The VIMOS Ultra Deep Survey. Luminosity and stellar mass dependence of galaxy clustering at z~3
A. Durkalec, O. Le Févre, A. Pollo, G. Zamorani, B.C. Lemaux, B. Garilli, S. Bardelli, N. Hathi, A. Koekemoer, J. Pforr, E. Zucca
AApril 16, 2018
The VIMOS Ultra Deep Survey
Luminosity and stellar mass dependence of galaxy clustering at z ∼ (cid:63) A. Durkalec , O. Le Fèvre , A. Pollo , , G. Zamorani , B. C. Lemaux , B. Garilli , S. Bardelli , N. Hathi , ,A. Koekemoer , J. Pforr , , and E. Zucca National Centre for Nuclear Research, ul. Hoza 69, 00-681, Warszawa, Poland, e-mail: [email protected] Aix Marseille Université, CNRS, LAM (Laboratoire d’Astrophysique de Marseille) UMR 7326, 13388, Marseille, France Astronomical Observatory of the Jagiellonian University, Orla 171, 30-001 Cracow, Poland INAF - Osservatorio Astronomico di Bologna, Via Gobetti 93 /
3, 40129 Bologna - Italy Department of Physics, University of California, Davis, One Shields Ave., Davis, CA 95616, USA INAF–IASF Milano, via Bassini 15, I–20133, Milano, Italy ESA / ESTEC SCI-S, Keplerlaan 1, 2201 AZ, Noordwijk Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USAApril 16, 2018
ABSTRACT
We present the study of the dependence of galaxy clustering on luminosity and stellar mass in the redshift range 2 < z < . .
92 deg .We measure the two-point real-space correlation function w p ( r p ) for four volume-limited sub-samples selected by stellar mass andfour volume-limited sub-samples selected by M UV absolute magnitude. We find that the scale dependent clustering amplitude r significantly increases with increasing luminosity and stellar mass. For the least luminous galaxies ( M UV < − .
0) we measure acorrelation length r = . ± . h − Mpc and slope γ = . ± .
07, while for the most luminous ( M UV < − . r = . ± . h − Mpc and γ = . ± .
25. This corresponds to a strong relative bias between these two sub-samples of ∆ b / b ∗ = .
43. Fittinga 5-parameter HOD model we find that the most luminous ( M UV < − .
2) and massive ( M (cid:63) > h − M (cid:12) ) galaxies occupy themost massive dark matter haloes with (cid:104) M h (cid:105) = . h − M (cid:12) . Similar to the trends observed at lower redshift, the minimum halo mass M min depends on the luminosity and stellar mass of galaxies and grows from M min = . h − M (cid:12) to M min = . h − M (cid:12) from thefaintest to the brightest among our galaxy sample, respectively. We find the di ff erence between these halo masses to be much morepronounced than is observed for local galaxies of similar properties. Moreover, at z ∼
3, we observe that the masses at which a halohosts, on average, one satellite and one central galaxy is M ≈ M min over all luminosity ranges, significantly lower than observed at z ∼ b g , HOD ( > L ) = . + . L / L ∗ ) . . We conclude our studywith measurements of the stellar-to-halo mass ratio (SHMR). We observe a significant model-observation discrepancy for low-massgalaxies, suggesting a higher than expected star formation e ffi ciency of these galaxies. Key words.
Cosmology: observations – large-scale structure of Universe – Galaxies: high-redshift – Galaxies: clustering
1. Introduction
The large structure of the Universe consists of two main ele-ments: the luminous, baryonic matter (e.g., in the form of stars,gas and dust) and the dominant underlying dark matter (DM).The properties and evolution of the former components can, andhave been, directly mapped with the use of large sky surveys,both at local and high redshifts using a variety of observationsat di ff erent wavelengths. As for the second, dark matter compo-nent, the situation is less clear. Direct observations are currentlydi ffi cult, but in the paradigm of the Λ CDM cosmology the vis-ible baryonic matter indirectly traces the dark matter structure.If we assume that all galaxies are hosted by dark matter haloes(White & Rees 1978), the information about the underlying darkmatter distribution can be extracted, e.g., using the mean occu-pation of galaxies in dark matter haloes. However, the relation (cid:63)
Based on data obtained with the European Southern ObservatoryVery Large Telescope, Paranal, Chile, under Large Program 185.A-0791. between these two components is not straightforward. In partic-ular, the spatial distribution of baryonic matter is biased with re-spect to that of dark matter, which is a result of additional physicsof the baryonic component, like star formation, supernova feed-back and galaxy merging, that regulate formation and evolutionof galaxies (see, e.g., Kaiser 1984; Bardeen et al. 1986; Mo &White 1996; Kau ff mann et al. 1997). It has been shown that thedi ff erence between the luminous and dark matter distributionsdepends both on the epoch of galaxy formation and the physicalproperties of galaxies (e.g., Fry 1996; Tegmark & Peebles 1998).Therefore, studies of the evolution of the luminous-dark mat-ter relation (called bias ), and its dependence on various galaxyproperties (like luminosity, stellar mass or colour) are crucial,because they can provide us with valuable information for in-vestigating the nature of the underlying dark matter distributionand, in the wider perspective, understanding the evolution of theaccelerating universe.There are various methods used to infer the properties of thedark matter through the observations of the luminous compo- Article number, page 1 of 20 a r X i v : . [ a s t r o - ph . GA ] D ec & A proofs: manuscript no. arxiv_final nent. The most direct ones involve gravitational lensing (Zwicky1937), which is a unique observational technique that allowsto probe both the nature and distribution of dark matter (e.g.,Van Waerbeke et al. 2000; Metcalf & Madau 2001; Moustakas& Metcalf 2003; Hoekstra et al. 2004; Massey et al. 2007; Fuet al. 2008; Rines et al. 2013). The gravitational lensing obser-vations, however, are usually possible only for a special set ofcircumstances, as the objects available for exploration are lim-ited by the geometry of lens and sources (see, e.g., Blandford& Narayan 1992; Meylan et al. 2006). Other methods for study-ing dark-luminous matter relations are applied on the scales ofindividual galaxies, where e.g., studies of rotation curves (Ru-bin et al. 1978) of stars or gas clouds within individual galax-ies are used to explore the hosting dark matter halo masses anddensity profiles, improving the understanding of the role of darkmatter haloes in galaxy formation and evolution (e.g., Genzelet al. 2017; Dekel et al. 2017; Katz et al. 2017). On the largescales considered in this work the most e ff ective methods makeuse of statistical tools. Among them the most extensively usedone is galaxy clustering based on galaxy correlation functionmeasurements, which allows to understand the time evolutionof luminous-dark matter relation and its dependence on galaxyproperties.The galaxy correlation function is a simple, yet powerful sta-tistical tool (Peebles 1980) and it can be modelled using, amongothers, the two parameter power-law ξ ( r ) = ( r / r ) − γ (Davis &Peebles 1983) model or can be modelled from Halo Occupa-tion Distribution models (HOD, Seljak 2000; Peacock & Smith2000; Magliocchetti & Porciani 2003; Zehavi et al. 2004; Zhenget al. 2005). In the HOD framework, the theoretical descriptionof the correlation function di ff ers for di ff erent scales r , account-ing for the fact that the clustering of galaxies residing in thesame halo di ff ers from clustering between galaxies residing inthe separate haloes. For small scales ( r (cid:54) . h − Mpc) the one-halo term is dominant, as it describes exclusively the clusteringof galaxies that reside within a single dark matter halo. On theopposite side, on large scales ( r (cid:62) h − Mpc), the two-halo termis dominant, which describes the clustering of galaxies residingin separate dark matter haloes.Using these two prescriptions of the galaxy correlation func-tion it has been shown that galaxy clustering, and by exten-sion the galaxy-dark matter relation, strongly depends on vari-ous galaxy properties. In general, at local ( z ∼
0) and intermedi-ate ( z <
2) redshifts, luminous and massive galaxies tend to bemore strongly clustered than their less luminous and less mas-sive counterparts. Additionally, it has been found that the clus-tering strength varies as a function of morphology, colour andspectral type. Galaxies with bulge dominated morphologies, redcolours, or spectral types indicating old stellar populations alsoexhibit stronger clustering and a preference for dense environ-ments (e.g., Norberg et al. 2002; Pollo et al. 2006; de la Torreet al. 2007; Coil et al. 2008; Meneux et al. 2006, 2008, 2009; Ab-bas et al. 2010; Hartley et al. 2010; Zehavi et al. 2011; Couponet al. 2012; Mostek et al. 2013; Marulli et al. 2013; Beutler et al.2013; Guo et al. 2015; Skibba et al. 2015). These studies are ingood agreement with the hierarchical theory of galaxy formationand evolution (White et al. 1987; Kau ff mann et al. 1997; Bensonet al. 2001).A lot of e ff ort has been put into testing whether or not sim-ilar clustering dependencies can be observed at high redshift( z > ff erence between the cluster-ing of massive, luminous and faint galaxies has been found (e.g.,Daddi et al. 2003; Adelberger et al. 2005; Le Fèvre et al. 2005;Ouchi et al. 2005; Lee et al. 2006; Hildebrandt et al. 2009; Wake et al. 2011; Lin et al. 2012; Bielby et al. 2014). However, mostof these observational constraints su ff er from a combination ofmany types of selection biases, due to the limited sample sizeand volume explored of galaxy surveys performed at z >
2. Un-til now, high redshift samples have been either too small to allowa subdivision into galaxy classes or they targeted special types ofgalaxies (like extremely massive red objects or sources selectedusing a Lyman-break or B z K technique) that cannot be easilyrelated to galaxy populations at lower redshifts. Therefore, theoverall picture of the possible dependence of galaxy clusteringon luminosity and stellar mass at these high redshifts is still dif-ficult to establish.In this paper we attempt to overcome some of these di ffi -culties and provide improved constraints on the dependence ofgalaxy clustering with luminosity and stellar mass at high red-shifts. We compute the projected two-point correlation function w p ( r p ) for galaxy samples limited in luminosity and stellar massin the redshift range 2 < z < . ∼ L ∗ ) luminosity,that are relatively easy to compare to low-redshift objects. Con-sequently, we are able to present reliable correlation functionmeasurements, with power-law and HOD fitting, as well as mea-surements of the galaxy bias, and satellite fraction at z ∼
3, anddiscuss all these results in terms of the current scenario of thedensity field evolution. Additionally, the comparison betweenVUDS clustering measurements with similar studies performedat lower redshifts allows us to put constraints on the cosmic evo-lution of the relationship between DM and galaxy properties,hence between gravity and cosmology on one side and processesassociated with baryonic physics on the other side.The paper is organized as follows. In Section 2 we brieflydescribe the properties of the VUDS survey and our selectedsamples. The methods used to measure the correlation functionand derive power-law and HOD fits are presented in Section 3.Results and comparison of our findings to other works are de-scribed in Section 4. We discuss the luminosity and stellar massdependence, as well as the redshift evolution of galaxy cluster-ing, galaxy bias, halo mass, satellite fraction and stellar-to-halomass ratio in Section 5, before concluding in Section 6.Throughout all this paper, we adopt a flat Λ CDM cosmolog-ical model with Ω m = . Ω Λ = . h = H /
100 to ease the comparison with previous works. We re-port correlation length measurements in comoving coordinatesand express magnitudes in the AB system.
2. Data
Our galaxy sample is drawn from the VIMOS Ultra Deep Sur-vey (VUDS). Details about the survey strategy, target selection,as well as data processing and redshift measurements are pre-sented in Le Fèvre et al. (2015). Below we provide only a briefsummary of these survey features, that are relevant to the studyof the galaxy clustering presented in this paper.
