The VIMOS Public Extragalactic Redshift Survey (VIPERS) : measuring non-linear galaxy bias at z $\sim$ 0.8

A & A 594, A 62 (2016)

D O I: 10.1051/0004-6361/201424448

© E S O 2016

A&stronomy Astrophysics

The VIMOS Public Extragalactic Redshift Survey (VIPERS)

Measuring non-linear galaxy bias at z ~ 0.8*

C. Di Porto1, E. Branchini2,3’4, J. Bel5, F. Marulli6,117, M. Bolzonella1, O. Cucciati1, S. de la Torre8, B. R. Granett5, L. Guzzo5,11, C. Marinoni12, L. Moscardini6,117, U. Abbas15, C. Adami8, S. Arnouts8,16, D. Bottini17, A. Cappi1,18, J. Coupon19, I. Davidzon6,1, G. De Lucia9, A. Fritz17, P. Franzetti17, M. Fumana17, B. Garilli8,17, O. Ilbert8, A. Iovino5,

J. Krywult20, V. Le Brun8, O. Le Fevre8, D. Maccagni17, K. Małek21, H. J. McCracken22, L. Paioro17, M. Polletta17, A. Pollo13,14, M. Scodeggio17, L. A. M. Tasca8, R. Tojeiro23, D. Vergani24, A. Zanichelli25, A. Burden23,

A. Marchetti5,27, D. Martizzi28, Y. Mellier22, R. C. Nichol23, J. A. Peacock26, W. J. Percival23, M. Viel9,10, M. Wolk22, and G. Zamorani1

(Affiliations can be found after the references) Received 22 June 2016 / Accepted 12 April 2016

ABSTRACT

Aims. We use the first release of the VImos Public Extragalactic Redshift Survey of galaxies (VIPERS) of ~50 000 objects to measure the biasing relation between galaxies and mass in the redshift range z = [0.5,1.1].

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 effects. With this procedure we constrain galaxy bias and its deviations from linearity down to scales as small as 4 h-1 Mpc and out to z = 1.1.

Results. We detect small (up to 2%) but statistically significant (up to 3 © deviations from linear bias. The mean biasing function is close to linear in regions above the mean density. The mean slope of the biasing relation is a proxy to the linear bias parameter. This slope increases with luminosity, which is in agreement with results of previous analyses. We detect a strong bias evolution only for z > 0.9, which is in agreement with some, but not all, previous studies. We also detect a significant increase of the bias with the scale, from 4 to 8 h-1 Mpc, now seen for the first time out to z = 1. The amplitude of non-linearity depends on redshift, luminosity, and scale, but no clear trend is detected. Owing to the large cosmic volume probed by VIPERS, we find that the mismatch between the previous estimates of bias at z ~ 1 from zCOSMOS and VVDS-Deep galaxy samples is fully accounted for by cosmic variance.

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.

Key words. cosmological parameters - dark matter - large-scale structure of Universe

1. Introduction

G alaxies do n o t p erfectly trace m ass. T he long know n p ro o f is th at galaxy clustering depends on properties o f galaxies such as lum inosity, colour, m orphology, stellar m ass, and so on (e.g. Szapudi e t al. 2 0 0 0 ; H aw kins e t al. 2 0 0 1 ; N orberg et al.

20 0 1 , 2 0 0 2 ; Z e h a v ie ta l. 2 0 0 2 , 2 0 1 1 ; M e n e u x e ta l. 2 0 0 9 ; M arulli e t al. 2013) and n o t solely on th e underlying m ass d is­

tribution. D ifferences in clustering properties are caused by the physical processes th at regulate th e form ation an d evolution o f

* Based on observations collected at the European Southern Ob­

servatory, Paranal, Chile, under programmes 182.A-0886 (LP) at the Very Large Telescope, and also based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT), which is operated by the National Research Council (NRC) of Canada, the Institut National des Science de l’Univers of the Centre National de la Recherche Scien- tifique (CNRS) of France, and the University of Hawaii. This work is based in part on data products produced at TERAPIX and the Canadian Astronomy Data Centre as part of the Canada-France-Hawaii Telescope Legacy Survey, a collaborative project of NRC and CNRS. The VIPERS web site is h t t p : / / v i p e r s . i n a f . i t /

galaxies and should disappear w hen averaging over scales m uch larger than those affected b y these processes.

