The VIMOS Public Extragalactic Redshift Survey (VIPERS). Measuring nonlinear galaxy bias at z~0.8
C. Di Porto, E. Branchini, J. Bel, F. Marulli, M. Bolzonella, O. Cucciati, S. de la Torre, B. R. Granett, L. Guzzo, C. Marinoni, L. Moscardini, U. Abbas, C. Adami, S. Arnouts, D. Bottini, A. Cappi, J. Coupon, I. Davidzon, G. De Lucia, A. Fritz, P. Franzetti, M. Fumana, B. Garilli, O. Ilbert, A. Iovino, J. Krywult, V. Le Brun, O. Le Fevre, D. Maccagni, K. Malek, H. J. McCracken, L. Paioro, M. Polletta, A. Pollo, M. Scodeggio, L. A. M. Tasca, R. Tojeiro, D. Vergani, A. Zanichelli, A. Burden, A. Marchetti, D. Martizzi, Y. Mellier, R. C. Nichol, J. A. Peacock, W. J. Percival, M. Viel, M. Wolk, G. Zamorani
AAstronomy & Astrophysics manuscript no. arXiv_revised c (cid:13)
ESO 2016July 8, 2016
The VIMOS Public Extragalactic Redshift Survey (VIPERS) (cid:63)
Measuring non-linear galaxy bias at z ∼ . C. Di Porto , E. Branchini , , , J. Bel , F. Marulli , , , M. Bolzonella , O. Cucciati , S. de la Torre , B. R. Granett ,L. Guzzo , , C. Marinoni , L. Moscardini , , , U. Abbas , C. Adami , S. Arnouts , , D. Bottini , A. Cappi , ,J. Coupon , I. Davidzon , , G. De Lucia , A. Fritz , P. Franzetti , M. Fumana , B. Garilli , , O. Ilbert , A. Iovino ,J. Krywult , V. Le Brun , O. Le Fèvre , D. Maccagni , K. Małek , H. J. McCracken , L. Paioro , M. Polletta ,A. Pollo , , M. Scodeggio , L. A. M. Tasca , R. Tojeiro , D. Vergani , A. Zanichelli , A. Burden ,A. Marchetti , , D. Martizzi , Y. Mellier , R. C. Nichol , J. A. Peacock , W. J. Percival , M. Viel , , M. Wolk ,and G. Zamorani (A ffi liations can be found after the references) Received –; accepted –
ABSTRACT
Aims.
We use the first release of the VImos Public Extragalactic Redshift Survey of galaxies (VIPERS) of ∼ ,
000 objects to measure the biasingrelation between galaxies and mass in the redshift range z = [0 . , . Methods.
We estimate the 1-point distribution function [PDF] of VIPERS galaxies from counts in cells and, assuming a model for the mass PDF,we infer their mean bias relation. The reconstruction of the bias relation is performed through a novel method that accounts for Poisson noise,redshift distortions, inhomogeneous sky coverage. and other selection e ff ects. With this procedure we constrain galaxy bias and its deviations fromlinearity down to scales as small as 4 h − Mpc and out to z = . Results.
We detect small (up to 2 %) but statistically significant (up to 3 σ ) deviations from linear bias. The mean biasing function is close tolinear in regions above the mean density. The mean slope of the biasing relation is a proxy to the linear bias parameter. This slope increases withluminosity, which is in agreement with results of previous analyses. We detect a strong bias evolution only for z > .
9, which is in agreement withsome, but not all, previous studies. We also detect a significant increase of the bias with the scale, from 4 to 8 h − Mpc , now seen for the first timeout to z =
1. The amplitude of non-linearity depends on redshift, luminosity, and scale, but no clear trend is detected. Owing to the large cosmicvolume probed by VIPERS, we find that the mismatch between the previous estimates of bias at z ∼ Conclusions.
The results of our work confirm the importance of going beyond the over-simplistic linear bias hypothesis showing that non-linearities can be accurately measured through the applications of the appropriate statistical tools to existing datasets like VIPERS.
1. Introduction
Galaxies do not perfectly trace mass. The long known proof isthat galaxy clustering depends on properties of galaxies such asluminosity, colour, morphology, stellar mass, and so on (e.g. Sza-pudi et al. 2000; Hawkins et al. 2001; Norberg et al. 2001, 2002;Zehavi et al. 2002, 2011; Meneux et al. 2009; Marulli et al. 2013)and not solely on the underlying mass distribution. Di ff erencesin clustering properties are caused by the physical processes thatregulate the formation and evolution of galaxies and should dis-appear when averaging over scales much larger than those af-fected by these processes.Modelling the physics of galaxy formation, or at least its im-pact on the bias relation, is of paramount importance to extractcosmological information from the spatial distribution of galax- (cid:63) Based on observations collected at the European Southern Ob-servatory, Paranal, Chile, under programmes 182.A-0886 (LP) at theVery Large Telescope, and also based on observations obtained withMegaPrime / MegaCam, a joint project of CFHT and CEA / DAPNIA, atthe Canada-France-Hawaii Telescope (CFHT), which is operated by theNational Research Council (NRC) of Canada, the Institut National desScience de l’Univers of the Centre National de la Recherche Scien-tifique (CNRS) of France, and the University of Hawaii. This work isbased in part on data products produced at TERAPIX and the CanadianAstronomy Data Centre as part of the Canada-France-Hawaii TelescopeLegacy Survey, a collaborative project of NRC and CNRS. The VIPERSweb site is http: // vipers.inaf.it / . ies. Indeed, the large-scale structure of the Universe as traced bygalaxies is one of the most powerful cosmological probes as tes-tified by the increasing number of large galaxy redshift surveyseither ongoing, such as Boss (Anderson et al. 2012), DES , andVIPERS (Guzzo et al. 2014) or those planned for the near future,such as eBOSS , DESI (Schlegel et al. 2011), and Euclid (Lau-reijs et al. 2011) . These surveys are designed to address severalimportant questions both in cosmology and in galaxy evolutiontheory. Chief among them is the origin of the accelerated expan-sion of the Universe.It has recently been realised that geometry tests based onstandard candles and standard rulers can trace the expansion his-tory of the Universe but cannot identify the cause of the accel-erated expansion, which can be obtained either by advocatinga dark energy component or by modifying the gravity theory(e.g. Wang 2008). To break this degeneracy one needs indepen-dent observational tests. These are provided by the build-up ofstructures over cosmic time (Guzzo et al. 2008). The analysis oflarge-scale structures in galaxy distribution allows us to performthese two tests at one time. The baryonic acoustic oscillationpeaks in the two point statistics provide a standard ruler to per-form geometry test (e.g. Seo & Eisenstein 2003; Percival et al. http: // / future / http: // / Article number, page 1 of 22 a r X i v : . [ a s t r o - ph . C O ] J u l & A proofs: manuscript no. arXiv_revised N -body simulations. Thetwo most popular schemes to populate halos with galaxies arehalo occupation distribution (HOD; e.g. Cooray & Sheth 2002;Zheng et al. 2005) and sub-halo abundance matching (SHAM;e.g. Vale & Ostriker 2004; Conroy et al. 2006). The second cat-egory is represented by physical models in which the processesthat regulate the evolution of baryons are explicitly considered tolink them to the host dark matter structures. This approach is atthe heart of the semi-analytic models of galaxy formation (e.g.White & Frenk 1991; Bower et al. 2006; De Lucia & Blaizot2007). In most cases these models have been used to estimategalaxy bias from clustering statistics such as galaxy counts or2-point correlation functions. The results indicate that the accu-racy in both types of models is one of the main limitations inconstraining dark energy or modified gravity from current and,even more so, future observational campaigns (Contreras et al.2013).Alternatively, one can adopt a purely phenomenological ap-proach and use an operational definition of the bias in terms ofmap between the density fluctuations of mass, δ and galaxies, δ g smoothed on the same scale. This approach assumes that galaxybias is a local process that depends on the local mass densityonly. Many studies further assume that the bias relation is lin-ear and deterministic, so that galaxy bias can be quantified bya single linear bias parameter b : δ g = b δ . The concept of linearbias has played an important role in cosmology and many resultshave been obtained using this assumption, which is known to beunphysical as it allows negative densities. Also, this assumption-has no justification at the relatively small scales of interest tothe study of galaxy formation processes, which depend on manyphysical parameters and on large scales due to the presence ofneutrinos (Villaescusa-Navarro et al. 2014). In fact, the bias isconstant only on scales larger than about 40 h − Mpc (Manera &Gaztañaga 2011). Indeed, galaxy bias can be more convenientlydescribed within a probabilistic framework as proposed by Dekel& Lahav (1999) and recently reformulated in the context of thehalo model (Cacciato et al. 2012a).From the phenomenological viewpoint, bias has been exten-sively investigated from counts in cells statistics, weak gravita-tional lensing, and galaxy clustering. The latter is probably mostpopular approach. It is typically based on 2-point statistics andon the assumption of linear bias (Norberg et al. 2001, 2002; Ze-havi et al. 2005; Coil et al. 2006; Basilakos et al. 2007; Nuzaet al. 2013; Arnalte-Mur et al. 2014; Skibba et al. 2014; Marulliet al. 2013). A comparatively smaller number of studies searched for deviations from the linear and deterministic bias either us-ing 2-point (Tegmark & Bromley 1999) or higher order statis-tics (Verde et al. 2002; Gaztañaga et al. 2005; Kayo et al. 2004;Nishimichi et al. 2007; Swanson et al. 2008).Gravitational lensing in the weak field regime has also beenexploited to constrain galaxy bias. In particular, within thelimit of scale-independent bias on large scales, weak lensing andgalaxy clustering can be combined to estimate the linear bias pa-rameter in a manner which is independent of the amplitude ofdensity fluctuations (Amara et al. 2012; Pujol et al. 2016; Changet al. 2016). On smaller scales weak lensing was also used tomeasure the scale dependence of galaxy bias (Hoekstra et al.2002; Simon et al. 2007; Jullo et al. 2012; Comparat et al. 2013),although this e ff ect is degenerate with bias stochasticity, i.e. thefact that galaxy bias might not be solely determined by the localmass density.The most natural way to study a possible scale dependence(or non-linearity) of galaxy bias is in a probabilistic frameworkby means of counts in cells statistics (Sigad et al. 2000) sincein this case one can separate deviations from linear bias and thepresence of an intrinsic scatter in the bias relation. This approachwas used to estimate the bias of galaxies in the PSC z (Branchini2001), VVDS (Marinoni et al. 2005, hereafter M05), and zCOS-MOS (Kovaˇc et al. 2011, hereafter K11) catalogues as well asthe relative bias of blue versus red galaxies in the 2 degrees fieldgalaxy redshift survey (2dFGRS; Colless et al. 2001; Wild et al.2005). Despite some disagreement, results obtained at low red-shift ( z < .
5) generally indicate that, at least for some typesof galaxies, the bias is stochastic, scale dependent and, there-fore, non-linear. However. The situation at z > . . × . ff ec-tive linear bias parameter showed little evolution with redshift.In contrast, the biasing relation of zCOSMOS galaxies measuredby K11 over a region of about 1.52 deg turned out to be closeto linear and rapidly evolving with the redshift. The tension be-tween these results is paralleled by the observed di ff erences inthe spatial correlation properties of the two samples, with the2-point correlation function in zCOSMOS systematically higherthan that of VVDS galaxies (see e.g. Meneux et al. 2009). Owingto the large cosmic variance in the two samples, a rather smallgalaxy sample was proposed as the source of this mismatch, soa larger galaxy sample should be used to settle the issue.The Vimos Public Extragalactic Redshift Survey [VIPERS](Guzzo et al. 2014) has a depth similar to the zCOSMOS surveybut with a much larger area of 24 deg . Its volume is comparableto that of 2dFGRS and is large enough to significantly reducethe impact of the cosmic variance (see Appendix in Fritz et al.2014). We adopt the same approach as M05 and K11 and esti-mate galaxy bias from counts in cells. To do so we use a novelestimator that accounts for the e ff ect of discrete sampling, allow- Article number, page 2 of 22. Di Porto et al.: The VIMOS Public Extragalactic Redshift Survey (VIPERS) Measuring non-linear galaxy bias at z ∼ . ing us to use small cells and probe unprecedented small scalesthat are more a ff ected by the physics of galaxy formation.The layout of the paper is as follows. In Section 2 we de-scribe both the real and mock datasets used in this work. In Sec-tion 3 we introduce the formalism used to characterise galaxybias and the estimators used to measure this bias from a galaxyredshift survey. In Section 4 we assess the validity of the es-timator and use mock galaxy catalogues to gauge random andsystematic errors. We present our results in Section 5 and com-pare these with those of other analyses in Section 6. The mainconclusions are drawn in Section 7Throughout this paper we assume a flat Λ CDM universe( Ω m , Ω Λ , σ ) = (0.25; 0.75; 0.9). Galaxy magnitudes are givenin the AB system and, unless otherwise stated, computed assum-ing h ≡ H /
100 km s − Mpc − = σ haslittle impact on our analysis since our results can be rescaled todi ff erent values of σ that are more consistent with current cos-mological constraints. The dependence of the magnitude upon h is expressed as M = M h − h ), where M h is the absolutemagnitude computed for a given h value.