Article number, page 2 of 20. Durkalec et al.: Luminosity and stellar mass dependence of galaxy clustering at z ∼ D ec li n a ti on ( J ) [ d e g ] Right Ascension (J2000) [deg]
P01P02P03 P04P05P06 P07P08 -4.8-4.6-4.4-4.2 36 36.3 36.6 36.9 D ec li n a ti on ( J ) [ d e g ] Right Ascension (J2000) [deg]
P01 P02P03 P04 P05 -28.1-28-27.9-27.8-27.7-27.6-27.5 53 53.1 53.2 53.3 D ec li n a ti on ( J ) [ d e g ] Right Ascension (J2000) [deg]
P01P02
Fig. 1: Spatial distribution of galaxies with spectroscopic redshifts 2 < z < . left panel), VVDS-02h ( central panel) and ECDFS ( right panel). The blue crosses indicate VIMOS pointing centres.Table 1: Properties of the galaxy sample in the range 2 < z < . N g z median S e f f [deg ]COSMOS 1605 2.79 0.50VVDS-02h 1237 2.63 0.31ECDFS 3¯94 2.57 0.11 Total N g a l Redshift z spec
Full sampleCOSMOS fieldVVDS-02h fieldECDFS field
Fig. 2: Redshift distribution of the VUDS galaxy sample in theredshift range 2 . < z < . ∼
10 000 galax-ies in the redshift range 2 < z < + . The survey covers a to-tal area of ∼ across three independent fields (see Fig. 1),thus reducing the e ff ect of cosmic variance, which is importantfor galaxy clustering measurements. The majority ( ∼ z phot + σ (cid:62) z >
2, exceeds greatly the number of spectroscopicallyconfirmed galaxies from all previous surveys, allowing, e.g., forselection of various volume-limited sub-samples characterisedby di ff erent galaxy properties. Moreover, due to its target selec-tion method, VUDS can be considered as a largely representa-tive sample of star-forming galaxies with luminosities close tothe characteristic luminosity, i.e., ∼ . L ∗ < L UV < L ∗ (Cas-sata et al. 2013) observed at redshift z >
2. However, the dustygalaxy population at high redshift in VUDS sample is almostcertainly underrepresented.The core engine for redshift measurement in VUDS is thecross-correlation of the observed spectrum with the referencetemplates using the EZ redshift measurement code (Garilli et al.2010). At the end of the process, each redshift is assigned a flag,that expresses the reliability of the measurement (for details seeLe Fèvre et al. 2015). In our study we are using only the mostreliable objects, with the high 75 − z f lag = , , , and 9). It isworth to mention, that the galaxies assigned with lower flags,e.g., z f lag =
1, do not appear to occupy a distinct region of stel-lar mass / M UV phase space relative to the z f lag ≥ Article number, page 3 of 20 & A proofs: manuscript no. arxiv_final + (GalaxyObserved-Simulated SED Interactive Program) software, whichperforms a joint fitting of both spectroscopy and multi-wavelengths photometry data with stellar population models, asdescribed in detail by Thomas et al. (2016). We note that thisstellar mass measurement method di ff ers from the commonlyused ones based on, e.g., SED fitting on the multi-wavelengthphotometry. We decided to use the GOSSIP + stellar masses,due to the larger number of reliable M (cid:63) measurements avail-able for the VUDS sample. Based on the tests performed prior tothis study, we observe no noticeable di ff erence in the correlationfunction shape and / or correlation amplitude when GOSSIP + de-rived or Le Phare derived stellar masses are used. Our full sample consists of 3236 objects with reliable spectro-scopic redshifts in the range 2 < z < . (the sum of VIMOS slitmask outline af-ter accounting for overlaps, see Fig. 1), which corresponds to avolume of ∼ . × Mpc . The spatial distribution of galax-ies in each field is presented in Fig. 1, while Fig. 2 shows theirredshift distribution. The general properties of the whole sam-ple, including the number of galaxies, median redshift and thee ff ective area, are listed in Tab. 1.For the following analysis we selected four volume-limitedluminosity sub-samples, with the selection cuts made in the UV-rest frame absolute magnitudes, computed at a rest wavelengthof 1500Å ( M UV , , also denoted as M FUV and further in thiswork simplified to M UV ), and four stellar mass sub-samples,in order to study respectively the luminosity and stellar massdependence of the galaxy clustering within the redshift range2 < z < .
5. We chose this specific redshift range to be able tostudy galaxy clustering of as faint galaxies as possible, and atthe same time, to maintain volume completeness and large num-ber of galaxies in the various sub-samples. On the other side,the choice of the UV wavelength for the luminosity selection isdriven by the fact that VUDS is an optically selected survey. Thefull wavelength coverage of VUDS is 3650 - 9350 Å, which cor-responds to the UV rest frame wavelength coverage at redshift ∼ ffi cient for a reliable measurement of thecorrelation function (based on tests performed on VUDS dataprior to this research, see Durkalec et al. 2015b). Selection cutsfor di ff erent sub-samples are shown in Fig. 3. Additionally, gen-eral properties of these sub-samples including number of galax-ies, median redshifts, UV median absolute magnitudes M medUV andmedian stellar masses log M med (cid:63) of each sub-sample are listed inTab. 2 and Tab. 3.To account for the mean brightening of galaxies due to theirevolution and to ease the comparison between measurementsbased on samples from various epochs, we normalized the abso-lute magnitudes and stellar masses, at each redshift, to the cor-responding value of the characteristic absolute magnitude M ∗ UV of the Schechter luminosity function in the UV band or to thecharacteristic stellar mass log M ∗ respectively. Therefore, for theabsolute magnitudes we compute M UV = M (cid:48) UV − ( M ∗ UV − M ∗ UV , ),where M (cid:48) UV is the original (not corrected) absolute magnitude inthe UV filter, M ∗ UV is the characteristic absolute magnitude and Table 2: Properties of the galaxy luminosity sub-samples, as usedin this study.Sample M maxUV N g z median M medUV M min (cid:63) N g z median log M med (cid:63) M ∗ UV , is the characteristic luminosity for galaxies at z =
0. Sim-ilar correction is applied for the stellar masses. The values of thecharacteristic absolute magnitudes have been estimated based onthe work of Bouwens et al. (2015); Mason et al. (2015); Hagenet al. (2015); Finkelstein et al. (2015); Sawicki & Thompson(2006), while the characteristic stellar masses have been takenfrom Ilbert et al. (2013) and Pérez-González et al. (2008). Thedetails of the methods used to determine the values of character-istic absolute magnitudes and stellar masses at a given redshiftare presented in Appendix A.
3. Measurement methods
This work is an extension of our previous studies presented inDurkalec et al. (2015b), and all the methods used in this studyto quantify the galaxy clustering are similar to those presentedtherein. This includes computation techniques, error estimations,analysis of systematics in the correlation function measurementsand correction methods. For a detailed description we refer thereader to Sec. 3 of Durkalec et al. (2015b), while below we pro-vide a short summary of the procedures.We measure the real-space correlation function ξ ( r p , π ) ofthe combined data from three independent VUDS fields throughthe Landy-Szalay estimator (Landy & Szalay 1993). The di ff er-ences in size and galaxy numbers between the fields have beenaccounted for by an appropriate weighting scheme. In particu-lar, each pair was multiplied by the number of galaxies per unitvolume. ξ ( r p , π ) = n field (cid:88) i = w i ( GG i − GR i + RR i ) / n field (cid:88) i = w i RR i , (1)where w i = (cid:16) N g , i / V i (cid:17) and GG , GR , RR are the number of dis-tinct galaxy-galaxy, galaxy-random and random-random pairswith given separations lying in the intervals of ( r p , r p + dr p )and ( π, π + d π ), respectively. Integrating the measured ξ ( r p , π )along the line of sight gives us the two-point projected corre-lation function w p ( r p ), which is the two-dimensional counter-part of the real-space correlation function, free from the redshift- Article number, page 4 of 20. Durkalec et al.: Luminosity and stellar mass dependence of galaxy clustering at z ∼ -22-21-20-19-18 2 2.5 3 3.5 4 M ’ UV - ( M * UV - M * UV , ) Redshift z spec l og ( M ’ ★ - ( M * - M *0 )) [ h - M O • ] Redshift z spec
Fig. 3: Construction of the volume-limited galaxy sub-sampleswith di ff erent luminosity ( upper panel) and stellar mass ( lower panel). In both figures grey dots represent the distribution ofVUDS galaxies as a function of spectroscopic redshift z . At eachredshift UV-band absolute magnitudes and stellar masses arenormalized to the characteristic absolute magnitudes, or to thecharacteristic stellar mass, respectively (see Sec. 2.2). The dif-ferent colour lines delineate the selection cuts for selected UVabsolute magnitude and stellar mass sub-samples as defined inTab. 2 and Tab. 3. The dashed black line represents the evolutionof the not corrected characteristic UV absolute magnitude M ∗ UV (upper panel), or characteristic stellar mass M ∗ (lower panel).The grey line indicates the volume limit of the VUDS sample.space distortions (Davis & Peebles 1983). w p ( r p ) = (cid:90) ∞ ξ (cid:16) r p , π (cid:17) d π. (2)In practice, a finite upper integral limit π max has to be used in or-der to avoid adding uncertainties to the result. A value that is toosmall results in missing small-scale signal of correlation function w p ( r p ), while a value that is too large has the e ff ect of inducingan unjustified increase in the w p ( r p ) amplitude (see, e.g., Guzzo et al. 1997; Pollo et al. 2005). After performing a number oftests for di ff erent π max , we find that w p ( r p ) is insensitive to thechoice of π max in the range 15 < π max < h − Mpc. Therefore,we choose π max = h − Mpc, which is the maximum value forwhich the correlation function measurement was not appreciablya ff ected by the mentioned uncertainties.All correlation function measurements presented in this pa-per have been corrected for the influence of various systematicsoriginating in the VUDS survey properties, by introducing thecorrection scheme developed in Durkalec et al. (2015b). In par-ticular, we accounted for the galaxies excised from the observa-tions due to the VIMOS layout and other geometrical constraintsintroduced by the target selection (see Fig. 1). Also, the correct-ing scheme addresses the possible underestimation of the corre-lation function related to the small fraction of incorrect redshiftspresent in the sample, as well as small scale underestimationsobserved in tests based on the VUDS mock catalogues.To estimate the two-point correlation function errors we ap-ply a combined method (see Durkalec et al. 2015b), whichmakes use of the so-called blockwise bootstrap re-samplingmethod with N boot =
100 (Barrow et al. 1984) coupled to N mock =
66 independent VUDS mock catalogues (see Durkalecet al. 2015b; de la Torre et al. 2013, for details about mocks),similar to the method proposed by Pollo et al. (2005). The asso-ciated covariance matrix C ik between the values w p on i th and k th scale has been computed using: C ik = (cid:68)(cid:16) w jp ( r i ) − (cid:104) w jp ( r i ) (cid:105) j (cid:17) (cid:16) w jp ( r k ) − (cid:104) w jp ( r k ) (cid:105) j (cid:17)(cid:69) j (3)where " (cid:104)(cid:105) " indicates an average over all bootstrap or mock re-alizations, the w jp ( r k ) is the value of w p computed at r p = r i in the cone j , where 1 < j < N mock for the VUDS mocks and1 < j < N boot for the bootstrap data.Throughout this study we use two approximations of theshape of the real-space correlation function. The first one is apower-law function ξ ( r ) = ( r / r ) − γ , where r and γ are the cor-relation length and slope, respectively. With this parametrization,the integral in Eq. 2 can be computed analytically and w p ( r p ) canbe expressed as w p ( r p ) = r p (cid:32) r r p (cid:33) γ Γ (cid:16) (cid:17) Γ (cid:16) γ − (cid:17) Γ (cid:16) γ (cid:17) , (4)where Γ is the Euler’s Gamma Function. Despite of its simplic-ity, a power-law model remains an e ffi cient and simple approxi-mation of galaxy clustering properties.A second, more detailed description of the real-space corre-lation function, used here, has been done in the framework of theHalo Occupation Distribution (HOD) models. Following a com-monly used, analytical prescription, we parametrized the halooccupation model in the way used, e.g., by Zehavi et al. (2011)and motivated by Zheng et al. (2007). The mean halo occupationfunction (cid:104) N g ( M h ) (cid:105) , i.e., the number of galaxies that occupy thedark matter halo of a given mass is the sum of the mean occupa-tion functions for the central and satellite galaxies, (cid:104) N g ( M h ) (cid:105) = (cid:104) N cen ( M h ) (cid:105) + (cid:104) N sat ( M h ) (cid:105) , (5)where, (cid:104) N cen ( M h ) (cid:105) = (cid:34) + erf (cid:32) log M h − log M min σ log M (cid:33)(cid:35) (6) (cid:104) N sat ( M h ) (cid:105) = (cid:104) N cen ( M h ) (cid:105) × (cid:32) M h − M M (cid:48) (cid:33) α . (7) Article number, page 5 of 20 & A proofs: manuscript no. arxiv_final
10 100 0.1 1 10 M fUV < -19.0M fUV < -19.5 M fUV < -20.0M fUV < -20.2 w p (r p ) r p [h -1 Mpc]
10 100 0.1 1 10 log M ★ > 8.75log M ★ > 9.25 log M ★ > 9.75log M ★ > 10.0 w p (r p ) r p [h -1 Mpc]
Fig. 4: Projected correlation functions for volume-limited samples corresponding to di ff erent luminosity (left panel, circles) andstellar mass (right panel, squares) bins, as labelled.This model includes five free parameters, two of which representcharacteristic halo masses, that describe the mass scales of ha-los hosting central galaxies and their satellites. The characteristicmass M min is the minimum mass needed for half of the haloes tohost one central galaxy above the assumed luminosity (or mass)threshold, i.e., (cid:104) N cen ( M min ) (cid:105) = .