M odelling the physics o f galaxy form ation, o r at least its im ­ p act on th e b ia s relation, is o f p aram ount im portance to extract cosm ological inform ation from the spatial distribution o f g alax ­ ies. Indeed, the large-scale structure o f the U niverse as traced by galaxies is one o f the m o st pow erful cosm ological probes as te s­

tified by the increasing n um ber o f large galaxy red sh ift surveys either ongoing, such as B oss (A nderson e ta l. 2012) , D E S 1, and V IPER S (G uzzo et al. 2014) or those plan n ed fo r the n ear fu ­ ture, such as eB O S S2, D E S I (S chlegel e t al. 2011) , and E uclid (L aureijs e ta l. 2011)3. T hese surveys are designed to address several im portant questions both in cosm ology and in galaxy evolution theory. C h ie f am ong them is the origin o f the accel­

erated expansion o f the U niverse.

It has recently been realised th at geom etry tests b ased on standard candles and standard rulers can trace th e expansion h is­

tory o f the U niverse b u t cannot identify th e cause o f the accel­

erated expansion, w hich can b e obtained either b y advocating 1 w w w .darkenergysurvey.com

2 h t t p : / / w w w . s d s s 3 . o r g / f u t u r e / 3 h t t p : / / w w w . e u c l i d - e c .o r g /

a dark energy com ponent o r by m odifying th e gravity theory (e.g. W ang 2008) . To b rea k this degeneracy one needs in d ep en ­ dent observational tests. T hese are p rovided b y th e build-up o f structures over cosm ic tim e (G uzzo et al. 2008) . T he analysis o f large-scale structures in galaxy distribution allow s us to perform these tw o tests a t one tim e. T he b aryonic acoustic oscillation peaks in the tw o p o in t statistics provide a standard ru ler to p er­

form geom etry test (e.g. Seo & E isenstein 2 0 0 3 ; Percival et a l 2 0 0 7 ; G aztanaga et al. 2 0 0 9 ; R e id e ta l. 2012) w hereas th e ap ­ p aren t radial distortions in galaxy clustering caused b y p eculiar m otions th at are gravitationally induced allow us to m easure the rate at w hich cosm ic structures grow. Since both tests rely on baryonic structures, the know ledge o f the bias relation is m a n d a­

tory to p robe the underlying m ass distribution and set co sm o ­ logical constraints. N otw ithstanding, a clustering statistics that is in p rinciple bias insensitive has been recen tly p roposed by B el & M arinoni (2014) an d applied to V IPER S d ata (B el et al.

2014) .

G alaxy bias is n o t ju s t a nuisance p aram eter in the quest for the w orld m odel. This bias also represents an o pportunity to co n ­ strain m odels o f g alaxy evolution as it encodes im portant infor­

m ation ab o u t the physical processes th at regulate th e evolution o f stars and galaxies. T herefore, it is im portant to m odel g alaxy bias by establishing its lin k to the relevant astrophysical processes that regulate galaxy evolutions.In a recen t review, B augh (2013) has classified galaxy evolution m odels into tw o categories. The so-called em pirical m odels belong to the first category. T hese authors use theoretically m otivated relations to m odel galaxy distribution from halos extracted from N -b o d y sim ulations. The tw o m o st p o pular schem es to populate halos w ith galaxies are halo occupation distribution (H O D ; e.g. C ooray & S heth 2 0 0 2 ; Zheng et al. 2005) an d sub-halo abundance m atching (SH A M ; e.g. Vale & O striker 2 0 0 4 ; C onroy et al. 2006) . T he second c a t­

egory is rep resen ted by p hysical m odels in w hich the processes that reg u late the evolution o f baryons are explicitly considered to lin k them to the host dark m a tte r structures. This approach is at th e h eart o f the sem i-analytic m odels o f galaxy form ation (e.g.

