful to M. Hilker for his constant help and support of this program. Italian participation to VIPERS has been funded by INAF through PRIN 2008 and 2010 programs. D.M. gratefully acknowledges financial support of INAFOABrera. L.G., A.J.H., and B.R.G. acknowledge support of the European Research Council through the Darklight ERC Advanced Research Grant (#291521). A.P., K.M., and J.K. have been supported by the National Science Centre (grants UMO2012/07/B/ST9/04425 and UMO2013/09/D/ST9/04030), the PolishSwiss Astro Project (cofinanced by a grant from Switzerland, through the Swiss Contribution to the enlarged European Union), and the European Associated Laboratory Astrophysics PolandFrance HECOLS. K.M. was supported by the Strategic Young Researcher Overseas Visits Program for Accelerating Brain Circulation (#R2405). O.L.F. acknowledges support of the European Research Council through the EARLY ERC Advanced Research Grant (#268107). G.D.L. acknowledges financial support from the European Research Council under the European Community’s Seventh Framework Programme (FP7/20072013)/ERC grant agreement # 202781. W.J.P. and R.T. acknowledge financial support from the European Research Council under the European Community’s Seventh Framework Programme (FP7/20072013)/ERC grant agreement #202686. W.J.P. is also grateful for support from the UK Science and Technology Facilities Council through the grant ST/I001204/1. E.B., F.M. and L.M. acknowledge the support from grants ASIINAF I/023/12/0 and PRIN MIUR 20102011. L.M. also acknowledges financial support from PRIN INAF 2012. Y.M. acknowledges support from CNRS/INSU (Institut National des Sciences de l’Univers) and the Programme National Galaxies et Cosmologie (PNCG). C.M. is grateful for support from specific project funding of the Institut Universitaire de France and the LABEX OCEVU. SdlT acknowledges the support of the OCEVU Labex (ANR11LABX0060) and the A*MIDEX project (ANR11IDEX000102) funded by the “Investissements d’Avenir” French government program managed by the ANR.
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All Tables
Real space values of σ_{8} and S_{3} of galaxies with M_{B}(z = 1.1) − 5log (h) ≤ −20.50 vs. those expected from secondorder perturbation theory (with Millennium and Planck cosmological parameters).
All Figures
Fig. 1 Comparison between mock catalogues with a sampling rate of 100% and without gaps (red triangles), and with a sampling rate and gaps as in VIPERS (blue triangles). The subsamples are limited at M_{B}(z = 1.1) − 5log (h) ≤ −20.5. From top to bottom: volumeaveraged two, three, and fourpoint correlation functions, normalized skewness S_{3} and kurtosis S_{4} in redshift space. First column: 0.5 ≤ z< 0.7; second column: 0.7 ≤ z< 0.9; third column: 0.9 ≤ z< 1.1. 

Open with DEXTER  
In the text 
Fig. 2 Fractional difference of the average , , , S_{3} and S_{4} (from top to bottom) for the same set of mock catalogues as defined in Fig. 1, i.e. with 100% sampling rate and without gaps, and with sampling rate and gaps as in VIPERS. The subsamples are limited at M_{B}(z = 1.1) − 5log (h) ≤ −20.5. Red triangles: 0.5 ≤ z< 0.7; blue squares: 0.7 ≤ z< 0.9; green hexagons: 0.9 ≤ z< 1.1. 

Open with DEXTER  
In the text 
Fig. 3 Comparison between mock catalogues with 100% sampling rate and without gaps in real space (red squares) and redshift space (blue triangles). The subsamples are limited at M_{B}(z = 1.1) − 5log (h) ≤ −20.5. First column: 0.5 ≤ z< 0.7; second column: 0.7 ≤ z< 0.9; third column: 0.9 ≤ z< 1.1. 

Open with DEXTER  
In the text 
Fig. 4 Volumeaveraged twopoint correlation functions as a function of the B absolute luminosity (upper panels) and stellar mass (lower panels). The limits of the subsamples in absolute magnitude and stellar mass are shown in the left upper and lower panels. Black triangles: M_{B}(z = 1.1) − 5log (h) ≤ −19.5 (log (M_{∗}/M_{⊙}h^{2} ≥ 9.0 M_{⊙}); red squares: M_{B}(z = 1.1) − 5log (h) ≤ −20.0 (log (M_{∗}/M_{⊙}h^{2} ≥ 9.5 M_{⊙}); blue pentagons: M_{B}(z = 1.1) − 5log (h) ≤ −20.5 (log (M_{∗}/M_{⊙}h^{2} ≥ 10.0 M_{⊙}); green hexagons: M_{B}(z = 1.1) − 5log (h) ≤ −21.0 (log (M_{∗}/M_{⊙}h^{2} ≥ 10.5 M_{⊙}); magenta heptagons: M_{B}(z = 1.1) − 5log (h) ≤ −21.5 (log (M_{∗}/M_{⊙}h^{2} ≥ 11.0 M_{⊙}). Dotted lines: realspace for the M_{B}(z = 1.1) − 5log (h) ≤ −20.5 (log (M_{∗}/M_{⊙}h^{2} ≥ 10.0 M_{⊙}) subsamples in the redshift bin [0.5, 0.7], predicted from the powerlaw fit of ξ_{2} in Marulli et al. (2013). 