2. Datasets
The results in this paper are based on the first release of theVIPERS galaxy catalogue (Garilli et al. 2014). Random and sys-tematic errors were computed using a set of simulated galaxycatalogues mimicking the real catalogue and its observationalselections. Both, the real and mock samples are described in thisSection.
The VIMOS Public Extragalactic Redshift Survey is an ongo-ing ESA Large Programme aimed at measuring spectroscopicredshifts for about 10 galaxies at redshift 0 . < z < . . VIPERScovers 24 deg on the sky, divided over two areas within theW1 and W4 CFHTLS fields. Galaxies are selected to a limit of I AB < .
5, further applying a simple and robust colour prese-lection to e ffi ciently remove galaxies at z < .
5. This colour cutand the adopted observing strategy (Scodeggio et al. 2009) al-low us to double the galaxy sampling rate with respect to a puremagnitude-limited sample. At the same time, the area and depthof the survey result in a relatively large volume, 5 × h − Mpc , which is analogous to that of the 2dFGRS at z ∼ . = σ v = + z ) km s − .The full VIPERS area of 24 deg is covered through a mo-saic of 288 VIMOS pointings. A complete description of thesurvey construction, from the definition of the target sample tothe actual spectra and redshift measurements, is given in Guzzoet al. (2014). The dataset used in this and other papers of theearly science release represent the VIPERS Public Data Release1 (PDR-1) catalogue that includes 55359 redshifts (27935 in W1and 27424 in W4), i.e. 64% of the final survey in terms of cov-ered area (Garilli et al. 2014). A quality flag was assigned to http: // terapix.iap.fr / cplt / oldSite / Descart / CFHTLS-T0005-Release.pdf each object in the process of determining their redshift from thespectrum, which quantifies the reliability of the measured red-shifts. In this analysis, we use only galaxies with flags 2 to 9.5,which corresponds to a sample with a redshift confirmation rateof 90%.Several observational e ff ects need to be taken into account toinvestigate the spatial properties of the underlying population ofgalaxies. i) Selection e ff ects along the radial direction are driven by theflux limit nature of the survey and, at z < .
6, by the colour pres-election strategy. We use volume-limited (luminosity-complete)galaxy subsamples that we obtain by selecting galaxies brighterthan a given magnitude threshold in a given redshift interval.We adopted a redshift-dependent luminosity cut of the form M B ( z ) = M − z that should account for the luminosity evolutionof galaxies (e.g. Zucca et al. 2009). The value of the thresholdis set to guarantee that the selected sample is >
90 % completewithin the given redshift interval. In this sense each subsampleis volume limited and luminosity complete. This z -dependent lu-minosity cut is very popular and has been adopted in other pa-pers (see e.g. K11). However, other works used di ff erent typesof cuts, either ignoring any dependence on redshift (such as inM05; Coil et al. 2008) or assuming a di ff erent functional formfor the redshift evolution (e.g. Arnalte-Mur et al. 2014). Adopt-ing an incorrect luminosity evolution would generate a spuriousradial gradient in the mean density of the objects and a wrong z − dependence in the galaxy bias. To minimise the impact of thispotential bias, we carry out our analysis in relatively narrow red-shift bins, so that adopting any of the aforementioned luminos-ity cuts would produce similar results, as we verified. The ro-bustness of our result to the choice of the magnitude cut can betested a posteriori . Figure 16 shows that the di ff erence betweenestimates obtained with a z -dependent cut (filled red dot) andwith a z -independent cut (open red dot) are smaller than the totalrandom errors.Selection e ff ects induced by the colour preselection strategywere determined from the comparison between the spectroscopicand photometric samples (Guzzo et al. 2014; de la Torre et al.2013; Fritz et al. 2014) and are accounted for by assigning toeach galaxy an appropriate statistical weight dubbed colour sam-pling rate (CSR). ii) The surveyed area presents regular gaps due to the specificfootprint of the VIMOS spectrograph that creates a pattern ofrectangular regions, called pointings, separated by gaps where nospectra are taken. Superimposed on this pattern are unobservedareas resulting from bright stars and technical and mechanicalproblems during observations. We discuss our strategy to takeinto account this e ff ect in our counts in cells analysis in the fol-lowing (see Cucciati et al. 2014, for a more detailed study). iii) In each pointing, slits are assigned to a number of po-tential targets that meet the survey selection criteria (Bottiniet al. 2005). Given the surface density of the targeted popula-tion, the multiplex capability of VIMOS, and the survey strat-egy, a fraction of about 45% of the parent photometric samplecan be assigned to slits. We define the fraction of targets thathave a measured spectrum as the target sampling rate (TSR) andthe fraction of observed spectra with reliable redshift measure-ment as the spectroscopic sampling sate (SSR). Both functionsare roughly independent of galaxy magnitude except the SSR,which decreases for I AB > .
0, as shown in Fig. 12 of Guzzoet al. (2014).All these selection e ff ects are thoroughly discussed andquantitatively assessed by de la Torre et al. (2013). We makeno attempt to explicitly correct for these e ff ects individually. In- Article number, page 3 of 22 & A proofs: manuscript no. arXiv_revised stead, we assess their impact on the estimate of galaxy bias inSection 4 using the mock galaxy catalogues described below.For the scope of our analysis, the main advantages ofVIPERS are the relatively dense sampling of tracers, which al-lows us to probe density fluctuations down to scales comparableto those a ff ected by galaxy evolution processes, and the largevolume that, as discussed in the previous Section, allows us toreduce the impact of cosmic variance considerably with respectto previous estimates of galaxy bias at z ∼ z = [0 . , .
1] since the number density ofobjects at larger distances is too small to permit a robust estimateof galaxy bias. To investigate the possible dependence of galaxybias on luminosity and redshift, we partitioned the catalogue intosubsamples by applying a series of cuts in both magnitude andredshift.The complete list of subsamples considered in this work ispresented in Table 1. We considered three redshift bins ( z = [0 . , . , [0 . , . , [0 . , . ff erent luminositycuts that we obtained by compromising between the need ofmaximising both completeness and number of objects. Di ff er-ent luminosity cuts within each redshift bin allow us to study theluminosity dependence of galaxy bias at di ff erent redshifts. Themagnitude cuts, M B = − . − z − h ) and − . − z − h ) , that run across the whole redshift range are used to investigatea possible evolution of galaxy bias. In the Table the subsamplesare listed in groups. The first three groups indicate subsamplesin the three redshift bins. The last group indicates subsamplesthat are designed to match the luminosity cuts performed by K11(M B = − . − z − h = . = − . − z − h ))and by M05 (M B = − . − h ). The most conservative cutM B = − . − z − h ) guarantees 90 % completeness outto z = h − Mpc , the number of such independentcells is N = , , z = [0 . , . , [0 . , . , [0 . , . We considered a suite of mock galaxy catalogues mimicking thereal PDR-1 VIPERS catalogue to assess our ability to measurethe mean biasing function and evaluate random and systematicerrors.We used two di ff erent types of mock galaxy catalogues. Webased the bulk of our error analysis on the first mock galaxy cat-alogue, which is described in detail in de la Torre et al. (2013).In this set of mocks, synthetic galaxies are obtained by applyingthe HOD technique to the dark matter halos extracted from theMultiDark N-body simulation (Prada et al. 2012)of a flat Λ CDMuniverse with ( Ω m , Ω Λ , Ω b , h , n , σ ) = (0.27; 0.73; 0.0469; 0.7;0.95; 0.82). Since the resolution of the parent simulation wastoo poor to simulate galaxies in the magnitude range sampledby VIPERS, de la Torre & Peacock (2013) applied an originaltechnique to resample the halo field to generate sub-resolutionhalos down to a mass of M = h − M (cid:12) . These halos wereHOD populated with mock galaxies by tuning the free parame-ters to match the spatial 2-point correlation function of VIPERSgalaxies (de la Torre et al. 2013). Once populated with HOD Table 1.
VIPERS subsamples. z -range M B - cut n VIPERS n mock M B − h ) 10 − h Mpc − − h Mpc − − . − z − . − z − . − z − . − z − . − z − . − z − . − z − . − z − . − z − . − z − . − z − . − z − . Notes.
Col. 1: redshift range. Col. 2: B-band magnitude cut (computedfor h = Parent mock catalogue the number density is afactor ∼ . ∼
40 % more objectson average.
Fig. 1.
Luminosity selection as a function of redshift. The black dotsshow the W1 and W4 VIPERS galaxies (with spectroscopic redshiftflag between 2 and 9.5). Yellow lines represent the principal magnitudecuts applied in every redshift bin. The green line represents the cut M = − . − z made to compare our results to those of K11. galaxies, the various outputs were rearranged to obtain 26 and31 independent light cones mimicking the W1 and W4 fieldsof VIPERS and their geometry, respectively. In our analysis weconsidered 26 W1 + W4 mock samples. They constitute our setof
Parent mock catalogues, as opposed to the
Realistic mock
Article number, page 4 of 22. Di Porto et al.: The VIMOS Public Extragalactic Redshift Survey (VIPERS) Measuring non-linear galaxy bias at z ∼ . catalogues that we obtain from the Parent set by applying thevarious selection e ff ects (VIPERS footprint mask besides TSR,SSR, and CSR) and by adding Gaussian errors to the redshiftsto mimic the random error in the measured spectroscopic red-shifts. The mock catalogues were built assuming a constant SSRwhereas, as we pointed out, this is a declining function of theapparent magnitude. However, the dependence is weak and onlya ff ects faint objects, i.e. preferentially objects at large redshifts.For this reason we decided to explicitly include this dependenceby selectively removing objects, starting from the faintest andmoving towards brighter objects until we match the observedSSR(m) (Guzzo et al. 2014).The average galaxy number densities in the mocks are listedin Column 4 of Table 1. For z ≤ . z = [0 . , . R = h − Mpc , the second-order statistics of simulated galaxies andthe variance of the galaxy density field are underestimated by ∼
10 % (Bel et al. 2014). Therefore, to check the robustness ofour bias estimate to the galaxy model used to generate the mockcatalogues and to the underlying cosmological model, we con-sidered a second set of mocks. These were obtained from theMillennium N -body simulation (Springel et al. 2005) of a flat Λ CDM universe with ( Ω m , Ω Λ , Ω b , h , n , σ ) = (0.25; 0.75; 0.045;0.73; 1.00; 0.9) and using the semi-analytic technique of De Lu-cia & Blaizot (2007), an alternative to the HOD. As a result ofthe limited size of the computational box, it was possible to cre-ate light cones with an angular size of 7 × , i.e. smallerthan the individual W1 and W4 fields. Overall, we considered26 +
26 reduced versions of the W1 + W4 fields. From these lightcones we created a corresponding number of
Realistic mock cat-alogues.Robustness tests that involve both types of mock catalogueswere restricted to a limited number of samples (one for eachredshift bin). In these tests we simply compared the errors in thebias estimates after accounting for the larger cosmic variance inthe Millennium mocks. Since these robustness tests turned out tobe successful in the sense that errors estimated with the two setsof mocks turned out to be consistent with each other, we do notmention these mocks again and, for the rest of the paper, fullyrely on the error estimates obtained from the HOD mocks.