W hite & F re n k 1991; B ow er e t a l. 2 0 0 6 ; D e L u cia & B laizot 2007) . In m o st cases these m odels have been used to estim ate galaxy bias from clustering statistics such as galaxy counts or 2-point correlation functions. T he results indicate th at the accu ­ racy in both types o f m odels is one o f th e m ain lim itations in constraining d ark energy o r m odified gravity from current and, even m o re so, future observational cam paigns (C ontreras et al.

2013) .

A lternatively, one can ad o p t a p urely phenom enological ap ­ proach and use an operational definition o f the bias in term s o f m ap betw een the density fluctuations o f m ass, 6 and g alax ­ ies, 6 g sm oothed on the sam e scale. This approach assum es that galaxy bias is a lo c al process th at depends on th e lo c al m ass density only. M any studies further assum e th at the bias relation is lin ear and determ inistic, so th at galaxy bias can b e qu an ti­

fied b y a single linear bias p aram eter b: 6 g = b6. T he co n ­ cept o f lin ear bias has p lay ed an im portant ro le in cosm ology and m any results have been obtained using this assum ption, w hich is know n to be unphysical as it allow s negative densities.

A lso, this assum ptionhas no ju stification at the relatively sm all scales o f interest to the study o f galaxy form ation processes, w hich depend on m any physical param eters and on larg e scales due to the presence o f neutrinos (V illaescusa-N avarro et al.

2014) . In fact, the bias is co nstant only on scales larger than about 40 h -1 M pc (M a n era & G aztanaga 2011) . Indeed, galaxy bias can be m o re conveniently described w ithin a probabilistic fram ew ork as p roposed by D ekel & L ahav ( 1999) and recently

reform ulated in the context o f the halo m odel (C acciato e t al.

2012) .

F ro m the phenom enological view point, bias has been exten­

sively investigated from counts in cells statistics, w eak grav i­

tational lensing, and galaxy clustering. T he la tte r is probably m o st p o pular approach. It is typically b ased on 2-p o in t statis­

tics and on the assum ption o f linear bias (N orberg et al. 20 0 1 , 2 0 0 2 ; Z ehavi et al. 2 0 0 5 ; C oil et al. 2 0 0 6 ; B asilakos et al. 2 0 0 7 ; N u z a e ta l. 2 0 1 3 ; A rnalte-M ur et al. 2 0 1 4 ; S k ib b a e ta l. 2 0 1 4 ; M a r u llie ta l. 2013) . A com paratively sm aller n um ber o f stud­

ies searched for deviations from the linear an d determ inistic bias either using 2-point (T egm ark & B rom ley 1999) o r h igher order statistics (Verde e t a l. 2 0 0 2 ; G aztanaga et al. 2 0 0 5 ; K ayo e ta l.

2 0 0 4 ; N ishim ichi et al. 2 0 0 7 ; Sw anson e t al. 2008) .

G ravitational lensing in the w eak field regim e has also been exploited to constrain galaxy bias. In particular, w ithin the lim it o f scale-independent bias on la rg e scales, w eak lensing and galaxy clustering can b e com bined to estim ate the lin e ar bias p aram eter in a m anner w hich is independent o f the a m ­ p litude o f density fluctuations (A m ara e t a l. 2 0 1 2 ; P ujol e ta l.

2 0 1 6 ; C hang e t al. 2016) . O n sm aller scales w eak lensing was also used to m easure the scale dependence o f galaxy bias (H oekstra e t al. 2 0 0 2 ; S im on e t a l. 2 0 0 7 ; J u ll o e ta l. 2 0 1 2 ; C om parat et al. 2013), although this effect is degenerate w ith bias stochasticity, i.e. the fact th at galaxy bias m ig h t n o t b e solely determ ined b y the lo c al m ass density.

T he m o st natural w ay to study a p ossible scale dependence (or non-linearity) o f galaxy bias is in a p robabilistic fram ew ork by m eans o f counts in cells statistics (S igad et al. 2000) since in this case one can separate deviations from lin e ar bias and the p resence o f an intrinsic scatter in the bias relation. This ap ­ proach was u sed to estim ate the bias o f galaxies in the PSCz (B ranchini 2001) , V V D S (M arinoni e t al. 2 0 0 5 , h ereafter M 05), and zC O SM O S (K o v a c e ta l. 20 1 1 , hereafter K 11) catalogues as w ell as the relative bias o f blu e versus re d galaxies in the 2 degrees field galaxy red sh ift survey (2dFG R S; C olless e ta l.