Open with DEXTER  
In the text 
Fig. 5 Volumeaveraged threepoint correlation functions as a function of the B absolute luminosity (upper panels) and stellar mass (lower panels). The limits of the subsamples in absolute magnitude and stellar mass are shown in the left upper and lower panels (symbols and colours are the same as in Fig. 4). 

Open with DEXTER  
In the text 
Fig. 6 Volumeaveraged fourpoint correlation functions as a function of the B absolute luminosity (upper panels) and stellar mass (lower panels). The limits of the subsamples in absolute magnitude and stellar mass are shown in the left upper and lower panels (symbols and colours are the same as in Fig. 4). 

Open with DEXTER  
In the text 
Fig. 7 Scaling relation of the volumeaveraged two and threepoint correlations as a function of the B absolute luminosity (upper panels) and stellar mass (lower panels). The limits of the subsamples in absolute magnitude and stellar mass are shown in the left upper and lower panels (symbols and colours are the same as in Fig. 4). The dashed line represents the scaling relation . 

Open with DEXTER  
In the text 
Fig. 8 Scaling relation of the volumeaveraged two and fourpoint correlations as a function of the B absolute luminosity (upper panels) and stellar mass (lower panels). The limits of the subsamples in absolute magnitude and stellar mass are shown in the left upper and lower panels (symbols and colours are the same as in Fig. 4). The dashed line represents the scaling relation . 

Open with DEXTER  
In the text 
Fig. 9 Normalized skewness S_{3} as a function of the B absolute luminosity (upper panels) and stellar mass (lower panels). The limits of the subsamples in absolute magnitude and stellar mass are shown in the left upper and lower panels (symbols and colours are the same as in Fig. 4). We note that black points corresponding to galaxies with M< −19.50 or M_{∗} ≥ 9.0 M_{⊙} are only plotted for the first redshift bin ([0.5, 0.7]), but most of them are not visible as they lie below the points of the other samples. The dashed line corresponds to S_{3} = 2. In the top panels, the dotted line is the prediction from secondorder perturbation theory for the matter distribution in real space; the solid line is the prediction of S_{3} for galaxies with M_{B}(z = 1.1) − 5log (h) ≤ −20.5 (to be compared to blue pentagons): it was obtained from the matter values, converted to redshift space and divided by the corresponding linear bias factor. 

Open with DEXTER  
In the text 
Fig. 10 Normalized skewness S_{4} as a function of the B absolute luminosity (upper panels) and stellar mass (lower panels). The limits of the subsamples in absolute magnitude and stellar mass are shown in the left upper and lower panels (symbols and colours are the same as in Fig. 4). We note that black points corresponding to galaxies with M< −19.50 or M_{∗} ≥ 9.0 M_{⊙} are only plotted for the first redshift bin ([0.5, 0.7]), but most of them are not visible as they lie below the points of the other samples. The dashed line corresponds to S_{4} = 8. In the top panels, the dotted line is the prediction from secondorder perturbation theory for the matter distribution in real space; the solid line is the prediction of S_{4} for galaxies with M_{B}(z = 1.1) − 5log (h) ≤ −20.5 (to be compared to blue pentagons): it was obtained from the matter values, converted to redshift space and divided by the corresponding linear bias factor. 

Open with DEXTER  
In the text 
Fig. 11 σ_{8g}(upper panel) and S_{3g} (lower panel) as a function of redshift. Colours represent the different absolute magnitude limits as in previous figures. The open hexagon represents the corresponding value for 2dFGRS galaxies in the absolute magnitude M_{bJ} range [–21, –20] (Croton et al. 2004b). Black open squares are the values for VVDS galaxies brighter than M_{B} = −21 (Marinoni et al. 2005). For better visibility, points corresponding to different redshifts are slightly shifted in magnitude. 

Open with DEXTER  
In the text 
Fig. 12 σ_{8g} (upper panel) and S_{3g} at R = 8 h^{1} Mpc (lower panel) as a function of galaxy luminosity for the three redshift bins. Red triangles: 0.5 ≤ z< 0.7; blue squares: 0.7 ≤ z< 0.9; green hexagons: 0.9 ≤ z< 1.1. For better visibility, points corresponding to different redshifts are slightly shifted in magnitude. 

Open with DEXTER  
In the text 
Fig. 13 Linear bias b as a function of redshift. Top panel: M_{B} ≤ −20.0(z = 1.1) + 5log (h); middle panel: M_{B} ≤ −20.5(z = 1.1) + 5log (h); bottom panel: M_{B} ≤ −21.0(z = 1.1) + 5log (h). Red hexagons: our estimates of b = σ_{8g}/σ_{8m}. Blue squares: estimates of Marulli et al. (2013). Magenta triangles: Di Porto et al. (2014). 

Open with DEXTER  
In the text 
Fig. 14 Estimates of b = σ_{8g}/σ_{8m} (solid lines, filled symbols) and b′ ≡ S_{3m}/S_{3g} (dotted lines, open symbols); b′ = b under the assumption of linear bias. Red lines with triangles: 0.5 ≤ z< 0.7; blue lines with squares: 0.7 ≤ z< 0.9; green lines with hexagons: 0.9 ≤ z< 1.1. 

Open with DEXTER  
In the text 
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