3. Theoretical background
In this section we briefly describe the formalism proposed byDekel & Lahav (1999) and the method that we use to estimatebias from galaxy counts. The key step is the procedure to esti-mate the galaxy PDF, P ( δ g ), from the measured probability ofgalaxy counts in cells, P ( N g ). We review some of the techniquesproposed to perform this crucial step and describe in detail thetechnique used in this work. Dekel & Lahav (1999) proposed a probabilistic approach togalaxy bias in which non-linearity and stochasticity are treatedindependently. In this framework, galaxy bias is described bythe conditional probability of galaxy over-density, δ g , given themass over-density δ : P ( δ g | δ ). Both quantities are smoothed onthe same scale and treated as random fields. If biasing is a localprocess then P ( δ g | δ ) fully characterises galaxy bias. Key quanti-ties formed from the conditional probability are the mean biasingfunction b ( δ ) δ ≡ (cid:104) δ g | δ (cid:105) = (cid:90) P ( δ g | δ ) δ g d δ g , (1)and its non-trivial second-order momentsˆ b ≡ (cid:104) b ( δ ) δ (cid:105) σ ˜ b ≡ (cid:104) ( b ( δ ) δ ) (cid:105) σ , (2)where σ ≡ (cid:104) δ (cid:105) is the variance of the mass over-density fieldon the scale of smoothing. The quantity ˆ b represents the slopeof the linear regression of δ g against δ and is the natural gen-eralisation of the linear bias parameter. The ratio ˜ b / ˆ b quanti-fies the deviation of the mean biasing function from a straightline. It measures the non-linearity of the mean biasing relationand, in realistic cases, is close to unity. In the limit of linear anddeterministic bias, the two moments ˆ b and ˜ b coincide with the(constant) mean biasing function b ( δ ) = b LIN , where b LIN is thefamiliar linear bias parameter. We note that ˆ b is sensitive to themass variance and scales as ˆ b ∝ σ − . On the contrary, the mo-ments’ ratio is very insensitive to it, ˜ b / ˆ b ∝ σ . (Sigad et al.2000). These scaling relations are used in Section 5 to compareresults obtained assuming di ff erent values of σ . There are otheruseful parameters related to galaxy bias that can be measuredfrom the data. One is the ratio of variances b var ≡ ( σ g /σ ) inwhich σ g is measured from counts in cells and σ depends on theassumed cosmological model. Another quantity is the inverse re-gression of δ over δ g , b inv ≡ σ g / (cid:104) δ g δ (cid:105) that requires an estimateof the galaxy and the mass density fields (Sigad et al. 1998). Inthe case of non-linear deterministic bias these quantities di ff erfrom ˆ b . Specifically, if the non-linearity parameter ˜ b / ˆ b is larger(smaller) than unity then they are biased high (low) with respectto ˆ b (Dekel & Lahav 1999).In this paper we focus on the ˆ b parameter, a choice that al-lows us to compare our results with those of K11 (but not withM05, in which the focus is instead on ˜ b ). Fortunately, as we shallsee, the small degree of non-linearity makes these two choicesalmost equivalent.If bias is deterministic, then it is fully characterised by themean biasing function b ( δ ) δ . However, we do not expect this tobe the case since galaxy formation and evolution are regulatedby complex physical processes that are not solely determined bythe local mass density. Therefore, for a given value of δ there is awhole distribution of δ g about the mean b ( δ ) δ . This scatter, oftenreferred to as bias stochasticity, is contributed by two sources:shot noise due to the discrete sampling of a continuous underly-ing density field and those astrophysical processes relevant to theformation and evolution of galaxies that do not depend (solely)on the local mass density.Previous studies (Branchini 2001; Marinoni et al. 2005; Vielet al. 2005; Kovaˇc et al. 2011) that, like this one, used the galaxy1-point PDF to recover the biasing function ignored the impactof stochasticity and assumed a deterministic bias. We aim to im-prove the accuracy of the bias estimator by taking bias stochas-ticity into account and we do this by assuming that shot noise is Article number, page 5 of 22 & A proofs: manuscript no. arXiv_revised the only source of stochasticity. This simplifying assumption canbe justified theoretically by both numerical and analytic argu-ments. Numerical experiments in which semi-analytic galaxiesare used to probe the mass density field in samples mimickingSDSS (Szapudi & Pan 2004) (see Figs. 11 and 16) and 2MRS(Nusser et al. 2014) (see Fig. 1), i.e. two surveys with galaxynumber densities similar to that of VIPERS, do indeed show thatshot noise is the dominant source of scatter. More specifically,Poisson noise accounts for the scatter in the δ g versus δ relationexcept at large over-density where the relation is over-dispersed.Analytic arguments in the framework of the halo model also con-firm that the main source of stochasticity is shot noise with thehalo-halo scatter providing a significant contribution for faint ob-jects alone (Cacciato et al. 2012b). Assessing the impact of thisshot noise only assumption is not simple, but some argumentscan be made to quantify the systematic e ff ect of underestimatingstochasticity.An upper limit can be obtained when stochasticity is ignoredaltogether. In the case of linear and stochastic bias, for example,˜ b and ˆ b would be equal whereas b inv would be systematicallylarger by about 10% (Somerville et al. 2000). The more realis-tic case of a non-linear and stochastic bias was considered bySigad et al. (2000) using numerical simulations again. In thiscase, the e ff ect of ignoring stochasticity is that of overestimat-ing both ˜ b and ˆ b . The amplitude of the e ff ect depends on boththe cosmological model assumed and the scale considered. Toobtain estimates relevant to our analysis we repeated the Sigadet al. (2000) test in Section 4.1. The results, which we anticipatehere, indicate that ˜ b and ˆ b are overestimated by 8(4)% on a scaleof 4(8) h − Mpc . As for the ratio, ˜ b / ˆ b we also confirm that it isremarkably insensitive to stochasticity and, as expected, to themodel adopted (Sigad et al. 2000).Analyses of the datasets may also constrain the size of thee ff ect. Galaxy clustering, higher order statistics, or gravitationallensing generally indicate that galaxy bias cannot be linear anddeterministic. However, as we anticipated in the introduction, itis not possible to disentangle the e ff ects induced by non-linearityand stochasticity, except for the case of relative bias betweentwo types of tracers. With respect to this, the largest stochastic-ity σ b / ˆ b = .
44 so far was measured by Wild et al. (2005). Ifignored, this would induce a systematic error of ∼
20% on therelative ˆ b moments.Overall, the variety of evidenceindicates that if stochasticityis ignored then σ b and ˆ b are overestimated by 10-20 %, whereastheir ratio is una ff ected. However, we stress that in our workstochasticity is, at least in large part, taken into account. There-fore, we expect that our assumption that shot noise is the onlysource of bias stochasticity generates systematic errors well be-low the 10 % level. b ( δ ) δ Under the hypothesis that bias is deterministic and monotonicthe mean biasing function, b ( δ ) δ , can be estimated by compar-ing the PDFs of the mass and of the galaxy over-density. We let C ( δ ) ≡ P ( > δ ) and C g ( δ g ) ≡ P ( > δ g ) be the cumulative probabil-ity distribution functions [CDFs] obtained by integrating the twoPDFs. Monotonicity guarantees that the ranking of the fluctua-tions δ and δ g is preserved and b ( δ ) δ can be obtained by equatingthe two CDFs at the same percentile, b ( δ ) δ = C − g ( C ( δ )) , (3)where C − g indicates the inverse function of C g . Equation 3 provides a practical recipe to estimate galaxy biasfrom observed counts in cells of a given size. It requires threeingredients: the galaxy over-density δ g , its PDF, and that of δ . δ g can be estimated from galaxy counts in cell, N g as1 + δ g = N g / (cid:104) N g (cid:105) , (4)where (cid:104) N g (cid:105) represents mean over all counts. From Eq. 4 one canform the galaxy PDF, P ( δ g ) and the count probability P ( N g ). Thebiasing function can then be obtained by comparing C g ( δ g ) witha model C ( δ ).This simple bias estimator has been used by several authors(Sigad et al. 2000; Branchini 2001; Marinoni et al. 2005; Vielet al. 2005; Kovaˇc et al. 2011). It is potentially a ff ected by sev-eral error sources that should be systematically investigated. Thefirst error source is shot noise that a ff ects the estimate of δ g from N g . Shot noise induces stochasticity in the bias relation in con-trast with the hypothesis of deterministic bias. Stochasticity af-fects the estimate of b ( δ ) δ from Equation 3, especially at largevalues of δ g , where the CDF flattens and the evaluation of the in-verse function C − g becomes noisy. A second issue is the massPDF for which no simple theoretical model is available. Thelast error source is redshift distortions. Galaxy over-densities arecomputed using the redshift of the objects rather than distances.This induces systematic di ff erences between densities evaluatedin real and redshift space (Kaiser 1987).All these issues potentially a ff ect the estimate of galaxy biasand should be properly quantified and accounted for. In the nextsection, we review some existing estimators designed to min-imise the impact of the shot noise and propose a new estimatorthat we apply in this paper. We investigate the performance ofthis new strategy in Section 4. P ( N g ) to P ( δ g ) ... The probability of galaxy counts, P ( N g ), can be expressed as P ( N g ) = (cid:90) + ∞− P ( δ g ) P ( N g | δ g ) d δ g , (5)where the conditional probability function P ( N g | δ g ) specifies theway in which discrete galaxies sample the underlying, continu-ous field. The common assumption that galaxies are a local Pois-son process implies that P ( N g | δ g ) = (cid:104) (cid:104) N g (cid:105) (1 + δ g ) (cid:105) N g e −(cid:104) N g (cid:105) (1 + δ g ) N g ! . (6)The Poisson model provides a good match to numerical exper-iments except at large densities where a negative binomial dis-tribution seems to provide a better fit (Sheth 1995; Somervilleet al. 2001; Casas-Miranda et al. 2002). In this work we adopt thePoisson model. However, di ff erent forms for P ( N g | δ g ) could beconsidered as well.The following strategies have been proposed to estimate P ( δ g ) from P ( N g ) using Equation 5: – Richardson-Lucy deconvolution . Szapudi & Pan (2004) pro-posed this iterative, non-parametric method to reconstruct P ( δ g ) by comparing the observed P ( N g ) to that computedfrom Eq. 5 at each step of the iteration, starting from an ini-tial guess for P ( δ g ). – Skewed lognormal model fit . This parametric method wasalso implemented by Szapudi & Pan (2004). In this approachone assumes a skewed lognormal form for P ( δ g ) and then de-termines the four free parameters of the model by minimisingthe di ff erence between Eq. 5 and the observed P ( N g ). Article number, page 6 of 22. Di Porto et al.: The VIMOS Public Extragalactic Redshift Survey (VIPERS) Measuring non-linear galaxy bias at z ∼ . – Gamma expansion [ Γ E ]. Among the various forms proposedto model the galaxy PDF, the Gamma expansion, defined byexpanding the Gamma distribution on a basis of Laguerrepolynomials (Mustapha & Dimitrakopoulos 2010) capturesthe essential features of the galaxy density field. The expan-sion coe ffi cients directly depend on the moments of the ob-served counts. Because of this, the full shape of the galaxyPDF can be recovered directly from the observed P ( N g ) withno need to integrate Eq. 5.Szapudi & Pan (2004) have tested the ability of the first twomethods in reconstructing the PDF of halos and mock galax-ies obtained from N -body simulations. They showed that a suc-cessful reconstruction can be obtained when the sampling is (cid:104) N g (cid:105) ≥ .
1; safely a factor 3 smaller than the smallest meangalaxy density in our VIPERS subsamples. Bel et al. (2016)extensively tested the Γ E -method and showed, using the samemock catalogues as in this paper, that this method reconstructsthe PDF of a VIPERS-like galaxy distribution with an accuracythat is superior to that of the other methods. This comes at theprice of discarding counts in cells that overlap the observed ar-eas by less than 60 %, which is a constraint that further reducesdeviations from the Poisson sampling hypothesis.To illustrate the performance of the Γ E -method we plot, inFigure 2, the galaxy PDFs Γ E -reconstructed from the 26 Re-alistic mock VIPERS subsamples with galaxies brighter than M B = − . − z − log ( h ) in the range z = [0 . , . σ scatter. The scatter for cells of R = h − Mpc islarger than for R = h − Mpc and is driven by the limited num-ber of independent cells rather than sparse sampling.The reconstruction is compared with the “reference” PDF(solid, red line) obtained by averaging over the PDFs recon-structed, with the same Γ E method, from the Parent mock cata-logues. We regard this as the “reference” PDF since, as shown bySzapudi & Pan (2004) and checked by us, when the sampling isdense, all the above reconstruction methods recover the PDF ofthe mass, P ( N g ) and the mean biasing function very accurately.In the plot we show P (1 + δ g )(1 + δ g ) to highlight the low- andhigh-density tails, where the reconstruction is more challenging.The reconstructed PDF underestimates the reference PDF in thelow- and high-density tails and overestimates it at δ ∼
0. Sys-tematic deviations in the low- and high-density tails are to be ex-pected since the probability of finding halos, and therefore mockgalaxies, in these regimes significantly deviates from the proba-bility expected for a Poisson distribution. However, these di ff er-ences are well within the 1- σ uncertainty strip as shown in thebottom panels of each plot.The Γ E method used to reconstruct the galaxy PDF from dis-crete counts is implemented as follows: – We consider as the input dataset one of the volume-limited,luminosity complete subsamples listed in Table 1. The po-sition of each object in the catalogue is specified in redshiftspace, i.e. by its angular position and measured spectroscopicredshift. – Spherical cells are thrown at random positions within the sur-veyed region. We consider cells with radii R = , , and 8 h − Mpc . The smallest radius is set to guarantee (cid:104) N g (cid:105) ≥ . P ( N g ) at large N g . We only consider cells that overlapby more than 60 % with the observed areas. This constraintreduces deviations from Poisson statistics (Bel & Marinoni2014). Counts in the partially overlapping cells are weighted Fig. 2.