5, whereas the second charac-teristic mass M is the mass of haloes that on average have oneadditional satellite galaxy above the luminosity (or mass) thresh-old, i.e., (cid:104) N sat ( M ) (cid:105) =
1. Note that M is di ff erent from M (cid:48) fromEq. 7. However, both quantities are related to each other and inmost cases M ∼ M (cid:48) (see Tab. 4). The remaining three free pa-rameters are: σ log M - related to the scatter between the galaxyluminosity (or stellar mass) and halo mass M h , the cuto ff massscale M , and the high-mass power-law slope α of the satellitegalaxy mean occupation function.The HOD parameter space for each galaxy sample has beenexplored by using the Population Monte Carlo (PMC) technique(Wraith et al. 2009; Kilbinger et al. 2011), using the full covari-ance error matrix, as described in Durkalec et al. (2015b). Fromthe best-fitting HOD parameters we derived quantities describ-ing the halo and galaxy properties, like the average host halomass (cid:104) M h (cid:105) , (cid:104) M h (cid:105) ( z ) = (cid:90) dM h M h n ( M h , z ) (cid:104) N g ( M h ) (cid:105) n g ( z ) , (8)the large-scale galaxy bias b g , b g ( z ) = (cid:90) dM h b h ( M h ) n ( M h , z ) (cid:104) N g ( M h ) (cid:105) n g ( z ) , (9)and the fraction of satellite galaxies per halo f s f s = − (cid:90) dM h n ( M h , z ) N c ( M h ) n g ( z ) , (10)where n ( M h , z ) is the dark matter mass function, b h ( M h , z ) is thelarge-scale halo bias, and n g ( z ) represents the number density ofgalaxies, n g ( z ) = (cid:90) dM h n ( M h , z ) (cid:104) N g ( M h ) (cid:105) . (11)
4. Results
The two-point projected correlation function w p ( r p ) has beenmeasured in four volume-limited luminosity sub-samples andfour stellar mass sub-samples selected from a total number of3236 spectroscopically confirmed VUDS galaxies observed inthe redshift range 2 < z < .
5. The composite correlation func-tions (from three VUDS fields, see Sec. 3) measured for eachof these luminosity and stellar mass sub-samples are presentedin Fig. 4, while the associated best power-law and HOD fits areshown in Fig. 5.In the case of luminosity limited sub-samples the minimumscale r p that can be reliably measured varies slightly for di ff erentgalaxy sub-samples. For the two faintest sub-samples we mea-sure a correlation signal on scales 0 . < r p < h − Mpc, whilefor the more luminous sub-samples it can be measured only onscales 0 . < r p < h − Mpc. We set these particular limits afterperforming a range of tests on correlation function measured foreach of VUDS luminosity sub-samples (see Sec. 2.2). The lower r p limit is set at the lowest scale for which (1) we are able tomeasure a correlation function signal, i.e., w p ( r p ) has a positivevalue, and / or (2) we are able to reliably correct (see Durkalecet al. 2015b, for details about the used correction methods) theunderestimation of the correlation function that occurs due tomissing close galaxy pairs (result of the low number of galaxiesin the sample and / or VIMOS limitations and positions of spec-tral slits). The maximum scale limit of r p has been chosen as aresult of similar tests, and under the same conditions. This time,however, the distant galaxy pairs at large r p are missing due tothe finite size of VUDS fields.In practice, we therefore limit our measurement to scales forwhich the number of galaxy-galaxy pairs in VUDS data is suf-ficient to measure correlation function with uncertainties that donot exceed the value of w p ( r p ), and are not a ff ected by volumee ff ects. Article number, page 6 of 20. Durkalec et al.: Luminosity and stellar mass dependence of galaxy clustering at z ∼ M UV < -19.0M UV < -19.5 M UV < -20.0M UV < -20.2 w p (r p ) r p [h -1 Mpc] M UV < -19.0M UV < -19.5M UV < -20.0M UV < -20.2 γ r [h -1 Mpc] logM ★ > 8.75logM ★ > 9.25 logM ★ > 9.75logM ★ > 10.0 w p (r p ) r p [h -1 Mpc] logM ★ > 8.75logM ★ > 9.25logM ★ > 9.75logM ★ > 10.0 γ r [h -1 Mpc]
Fig. 5: Projected two-point correlation function w p ( r p ) associated with the best-fitting power-law function ( left side) and best-fitpower-law parameters r and γ along with 68.3% and 95.4% joint confidence levels ( right side) in four UV absolute magnitudesub-samples ( upper panel) and four stellar mass sub-samples ( lower panel). The symbols and error bars (see Sec. 3 for the errorestimation method) denote measurements of the composite correlation function for di ff erent luminosity (circles) and stellar mass(squares) sub-samples selected from VUDS survey in the redshift range 2 < z < .
5. For clarity, o ff sets are applied both to the datapoints and best-fitting curves of the w p ( r p ), i.e., the values of w p ( r p ) and associated best-fits for galaxy sub-samples with increasingluminosity and stellar masses have been staggered by 0.5 dex each. Error contours on the fit parameters are obtained taking intoaccount the full covariance matrix. The 68.3% and 95.4% joint confidence levels are defined in terms of the corresponding likelihoodintervals that we obtain from our fitting procedure. The best power-law fits of w p ( r p ), parametrized with two freeparameters r and γ (see Sec. 3), are presented in the left panel ofFig. 5. The best-fitting parameters for all luminosity and stellarmass sub-samples are listed in Tab. 4 and their 68.3% and 95.4%joint confidence levels are shown in the right panel of Fig. 5.At redshift z ∼ r . We find that r rises from r = . ± . h − Mpc forthe least luminous galaxy sub-sample (with M medUV = − .
84) to r = . ± . h − Mpc for the most luminous galaxies (with M medUV = − . M UV < − . M UV < − . r p > h − Mpc, which results in a subtleincrease in r between these sub-samples (see Tab. 4). The rapidgrowth in the correlation length, by ∆ r ∼ h − Mpc, can beobserved afterwards for the brightest galaxies ( M UV < − . Article number, page 7 of 20 & A proofs: manuscript no. arxiv_final M UV < -19.0M UV < -19.5 M UV < -20.0M UV < -20.2 w p (r p ) r p [h -1 Mpc] UV < -19.0M UV < -19.5M UV < -20.0M UV < -20.2 N ( M ) M h [h -1 M O• ] logM ★ > 8.75logM ★ > 9.25 logM ★ > 9.75logM ★ > 10.0 w p (r p ) r p [h -1 Mpc] ★ > 8.75logM ★ > 9.25logM ★ > 9.75logM ★ > 10.0 N ( M ) M h [h -1 M O• ] Fig. 6: Projected two-point correlation function w p ( r p ) associated with the best-fitting HOD models ( left side) and evolution ofthe halo occupation function of the best-fit HOD model ( right side) in four UV absolute magnitude sub-samples ( upper panel)and four stellar mass sub samples ( lower panel). The symbols and error bars (see Sec. 3 for the error estimation method) denotemeasurements of the composite correlation function for di ff erent luminosity (circles) and stellar mass (squares) sub-samples selectedfrom the VUDS survey in the redshift range 2 < z < .
5. For clarity, o ff sets are applied to both the data points and best-fitting curvesof the w p ( r p ), i.e., the values of w p ( r p ) and associated best-fits for the galaxy sub-samples with increasing luminosity and stellarmasses have been staggered by 0.5 dex each.A similar behaviour occurs for the galaxies selected accord-ing to their stellar masses, with the correlation length increas-ing from r = . ± . h − Mpc for the least massive sub-sample (log M med (cid:63) = . h − Mpc) to r = . ± . h − Mpc measured for the most massive ones (log M med (cid:63) = . h − Mpc). However, in this case the change in the correlation func-tion between sub-samples of increasing stellar mass appears tobe smoother.The second of the two free parameters, the slope γ , has alsoa tendency to grow with increasing luminosity and stellar mass.We find that for the luminosity selected sub-samples the valueof γ rises from γ = . ± .
07 for the faint galaxies to γ = . ± .
25 for the brightest ones. Similarly, the slope of thepower-law fit changes from γ = . ± .
06 to γ = . ± . γ is likely related to the continuously stronger one-haloterm measured for sub-samples with increasing luminosities andstellar masses, as discussed below. In the left panel of Fig. 6 we present the measurements of theprojected real-space correlation function w p ( r p ) and the best-fitting HOD models for the four volume limited UV absolutemagnitude (upper panel) and stellar mass (lower panel) sub-samples at redshift z ∼
3. As shown, for all selected galaxy sam-ples the best-fitting HOD models reproduce the measurementsof the projected correlation function well. However, it is notice-
Article number, page 8 of 20. Durkalec et al.: Luminosity and stellar mass dependence of galaxy clustering at z ∼ able that in all cases there are some deviations with respect tothe model, which predicts correlation function values at largescales ( r p > h − Mpc) lower than measured. Given the mea-surement errors, these deviations are more significant for the twoleast massive and least luminous sub-samples. We verified thatthese deviations are mostly driven by the behaviour of the cor-relation function measured in the COSMOS field, the field withthe most galaxies distributed over the largest area in our sample(it comprises of ∼
50% of our galaxy sample, see Tab. 1), hencewith a significant influence on the combined correlation func-tion. The flattening of w p ( r p ) measured for the COSMOS fieldat large separations r p > h − Mpc can be explained by the pres-ence of an extremely large structure in the COSMOS field whichspans a size comparable to that covered by VUDS-COSMOS(see Appendix B and Cucciati et al. 2017, in prep).In Tab. 4 we list the values of the best-fitting HOD parame-ters (inferred using the full error covariance matrix), with their1 σ errors. Similarly to what is seen at lower redshifts (e.g., Ze-havi et al. 2011; Abbas et al. 2010; Zheng et al. 2007) we ob-serve a mass growth of the dark matter haloes hosting galaxieswith rising luminosity and stellar mass. The minimum halo mass M min , for which at least 50% of haloes host one central galaxy,increases from M min = . ± . h − M (cid:12) to M min = . ± . h − M (cid:12) for galaxies with the median UV absolute magnitude M medUV = − .
84 and M medUV = − .
56, respectively. At the sametime, for galaxy sub-samples selected according to stellar mass, M min grows from M min = . ± . h − M (cid:12) to M min = . ± . h − M (cid:12) for galaxies with log M med (cid:63) = . h − M (cid:12) to log M med (cid:63) = . h − M (cid:12) .We also observe a growth of another characteristic halomass, M , with the luminosity and stellar mass of galaxies. Thelimiting mass of dark matter halo hosting on average one addi-tional satellite galaxy above the luminosity (or mass) thresholdincreases from M = . ± , h − M (cid:12) for the faintest galaxysub-sample to M = . ± . h − M (cid:12) for the most luminousgalaxies. Similarly, for the stellar mass selected sub-samples M rises from M = . ± . h − M (cid:12) to M = . ± . h − M (cid:12) from the less to the most massive galaxy sub-samples, respec-tively.These changes, both, of the minimum M min and ’satellite’ M masses of dark matter haloes hosting galaxies with di ff erentproperties are in agreement with the predictions of the hierarchi-cal scenario of structure formation as discussed in Sec. 5.3.Additionally, we observe an increase with luminosity of thehigh-mass slope α of the satellite occupation in the UV abso-lute magnitude selected galaxy sub-samples. For the two bright-est sub-samples ( M UV < − . M UV < − . α is no-ticeably higher α = . ± .