2 0 0 1 ; W ild e t al. 2005) . D espite som e disagreem ent, results o b ­ tained at low red sh ift (z < 0.5) generally indicate that, a t least for som e types o f galaxies, the bias is stochastic, scale d e­

p en d en t and, therefore, non-linear. However. T he situation at z > 0.5 is less clear. G ravitational lensing studies either focused on very b rig h t objects to p robe th e b aryonic acoustic o scilla­

tions (C om parat et al. 2013) o r on galaxies in th e C O SM O S field (Jullo et al. 2012) ; these studies found no evidence for stochas­

ticity but, in the case of J u ll o e ta l. (2012), detected a signifi­

cant scale dependence o f galaxy bias. This conflicting evidence shows a la ck o f accuracy in current estim ators fo r galaxy bias th at is a serious w arning for precision cosm ology. This is esp e­

cially true considering th at this is the ran g e that w ill b e probed b y nex t generation surveys th at have the p otential to trace both the red sh ift an d scale dependence o f galaxy bias (D i Porto e t al.

2 012a,b )

T he results obtained so far th at focus on counts in cells p ro ­ vide som e conflicting evidence. In M 05 authors analysed g alax ­ ies in the V V D S -D eep catalogue over an area o f 0 .4 x 0.4 deg and found significant deviations from linearity. T he estim ated effective lin e ar bias param eter show ed little evolution w ith red- shift. In contrast, the biasing relatio n o f zC O SM O S galaxies m easured b y K11 over a region o f abo u t 1.52 deg2 turned out to b e close to lin e ar and rapidly evolving w ith the red- shift. T he tension betw een these results is p aralleled by th e o b ­ served differences in the spatial correlation properties o f the tw o sam ples, w ith th e 2-point correlation function in zC O S- M O S system atically h igher than th at o f V V D S galaxies (see e.g.

A62, page 2 of 22

C. Di Porto et al.: The VIMOS Public Extragalactic Redshift Survey (VIPERS). Measuring non-linear galaxy bias at z ~ 0.8

M eneux et al. 2009) . O w ing to the large cosm ic variance in the tw o sam ples, a rath er sm all galaxy sam ple w as p roposed as the source of this m ism atch, so a larger galaxy sam ple should be used to settle the issue.

T he Vim os Public E x tragalactic R edshift S urvey [VIPERS]

(G uzzo e t al. 2014) has a d epth sim ilar to th e zC O SM O S survey but w ith a m u c h larger area o f 24 deg2. Its volum e is com parable to that o f 2dFG R S an d is large enough to significantly reduce the im p act o f th e cosm ic variance (see A ppendix in F ritz et al.

2014) . W e ado p t the sam e approach as M 05 and K11 and e sti­

m a te galaxy bias from counts in cells. To do so w e use a novel estim ator th at accounts for the effect o f discrete sam pling, allow ­ ing us to use sm all cells an d p robe unprecedented sm all scales that are m o re affected b y the physics o f galaxy form ation.

T he layout o f th e p ap e r is as follow s. In Sect. 2 w e describe both th e real and m o c k datasets used in this w ork. In Sect. 3 w e introduce th e form alism used to characterise galaxy bias and the estim ators used to m easu re this bias from a galaxy red sh ift survey. In Sect. 4 w e assess the validity o f th e estim ator and use m o ck galaxy catalogues to gauge random and system atic errors.

We presen t ou r results in Sect. 5 and com pare these w ith those o f other analyses in Sect. 6 . T he m ain conclusions are draw n in Sect. 7

T hroughout this pap er w e assum e a flat A C D M universe (Q m, Q A, ^ 8) = (0.25; 0.75; 0.9). G alaxy m agnitudes are given in the AB system and, unless otherw ise stated, com puted assu m ­ ing h = H o /100 k m s-1 M p c-1 = 1. T he high value o f ^ 8 has little im p act on ou r analysis since ou r results can be rescaled to different values o f ^ 8 that are m o re consistent w ith cu rrent c o s­

m ological constraints. T he dependence o f the m ag n itu d e upon h is expressed as M = M h - 5 log(h), w here M h is the absolute m agnitude com puted for a given h value.