Reconstructed PDF of the mock VIPERS galaxies measured incells of R = h − Mpc (top) and R = h − Mpc (bottom). The bluesolid curve represents the reference galaxy PDF obtained by averag-ing over the PDFs reconstructed from the
Parent mocks using the Γ E method. The blue dashed curve shows the average PDF reconstructedfrom the Realistic mocks using the same method. The blue shaded re-gion represents the 1- σ scatter among the 26 Realistic mocks. We plot P (1 + δ g )(1 + δ g ) to highlight the performance of the reconstruction athigh and low over-densities. We note the di ff erent Y-ranges in the twopanels. The bottom panels in each plot show the di ff erence ∆ p betweenthe reconstructed and reference PDFs in units of the random error σ p .Horizontal, dashed lines indicate systematic errors equal to 1- σ p ran-dom uncertainties. by the fraction f of the surveyed volume in the cell: N g / f .The probability function P ( N g ) is then computed from thecounts frequency distribution. – We use the measured P ( N g ) and its moments to model thegalaxy PDF with the Γ E method that we compute using allfactorial moments up to the sixth order. P ( δ g ) to b ( δ ) δ . To estimate the mean biasing function from the galaxy PDF, wesolve Equation 3. To do so, we assume that shot noise is themain source of stochasticity and that a reliable model for themass PDF is available. Despite its conceptual simplicity, thisprocedure requires several non-trivial steps that we describe be-low. The uncertainties introduced in each step are estimated inthe next section. The procedure is as follows: – We start from the galaxy PDF estimated from the measured P ( N g ), as described in the previous section. – We assume a model PDF for the mass density field in red-shift space. Rather than adopting some approximated, ana-lytic model, we measure the mass PDF directly from a darkmatter only N -body simulation with the same characteris-tics and cosmological model as the Millennium run (Springel Article number, page 7 of 22 & A proofs: manuscript no. arXiv_revised et al. 2005), thatis not based on the same model used to buildthe HOD-mock VIPERS catalogues. The use of an incorrectmass PDF is yet another possible source of systematic er-rors that we quantify in Section 4. However, this error is ex-pected to be small since ˆ b and ˜ b are mainly sensitive to σ and their ratio is largely independent of the underlying cos-mology (Sigad et al. 2000). – After computing the cumulative distribution function fromthe mass and galaxy PDFs, we use Eq. 3 to estimate the meanbiasing function. – We determine the maximum over-density δ MAX at which thereconstructed mean biasing function can be considered re-liable. To estimate δ MAX we compare the measured P ( N g )with the estimated P ( N g ) following the procedure describedin Section 4.4.4 . – We estimate the second-order moments ˆ b and ˜ b and their ra-tio by integrating over all δ up to δ MAX ˆ b = σ − (cid:90) δ MAX − b ( δ ) δ P ( δ ) d δ , ˜ b = σ − (cid:90) δ MAX − ( b ( δ ) δ ) P ( δ ) d δ . (7)and test the robustness of the result with respect to the choiceof δ MAX .
4. Error sources
In this Section we review all possible sources of uncertainty thatmight a ff ect the recovery of the biasing function and assess theiramplitude using mock catalogues. In this process we need toconsider a reference biasing function to compare with the resultsof the reconstruction. This could be estimated directly from thedistribution of the dark matter particles and mock galaxies withinthe simulation box. However, we use the mean biasing func-tion obtained from the Parent mocks as reference. We justifythis choice as follows. First, Szapudi & Pan (2004) showed thatwhen the sampling is dense both the Richardson-Lucy and theskewed lognormal fit methods recover the mean biasing func-tion with high accuracy. Second, in Section 3.3 we found thatwhen the sampling is dense the Γ E method accurately recoversthe mean biasing function in the Parent mocks.
Most of the previous estimates of the mean biasing function didnot attempt to account for shot noise directly. This choice canhamper the recovery of b ( δ ) δ when the sampling is sparse. Toestimate errors induced by ignoring shot noise and quantify thebenefit of using the Γ E method we compared the biasing func-tions reconstructed using both procedures. The result of this testis shown in Figure 3. The red curve represents the reference bi-asing function obtained by averaging over the Parent mocks. Ineach mock the biasing function was estimated from the galaxyPDF using the Γ E method. The blue dashed curve represents thesame quantity estimated from the 26 Realistic mocks using the Γ E method. The blue band represents the 2- σ scatter. For nega-tive values of δ g the reconstructed biasing function is below thereference biasing function, but the trend is reversed for δ g > σ scatter (horizontal dashed line in the bottom sub-panels).On the contrary, the biasing function obtained from the “direct” Fig. 3.
Mean biasing function of mock VIPERS galaxies computedfrom counts in cells of R = h − Mpc (bottom panel) and R = h − Mpc (top panel). The magnitude cut and redshift range of the mockVIPERS subsample, indicated in the plot, are the same as Figure 2.Solid red curve: reference biasing function obtained from the
Par-ent mock catalogues. Blue dashed curve and blue-shaded region: av-erage value and 2- σ scatter of the biasing function reconstructed fromthe Realistic mocks using the Γ E method.Brown dot-dashed curve andorange-shaded band: average value and 2- σ scatter of the biasing func-tion reconstructed from the Realistic mocks using a ’direct’ estimate ofthe galaxy PDF. Bottom sub-panels: di ff erence ∆ p between the recon-structed and reference PDFs in units of the random error σ p . Dashedlines indicate systematic errors equal to 1- σ p random errors. estimate of δ g (brown dot-dashed curve and the corresponding2- σ scatter, orange band) is significantly di ff erent from the ref-erence function. The discrepancy increases at low densities andfor small spheres, i.e. when the counts per cell decrease and theshot noise is large. Another key ingredient of the mass reconstruction is the massPDF. In principle this quantity could be obtained from galaxypeculiar velocities or gravitational lensing. However, in practice,errors are large and would need to be averaged out over scalesmuch larger than the size of the cells considered here. For thisreason we need to rely on theoretical modelling. Coles & Jones(1991) and Kofman et al. (1994) found that the mass PDF can beapproximated by a lognormal distribution and this model was in-deed adopted in previous reconstructions of the biasing function(e.g. M05, Wild et al. (2005), K11).However, the lognormal approximation is known to performpoorly in the high- and low-density tails and for certain spec-tra of density fluctuations. An improvement over the lognor-mal model is represented by the skewed lognormal distribution(Colombi 1994). This model proved to be an excellent approx-
Article number, page 8 of 22. Di Porto et al.: The VIMOS Public Extragalactic Redshift Survey (VIPERS) Measuring non-linear galaxy bias at z ∼ . imation to the PDF of the dark matter measured from N -bodyexperiments over a wide range of scales and of over-densities(Ueda & Yokoyama 1996). The impact of adopting either modelfor the mass PDF can be appreciated in Fig. 4. The solid redcurves represent the same biasing functions shown in Fig. 3 ob-tained from the galaxy PDFs of the Parent mocks and from amass PDF obtained directly from an N -body simulation with thesame cosmological parameter and size as the Millennium sim-ulation using the output corresponding to z = .
8. As in theprevious test, we consider the red solid curve as the referencebiasing function. The brown dot-dashed curve shows the meanbiasing function reconstructed assuming a lognormal model forthe mass PDF, i.e. a lognormal fit to the PDF measured fromthe N -body simulation. The curve represents the average among26 mocks and the orange band is the 2- σ scatter. For R = h − Mpc , the biasing function is systematically below the refer-ence whereas for R = h − Mpc is above the reference at bothhigh and low densities. The mismatch is very large and signifi-cantly exceeds the 1- σ scatter (bottom sub-panels). The skewedlognormal model (blue dashed curve) performs significantly bet-ter with di ff erences well below 1- σ except at very negative δ values.We conclude that, for the practical purpose of reconstruct-ing galaxy bias, the mass PDF measured from N -body data anda skewed lognormal fit perform equally well. The main advan-tage of using the latter would be the possibility of determiningthe four parameters of the fit experimentally. Since, however, theparameters are poorly constrained by observations, we decidedto adopt the mass PDFs from N -body simulations. This choiceintroduces a dependence on the cosmological model, however,that is mostly captured by one single parameter, σ , for which ˆ b and ˜ b exhibit a linear dependent. With respect to this, the massPDF used to obtain the biasing functions in Figure 4 is not the true mass PDF since it is obtained from an N -body simulationthat uses a cosmological model that is di ff erent from the modelused to produce the mock catalogues. We did this on purposeto mimic the case of the real analysis for which the underlyingcosmological model is not known. Galaxy positions are measured in redshift space, i.e. usingthe observed redshift to estimate the distance of the objects.The presence of peculiar velocities induces apparent radialanisotropies in the spatial distribution of galaxies and, as a con-sequence, modifies the local density estimate and their PDF(Kaiser 1987). However, our goal is to reconstruct the mean bi-asing function in real space without redshift distortions. Consid-ering the di ffi culties and uncertainties in determining the galaxyPDF in real space, one could instead consider the galaxy andmass PDFs both measured in redshift space under the assump-tion that peculiar velocities induce similar distortions in the spa-tial distribution of both dark matter and galaxies so that theycancel out when estimating the mean biasing relation from Eq. 3.In the limit of the Gaussian field, linear perturbation theory andno velocity bias, the cancelation is exact. However, non-lineare ff ects have a di ff erent impact on the mass and galaxy densityfields and induce di ff erent distortions in their respective PDFs.To assess the impact of these e ff ects we compared the mean bi-asing function of mock galaxies reconstructed from PDFs esti-mated in real and redshift space.The results are shown in Figure 5. The solid red curve repre-sents the mean biasing function of galaxies in the Realistic mock
Fig. 4.
Solid red curve: reference mean biasing function of Fig. 3 com-puted using the mass PDF from N -body simulations. Brown dot-dashedcurve and orange band: biasing function obtained using a lognormal fitto the mass PDF and 2- σ scatter from the mocks. Blue dashed curveand blue band: biasing function obtained using a skewed lognormal fitto the mass PDF and 2- σ scatter from the mocks. Bottom panels: dif-ference ∆ p between the reconstructed and reference PDFs in units of therandom error σ p . Dashed lines indicate systematic errors equals to 1- σ p random errors. catalogues estimated using the PDFs of galaxies and mass in realspace. The blue dashed line shows the same function estimatedin redshift space. Both curves are obtained by averaging over the26 mocks and the blue band represents the 2- σ scatter in red-shift space. The redshift space biasing function underestimatesthe true biasing function in low-density regions and overesti-mates it at high densities, i.e. in the presence of highly non-linearflows. The di ff erence is systematic but its amplitude is withinthe 2- σ random errors estimated by adding in quadrature thescatter among mocks in real and redshift space (bottom panelsin each plot). The biasing functions shown in Figure 5 repre-sents a demanding test in which we consider the smallest cellsof 4 h − Mpc where deviations from linear motions are larger.The discrepancy decreases if the size of the cell increases.These systematic di ff erences induce errors in the estimatedmoments ˆ b and ˜ b . To quantify the e ff ect we computed the mo-ments as a function of δ (i.e. by varying δ MAX in Eq. 7) both inreal and redshift space. The results are shown in Figure 6. Theplots show the per cent di ff erence between the moments mea-sured in redshift versus real space. The panels and curves referto the same redshift bins and magnitude cuts as in Figure 5. Sys-tematic errors induced by redshift distortions are ∼ b andfor ˜ b (not shown) and one order of magnitude smaller for ˜ b / ˆ b .They provide the main contribution to the total systematic errorslisted in Table 2 and are of the same size, although somewhatsmaller than the random errors. Article number, page 9 of 22 & A proofs: manuscript no. arXiv_revised
Fig. 5.
Mean biasing function estimated in real space (solid, red curve)and redshift space (dashed blue curve and its 2- σ uncertainty band).Counts are performed in spherical cells with a radius of 4 h − Mpc . Theluminosity cut and the redshift range is indicated in each panel. Thewidth of each band represents the scatter among mocks. In the bottompart of each plot we show the di ff erence ∆ p between the reconstructedand reference PDFs in units of σ TOT , where σ TOT accounts for the rms scatter in both the real- and redshift-space mocks. Dashed lines indicatewhere systematic errors equal to 1- σ TOT random errors.