23, than observed for the faintergalaxy populations, where α takes values around unity. This ob-served di ff erence is likely related to the more pronounced one-halo term for the most luminous galaxy sample. It indicates thatsatellite galaxies are more likely to occupy most massive darkmatter haloes. The situation is less clear for the stellar mass se-lected sub-samples, where, given the measurement uncertainties,we do not observe any significant change in the slope α for thefour di ff erent stellar mass sub-samples.All these di ff erences in the HOD parameter values mea-sured for galaxy populations with di ff erent luminosities and stel-lar masses are reflected in the evolution of the halo occupationfunction presented in the right panels of Fig. 6. The halo occu-pation function shifts towards higher halo masses when goingtowards brighter and more massive galaxy sub-samples show-ing that more luminous and more massive galaxies occupy, re-spectively, more massive haloes. For the luminosity selected sub- samples this shift of the halo occupation function is rather con-tinuous, while for the stellar mass selected galaxies there is arapid 1 dex increase in halo masses moving from the two leastmassive to the two most massive galaxy populations.Such a rapid shift in the halo mass related to a relativelysmall change in the stellar mass has not been reported in the lit-erature. At z ∼ M min ∼ . M (cid:12) to M min ∼ . M (cid:12) ) and ’satellite’ halo masses (from M ∼ . M (cid:12) to M = . M (cid:12) ) for galaxies with stellar masses ranging from M thresh (cid:63) = . M (cid:12) to M thresh (cid:63) = . M (cid:12) . Similarly, at z ∼ . ∆ log M min = . M (cid:12) for sub-samples of galaxieswith stellar masses from M thresh (cid:63) = . M (cid:12) to M thresh (cid:63) = . M (cid:12) . These studies, however, do not cover stellar masses smallerthan M thresh (cid:63) ∼ M (cid:12) , which is the threshold limit of the mostmassive galaxy sub-sample used in this work.The presence of the halo mass discontinuity with respect tothe increasing stellar mass of galaxies and lack of such disconti-nuity observed for luminosity selected sub-samples suggests thatthe relationship between the luminosity of a galaxy and the cor-responding halo mass significantly di ff ers from the relationshipbetween its stellar mass and the mass of the dark matter halo.This in turn implies that the processes determining the galaxyluminosity, even if related to the evolution of the hosting halo,could be more complex than the relation between the halo andgalaxy stellar mass.The observed discontinuity in halo mass, with respect tosmall di ff erence in stellar mass, directly influence the observedstellar-to-halo mass relation. In particular we observe that, at z ∼
3, low mass end of this relation deviates from the theo-retical predictions by, e.g., Behroozi et al. (2013) and Mosteret al. (2013). We discuss the possible implications of this resultin more details in Sec. 5.5.
5. Discussion
Our most important conclusion is that at redshift z ∼ r from faint and low massive samples to the mostluminous and the most massive ones. This implies that at highredshift the most luminous and most massive galaxies are morestrongly clustered than their fainter and less massive counter-parts, with a higher clustering observed on both small and largespatial scales.This luminosity and stellar mass dependence of galaxy clus-tering can be explained in the framework of the hierarchicalmass growth paradigm. In this scenario, the mass overdensitiesof the density field collapsed overcoming the cosmological ex-pansion. The initially stronger overdensities grew faster, hencetheir stronger clustering pattern imprinted in the dark matter den-sity field. With time, the resulting dark matter haloes mergedtogether, forming larger haloes, which served as the environ-ment where galaxies formed and evolved (Press & Schechter1974; White 1976). The strongest and most clustered overden-sities produced the largest haloes, containing the correspondingamount of baryons, which - in turn - agglomerated to produce thelargest and the most massive (consequently also the most lumi-nous) galaxies. This behaviour is reflected in the N-body simula- Article number, page 9 of 20 & A proofs: manuscript no. arxiv_final tions complemented by the semi-analytical models which showthat the galaxy luminosity and stellar mass are tightly correlatedwith the mass of their haloes. In consequence, the clustering ofa particular galaxy sample is expected to be largely determinedby the clustering of haloes that host these galaxies (Conroy et al.2006; Wang et al. 2007).This simple picture, however, complicates when we need totake the evolution of galaxies, driven by baryonic physics, intoaccount. This makes more di ffi cult to predict how exactly lumi-nosity and stellar mass dependence of galaxy clustering changeswith time. In particular, the star formation occurs only after bary-onic matter reaches a certain critical density and proceeds in adi ff erent way depending, e.g., on the initial galaxy mass, halomass, and interactions with other galaxies (see, e.g., White &Rees 1978; De Lucia et al. 2007; López-Sanjuan et al. 2011;Tasca et al. 2014). Therefore, the evolution of luminosity andstellar mass clustering dependence is not only related to thegrowth of dark matter halo masses, but also to the physics ofbaryons that make up the galaxies. We expect that for the mostmassive galaxies occupying the most massive dark matter haloesthe build up of stellar mass is eventually limited by various feed-back e ff ects (e.g., Blanton et al. 1999), while the less massivegalaxies occupying less massive dark matter haloes continue toform stars (downsizing, see e.g., De Lucia et al. 2006). In con-sequence, we expect to observe a strong luminosity and stellarmass dependence of galaxy clustering at z ∼ ff erential evo-lution between low and high luminosity galaxies or low and highstellar mass galaxies from z ∼ z ∼ ff erent red-shifts are di ffi cult. The clustering amplitudes observed at di ff er-ent epochs cannot be easily related due to the di ff erences in theselection methods used to sample galaxies in di ff erent surveys,which in turn results in sampling di ff erent galaxy populations atdi ff erent redshifts. Still, we find that our results - a higher clus-tering amplitude observed for more luminous galaxies on bothsmall and large spatial scales r p - are consistent with the re-sults based on the data from low (e.g., the SDSS survey - Zehaviet al. 2011, Guo et al. 2015, the 2dF survey Norberg et al. 2002)and intermediate (e.g., the DEEP2 survey - Coil et al. 2006, theVVDS survey Pollo et al. 2006, Abbas et al. 2010, the zCOS-MOS - Meneux et al. 2009, the VIPERS survey Marulli et al.2013) redshift ranges. For example, based on the large SDSS z ∼ ∆ r ∼ . h − Mpc between galax-ies with M r < − . M r < − .
0. Moreover, similarly toour work, the luminosity dependence is more pronounced forbright samples, and less significant for the fainter ones (see Sec.4.1). At intermediate redshift ranges, e.g., Marulli et al. (2013)analysing data from the VIPERS survey, found that at z ∼ r = . ± . h − Mpcto r = . ± . h − Mpc for galaxies with M B < − . M B < − .
5, respectively. Consistently with these find-ings at lower redshifts, also at z ∼
3, we find a ∆ r ∼ . M UV < − .
0) and the brightest( M UV < − .
2) galaxies and a ∆ r ∼ . ff erent galaxy populations, we are not able todraw a detailed conclusion whether or not luminosity and stel-lar mass clustering dependence is stronger (or weaker) at highredshift in comparison to the local universe. What can be safelysaid, however, is that dependence of clustering with luminosity b / b * L med /L * b / b * L med /L *Tegmark et al. 2004Norberg et al. 2002Pollo et al. 2006This work Fig. 7: The relative bias b / b ∗ (see Eq. 12) for the selected VUDSluminosity sub-samples at z ∼ L ∗ as a reference point. The results from thiswork are compared to similar studies at lower redshift ranges:at z ∼ . z ∼ . z ∼
3, as it is observedat z ∼ ∆ r of the same order of magnitude at both redshifts),and therefore much of the processes which produced luminosityand stellar mass clustering dependence must have been at workat significantly higher redshift than z ∼ Using the best-fitting power-law parameters r and γ we interpretour results in terms of the relation between the distribution ofgalaxies and the underlying dark matter density field for galaxypopulations with di ff erent luminosities. We compare the valuesof the relative galaxy bias b / b ∗ measured from the VUDS sur-vey to the bias of galaxy populations with di ff erent luminositiesmeasured at lower redshift ranges, taken from the literature.The relative bias parameter, b / b ∗ , is based on the amplitudeof the correlation function relative to that of L ∗ galaxies and canbe defined as the relative bias of the generic i th sample with agiven median luminosity L med , with respect to that correspondingto L ∗ , as b i b ∗ = (cid:118)(cid:117)(cid:117)(cid:117)(cid:116) (cid:16) r i (cid:17) γ i (cid:16) r ∗ (cid:17) γ ∗ r γ ∗ − γ i . (12)In our study we use a fixed scale r = h − Mpc (see also Meneuxet al. 2006, for a slightly di ff erent definition). To apply this for-mula, first we need to estimate the values of r ∗ and γ ∗ for M ∗ UV galaxies. We obtain them through a linear fit to the relation be-tween correlation length and absolute magnitude of the samplenormalised to the characteristic absolute magnitude at median Article number, page 10 of 20. Durkalec et al.: Luminosity and stellar mass dependence of galaxy clustering at z ∼ b g , HOD L thresh /L *This workZehavi et al. 2011 b g , HOD M thresh /M * This workSkibba et al. 2015Mostek et al. 2013
McCracken et al. 2015 b g , HOD M thresh /M * Fig. 8: Large scale galaxy bias b g , HOD as a function of luminosity, with L ∗ as a reference point ( left panel) and as a function of stellarmass, with M ∗ as a reference point ( right panel). In both plots the coloured points indicate measurements at z ∼ b g , HOD ( > L ) (Eq. 13) and bias-stellar mass dependence b g , HOD ( > M ) (Eq. 14) as described in Sec. 5.2. In the left panel the black circles show the results from Zehavi et al. (2011) at z ∼ b g measurements at z ∼ . z ∼ ff erent samples in the redshift range 0 . < z < . r ( M UV − M ∗ UV ) and γ ( M UV − M ∗ UV ) measured in thiswork.Fig. 7 shows the relative bias measured for the VUDS galax-ies with the luminosities sampled at z ∼ b / b ∗ = . + . L / L ∗ from Norberg et al. (2002)and b / b ∗ = . + . L / L ∗ − . M − M ∗ ) based on the SDSSsample (Tegmark et al. 2004).In each luminosity sub-sample the relative bias at z ∼ L med / L ∗ < L med / L ∗ ra-tios. However, none of our sub-samples has L med > L ∗ , thus wecannot exclude the possibility that for galaxies with L med > L ∗ the relative bias would be higher, which is very likely, takinginto account the trend visible in Fig. 7. Additionally, we observethat the value of b / b ∗ rises more steeply with L med / L ∗ for highredshift galaxies than observed locally. At z ∼ b / b ∗ = . ± .
03 at low luminositiesto b / b ∗ = . ± .
11 for the high luminosity sub-sample. Polloet al. (2006) found a similar steep growth of the relative bias forgalaxies observed at z ∼
1. At z ∼
0, instead, b / b ∗ increases onlyby ∼ . L med / L ∗ interval, following the model fromNorberg et al. (2002). This appears to be an indication that go-ing back in time the bias contrast of the most luminous galaxieswith respect to the rest of the population becomes stronger and isconsistent with the fact that fainter galaxies are found to be sig-nificantly less biased tracers of the mass than brighter galaxieseven at high redshifts. However, we need to take into account thepossibility that the observed strengthening of b / b ∗ relation withluminosity at higher redshifts can also be partially attributed toa more pronounced one-halo term at higher z making the power-law fit of the clustering measurement less reliable. In order to break this ambiguity, we use the best-fitting pa-rameters of the HOD model to estimate the large scale galaxybias b g , HOD , using Eq. 9. The results obtained for the luminosityand stellar mass sub-samples at z ∼ b g , HOD mea-sured at z ∼ z > ff ectivelypreventing the formation of new stars (e.g., Blanton et al. 1999)and resulting in galaxy formation systematically moving to lessdense, hence less biased, regions.In addition to the redshift dependence of galaxy bias and inagreement with previous studies at lower redshifts (e.g., Nor-berg et al. 2002; Tegmark et al. 2004; Meneux et al. 2008; Ze-havi et al. 2011; Mostek et al. 2013) we observe a clear luminos-ity and stellar mass bias dependence, with the brightest and mostmassive galaxies being the most biased ones.In the left panel of Fig. 8 we show the large scale galaxybias b g , HOD as a function of luminosity and compare it with thesimilar results from Zehavi et al. (2011). As presented, at z ∼ Article number, page 11 of 20 & A proofs: manuscript no. arxiv_final l og M [ h - M O • ] L thresh /L *This work, M min This work, M M ( z = ) M m i n ( z = ) l og M [ h - M O • ] M thresh /M *This work, M min This work, M M ( z = . ) M m i n ( z = . ) Fig. 9: Characteristic halo masses from the best-fitting HOD models of the correlation function selected in luminosity versus L thresh / L ∗ ( left panel) and selected in stellar mass versus M thresh / M ∗ ( right panel). Minimum halo masses M min for which 50% ofhaloes host one central galaxy above the threshold limit (filled symbols) and masses of haloes which on average host one additionalsatellite galaxy M (open symbols) observed at z ∼ z ∼
0, forthe luminosity selected galaxies, and by Skibba et al. (2015) at z ∼ .
5, for the stellar mass selected galaxies (dotted and dashedlines).with luminosities L (cid:54) L ∗ and then rises at brighter luminosities.According to Zehavi et al. (2011) this relation is best fitted bythe functional form b g ( > L ) × ( σ / . = . + . L / L ∗ ) . .We adopt a similar formula to model the galaxy bias-luminosityrelation, and at z ∼ b g , HOD ( > L ) is best fitted by b g , HOD ( > L ) = . + . L / L ∗ ) . (13)represented by a solid line in the left panel of Fig. 8. Here L is theUV luminosity and L ∗ corresponds to the characteristic absolutemagnitude M ∗ UV ( z =
3) obtained as described in Appendix A.Our estimate of the dependence of the large scale bias on galaxyluminosity is nearly flat for galaxies with luminosities L (cid:54) . L ∗ and rises very sharply for brighter ones. Therefore, in agreementwith the analysis of the relative bias discussed above, this sug-gests that the bias contrast between bright and faint galaxies be-comes stronger when going back in time.In the right panel of Fig. 8 we also present the large scalegalaxy bias measurements for the stellar mass selected sub-samples. We compare our results with the similar measurementsat z ∼ . z ∼ . < z < . b g values at z ∼ b g , HOD = . ± .
58 measured for galaxies with M med = . h − M (cid:12) to b g , HOD = . ± .
99 for the most massive galaxysub-sample with M med = . h − M (cid:12) . These galaxy bias val-ues are also in excellent agreement with measurements based onN-body simulations performed by Chiang et al. (2013), who at z = b g = .
24 and b g = .
71 for galaxies with stellarmasses M > M (cid:12) and M > M (cid:12) , respectively. Like for the luminosity selected galaxies, we made an attempt to modelthis bias-stellar mass relation at z ∼
3. We find that the best fit-ting function, represented in Fig. 8 by a solid line, is given by b g , HOD ( > M ) = . + . M / M ∗ ) . , (14)where M is the galaxy stellar mass and M ∗ is the characteristicstellar mass at z ∼ In Fig. 9 we show the values of two characteristic halo masses, M min and M , in terms of the sample threshold luminosity (leftpanel) and stellar mass (right panel) relative to the characteristicluminosity and stellar mass, L thresh / L ∗ and M thresh / M ∗ respec-tively, at di ff erent redshifts. The minimum halo mass needed forhalf of the haloes to host one central galaxy above the luminosityor stellar mass threshold M min (filled symbols) and the mass ofhaloes with on average one additional satellite galaxy above theluminosity or stellar mass threshold M (open symbols), mea-sured at z ∼ z ∼ M min and M at z ∼ z ∼ Article number, page 12 of 20. Durkalec et al.: Luminosity and stellar mass dependence of galaxy clustering at z ∼ f s L thresh /L *This work, z~3Zehavi et al. 2011 z ~ 0 f s M thresh /M *This work, z~3Skibba et al. 2015 z~0.5 Fig. 10: Satellite fraction f s as a function of threshold luminosity, with L ∗ as a reference point ( left panel) and as a function ofthreshold stellar mass, with M ∗ as a reference point ( right panel). Results obtained in this work at z ∼ z ∼ z ∼ . L thresh / L ∗ . This implies that at high red-shift the bright and most massive galaxies are much more likelyto occupy the most massive dark matter haloes. Combining thiswith the earlier observation that a lower mass dark matter halois needed to host a galaxy of higher luminosity / stellar mass athigher redshift, suggests that the processes responsible for thefollowing increase of the mass of the halo and the stellar mass ofthe galaxies operate on di ff erent timescales and are both stellarmass and epoch dependent.With the increasing M min we observe a proportional growthof M . At all luminosities the values of M min and M present anapproximately constant ratio M / M min ≈ z ∼ M / M min ≈ .
5. For comparison, in the local universe the ratiobetween M and M min is higher. At z ∼ M ≈ M min for the SDSS galaxies selected bytheir r-band absolute magnitude, while at intermediate redshift z ∼ ≈
15 and McCracken et al. (2015) reportvalues of ≈
10 for galaxies at 1 . < z < .
0. These results, com-bined with our observations at z ∼
3, can be interpret as evidencethat (1) at higher redshifts, dark matter haloes consist of manyrecently accreted satellites, and (2) the M / M min ratio evolveswith redshift, with smaller values observed at higher redshifts,which is in agreement with other studies (e.g., de la Torre et al.2013; Skibba et al. 2015; McCracken et al. 2015) and can beexplained by the relation between halo versus galaxy merging(Conroy et al. 2006; Wetzel et al. 2009). The dark matter halomergers create an infall of satellite galaxies onto a halo, whilethe galaxy major mergers destroy them. If the halo mergers oc-cur more often than the galaxy mergers, we can expect a largesatellite population, resulting in a small M / M min ratio. According to the high-resolution N-body simulations per-formed by Wetzel et al. (2009), at z > . ff ectively galaxies) is significantly lower than that ofhaloes. For example, at z ∼ M h ≈ h − M (cid:12) are expected to experience ∼ . / Gyr, compared to only ∼ . / Gyr expected for sub-haloes (based on a prelim-inary VUDS sample Tasca et al. 2014, find an even lower valueof major galaxy mergers, 0 .
17 mergers / Gyr). This implies that at z > . z < . z ∼ M / M min , while a smaller halo occupationat lower redshift corresponds to its increase with time.From the observational side the galaxy major merger ratehas been shown to rapidly rise from z ∼ z ∼ . z > z ∼ . − M / M min at z ∼ We compute the fraction of satellite galaxies per halo f s for allluminosity and stellar mass sub-samples using the HOD best-fitting parameters (Eq. 10). The results, as a function of thresholdluminosity, with L ∗ as a reference point (left panel) and thresh-old stellar mass, with M ∗ as a reference point (right panel), are Article number, page 13 of 20 & A proofs: manuscript no. arxiv_final shown in Fig. 10. We compare our measurements at z ∼
3, rep-resented by filled symbols, with similar results obtained at z ∼ z ∼ . z ∼ ∼
60% forthe faintest galaxy population to ∼
20% for the brightest ones. Asmaller value of f s for the brightest galaxies does not necessarilymean that there are no other satellite galaxies occupying a darkmatter halo, but rather that there are no bright satellite galaxies.Therefore, our results would suggest that, at high redshift it ismore probable that a dark matter haloes host faint satellite galax-ies, rather than very bright ones. A similar, however less steep,trend is present in the local universe (Zehavi et al. 2011). Forgalaxies selected according to their stellar mass the situation isless clear. Taking into account the uncertainties of our measure-ment we are not able to determine if f s changes with the stellarmass of galaxies, as observed at lower redshift ranges (Skibbaet al. 2015). At face value our data suggest the possible presenceof a small drop, by ∆ f s ∼ .
1, from the least massive to the mostmassive galaxies, but it is not a significant change (at the level of0 . σ ).From the perspective of the redshift evolution, we observethat the satellite fraction of the two faintest galaxy sub-samplesand of all the stellar mass selected galaxy sub-samples is higherat z ∼ L ∼ L ∗ athigh redshift can be explained using the same reasoning as pre-sented in Sec. 5.3. It suggests that the infall of the satellite galax-ies, as a result of halo mergers, onto a dark matter halo is fasterthan their destruction via galaxy major mergers (Wetzel et al.2009). Therefore, the subhaloes that form after halo mergers arelikely to remain intact and this leads to a large number of satellitegalaxies at high redshift, resulting in the measured high satellitefraction. It is necessary to mention, however, that this conclusionapplies to star forming galaxies, with L ∼ L ∗ , as the used datasample does not include a population of faint galaxies at z ∼ In this section we focus on the relationship between halo massand stellar mass of each galaxy sample, in the literature simplyreferred to as the stellar-to-halo mass relation (SHMR, see e.g.,Behroozi et al. 2010, 2013; Moster et al. 2013; Leauthaud et al.2012; Yang et al. 2012; Durkalec et al. 2015a).In Fig. 11 we present the SHMR at z ∼ M min as the onethat represents the halo mass associated with the threshold stellarmasses M thresh (cid:63) of the galaxy sub-samples. The errors associatedwith the stellar mass threshold limit are computed as the averageof the errors on M (cid:63) for each stellar mass sub-sample separately.We compare our results with the z = M (cid:63) > . M (cid:12) our results are in agreement with these models. How- l og M ★ [ M O • ] log M h [M O• ] Moster+13Behroozi+13Yang+12Yang+12 (SMF2) l og M ★ [ M O • ] log M h [M O• ] Ishikawa+17This work
Fig. 11: Stellar mass - halo mass relation (SHMR) of centralgalaxies obtained for di ff erent stellar mass selected sub-samplesat z ∼ M min , while the associated stellar massesof the galaxies are represented by the threshold limits M thresh (cid:63) ofeach sub-sample. The measurements from this work are com-pared with the results based on the z = z = ff erent lines, as labelled. Yang et al.(2012) paper includes best-fit SHMR models for two di ff erentstellar mass functions and we plot both of them. The blue shadedarea corresponds to the 68% confidence limits of Behroozi et al.(2013).ever, for galaxies with low stellar masses ( M (cid:63) < . M (cid:12) ),there is a striking di ff erence between our z ∼ M h = . M (cid:12) hosting galax-ies with minimum stellar masses of M thresh (cid:63) = . M (cid:12) , whilemodel predictions by Behroozi et al. (2013) place the samegalaxies in much more massive haloes of M h ∼ M (cid:12) . Inother words, we observe that the low-mass galaxies at z ∼ ffi ciently than it is expected from thesemodels, that all assume a much steeper decrease of the e ff ectivestar formation with decreasing halo mass.Such discrepancies between model predictions and observa-tional constraints at high redshift have not been reported beforein the literature. E.g., in our previous studies (Durkalec et al.2015a) based on the preliminary VUDS observations and forsub-samples covering a wider redshift range (2 . < z < . z ∼ z =
3, Ishikawa et al. (2017) reports an excellentagreement of their SHMR measurements with the model predic-tions by Behroozi et al. (2013) for a large Lyman break galaxy