Considering the absolute and relative size of these errors, weperform our analysis in redshift space. Di ff erent sources of errors a ff ect the recovery of the biasingfunction. One error source is cosmic variance due to the finitevolume of the sample. This source dominates the error budget ofthe M05 and K11 analyses.The other sources are the shot noise induced by discrete sam-pling and the limited number of independent cells used to buildthe probability of galaxy counts P ( N g ). In the VIPERS survey,which is based on a single-pass strategy, sparse sampling is moreof an issue than in the M05 and K11 cases. The cumulative ef-fect of the single pass strategy and colour preselection reducesthe sampling rate to ∼
35 % on average with significant vari-ations across quadrants. The survey geometry, characterised bygaps and missing quadrants that occupy ∼
25 % of the would-becontinuous field, further dilutes the sampling (we consider cellsthat overlap up to 40 % with unobserved regions) and limits thenumber of independent cells that can be accommodated withinthe survey. Our PDF reconstruction strategy is designed to min-imise these e ff ects that, nevertheless, induce random and system-atic errors that need to be estimated. We do this with the help ofboth the Parent and
Realistic mock catalogues. The former pro-vide the reference mean biasing function. Errors are estimated
Fig. 6.
Bottom panel: per cent di ff erence between the ˆ b values esti-mated in redshift space and in real space using spherical cells with aradius of 4 h − Mpc as a function of 1 + δ MAX (see Eq. 7). The di ff erentcurves refer to di ff erent redshift shells and magnitude cuts, as indicatedin the plot. Upper panel: per cent di ff erence in the estimated non-linearparameter ˜ b / ˆ b . Vertical dashed lines are drawn in correspondence tothe δ MAX values at which systematic errors are computed and listed inTable 2. by comparing the bias function reconstructed from the
Realistic mocks to the reference mocks. The procedure is detailed belowand the estimated errors are listed in Table 2.
To estimate the total random error σ RND , we proceed as follows.We reconstruct the mean biasing function in each of the
Real-istic mock catalogues, compute the average over the 26 mocksand, finally, estimate the scatter around the mean. The rms scat-ter provides an estimate of the total random error. All sources ofuncertainties contribute to this error (e.g. cosmic variance, shotnoise, and limited number of cells), which may a ff ect the recov-ery of the biasing function. Total random errors for ˆ b and ˜ b / ˆ b arelisted in columns 6 and 10 of Table 2, respectively. To assess the contribution of the cosmic variance, σ CV , to theerror budget, we proceed as for the estimate of total random er-rors using, however, the Parent catalogues rather than the
Realis-tic catalogues. Since errors in the bias reconstruction are mainlydriven by discrete sampling and in the
Parent catalogues thesampling is dense, the rms scatter among these mocks is domi-nated by cosmic variance. Cosmic variance contributions to er-rors in ˆ b and ˜ b / ˆ b are shown in columns 7 and 11 of Table 2,respectively. It turns out that the contribution of the cosmic vari- Article number, page 10 of 22. Di Porto et al.: The VIMOS Public Extragalactic Redshift Survey (VIPERS) Measuring non-linear galaxy bias at z ∼ . ance is of the same order as that of the sparse sampling and, un-like in the case of M05 and K11, it does not dominate the errorbudgets. Following K11, we compute systematic errors, σ SYS , as the av-erage o ff set of the bias estimates in the Realistic and the
Parent catalogues, i.e. σ SYS = (cid:104) X Realistic − X Parent (cid:105) , where X is either ˆ b or ˜ b / ˆ b and the mean is over the 26 pairs of mocks. These sys-tematic errors are plotted in the bottom panels of Fig. 3 (blue,dashed curves). Their amplitudes at δ MAX are listed in columns8 and 12 of Table 2. These systematic errors are of the sameorder as the random errors and as the errors induced by redshiftdistortions discussed in Section 4.3. These systematic errors in-clude those induced by redshift distortions. The fact that they areof the same order as those discussed in Section 4.3 indicates thatthey dominate the budget of systematic errors.Our systematic errors are similar to those estimated by K11(upper part of their Table 2) from the z COSMOS sample, whichis significantly small than VIPERS. As these errors do not seemto depend on the volume of the survey, we conclude that they canbe regarded as genuinely systematic. Systematic errors on ˆ b areon average positive, meaning that the mean slope of the recon-structed biasing function typically overestimates the true biasingfunction. For the non-linear bias, parameter systematic errors arepreferentially negative, indicating that the reconstruction proce-dure has the tendency to underestimate the non-linearity of thebiasing function. δ MAX
Our bias estimator becomes progressively less reliable as thedensity increases, for two reasons: first, the numerical solution toEq. 3 becomes unstable when the cumulative distribution func-tions approach unity, i.e. in correspondence of the high peaks ofthe mass and galaxy density fields. In this regime, small errors inthe estimated galaxy PDF propagate into large uncertainties in δ ;second, as anticipated in the previous Section, the scatter in the δ g versus δ relation is larger than Poisson. Our assumption thatEq. 6 is valid at all δ leads to underestimating the high-densitytail of the galaxy PDF and, consequently, the value of ˆ b .Our mock catalogues can be used to estimate the first type oferror, but cannot fully account for the second type of error sinceour mock galaxies are sampled from dark matter halos assum-ing Poisson statistics. We therefore take the alternative route ofreducing the impact of deviations from Poisson statistics at highdensities. We do this by setting a sensible maximum over den-sity value, δ MAX , at which we compute the bias moments. Thevalue of this threshold is computed as follows:1. We consider the di ff erence ∆ P between the ‘true’ P t ( N g )measured in the Realistic mock catalogues and the recon-structed P r ( N g ) estimated through Eq. 5.2. We search for the first N g value, N , at which ∆ P > σ P ,where σ P is the rms scatter in the mocks.3. We search for the first N g value, N , at which | ∆ P / P t ( N g ) | > . .
4. We take N MAX = Min [ N , N ] and compute the correspond-ing over-density in galaxy counts δ g , MAX = N MAX / (cid:104) N (cid:105) .5. We obtain the corresponding mass over-density δ MAX from δ g , MAX from the estimated mean biasing function. The largest over-density at which we search for a solution toEq. 3 is δ MAX , and this is also the over-density at which we es-timate the bias moments. This value is clearly model dependentsince it was estimated from the VIPERS mocks. An alternativeway of setting this threshold would be to look for wiggles in themean biasing function measured from real data, i.e. spuriousfeatures induced by instabilities in the reconstruction procedure.We found that this second criterion is less stringent as it produces δ MAX values larger than using mocks. We decided to adopt a con-servative approach and use the δ MAX thresholds estimated withthe first procedure.With this criterion we obtain di ff erent δ MAX for the di ff erentgalaxy subsamples considered in our analysis. This limits ourability to compare results. Since the value of δ MAX mainly de-pends on the radius of the cell, we use one single value for δ MAX for a given cell size, irrespective of the other parameters used todefine the subsample. These values, which are listed in Table 2,correspond to the minimum δ MAX among those computed for allsubsamples.All bias parameters presented in our work were computed atthese over-density values. To check the robustness of our resultsto δ MAX we also considered a second, less stringent thresholdobtained by taking the maximum value of δ MAX among those ofthe various subsamples for a given cell size. This second set of δ MAX that we denote as ¯ δ MAX , is also listed in Table 2 togetherwith the corresponding estimates for the bias moments (valuesin parenthesis).
5. Results
In this section we present the results of our analysis, focusing onthe dependence of the mean biasing function and its moments onvarious quantities. In Sections 5.1 and 5.2 we explore the bias de-pendence on magnitude and redshifts, respectively. In both caseswe fix the radius of the cells equal to 6 h − Mpc . The depen-dence on the cell size is investigated in Section 5.3. Results aresummarised in Section 5.4 and listed in Table 2.
The di ff erent solid curves in Figure 7 represent the mean bias-ing function of VIPERS galaxies reconstructed from counts incells of radius 6 h − Mpc for di ff erent magnitude cuts for threedi ff erent redshift shells (the three panels). We applied a smallhorizontal o ff set δ = .
015 to the curves to avoid overlappingerror bars. We plot (1 + δ ) in logarithmic units both to ease thecomparison with similar plots in the literature and to highlightdeviations from linearity in the low-density regions. Error barsrepresent the 2- σ random scatter computed from the Realistic mocks.The magnitude range that we are able to explore is set bycompeting constraints: the faint limit reflects the requirement ofmaximising the completeness of the sample whereas the brightlimit is set by requiring (cid:104) N g (cid:105) > . z = [0 . , .
7] itspans a range ∆ M B = . z = [0 . , . ∆ M B = . ff erentmagnitude cuts are well separated for δ g <
0. The separationreduces and then disappears with the redshift. This is not sur-prising since at z ≥ . R = [4 , Article number, page 11 of 22 & A proofs: manuscript no. arXiv_revised h − Mpc (see Table 2) and confirm the results obtained at lowerredshifts from galaxy clustering (e.g. Norberg et al. 2002; Ze-havi et al. 2005; Pollo et al. 2006; Coil et al. 2008; Skibba et al.2014; Arnalte-Mur et al. 2014; Marulli et al. 2013), gravitationallensing (e.g. Coupon et al. (2012)) and counts in cells (e.g. M05and K11).To further investigate galaxy bias in under-dense regions,we zoom into the δ < b value esti-mated at δ MAX , which is listed in Table 2. Since ˆ b only weaklydepends on the magnitude cut we only consider one represen-tative case per panel. The local slope of the biasing function isalways steeper than the best-fitting linear bias model. The hor-izontal, short-dashed line shows the δ g = − . δ TH , increases with the redshift and, to a lesser extent, with theluminosity. This trend, which was noticed by M05 and, with lesssignificance, by K11,has been interpreted as evidence that low-density regions are preferentially populated by low-luminositygalaxies. Also, the quantity δ TH has been regarded as the typi-cal mass over-density below which very few galaxies form.Figure 8 shows that galaxies can be found at mass over-densities well below δ TH . This low-density tail, together withthe steepness of the biasing function for δ > δ TH , shows thatthe biasing relation in the under-density region significantly de-viates from the linear prescription. Non-linearity increases whendecreasing the cell size. As we checked, for R = h − Mpc theslope of the biasing curves further increases well above δ TH . For R = h − Mpc , the di ff erence disappears and the two slopesstart to match. Still, the bias curves keep featuring a negative δ tail that cannot be matched by linear models.Figure 9 shows the second-order moment ˆ b (left panels) andthe ratio ˜ b / ˆ b (right panels) of the biasing functions shown inFig. 7. The same colour-code is used to indicate the magnitudecuts. Large filled symbols refer to measurements performed at δ MAX assuming σ = . ff set, smalleropen symbols refer to estimates performed at ¯ δ MAX . The valuesof the corresponding bias moments are listed in Table 2. Errorbars represent 1- σ total random uncertainties estimated from the Realistic mocks (see Table 2).In the left panels of Fig. 7, we notice that in the low red-shift bin, where the magnitude interval that we probe is larger, ˆ b increases with the luminosity. This dependence is much weakerfor z = [0 . , .
9] and completely absent at higher redshifts. Weshow results for cells of 6 h − Mpc . However, the same trend isalso seen for 4 and 8 h − Mpc .The right panels show the non-linear parameter ˜ b / ˆ b . Val-ues that di ff er from unity indicate deviations from linear bias(horizontal dashed line). A small but significant degree of non-linearity is present at all redshifts. We do not detect any signif-icant dependence on luminosity in any redshift bin and for anycell size.A common feature of the reconstructed mean biasing func-tions at z = [0 . , .
1] is the presence of some irregular behaviour(wiggling) at high over-densities. This is the typical fingerprintof an imperfect inversion (Eq. 3) discussed in Section 3.2 andone of the reasons for introducing the threshold δ MAX . Theseirregularities typically arise as a result of sampling rare, largeover-densities with a limited number of independent cells. Thee ff ect is most evident at large redshifts and for bright magnitudecuts, i.e. when the sampling is sparser. This a ff ects the shape ofthe reconstructed mean biasing function. However, the impact on Fig. 7.
Mean biasing function of VIPERS galaxies from counts in cellsof radius 6 h − Mpc as a function of the B-band magnitude cut in threeredshift ranges indicated in each panel. Curves with di ff erent coloursand line styles correspond to the di ff erent magnitude cuts indicated inupper panel. Error bars with matching colours represent the associated1- σ uncertainty intervals estimated from the mocks. A horizontal o ff set δ = .
015 was applied to avoid overlapping error bars. All biasingfunctions are plotted out to δ MAX . Fig. 8.
Zoom into the under-density range of Fig. 7. The horizontalshort-dashed line represents the over-density threshold δ g = − .
9. Thelong dashed line shows the linear biasing function δ g = ˆ b δ , for the ˆ b value corresponding to the M B − z − log ( h ) < . b LIN = σ rms scatter among the mocks. They areonly shown for the case M B − z − log ( h ) < . z ∼ . Fig. 9.