Article number, page 14 of 20. Durkalec et al.: Luminosity and stellar mass dependence of galaxy clustering at z ∼ (LBGs) sample. It is important to note, however, that galaxiesused in these studies do not reach the stellar mass range below M (cid:63) = . M (cid:12) , while the stellar mass limit of our least massivesub-sample is significantly smaller (10 . M (cid:12) ). The same limita-tion applies to the theoretical models of SHMR at high redshift,which are not constrained by observations at the low stellar massend (e.g., the SHMR model by Behroozi et al. 2013, at z = M (cid:63) = . M (cid:12) ).The SHMR is most commonly parametrized either with adouble power-law function (Behroozi et al. 2010; Yang et al.2012; Moster et al. 2013), or with a the five parameter functionproposed by Behroozi et al. (2013), which retains a power lawform for halo masses M h << . at z =
3. Our results sug-gest that, at high redshifts, this power-law shape is broken at thelow mass end below M h = M (cid:12) (see Fig. 11). In particular,according to our measurements, the stellar to halo mass ratio ishigher than predicted for this halo mass range. This is in agree-ment with the conclusion by Behroozi et al. (2013) who note thatthe low-mass end of the SHMR cannot be predicted by extrap-olating results from massive galaxies and fit with the power-lawfunction alone.A similar higher-than-expected stellar mass to halo mass ra-tio is observed for dwarf galaxies (e.g., Boylan-Kolchin et al.2012; Ferrero et al. 2012; Miller et al. 2014; Brook et al. 2014;Read et al. 2017). While the low-mass galaxy sub-samples usedin this paper are not as low mass as the dwarf galaxies observedin the local group (the minimum stellar mass of VUDS galaxiesused in our sample is M (cid:63) = . M (cid:12) while the masses of localdwarf galaxies are 10 − M (cid:12) ), the low mass observation-models discrepancy of SHMR we observe is consistent withthese low-mass low redshift samples and it is possible that simi-lar processes are behind it at high z for the higher mass galaxies.A possible explanation of the discrepancy between the ob-served SHMR of low mass galaxies and models may lie in theflaws of the abundance matching technique (used in the pre-sented theoretical models to infer SHMR), coupled with our stillpoor understanding of the feedback e ff ects that influence notonly the galaxy stellar mass assembly, but also on the mass distri-bution of the hosting dark matter haloes (e.g. Pontzen & Gover-nato 2012; Di Cintio et al. 2014; Ogiya & Mori 2014; Katz et al.2017). The abundance matching technique uses simulated darkmatter distributions. It is well known, however, that N-body sim-ulations predict a dark matter halo mass function much steeperthan the galaxy stellar mass function derived from observations(Press & Schechter 1974; Jenkins et al. 2001; Sheth et al. 2001;Springel et al. 2005). Moreover, this di ff erence increases whilemoving toward low, both stellar and halo, mass regimes (ourpoint of interest here). This is usually reconciled by assumingthat galaxy formation is directly connected to the halo mass andgalaxies do not form e ffi ciently in low mass haloes, which leadsto an overestimation of halo masses for the low mass galaxies,when the galaxies are matched with haloes under the assump-tion that dark matter-only simulations represent structure forma-tion and that every halo hosts a galaxy (which is the case in theabundance matching method). This overestimation of the halomasses derived by models, with respect to the observations, isthe one visible in Fig. 11 for the galaxies with M (cid:63) < . .The relation between dark matter halo mass and galaxy stel-lar mass is therefore not direct. It can be additionally influencedby, e.g., the strong feedback e ff ects, which a ff ect the star for-mation in low-mass galaxies more strongly than in more mas- M (cid:63) = . M (cid:12) in Durkalec et al. (2015a), M (cid:63) = . M (cid:12) inIshikawa et al. (2017) and M (cid:63) = M (cid:12) in McCracken et al. (2015) sive ones. In particular the strong positive feedback (either SNor AGN originated) would result in higher than expected starformation e ffi ciency of low-mass galaxies visible as the model-observation discrepancy for these galaxies in Fig. 11.At low redshifts the feedback e ff ects have been proposed asthe ones that have the major impact on the evolution of dwarfgalaxies (see, e.g., Ferrara & Tolstoy 2000; Fujita et al. 2004;Mashchenko et al. 2008; Sawala et al. 2011; Kawata et al. 2014;Oñorbe et al. 2015; Chen et al. 2016; Papastergis & Shankar2016). Our SHMR measurements, i.e., the higher than expectedstar formation e ffi ciency, suggest that at z ∼ ff ects have a significant influence on stellar mass assemblyin not only dwarf galaxies ( M (cid:63) < ), like it is observed locally,but also in more massive ones, which at z = ff ected. This conclusion can be supported by thefact that a strong feedback e ff ects, both positive and negative,has been observed in abundance in nearly all star-forming galax-ies at high z (e.g., Pettini et al. 2001; Shapley et al. 2003; Weineret al. 2009; Steidel et al. 2010; Jones et al. 2012; Newman et al.2012; Erb 2015; Talia et al. 2017; Le Fèvre et al. 2017).However, we note that other processes might be at work,hence this interpretation may not be the only one and that onlyfurther observations of low-mass high redshift galaxies mighthelp to resolve the problem. For example, positive feedbackmight not be su ffi cient to alleviate model to observations at low-mass end, and we need to account also for the possible existenceof ’dark haloes’, i.e., haloes that are completely devoid of stars(see, e.g., Sawala et al. 2013, 2015). A high number of suchhaloes would strongly a ff ect the accuracy of models based onthe abundance matching techniques. Also, regardless of the factthat introducing a strong positive feedback in low-mass galaxiesat high redshifts is physically motivated, it might not producethe correct star formation histories, resulting in a more numer-ous population of passive galaxies than it is observed locally,as suggested by, e.g., Fontanot et al. (2009); Weinmann et al.(2012) and Moster et al. (2013). We, therefore, conclude that amixture of both e ff ects, i.e., strong positive feedback e ff ects andhigh number of empty dark matter haloes is a possible explana-tion of the observed trends.
6. Summary and conclusions
In this paper we study the luminosity and stellar mass depen-dence of galaxy clustering at redshift z ∼ w p ( r p )in four volume-limited luminosity sub-samples, with the cutsmade in UV absolute magnitude, and four stellar mass sub-samples. Our measurements are quantified in the frameworkof two approximations. The first one is the power-law model ξ ( r ) = ( r / r ) − γ , with two free parameters. The second one isbased on the halo occupation distribution model (HOD), withfive free parameters.The main results and conclusions of our study can be sum-marised as follows: – We observe an increase of the correlation length r with theluminosity and stellar mass of the galaxy populations, indi-cating a luminosity and stellar mass dependence of galaxyclustering at z ∼
3. For UV luminosity selected sub-samples r rises from r = . ± . h − Mpc to r = . ± . h − Mpc over a threshold UV absolute magnitude rangingfrom M UV = − . M UV = − .
2. A similar trend isfound for stellar mass selected galaxy samples, where the
Article number, page 15 of 20 & A proofs: manuscript no. arxiv_final
Table 4: Best-fitting power-law and HOD parameters, with other derived parameters (as described in Sec. 3) for the luminosity andstellar mass sub-samples used in this work. For the power-law fit, the number of degrees of freedom (dof) is 6 (8 measured w p values minus the 2 fitted parameters), while for the HOD dof =
3. All masses are given in units of h − M (cid:12) and correlation length r is given in h − Mpc.Sample / Parameter M maxUV log M min (cid:63) − . − . − . − . .
75 9 .
25 9 .
75 10 . r . ± .
22 3 . ± .
32 3 . ± .
42 5 . ± .
50 3 . ± .
18 3 . ± .
30 3 . ± .
42 4 . ± . γ . ± .
07 1 . ± .
09 1 . ± .
23 1 . ± .
25 1 . ± .
06 1 . ± .
09 1 . ± .
19 1 . ± . M min . ± .
51 10 . ± .
57 10 . ± .
63 11 . ± .
62 9 . ± .
48 9 . ± .
62 11 . ± .
36 11 . ± . M (cid:48) . ± .
89 10 . ± .
88 11 . ± .
81 12 . ± .
50 10 . ± .
87 10 . ± .
88 11 . ± .
62 11 . ± . M . ± .
74 10 . ± .
60 11 . ± .
80 12 . ± .
48 10 . ± .
69 10 . ± .
65 11 . ± .
51 11 . ± . M . ± .
96 9 . ± .
19 8 . ± .
23 9 . ± .
12 8 . ± .
98 8 . ± .
97 9 . ± .
19 9 . ± . σ log M . ± .
13 0 . ± .
21 0 . ± .
20 0 . ± .
16 0 . ± .
16 0 . ± .
18 0 . ± .
16 0 . ± . α . ± .
25 0 . ± .
27 1 . ± .
35 1 . ± .
23 1 . ± .
23 1 . ± .
25 1 . ± .
27 1 . ± . (cid:104) M h (cid:105) . ± .
58 11 . ± .
45 12 . ± .
46 12 . ± .
71 11 . ± .
45 12 . ± .
42 11 . ± .
34 12 . ± . b g . ± .
26 2 . ± .
84 2 . ± .
25 3 . ± .
16 1 . ± .
58 2 . ± .
64 2 . ± .
67 2 . ± . f s . ± .
41 0 . ± .
31 0 . ± .
34 0 . ± .
30 0 . ± .
44 0 . ± .
31 0 . ± .
25 0 . ± . r = . ± . h − Mpcto r = . ± . h − Mpc over a relatively small stellarmass range ∆ log M (cid:63) = . h − M (cid:12) . Based on these obser-vations we conclude that at z ∼ z ∼
3. It indicates thatmechanisms which led to luminosity and stellar mass clus-tering dependence must have been at work at a significantlyhigher redshift than z ∼ – Based on the power-law approximation of the correlationfunction we interpret our results in terms of the relation be-tween the distribution of galaxies and the underlying darkmatter density field, called bias ( b ), relative to the b ∗ of the L ∗ galaxies. We note that at z ∼ b / b ∗ , in each luminosity sub-sample, are significantly lowerthan observed for the local and intermediate redshift rangesfor galaxies of similar properties. Additionally we observethat the relative galaxy bias grows with the increasing lumi-nosity of the sample from low values b / b ∗ = . ± .
03 atlow luminosities to b / b ∗ = . ± . b / b ∗ at z ∼ – Taking advantage of the HOD best-fitting parameters wemeasure the large scale galaxy bias b g , HOD . We interpret ourresults in terms of both redshift evolution, and as a functionof luminosity and stellar mass. As expected in the frame-work of the hierarchical scenario of structure formation andevolution, we observe that the b g , HOD measured at z ∼ b g , HOD depen-dence, with the brightest and most massive galaxies being themost biased ones. We find that the luminosity dependence ismuch steeper than observed in the local universe. The largescale galaxy bias grow by ∆ b g , HOD = .
16, while at z ∼ ∆ b g , HOD = .
09 over the same luminos-ity range. A similar growth is observed for stellar mass se-lected galaxies, with the large scale galaxy bias rising from b g , HOD = . ± .
58 to b g , HOD = . ± .
99 over the thresh-old stellar mass range of ∆ log M (cid:63) = .
25. Following Zehaviet al. (2011), we made an attempt to model the galaxy bias-luminosity and galaxy bias-stellar mass relation, and at z ∼ b g , HOD ( > L ) is best fitted by b g , HOD ( > L ) = . + . L / L ∗ ) . ,while for the stellar mass threshold samples the best fit is b g , HOD ( > M ) = . + . M / M ∗ ) . . – We report values of the best-fitting HOD parameters for allvolume limited UV absolute magnitude and stellar mass sub-samples at redshift z ∼
3. Similarly to what is seen at lowerredshift we observe a growth of the dark matter halo char-acteristic masses M min and M with rising luminosity andstellar mass of the galaxy population, indicating that brightand most massive galaxies are likely to occupy the most mas-sive dark matter haloes. Both quantities grow proportionallywith a scaling relation of M / M min ≈ M / M min ≈ . z ∼
0, where this ratio is reported to have val-ues of M / M min ≈ −
20 (Zehavi et al. 2011; McCrackenet al. 2015; Skibba et al. 2015), which suggests that at highredshift dark matter haloes contain mainly recently accretedsatellite galaxies. We discuss (1) the observed low value of
Article number, page 16 of 20. Durkalec et al.: Luminosity and stellar mass dependence of galaxy clustering at z ∼ M / M min at z ∼ z ∼
3. Our results are consistent with high resolutionN-body simulations (see Sec. 5.3). – We discuss further the satellite galaxies that occupy darkmatter haloes at z ∼ f s . Again our results have implications for the satellite abun-dances as a function of luminosity and stellar mass, but alsoas a function of redshift. At z ∼ f s ∼ ∼ f s ∼ − – Finally we focus on the stellar to halo mass relation (SHMR)obtained for di ff erent stellar mass sub-samples. We find thatour z ∼ M (cid:63) < . M (cid:12) , Fig. 11). This suggests that the low-massgalaxies are producing stars more e ff ectively than expected.We discuss the possibility that strong SNe or AGN feed-back e ff ects are at work, that would at least partly explainthe observed discrepancy between observations and modelsfor low-mass galaxies at z ∼ z > ffi ciency of the processes whichdrive the star formation and mass assembly in galaxies at thattime. Moreover, as shown in this paper, our results very wellcomplement lower-z measurements regarding the galaxy clus-tering dependencies. All of this information can be used, amongothers, as an input to improve galaxy formation models (likesemi-analytical models) and simulations (like the latest hydro-dynamical simulations), which are still uncertain at high red-shifts and need to be confronted by improved observational con-straints. Acknowledgements.
This work is supported by funding from the European Re-search Council Advanced Grant ERC-2010-AdG-268107-EARLY and by INAFGrants PRIN 2010, PRIN 2012 and PICS 2013. AD is supported by the Pol-ish National Science Centre grant UMO-2015 / / D / ST9 / / MegaCam, a joint project of CFHT and CEA / DAPNIA, at theCanada-France-Hawaii Telescope (CFHT) which is operated by the National Re-search Council (NRC) of Canada, the Institut National des Sciences de l’Universof the Centre National de la Recherche Scientifique (CNRS) of France, and theUniversity of Hawaii.
References
Abbas, U., de la Torre, S., Le Fèvre, O., et al. 2010, MNRAS, 406, 1306Adelberger, K. L., Steidel, C. C., Pettini, M., et al. 2005, ApJ, 619, 697Arnouts, S., Cristiani, S., Moscardini, L., et al. 1999, MNRAS, 310, 540Bardeen, J. M., Bond, J. R., Kaiser, N., & Szalay, A. S. 1986, ApJ, 304, 15Barrow, J. D., Bhavsar, S. P., & Sonoda, D. H. 1984, MNRAS, 210, 19PBehroozi, P. S., Conroy, C., & Wechsler, R. H. 2010, ApJ, 717, 379Behroozi, P. S., Wechsler, R. H., & Conroy, C. 2013, ApJ, 770, 57 Benson, A. J., Frenk, C. S., Baugh, C. M., Cole, S., & Lacey, C. G. 2001, MN-RAS, 327, 1041Beutler, F., Blake, C., Colless, M., et al. 2013, MNRAS, 429, 3604Bielby, R. M., Gonzalez-Perez, V., McCracken, H. J., et al. 2014, A&A, 568,A24Blandford, R. D. & Narayan, R. 1992, ARA&A, 30, 311Blanton, M., Cen, R., Ostriker, J. P., & Strauss, M. A. 1999, ApJ, 522, 590Bouwens, R. J. & Illingworth, G. D. 2007, in Astronomical Society of the PacificConference Series, Vol. 380, Deepest Astronomical Surveys, ed. J. Afonso,H. C. Ferguson, B. Mobasher, & R. Norris, 41Bouwens, R. J., Illingworth, G. D., Oesch, P. A., et al. 2015, ApJ, 803, 34Boylan-Kolchin, M., Bullock, J. S., & Kaplinghat, M. 2012, MNRAS, 422, 1203Brook, C. B., Di Cintio, A., Knebe, A., et al. 2014, ApJ, 784, L14Cassata, P., Le Fèvre, O., Charlot, S., et al. 2013, A&A, 556, A68Chen, J., Bryan, G. L., & Salem, M. 2016, MNRAS, 460, 3335Chiang, Y.-K., Overzier, R., & Gebhardt, K. 2013, ApJ, 779, 127Coil, A. L., Newman, J. A., Cooper, M. C., et al. 2006, ApJ, 644, 671Coil, A. L., Newman, J. A., Croton, D., et al. 2008, ApJ, 672, 153Conroy, C., Wechsler, R. H., & Kravtsov, A. V. 2006, ApJ, 647, 201Conselice, C. J., Rajgor, S., & Myers, R. 2008, MNRAS, 386, 909Coupon, J., Kilbinger, M., McCracken, H. J., et al. 2012, A&A, 542, A5Daddi, E., Röttgering, H. J. A., Labbé, I., et al. 2003, ApJ, 588, 50Davis, M. & Peebles, P. J. E. 1983, ApJ, 267, 465de la Torre, S., Guzzo, L., Peacock, J. A., et al. 2013, A&A, 557, A54de la Torre, S., Le Fèvre, O., Arnouts, S., et al. 2007, A&A, 475, 443De Lucia, G., Poggianti, B. M., Aragón-Salamanca, A., et al. 2007, MNRAS,374, 809De Lucia, G., Springel, V., White, S. D. M., Croton, D., & Kau ff mann, G. 2006,MNRAS, 366, 499de Ravel, L., Le Fèvre, O., Tresse, L., et al. 2009, A&A, 498, 379Dekel, A., Ishai, G., Dutton, A. A., & Maccio, A. V. 2017, MNRAS, 468, 1005Di Cintio, A., Brook, C. B., Macciò, A. V., et al. 2014, MNRAS, 437, 415Durkalec, A., Le Fèvre, O., de la Torre, S., et al. 2015a, A&A, 576, L7Durkalec, A., Le Fèvre, O., Pollo, A., et al. 2015b, A&A, 583, A128Erb, D. K. 2015, Nature, 523, 169Ferrara, A. & Tolstoy, E. 2000, MNRAS, 313, 291Ferrero, I., Abadi, M. G., Navarro, J. F., Sales, L. V., & Gurovich, S. 2012, MN-RAS, 425, 2817Finkelstein, S. L., Ryan, Jr., R. E., Papovich, C., et al. 2015, ApJ, 810, 71Fontanot, F., De Lucia, G., Monaco, P., Somerville, R. S., & Santini, P. 2009,MNRAS, 397, 1776Fry, J. N. 1996, ApJ, 461, L65Fu, L., Semboloni, E., Hoekstra, H., et al. 2008, A&A, 479, 9Fujita, A., Mac Low, M.-M., Ferrara, A., & Meiksin, A. 2004, ApJ, 613, 159Garilli, B., Fumana, M., Franzetti, P., et al. 2010, PASP, 122, 827Genzel, R., Schreiber, N. M. F., Übler, H., et al. 2017, Nature, 543, 397Guo, H., Zheng, Z., Zehavi, I., et al. 2015, MNRAS, 453, 4368Guzzo, L., Strauss, M. A., Fisher, K. B., Giovanelli, R., & Haynes, M. P. 1997,ApJ, 489, 37Hagen, L. M. Z., Hoversten, E. A., Gronwall, C., et al. 2015, ApJ, 808, 178Hartley, W. G., Almaini, O., Cirasuolo, M., et al. 2010, MNRAS, 407, 1212Hatfield, P. W., Lindsay, S. N., Jarvis, M. J., et al. 2016, MNRAS, 459, 2618Hathi, N. P., Ryan, Jr., R. E., Cohen, S. H., et al. 2010, ApJ, 720, 1708Hildebrandt, H., Pielorz, J., Erben, T., et al. 2009, A&A, 498, 725Hoekstra, H., Yee, H. K. C., & Gladders, M. D. 2004, ApJ, 606, 67Ilbert, O., Arnouts, S., McCracken, H. J., et al. 2006, A&A, 457, 841Ilbert, O., McCracken, H. J., Le Fèvre, O., et al. 2013, A&A, 556, A55Ilbert, O., Tresse, L., Zucca, E., et al. 2005, A&A, 439, 863Ishikawa, S., Kashikawa, N., Toshikawa, J., et al. 2017, ApJ, 841, 8Jenkins, A., Frenk, C. S., White, S. D. M., et al. 2001, MNRAS, 321, 372Jones, O. C., Kemper, F., Sargent, B. A., et al. 2012, MNRAS, 427, 3209Kaiser, N. 1984, ApJ, 284, L9Katz, H., Lelli, F., McGaugh, S. S., et al. 2017, MNRAS, 466, 1648Kau ff mann, G., Nusser, A., & Steinmetz, M. 1997, MNRAS, 286, 795Kawata, D., Gibson, B. K., Barnes, D. J., Grand, R. J. J., & Rahimi, A. 2014,MNRAS, 438, 1208Kilbinger, M., Benabed, K., Cappe, O., et al. 2011, arXiv1101.0950[ arXiv:1101.0950 ]Landy, S. D. & Szalay, A. S. 1993, ApJ, 412, 64Le Fèvre, O., Guzzo, L., Meneux, B., et al. 2005, A&A, 439, 877Le Fèvre, O., Lemaux, B. C., Nakajima, K., et al. 2017, ArXiv e-prints[ arXiv:1710.10715 ]Le Fèvre, O., Tasca, L. A. M., Cassata, P., et al. 2015, A&A, 576, A79Leauthaud, A., Tinker, J., Bundy, K., et al. 2012, ApJ, 744, 159Lee, K.-S., Giavalisco, M., Gnedin, O. Y., et al. 2006, ApJ, 642, 63Lilly, S. J., Le Fevre, O., Hammer, F., & Crampton, D. 1996, ApJ, 460, L1Lin, L., Dickinson, M., Jian, H.-Y., et al. 2012, ApJ, 756, 71López-Sanjuan, C., Le Fèvre, O., de Ravel, L., et al. 2011, A&A, 530, A20López-Sanjuan, C., Le Fèvre, O., Tasca, L. A. M., et al. 2013, A&A, 553, A78Magliocchetti, M. & Porciani, C. 2003, MNRAS, 346, 186 Article number, page 17 of 20 & A proofs: manuscript no. arxiv_final
Marulli, F., Bolzonella, M., Branchini, E., et al. 2013, A&A, 557, A17Mashchenko, S., Wadsley, J., & Couchman, H. M. P. 2008, Science, 319, 174Mason, C. A., Trenti, M., & Treu, T. 2015, ApJ, 813, 21Massey, R., Rhodes, J., Ellis, R., et al. 2007, Nature, 445, 286McCracken, H. J., Wolk, M., Colombi, S., et al. 2015, MNRAS, 449, 901McLure, R. J., Dunlop, J. S., Bowler, R. A. A., et al. 2013, MNRAS, 432, 2696Meneux, B., Guzzo, L., de la Torre, S., et al. 2009, A&A, 505, 463Meneux, B., Guzzo, L., Garilli, B., et al. 2008, A&A, 478, 299Meneux, B., Le Fèvre, O., Guzzo, L., et al. 2006, A&A, 452, 387Metcalf, R. B. & Madau, P. 2001, ApJ, 563, 9Meylan, G., Jetzer, P., North, P., et al., eds. 2006, Gravitational Lensing: Strong,Weak and MicroMiller, S. H., Ellis, R. S., Newman, A. B., & Benson, A. 2014, ApJ, 782, 115Mo, H. J. & White, S. D. M. 1996, MNRAS, 282, 347Mostek, N., Coil, A. L., Cooper, M., et al. 2013, ApJ, 767, 89Moster, B. P., Naab, T., & White, S. D. M. 2013, MNRAS, 428, 3121Moustakas, L. A. & Metcalf, R. B. 2003, MNRAS, 339, 607Newman, S. F., Genzel, R., Förster-Schreiber, N. M., et al. 2012, ApJ, 761, 43Norberg, P., Baugh, C. M., Hawkins, E., et al. 2002, MNRAS, 332, 827Oñorbe, J., Boylan-Kolchin, M., Bullock, J. S., et al. 2015, MNRAS, 454, 2092Ogiya, G. & Mori, M. 2014, ApJ, 793, 46Ouchi, M., Hamana, T., Shimasaku, K., et al. 2005, ApJ, 635, L117Papastergis, E. & Shankar, F. 2016, A&A, 591, A58Parsa, S., Dunlop, J. S., McLure, R. J., & Mortlock, A. 2016, MNRAS, 456,3194Peacock, J. A. & Smith, R. E. 2000, MNRAS, 318, 1144Peebles, P. J. E. 1980, The large-scale structure of the universePérez-González, P. G., Rieke, G. H., Villar, V., et al. 2008, ApJ, 675, 234Pettini, M., Shapley, A. E., Steidel, C. C., et al. 2001, ApJ, 554, 981Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. 2014, A&A, 571, A16Pollo, A., Guzzo, L., Le Fèvre, O., et al. 2006, A&A, 451, 409Pollo, A., Meneux, B., Guzzo, L., et al. 2005, A&A, 439, 887Pontzen, A. & Governato, F. 2012, MNRAS, 421, 3464Press, W. H. & Schechter, P. 1974, ApJ, 187, 425Read, J. I., Iorio, G., Agertz, O., & Fraternali, F. 2017, MNRAS, 467, 2019Reddy, N. A. & Steidel, C. C. 2009, ApJ, 692, 778Rines, K., Geller, M. J., Diaferio, A., & Kurtz, M. J. 2013, ApJ, 767, 15Robertson, B. E. 2010, ApJ, 713, 1266Rubin, V. C., Thonnard, N., & Ford, Jr., W. K. 1978, ApJ, 225, L107Sawala, T., Frenk, C. S., Crain, R. A., et al. 2013, MNRAS, 431, 1366Sawala, T., Frenk, C. S., Fattahi, A., et al. 2015, MNRAS, 448, 2941Sawala, T., Guo, Q., Scannapieco, C., Jenkins, A., & White, S. 2011, MNRAS,413, 659Sawicki, M. & Thompson, D. 2006, ApJ, 648, 299Schechter, P. 1976, ApJ, 203, 297Seljak, U. 2000, MNRAS, 318, 203Shapley, A. E., Steidel, C. C., Pettini, M., & Adelberger, K. L. 2003, ApJ, 588,65Sheth, R. K., Mo, H. J., & Tormen, G. 2001, MNRAS, 323, 1Skibba, R. A., Coil, A. L., Mendez, A. J., et al. 2015, ApJ, 807, 152Springel, V., White, S. D. M., Jenkins, A., et al. 2005, Nature, 435, 629Steidel, C. C., Erb, D. K., Shapley, A. E., et al. 2010, ApJ, 717, 289Talia, M., Brusa, M., Cimatti, A., et al. 2017, MNRAS, 471, 4527Tasca, L. A. M., Le Fèvre, O., Hathi, N. P., et al. 2015, A&A, 581, A54Tasca, L. A. M., Le Fèvre, O., López-Sanjuan, C., et al. 2014, A&A, 565, A10Tegmark, M., Blanton, M. R., Strauss, M. A., et