Second-order moments of the mean biasing functions shownin Fig. 7. Left panels: moment ˆ b . Right panels: non-linear parameter˜ b / ˆ b . The cell size is R = h − Mpc . Error bars indicate 1- σ scatterfrom the mocks. The redshift ranges and colour code are the same asin Fig. 7. Magnitude cuts are indicated in the plots. All values werecomputed assuming σ = .
9. The horizontal dashed line is plotted forreference and represents the case of no bias (left plot) and linear bias(right panels). Large filled symbols refer to measurements performedat δ MAX , and small, open symbols refer to estimates at ¯ δ MAX . the second moments ˆ b and ˜ b and, especially, ˜ b / ˆ b , is rather lim-ited. This is because bias moments are integral quantities (Equa-tion 7) weighted by the mass PDF, which peaks at δ ∼ b and ˜ b and furthersmoothed out when computing their ratio.Figure 10 demonstrates the validity of this conjecture. In theleft panels we show the values of ˆ b ( δ ) computed from equa-tion 7. Curves with di ff erent line styles refer to the di ff erentmagnitude cuts indicated in the plot. Error bars with matchingcolours indicate the 1- σ scatter from the mocks. In the interval z = [0 . , .
1] and for the brightest and sparsest sample, ˆ b ( δ ) flat-tens for δ >
3, i.e. well below δ MAX . Analogous considerationshold for the curve ˜ b / ˆ b ( δ ) shown in the right panels. These trendsare robust to the size of the cells. To explore the bias dependence on the redshift we set a mag-nitude cut M B = − . − z − h ) and estimated the meanbiasing function of brighter galaxies in the three redshift bins.This z -dependent magnitude cut is designed to account for lu-minosity evolution (Zucca et al. 2009), so that di ff erences in thegalaxy bias measured in the di ff erent z -bins can be interpretedas the result of a genuine evolution. The results of our analy-sis are shown in Figure 11. The plots are analogous to those ofFig. 7 and use the same symbols, colour scheme, and line style.However, we consider cells of di ff erent sizes in the three panels. Fig. 10.
Left: second-order moment ˆ b ( δ ) of the reconstructed meanbiasing functions shown in Fig 7. The cell size is R = h − Mpc . Dif-ferent line styles and colours indicate di ff erent luminosity cuts listed inthe plot. The redshift ranges and colour codes are the same as in Fig 7.Error bars represent the - σ scatter among the mocks. Right: similarplots showing the non-linear bias parameter ˜ b / ˆ b ( δ ). A horizontal o ff -set δ = .
015 was applied to avoid overlapping error bars. All curvesare plotted out to δ MAX . The biasing function shows little or no evolution in therange z = [0 . , . z = [0 . , . z = [0 . , . σ error bars.The red solid line, however, is separated fullyfrom the others,indicating that galaxy bias evolves significantly beyond z = . δ TH increases significantly with the red-shift, indicating that evolution shifts galaxy formation towardsregions of progressively lower density. At δ > ff ect ofevolution is that of increasing the slope of the biasing functionwith z . Since in this range the biasing is close to linear, an es-timate of b LIN would reveal a redshift evolution consistent withthat observed in several analyses, as detailed in Section 6. Thesame trend is evident in all panels, indicating that the bias evo-lution is similar in all explored scales.At high redshifts and for R = h − Mpc (bottom panel) thebiasing function is characterised by some irregularities at mod-erate values of δ . As pointed out, these have little impact on theestimated values of ˆ b and ˜ b / ˆ b . It is reassuring that these anoma-lies are only seen at high redshifts, confirming the fact that theyare induced by poor sampling of the counts probability. All thismakes us confident that bias evolution is a genuine feature.Figure 12 shows the values of ˆ b and ˜ b / ˆ b as a function of red-shift. The values were obtained by integrating the mean biasingfunctions in Figure 11 out to the values δ MAX and ¯ δ MAX (largeand small symbols, respectively). The corresponding values arelisted in Table 2. The colour code is the same as in Figure 11and is indicated in the plot. The mean slope of the curve, ˆ b (leftpanels), increases significantly beyond z = . Article number, page 13 of 22 & A proofs: manuscript no. arXiv_revised
Fig. 11.
Mean biasing function of VIPERS galaxies with M B < − . − z − h ) measured in di ff erent redshift bins, characterised by di ff erentcolours and line-styles, as indicated in the plot. Error bars represents the1- σ scatter in the mocks. The three panels refer to di ff erent cell sizeswith radii R = , , h − Mpc from top to bottom. A horizontal o ff set δ = .
015 was applied to avoid overlapping error bars. All curves areplotted out to δ MAX . little or no evolution at lower redshifts. This shows that the trendseen in Figure 11 is seen at all scales, indicating that the biasevolution at z > . B = − . − z − h )exhibits a small but significant degree of non-linearity at all red-shifts and scales explored in our analysis (right panels); this biasparameter, however, does not significantly evolve with redshift.These results are robust to the luminosity cut since they are alsofound for galaxies brighter than M B < − . − z − h ). In Fig. 13 we explore the dependence of the bias of VIPERSgalaxies on the radius of the cells down to a scale of 4 h − Mpc .In the plots we show the mean biasing function of VIPERSgalaxies brighter than M B = − . − z − h ) measured at R = , h − Mpc . Di ff erent scales are characterised bydi ff erent colours, as indicated in the plot. The panels show theresults in the three redshift shells. At negative over-density thecurves are remarkably similar, indicating that δ TH and the e ffi -ciency of galaxy formation do not depend on the scale in therange [4 , h − Mpc . At δ > b steadily increases with thecell radius, R , especially at high redshift. This trend may soundcounterintuitive; galaxies are expected to trace the mass with in-creasing accuracy on a larger scale and, consequently, galaxybias is expected to approach its linear value. This, however, oc-curs on scales much larger than those considered here (see e.g. Fig. 12.
Moments of the mean biasing functions shown in Fig. 11.Panels from top to bottom indicate cells of increasing size. Left panels:ˆ b . Right panels: ˜ b / ˆ b . Error bars indicate the 1- σ scatter from the mocks.All values were computed assuming σ = .
9. Large filled symbolsrefer to measurements performed at δ MAX , small open symbols refer toestimates at ¯ δ MAX . Fig. 13.
Mean biasing function of VIPERS galaxies with M B < − . − z − h ) computed from counts in cells with radii of 4 , , and 8 h − Mpc . Biasing functions at di ff erent scales are indicated with di ff er-ent colours and line styles, as indicated in the plots. Error bars representthe 2- σ rms scatter in the mocks. Di ff erent panels refer to di ff erent red-shift shells. An o ff set δ = .
015 was applied to avoid overlapping errorbars. Curves are plotted out to δ MAX .Article number, page 14 of 22. Di Porto et al.: The VIMOS Public Extragalactic Redshift Survey (VIPERS) Measuring non-linear galaxy bias at z ∼ . Fig. 14.
Moments of the mean biasing functions vs. the size of the cell.Panels from top to bottom refer to di ff erent redshift ranges indicated inthe plot. Left panels: ˆ b . Right panels: ˜ b / ˆ b . The parameters were com-puted assuming σ = .
9. Error bars indicate the 1- σ rms scatter fromthe mocks. Large, filled symbols refer to measurements performed at δ MAX , small open symbols refer to estimates at ¯ δ MAX . Wild et al. 2005). On the scales explored here the halo modelpredicts that the opposite trend should be observed (see e.g. Fig.4 of Zehavi et al. 2004). The reason is that in this range ofscales the contribution to galaxy clustering of the 1-halo term,which dominates on small scales, is comparable to that of the2-halo term, which dominates on large scales. The scale of thecrossover depends on galaxy type and redshift but it is expectedto be bracketed in the range probed by our analysis. This expla-nation is corroborated by the fact that the values of ˆ b measuredin the HOD mocks, designed following the halo model prescrip-tions, do show an increasing trend with the size of the cells.An increase of galaxy bias with the scale was already de-tected at lower redshifts from the analysis of galaxy clustering(Zehavi et al. 2005) and from weak lensing (Hoekstra et al. 2002;Simon et al. 2007). This is the first detection at relatively highredshift that exploits counts in cell statistics. A small, but signif-icant amount of non-linearity is detected at all redshifts. Unlikeˆ b , the non-linear parameter ˜ b / ˆ b seems to be scale independent.These results are robust to magnitude cut since similar trends forˆ b and ˜ b / ˆ b are also seen when one restricts the biasing analysis toobjects brighter than M B < − . − z − h ). We now summarise the results presented in this section. Overall,the biasing functions of the VIPERS subsample are in qualita-tive agreement with those of M05 and K11 with some intriguingdi ff erences. At moderate over-densities and out to δ MAX , our bi-asing functions are close to linear with a slope close to ˆ b ( δ MAX ).This is at variance with M05 and K11 whose biasing functionflattens at large δ , leading to an anti-bias signature. This fea-ture has been variously interpreted as evidence for quenching Fig. 15.
Bottom panel: comparison between ˆ b measured in variousmock subsamples and ˆ b measured in the VIPERS catalogue. Di ff er-ent symbols and colours refer to results obtained with cells of di ff erentsizes, as indicated in the plot. The subsamples were obtained by apply-ing the same magnitude and redshift cuts used in this section and indi-cated in Table 2. The error bars represent 1- σ scatter from the mocks.Top panel: comparison among the non-linear parameters ˜ b / ˆ b measuredin the mocks and in the real sub-catalogues .processes (Blanton et al. 2000), enhanced galaxy merging rate(Marinoni et al. 2005), and early galaxy formation (Yoshikawaet al. 2001) in high-density regions. We find a similar flatten-ing only if we push our analysis beyond δ MAX . However, its sta-tistical significance is less that 2 σ . A similar feature was alsodetected in simulations and interpreted as an artefact due to lim-itations of the bias estimator at high redshift (Sigad et al. 2000).Given the fact that all these works, including ours, use a similartechnique to measure galaxy bias, we suspect that the flatteningat large density is not a genuine e ff ect.At δ < b . Moreover, the galaxy den-sity remains positive below δ TH , indicating that galaxy formationis not entirely quenched even in very low-density environments.Table 2 lists the bias parameters measured in the VIPERSsubsamples of Table 1 together with random and systematic er-rors estimated from the mocks. We computed all of the param-eters by integrating the mean biasing function out to the value δ MAX listed in the Table. Altogether the results confirm the var-ious trends that we described in the previous sections: the valueof ˆ b increases with luminosity, scale, and with the redshift be-yond z = .
9. Deviations from linear biasing are small but typi-cally detected with significance larger than 1 σ . The non-linearbias parameter is, within the errors, independent of redshift, lu-minosity, or scale.We obtained errors from the VIPERS mock catalogues de-signed to match the 2-point statistics of real galaxies but nottheir abundance or their bias. One consequence of this is that Article number, page 15 of 22 & A proofs: manuscript no. arXiv_revised bright galaxies in the subsamples of the mocks are sparser thanthe real galaxies at z < . ff erent fromthe mock sample, then our error estimate would be a ff ected. Wecompared the values of ˆ b and ˜ b / ˆ b in the mock and in the realsamples to investigate this issue. The resulting scatter plots areshown in Figure 15. The di ff erent points represent the individualsubsamples considered in our analysis. Symbols with di ff erentcolours are used to highlight results obtained with di ff erent cellsizes. Error bars represent the rms scatter in the mocks. Most ofthe points deviates less than 2 σ from the expected value (blackdashed line), implying that our mocks are realistic and that ourerrors are indeed reliable.
6. Comparison with previous results
Several authors estimated the bias of galaxies in the same range, z = [0 . , . b LIN in this redshift range available in theliterature and compare them with our value of ˆ b .In these com-parisons all results were rescaled to the value σ = . In Figure 16 we plot the values of ˆ b and ˜ b / ˆ b obtained from ouranalysis as a function of redshift (filled and open red dots) andcompare these values to those obtained by M05 (green triangles)and K11 (blue squares) from counts in cells following a proce-dure similar to ours. We do not consider the results of the analy-ses of Simon et al. (2007) and Jullo et al. (2012) since these au-thors estimate the so-called correlation parameter that accountsfor both non-linearity and stochasticity.We only considered objects that, at a given redshift, spana similar range of magnitudes to avoid mixing evolution andluminosity dependence. For VIPERS we consider objects withM B < − . − z − h ). For zCOSMOS we consider objectsabove a similar cut-o ff , M B = − . − z − h ). For theVVDS-Deep sample, M05 use a redshift-independent luminos-ity threshold of M B = − . − h ) , which is comparable withthe above cut-o ff s in the range z = [0 . , . b and ˜ b / ˆ b shown in Fig-ure 16 were inferred from the published values of ˜ b and ˆ b / ˜ b .In addition, M05 do not provide the errors for ˜ b / ˆ b . The errorbars shown in the plot were extrapolated from the errors on ˆ b under the assumption that the ratio of the errors on ˆ b and thoseon ˜ b / ˆ b are the same for the two datasets. The comparison be-tween zCOSMOS and VIPERS shows that this assumption isapproximately valid. The zCOSMOS points are plotted at the centre of their redshift bins. In the VVDS case we added an o ff -set ∆ z = + .
02 to avoid overlapping. Finally, we restrict ourcomparison to counts in cells of R = h − Mpc since this is theminimum cell size considered by K11 and the only one commonto the three analyses.The values ˆ b of zCOSMOS galaxies (bottom panel) are inagreement with those of VIPERS galaxies. These values increasewith redshift in both cases. This trend is more evident in thezCOSMOS case, while for VIPERS the evolution is detectedonly with a significance of ∼ σ only. Our results do not matchthose of M05 at z = . , where the two samples overlap. Thesignificance of the discrepancy, however, is about 1 σ . A sim-ilar mismatch was observed between VVDS-Deep zCOSMOSand interpreted by K11 in terms of di ff erent clustering ampli-tude in the two datasets (McCracken et al. 2007; Meneux et al.2009; Kovaˇc et al. 2011). Indeed, zCOSMOS is characterised byprominent structures and large spatial coherence as opposed tothe VVDS Deep field. This di ff erence was interpreted as a man-ifestation of cosmic variance. The VIPERS survey was designedto reduce the impact of cosmic variance and solve these typesof controversies. In this specific case, the agreement betweenVIPERS and zCOSMOS galaxies suggests that the bias of thelatter is closer to the cosmic mean than that of the VVDS-Deepfield.The comparison among the non-linear bias parameters ofthe three galaxy samples (upper panel of Fig. 16) corroboratesthis conclusion. The values of ˜ b / ˆ b for zCOSMOS and VIPERSgalaxies agree with each other and significantly deviates fromunity. Thanks to the smaller error bars in VIPERS these devia-tions are now detected with higher statistical significance. Devi-ations from non-linear bias in the VVDS-Deep are larger than inVIPERS but the statistical significance for this mismatch is justabout 1 σ .Figure 17 is analogous to Figure 16. It shows the valuesof ˆ b and ˜ b / ˆ b for galaxy subsamples extracted from VIPERS(red dots) and zCOSMOS (blue squares) using magnitude cutsbrighter than before: M B = − . − z − h ) for VIPERSand M B = − . − z − h ) for zCOSMOS. Our resultsconfirm those obtained with the fainter samples; the values ofˆ b and ˜ b / ˆ b for VIPERS galaxies agree with those of zCOSMOSgalaxies in the redshift range in which the two analyses over-lap. Non-linearity is detected at more than 1- σ in the VIPERSsample alone. No comparison was made with the VVDS-Deepsample in this case since none of the subsamples analysed byM05 match these luminosity cuts. In Section 5 we saw that the bias of VIPERS galaxies devi-ates from linearity at all redshifts and on all scales explored.The amount of non-linearity, quantified by the parameter ˜ b / ˆ b ,is rather small, of the order of a few per cent. This means thatˆ b is reasonably similar to b LIN and, therefore, can be comparedwith the linear bias parameter computed in other analyses.In the following, we therefore compare the values of ˆ b com-puted in this work with the values of b LIN obtained from di ff erentdatasets in the same redshift range but using a variety of bias es-timators. Galaxy bias at these redshifts has been estimated fromboth galaxy clustering and weak lensing. The latter probe, how-ever, has either focused on bright objects used to trace baryonicacoustic oscillations (Comparat et al. 2013) or to explore biasdependence on the stellar mass (Jullo et al. 2012). Therefore, we Article number, page 16 of 22. Di Porto et al.: The VIMOS Public Extragalactic Redshift Survey (VIPERS) Measuring non-linear galaxy bias at z ∼ . Table 2.
Bias parameters of VIPERS galaxies and their errors. z -range M B - cut R δ MAX ˆ b σ ˆ b RND σ ˆ b CV σ ˆ b SYS ˜ b / ˆ b σ NL RND σ NL CV σ NL SYS M B ( z = − h h − Mpc0.5-0.7 − . − z − . − z − . − z − . − z − . − z − . − z − . − z − . − z − . − . − z − . − . − z − . − z − . − z − . − z − . − z − . − z − . − z − . − z − . − z − . − z − . − z − . − z − . − z − . − z − . − z − . − z − . − z − . − z − . − z − . − z − . − z − . Notes.
Col. 1: redshift range. Col. 2: z -dependent B-band magnitude cut. Col. 3: cell radius [ h − Mpc ]. Col. 4: maximum over-density consideredin the analysis δ MAX ; the value in parenthesis indicates ¯ δ MAX . Col. 5: estimated value of the bias moment ˆ b ; the values in parenthesis refer tomeasurements performed at ¯ δ MAX . Col. 6: total random error on ˆ b . Col. 7: cosmic variance contribution to ˆ b error. Col. 8: systematic error on ˆ b .Col. 9: estimated value of the non-linearity parameter ˜ b / ˆ b ; values in parenthesis refer to measurements performed at ¯ δ MAX . Col. 10: total randomerror on ˜ b / ˆ b . Col. 11: cosmic variance contribution to ˜ b / ˆ b error. Col. 12: systematic error on ˜ b / ˆ b . focus on the values of b LIN obtained from galaxy clustering inother datasets available in the literature.The results of these comparisons are shown in Figure 18 inwhich we plot the most recent estimates of both ˆ b and b LIN inthe three redshift bins as a function of the magnitude cut. Weconsider the reference scale of R = h − Mpc since this is thesize of the cells used to measure ˆ b in VIPERS (large red circles),VVDS-Deep (small orange pentagons), and zCOSMOS (small,light green circles), as shown in the previous section. Magni-tudes on the X-axis are specified in the B-band since this bandis used in most of the considered samples with the exception ofthe PRIMUS and CHFTLS-wide. For these two latter cases, weconsider the g -band magnitude and transform it into B-band ac-cording to the g − B versus z relation measured from the VIPERScatalogue. Finally, all results were normalised to σ = . b values obtained fromVIPERS, for the systematic errors listed in Table 2. Thereforethese values are slightly di ff erent from those shown in Figures 16and 17.The light blue asterisks represent the b LIN values obtainedfrom the Wide part of the Canada-France-Hawaii Legacy Survey (CFHTLS) (Coupon et al. 2012). In this case, the bias valueswere computed from ∼ × galaxies in the redshift interval z = [0 . , .
2] by fitting a Halo Occupation Distribution modelto the measured angular correlation function. These b LIN valueswere obtained by integrating the halo bias over the halo massfunction. Therefore they are integral quantities much like ˆ b ,which is computed by integrating over the mass PDF (Eq. 2).The b LIN values of CFHTLS galaxies agree well with our resultsin all redshift bins, including z = [0 . , . i <
23. This dataset, which covers five independentfields (including the COSMOS field), spans the redshift range z = [0 . , . z = [0 . , .
0] andplot the corresponding bias values in the middle panel. Thebias was estimated from the projected galaxy 2-point correla-tion function, w p , g ( r ), as b LIN ( r ) = (cid:112) w p , g ( r ) / w p , m ( r ), where theprojected 2-point correlation function of the matter, w p , m ( r ), wasmodelled assuming a flat Λ CDM model with σ = .
8. The
Article number, page 17 of 22 & A proofs: manuscript no. arXiv_revised
Fig. 16.
Comparison among the values of ˜ b / ˆ b (top panel) and ˆ b (bottompanel) for zCOSMOS galaxies (blue squares) for VVDS-Deep galaxies(green triangles) and VIPERS galaxies (filled red circles). All samplesare luminosity limited and the magnitude cuts are indicated in the plot.The open red circles represents a VIPERS subsample matching the mag-nitude cut and redshift range of the VVDS-Deep sample. Estimates forthe bias parameters of zCOSMOS are taken from K11, and those forVVDS-Deep galaxies are from M05. Fig. 17.
Same as figure 16, but referring to brighter VIPERS (M B < − . − z − h ), red dots) and zCOSMOS galaxies (M B < − . − z − h ), blue squares). bias of PRIMUS galaxies is systematically larger than that ofVIPERS. However, the significance of the mismatch is below σ .The purple hexagons in the plot show b LIN of ALHAM-BRA galaxies (Arnalte-Mur et al. 2014). The photometric red-shift survey ALHAMBRA covers seven independent fields, in-cluding DEEP2 and COSMOS. Photometric redshifts are accu-rate enough to measure the projected galaxy correlation func-tion at di ff erent redshifts and, from this, to estimate the bias.In Figure 18 we show the b LIN values estimated in three red-shift bins: [0 . , .
65] (top panel), [0 . , .
85] (middle panel),and [0 . , .
05] (bottom panel). We did not consider the interval z = [0 . , .
25] since it is largely beyond the VIPERS range. Weshow two sets of points. Small open hexagons represent the val-ues of b LIN obtained from the clustering of galaxies in all sevenfields (labelled ALHAMBRA + in the plot). Filled hexagons(labelled ALHAMBRA-) illustrate the e ff ect of removing two"outlier" fields, COSMOS and ELAIS-N1, which are charac-terised by a high degree of clustering. The bias of galaxies inALHAMBRA- agrees with that of VIPERS for z < .
9. In thelast redshift bin, for M B ( z = < − . − h ) the bias ofALHAMBRA- is ∼ . σ below that of VIPERS. However, thediscrepancy disappears when one considers ALHAMBRA + andseems to reappear, with a reverse sign, at higher luminosities.The green triangles show the b LIN values obtained from theprojected galaxy 2-point correlation function of galaxies brighterthan M B − h ) = − . − h ) at z = [0 . , .
1] in theDEEP2 survey (Coil et al. 2006). In the brightest magnitudebin, where the three samples overlap, we find that the bias ofDEEP2 galaxies is significantly smaller than that of VIPERS andALHAMBRA objects.To summarise, we find a good agreement between the valueof ˆ b measured in our work and those of b LIN estimated in a num-ber of surveys in the range z = [0 . , . b .In the outermost redshift shell not all the bias values mea-sured in di ff erent surveys agree with each other. The value of b LIN for DEEP2 and, to a lesser extent, for ALHAMBRA- galax-ies, are smaller than ˆ b from VIPERS. This mismatch may indi-cate either a genuine di ff erence in the clustering properties of thedi ff erent samples or deviations from linear bias highlighted bythe di ff erent bias estimators.To quantify the impact of non-linear bias, we compare ourˆ b values with the corresponding b LIN estimated from the verysame VIPERS subsamples considered here. Figure 19 comparesˆ b from VIPERS (red filled symbols) with b LIN from Marulliet al. (2013) (blue filled squares, also shown, for reference inFigure 18). The two estimates agree at all redshifts but the lastredshift bin where the bias of Marulli et al. (2013) matches thatof DEEP2 galaxies and, consequently, is significantly below ourˆ b value.Like most of the other measurements, Marulli et al. (2013)estimated b LIN from the projected 2-point correlation function.More precisely, they averaged the correlation signal over the in-terval r = [1 , h − Mpc . In the presence of a scaled dependentbias, a manifestation of which is a ˜ b / ˆ b ratio di ff erent from unity,it is not obvious which e ff ective scale of the bias is estimated byMarulli et al. (2013). In our comparison we implicitly assumedthat this scale is the same as the cell size, i.e. 8 h − Mpc . In fact,a small scale seems more appropriate, especially if one accountsfor the fact that errors in the projected correlation function in-creases with the pair separation. For this reason, we also show ˆ b Article number, page 18 of 22. Di Porto et al.: The VIMOS Public Extragalactic Redshift Survey (VIPERS) Measuring non-linear galaxy bias at z ∼ . measured in cells of R = h − Mpc (orange hexagons). In thiscase, the significance of the mismatch is significantly reduced.Decreasing the scale to R = h − Mpc (not shown) would bringthe two values into agreement at the price, however, of creatinga mismatch at lower redshifts.Focusing on the VIPERS sample, a more homogeneous com-parison can be performed considering the b LIN value obtained byCappi et al. (2015) from counts in cells of R = h − Mpc (brownasterisks, in the plots). In this case the results agree with ourswithin the (rather large) error bars.In the figure we also show the VIPERS linear bias estimatedby Granett et al. (2015) (green triangles) from a Bayesian re-construction of a Wiener filtered, adaptively smoothed galaxydensity field. The result agrees with that of Marulli et al. (2013).However, as in that case, it is di ffi cult to associate an e ff ectivescale to the filtering procedure and perform a homogeneous com-parison with our estimate.Therefore, all the bias estimates of the VIPERS galaxiesagree with each other at z < .