al. 2004, ApJ, 606, 702Tegmark, M. & Peebles, P. J. E. 1998, ApJ, 500, L79Thomas, R., Le Fèvre, O., Scodeggio, M., et al. 2016, ArXiv 1602.01841[ arXiv:1602.01841 ]Van Waerbeke, L., Mellier, Y., Erben, T., et al. 2000, A&A, 358, 30Wake, D. A., Whitaker, K. E., Labbé, I., et al. 2011, ApJ, 728, 46Wang, L., Li, C., Kau ff mann, G., & De Lucia, G. 2007, MNRAS, 377, 1419Weiner, B. J., Coil, A. L., Prochaska, J. X., et al. 2009, ApJ, 692, 187Weinmann, S. M., Pasquali, A., Oppenheimer, B. D., et al. 2012, MNRAS, 426,2797Wetzel, A. R., Cohn, J. D., & White, M. 2009, MNRAS, 395, 1376White, S. D. M. 1976, MNRAS, 177, 717White, S. D. M., Davis, M., Efstathiou, G., & Frenk, C. S. 1987, Nature, 330,451White, S. D. M. & Rees, M. J. 1978, MNRAS, 183, 341Wraith, D., Kilbinger, M., Benabed, K., et al. 2009, Phys. Rev. D, 80, 023507Yang, X., Mo, H. J., van den Bosch, F. C., Zhang, Y., & Han, J. 2012, ApJ, 752,41Zehavi, I., Weinberg, D. H., Zheng, Z., et al. 2004, ApJ, 608, 16Zehavi, I., Zheng, Z., Weinberg, D. H., et al. 2011, ApJ, 736, 59Zheng, Z., Berlind, A. A., Weinberg, D. H., et al. 2005, ApJ, 633, 791Zheng, Z., Coil, A. L., & Zehavi, I. 2007, ApJ, 667, 760Zwicky, F. 1937, ApJ, 86, 217 Article number, page 18 of 20. Durkalec et al.: Luminosity and stellar mass dependence of galaxy clustering at z ∼ -2-1.5-1-0.5 0 0.5 1 0 2 4 6 8 10 M * UV - M * UV , Redshift z
Bouwens et al. (2015)Mason et al. (2015)Hagen et al. (2015)Finkelstein et al. (2015)Hathi et al. (2010)Sawicki&Thomson (2006)Best fit -0.4-0.2 0 0.2 0.4 0.6 0.8 0 1 2 3 4 L og ( M * / M *0 ) Redshift z
Ilbert et al. (2013)Perez-Gonzalez et al. (2008)Best fit
Fig. A.1: A compilation of the values of Schechter characteristicUV galaxy luminosity M ∗ UV − M ∗ UV , (upper panel) and Schechtercharacteristic stellar mass log (cid:16) M ∗ ( z ) / M ∗ (cid:17) (lower panel). Thesymbols represent the measurements taken from various works(Bouwens et al. 2015; Mason et al. 2015; Hagen et al. 2015;Finkelstein et al. 2015; Ilbert et al. 2013; Hathi et al. 2010; Pérez-González et al. 2008; Sawicki & Thompson 2006) as describedin the legend. In each plot the solid red line shows the best-fittingexponential function given by Eq. A.1 and Eq. A.2 for the upperand lower panel, respectively. Appendix A: Correction for the luminosity andstellar mass function evolution
The mass, shape, number density of stars in the galaxies are con-stantly evolving with time. Consequently we observe the overallchanges in luminosity and stellar mass of the galaxy populationsat di ff erent epochs. The influence of these changes on the ab-solute magnitude and stellar mass of the galaxy population arereflected in the evolution of the luminosity and stellar mass func-tions, respectively. Particularly in the evolution of the M ∗ param-eter, from the best-fitted Schechter function (Schechter 1976), which describes the characteristic absolute magnitude (or stellarmass) of the galaxy population at given epoch.The luminosity and stellar mass functions have been ex-tensively studied in the literature, even at extremely high red-shift ranges (e.g., Lilly et al. 1996; Bouwens & Illingworth2007; Reddy & Steidel 2009; Robertson 2010; McLure et al.2013) and all the evidence to date suggests a brightening ofthe galaxy population when moving back in time. In the red-shift range 2 < z <
4, one of the most recent studies ofthe galaxy UV luminosity function from Parsa et al. (2016)(based on the combination of data from the Hubble UltraDeep Field (HUDF), CANDELS / GOODS-South, and UltraV-ISTA / COSMOS surveys), shows a brightening in the UV char-acteristic luminosity from M ∗ UV = − . ± .
07 at z ∼ . M ∗ UV = − . ± . z ∼ .
8. At even higher redshift ranges(4 < z <
8) Bouwens et al. (2015) finds that the characteris-tic UV galaxy luminosity does not change its value significantlyand at z ∼ . M ∗ UV = − . ± .
08, while at at z ∼ M ∗ UV = − . ± . ff erent epochs, we needto address the evolutionary brightening of galaxies. Hence, wenormalised the absolute magnitudes and stellar masses, at eachredshift, to the corresponding value of the characteristic lumi-nosity M ∗ UV or characteristic stellar mass log M ∗ Using measurements of the UV characteristic absolute mag-nitudes from Bouwens et al. (2015); Mason et al. (2015); Hagenet al. (2015); Finkelstein et al. (2015); Hathi et al. (2010) andSawicki & Thompson (2006), we construct the M ∗ UV ( z ) − M ∗ UV , function, as presented in the upper panel of Fig. A.1, where the M ∗ UV , is the characteristic luminosity for galaxies at z =
0. Thenthe best-fitting exponential function in form, M ∗ UV ( z ) − M ∗ UV , = − . + .
44 exp ( − z / . , (A.1)has been used to normalise the absolute magnitudes of galax-ies used in this paper. For each galaxy we take M UV = M (cid:48) UV − ( M ∗ UV − M ∗ UV , ), where M (cid:48) UV is the original (not corrected) abso-lute magnitude.We proceeded similarly to normalise the galaxy stellarmasses. We took the characteristic stellar masses measured byIlbert et al. (2013) and Pérez-González et al. (2008) in the red-shift range 0 < z < (cid:32) M ∗ ( z ) M ∗ (cid:33) = − .
18 exp ( − z / . + . , (A.2)has been used to normalise all stellar masses of the galaxies usedin this study. Appendix B: Tests of sample variation - a largestructure in the COSMOS field at z ∼ The correlation function measurements presented in this workare obtained from three independent VUDS fields (COSMOS,VVDS-02h and ECDFS). The di ff erences between these fields,like their angular size and number of galaxies, are accountedfor by using an appropriate weighting scheme (see Sec. 3). Thisweighting scheme favours the biggest and the most populatedfields in order to retrieve the best correlation function signal forall separations r p . At the same time, the di ff erences between the Article number, page 19 of 20 & A proofs: manuscript no. arxiv_final M fUV > -19.02 < z < 3.5 w p (r p ) r p [h -1 Mpc]
COSMOS fieldVVDS02h field
Fig. B.1: Projected two-point correlation function w p ( r p ) mea-sured independently for the M UV > − . ff erent fields yield in-formation about the cosmic variance.As a representative example in Fig. B.1 we show a com-parison of the independent correlation function measurementsfor the M UV > − . S e f f = .
11 deg ), the measure-ment of the correlation function in this field does not contributeto the final w p ( r p ) measurement at scales r p > h − Mpc onwhich the discussion below is focused.The most significant cosmic variance e ff ect appears at largeseparations r p > h − Mpc. At these large scales we observe asignificant di ff erence between the two correlation function mea-surements, as presented in Fig. B.1. The values of w p ( r p ) mea-sured at r p > h − Mpc for the COSMOS field are approxi-mately two times higher than the correlation function signal ob-tained for the VVDS-02h field. Naturally, this di ff erence has animpact on the overall composite correlation function measure-ments presented in this work. The COSMOS field contains ofthe largest number of galaxies spread across the biggest e ff ectivesurface (see Tab. 1). Therefore, the clustering results obtained forthis field have the biggest impact on the final composite correla-tion function measurements, and this results in the higher valesof the correlation function with respect to the best HOD modelsseen in Fig. 6. For all UV absolute magnitude and stellar massselected sub-samples, the correlation function measurement, atscales r p > h − Mpc, is higher on average by a factor of 1.7with respect to the HOD model.The flattening of the correlation function measured in theCOSMOS field at large scales is likely related to the existenceof an extremely large structure of galaxies, possibly a proto-supercluster or a massive filament, at z ∼ .
5, which spans a sizecomparable to to the entire filed covered by VUDS-COSMOS(Cucciati et al. 2017, in prep). This would be the first observa- tion of such a structure at high redshift. This hypothesis requiresfurther investigation and will be addressed in the dedicated fol-low up research.Ideally, to get the most robust measurements of the correla-tion function, one would exclude members of this structure fromthe measurements, however (1) the members of this structurehave not been fully identified yet and (2) this would significantlylower the sample statistic and probably make it impossible toperform correlation function measurements for the luminosityand stellar mass selected galaxy samples.5, which spans a sizecomparable to to the entire filed covered by VUDS-COSMOS(Cucciati et al. 2017, in prep). This would be the first observa- tion of such a structure at high redshift. This hypothesis requiresfurther investigation and will be addressed in the dedicated fol-low up research.Ideally, to get the most robust measurements of the correla-tion function, one would exclude members of this structure fromthe measurements, however (1) the members of this structurehave not been fully identified yet and (2) this would significantlylower the sample statistic and probably make it impossible toperform correlation function measurements for the luminosityand stellar mass selected galaxy samples.