9, a sign that galaxy bias islargely independent of scales. At higher redshifts we observesome discrepancies among the various estimates whose signifi-cance, however, is di ffi cult to assess since the di ff erent estimatesare sensitive to di ff erent scales. It is safe to conclude that thescale-dependence bias of VIPERS galaxies is more pronouncedat high redshifts, as confirmed by the results presented in Sec-tion 5.3, and that this can account for most of the discrepanciesseen in Fig. 19.An additional, though minor, source of discrepancy is incom-pleteness. At z ∼ B − h ) ∼ − . ff ect depends on the luminosity cut.We conclude that deviations from linear bias cannot be ignoredat high redshifts and that using b LIN as a proxy for galaxy biasleads to significant systematic errors.
7. Discussion and conclusions
The importance of characterising galaxy bias at intermediateredshifts stems from the need to infer the properties of the distri-bution of the mass from that of the galaxies. This will be espe-cially important in future redshift surveys aimed at an accurateestimate of the cosmological parameters. This has prompted sev-eral e ff orts to estimate galaxy bias at z > . ∼ ,
000 galaxies distributed overa much larger volume than its predecessors. This significantlyreduces the impact of cosmic variance that in previous studiesdominated the error budget. Secondly, we use a new techniqueto infer the mean biasing function from counts in cells that, un-der the hypothesis of local Poisson sampling, accounts and auto-matically corrects for shot noise. This improvement greatly in-creases our ability to recover the biasing function since Pois-
Fig. 18.
Comparison between the bias parameters ˆ b and b LIN obtainedfrom galaxy counts and galaxy clustering, respectively. Large red cir-cles: ˆ b of VIPERS galaxies. Blue squares: b LIN of VIPERS from Marulliet al. (2013). Green triangles: b LIN for DEEP2 galaxies from Coil et al.(2006). Brown crosses: b LIN for PRIMUS galaxies from Skibba et al.(2014). Light blue asterisks: b LIN for CHFTLS galaxies from Couponet al. (2012). Purple hexagons: b LIN for ALHAMBRA galaxies fromArnalte-Mur et al. (2014).Small light green dots: ˆ b for zCOSMOSgalaxies from Kovaˇc et al. (2011). Small light brown pentagons: ˆ b forVVDS-Deep galaxies from Marinoni et al. (2005). Values of ˆ b weremeasured on a scale R = h − Mpc . son noise is the main source of stochasticity in the bias rela-tion. Thirdly, owing to the size of the sample, we are able to ex-plore the bias dependence on magnitude, redshift, and scale withunprecedented accuracy. We postpone the investigation of addi-tional dependences on galaxy colour and stellar mass to a futureanalysis to be performed with the final VIPERS sample and newmock galaxy catalogues designed to mimic these galaxy proper-ties.The main results of our study are:The overall qualitative behaviour of the mean biasing func-tion of VIPERS galaxies is similar to that of zCOSMOS andVVDS-Deep galaxies as well as to that of the synthetic VIPERSgalaxies in the mock catalogues that we used to estimate er-rors. The shape of the mean biasing function is close to linearin regions above the mean density. It deviates from linear biasat δ <
0. More specifically, above the threshold δ TH at which δ g = − . , the bias function is significantly steeper than its meanslope ˆ b on scales smaller than 8 h − Mpc . For over-densities be-low δ TH the mean biasing function features a tail that cannot beaccounted for by linear biasing. The over-density threshold δ TH has been interpreted as a typical density scale below which veryfew galaxies form. In our analysis, we find that this threshold in-creases with the redshift and with the luminosity cut-of,f so thatat moderate redshifts low-density regions are typically populatedby faint galaxies. Article number, page 19 of 22 & A proofs: manuscript no. arXiv_revised
Fig. 19.
Comparison between the bias parameter ˆ b obtained fromour analysis on a scale of R = h − Mpc (large red circles), R = h − Mpc (small orange hexagons) and the linear bias parameters ofVIPERS galaxies, b LIN , obtained by Marulli et al. (2013) (blue squares),Cappi et al. (2015) (brown asterisks), and Granett et al. (2015) (greentriangles).
The biasing function shows small but significant deviationsfrom linearity at all redshifts, scales, and magnitude intervalsthat we explored. The parameter ˜ b / ˆ b that we use to quantify non-linearity neither seems to evolve with the redshift nor to dependon the luminosity. A scale dependence is observed at low red-shifts below 6 h − Mpc with only ∼ σ significance.We confirm that galaxy bias depends on luminosity. Themean slope ˆ b of the biasing function, a good proxy to linearbias given the small degree of non-linearity, increases with theluminosity threshold used to select the galaxy sample. The e ff ectis significant for z = [0 . , . z = [0 . , . ∆ M B = .
4. Thevalue of δ TH also increases with the magnitude, suggesting thatthe e ffi ciency of galaxy formation decreases with the luminosityof the object.We confirm that galaxy bias increases with redshifts as pre-dicted by most bias models and verified in other datasets. In ourcase we find evidence for a rapid evolution beyond z = .
9. Thisresult is highly significant and robust since it depends neitheron the scale nor on the luminosity of the objects. The statisticalsignificance of this result depends on the reliability of our erroranalysis, which is based on a mock galaxy catalogue designed tomatch the correlation properties of VIPERS galaxies. We veri-fied that mock catalogues are very realistic in the sense that theirbiasing function matches that of real objects remarkably well.Inthis analysis, we modelled all known sources of systematic er-rors, including the magnitude dependence of the spectroscopicsampling rate that was not originally included in the mock cat- alogues. We find no evidence for systematic errors that mightmimic a spurious evolution in the bias moments ˆ b and ˜ b .The value of ˆ b increases with the scale from 4 to 8 h − Mpc .We interpret this in the framework of the halo modelas the transition between the one-halo and two-halo contributionto galaxy bias. The same trend is seen in our mock catalogues,in which objects were extracted assuming the HOD model andin previous analyses performed at lower redshifts using bothgalaxy clustering (e.g. Zehavi et al. (2005)) and weak lensing(e.g. Hoekstra et al. (2002), Simon et al. (2007)). This is the firsttime that this e ff ect is detected at high redshifts with counts incells statistics.We compared our results with those of M05 and K11. Theseauthors performed an analysis similar to that presented here in asimilar range of redshifts. We limited the comparison on a scaleof 8 h − Mpc , which is common to the three analyses. We findthat the values of ˆ b of VIPERS and zCOSMOS galaxies agreewithin the errors. M05 find a smaller degree of biasing but thedi ff erence is of the order of 1 − σ . We conclude that the claimeddiscrepancy between K11 and M05 results is a manifestation ofcosmic variance.Deviations from linear biasing were also detected by M05and K11, although with a lower significance than in our case.Our results agree with those of K11. In M05 the degree of non-linearity is slightly larger than in our case but the discrepancyis barely larger than 1- σ . The bias non-linearity is sometimesexpressed in terms of the parameter b of the second-order Tai-lor expansion of δ (Fry & Gaztanaga 1993). Cappi et al. (2015)analysed the same VIPERS dataset using higher order statistics,which is a procedure that is less sensitive to non-linear bias thanours. They detected a deviation from linear bias at z ≤ . ∼ σ . Their b value turned out to be negative,in agreement with M05 and Marinoni et al. (2008) and, qualita-tively, with our results too.measured from the clustering of galaxies in recent galaxyredshift surveys (DEEP2, PRIMUS, CHFTLS-wide, and AL-HAMBRA). This comparison is qualitative since it assumes thatbias is linear, while our analysis has detected a small, but sig-nificant, degree of non-linearity in the bias of VIPERS galax-ies. The comparison is generally successful at z < . , wherewe find a very good agreement with all existing results. In thisredshift range our results provide additional evidence in favourof a luminosity-dependent bias and of a weak evolution. At z > .
9, where the spread among current results is large, ourresults favour the case of a significant bias evolution, in agree-ment with the CHFTLS-wide and ALHAMBRA analyses.Our results confirm the importance of going beyond the sim-plistic linear biasing hypothesis. Galaxy bias is a complicatedphenomenon. It can be non-deterministic, non-local, and non-linear. In this work we focused on deviations from linearity un-der the assumption that stochasticity is dominated by (and, con-sequently, accounted for) Poisson noise and that non-local ef-fects are smoothed out within the volume of our cells. While thevalidity and the impact of these assumptions can (and will) needto be tested, our results show that the application of an improvedstatistical tool to the new VIPERS dataset is already able to de-tect deviations from linear bias with 5 −
10 % accuracy.
Acknowledgments
We acknowledge the crucial contribution of the ESO sta ff forthe management of service observations. In particular, we aredeeply grateful to M. Hilker for his constant help and sup- Article number, page 20 of 22. Di Porto et al.: The VIMOS Public Extragalactic Redshift Survey (VIPERS) Measuring non-linear galaxy bias at z ∼ . port of this programme. Italian participation in VIPERS isfunded by INAF through PRIN 2008 and 2010 programmes.LG and BRG acknowledge support of the European ResearchCouncil through the Darklight ERC Advanced Research Grant( / / ERCgrant agreement n. 202781. WJP and RT acknowledge financialsupport from the European Research Council under the Euro-pean Community’s Seventh Framework Programme (FP7 / / ERC grant agreement n. 202686. WJP is also grateful forsupport from the UK Science and Technology Facilities Councilthrough the grant ST / I001204 /
1. EB, FM, and LM acknowledgethe support from grants ASI-INAF I / / / / INSU (In-stitut National des Sciences de l’Univers) and the ProgrammeNational Galaxies et Cosmologie (PNCG). CM is grateful forsupport from specific project funding of the Institut Universi-taire de France and the LABEX OCEVU. MV is supported byFP7-ERC grant “cosmoIGM” and by PD-INFN INDARK. CDPwishes to thank Pangea Formazione S.r.l. for supporting her inthe final stages of this work.
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2, I-40127 Bologna, Italy INFN - Sezione di Bologna, viale Berti Pichat 6 /
2, I-40127 Bologna,Italy Aix Marseille Université, CNRS, LAM (Laboratoired’Astrophysique de Marseille) UMR 7326, 13388, Marseille,France INAF - Osservatorio Astronomico di Trieste, via G. B. Tiepolo 11,I-34143 Trieste, Italy INFN - Istituto Nazionale di Fisica Nucleare, Via Valerio 2, I-34127Trieste, Italy Dipartimento di Fisica, Università di Milano-Bicocca, P.zza dellaScienza 3, I-20126 Milano, Italy Centre de Physique Théorique, UMR 6207 CNRS-Université deProvence, Case 907, F-13288 Marseille, France Astronomical Observatory of the Jagiellonian University, Orla 171,30-001 Cracow, Poland National Centre for Nuclear Research, ul. Hoza 69, 00-681Warszawa, Poland INAF - Osservatorio Astronomico di Torino, I-10025 Pino Torinese,Italy Canada-France-Hawaii Telescope, 65–1238 Mamalahoa Highway,Kamuela, HI 96743, USA INAF - Istituto di Astrofisica Spaziale e Fisica Cosmica Milano, viaBassini 15, I-20133 Milano, Italy Laboratoire Lagrange, UMR7293, Université de Nice Sophia-Antipolis, CNRS, Observatoire de la Côte d’Azur, 06300 Nice,France Institute of Astronomy and Astrophysics, Academia Sinica, P.O.Box 23-141, Taipei 10617, Taiwan Institute of Physics, Jan Kochanowski University, ul. Swietokrzyska15, 25-406 Kielce, Poland Department of Particle and Astrophysical Science, Nagoya Univer-sity, Furo-cho, Chikusa-ku, 464-8602 Nagoya, Japan Institute d’Astrophysique de Paris, UMR7095 CNRS, UniversitéPierre et Marie Curie, 98 bis Boulevard Arago, 75014 Paris, France Universitätssternwarte München, Ludwig-Maximillians Universität,Scheinerstr. 1, D-81679 München, Germany Max-Planck-Institut für Extraterrestrische Physik, D-84571 Garch-ing b. München, Germany Institute of Cosmology and Gravitation, Dennis Sciama Building,University of Portsmouth, Burnaby Road, Portsmouth, PO1 3FX,UK INAF - Istituto di Astrofisica Spaziale e Fisica Cosmica Bologna,via Gobetti 101, I-40129 Bologna, Italy INAF - Istituto di Radioastronomia, via Gobetti 101, I-40129Bologna, Italy SUPA - Institute for Astronomy, University of Edinburgh, Royal Ob-servatory, Blackford Hill, Edinburgh, EH9 3HJ, UK Università degli Studi di Milano, via G. Celoria 16, I-20130 Milano,Italy30