2. Datasets

T he results in this p ap e r are b ased on the first rele ase o f the V IPER S galaxy catalogue (G arilli e t al. 2014) . R andom and sys­

tem atic errors w ere com puted using a set o f sim ulated galaxy catalogues m im icking the rea l catalogue and its observational selections. B oth, the real an d m o c k sam ples are described in this Section.

2.1. R e a l d ata

T he V IM O S Public E xtragalactic R edshift Survey is an on g o ­ ing E S A L arge P rogram m e aim ed at m easuring spectroscopic redshifts for abo u t 105 galaxies at red sh ift 0.5 < z < 1.2 and beyond. T he galaxy target sam ple is selected from the “T 0005”

release o f the C anada-F rance-H aw aii Telescope L egacy Survey- W ide (C F H T L S -W ide) optical p hotom etric catalogue4. V IPERS covers 24 deg2 on the sky, divided over tw o areas w ithin the W 1 an d W 4 C F H T LS fields. G alaxies are selected to a lim it o f /ab < 22.5, further applying a sim ple and robust colour preselection to efficiently rem ove galaxies a t z < 0.5. This colour cu t and the adopted observing strategy (S codeggio et al.

2009) allow us to double th e galaxy sam pling ra te w ith r e ­ spect to a pure m ag n itude-lim ited sam ple. A t the sam e tim e, the area and depth o f th e survey resu lt in a relatively large volum e, 5 x 107 h -3 M p c3, w hich is analogous to th at o f the 2dF G R S at z ~ 0.1. V IPER S spectra are collected w ith the V IM O S m ulti-object spectrograph (L e F evre e t al. 2003) at 4 h t t p : / / t e r a p i x . i a p . f r / c p l t / o l d S i t e / D e s c a r t /

C FH TLS-T0005-R elease.pdf

m o derate resolution (R = 210) using th e L R R ed grism , p ro v id ­ ing a w avelength coverage o f 5 5 0 0 -9 5 0 0 A and a typical radial velocity error o f <rv = 141(1 + z) k m s-1 .

T he full V IPER S area o f 24 d eg2 is covered through a m osaic o f 288 V IM O S pointings. A com plete description o f the survey construction, from th e definition o f the target sam ple to the a c ­ tual spectra and red sh ift m easurem ents, is given in G uzzo e t al.

(2014). T he dataset u sed in this and o ther papers o f th e early sci­

ence rele ase rep resen t the V IPER S P ublic D ata R elease 1 (PD R- 1) catalogue th at includes 55 359 redshifts (2 7 9 3 5 in W 1 and 27 42 4 in W 4), i.e. 64% o f the final survey in term s o f covered area (G arilli et al. 2014) . A quality flag was assigned to each o b ­ je c t in the process o f determ ining th eir red sh ift from the spec­

trum , w hich quantifies th e reliability o f th e m easured redshifts.

In this analysis, w e use only galaxies w ith flags 2 to 9.5, w hich corresponds to a sam ple w ith a red sh ift confirm ation rate o f 90% .

Several observational effects need to be taken into account to investigate the spatial properties o f the underlying p opulation o f galaxies.

i) Selection effects along the rad ial direction are driven b y the flux lim it n atu re o f the survey and, a t z < 0.6, b y th e colour p reselection strategy. W e use volum e-lim ited (lum inosity- com plete) galaxy subsam ples th at w e obtain b y selecting galaxies brig h ter than a given m agnitude threshold in a given red sh ift interval. W e adopted a redsh ift-d ep en d en t lum in o s­

ity cu t o f the form M B(z) = M 0 - z th at should account for the lum inosity evolution o f galaxies (e.g. Z u cca e t al. 2009) . T he value o f th e threshold is set to guarantee th at the selected sam ple is > 90% com plete w ithin the given red sh ift interval.

In this sense each subsam ple is volum e lim ited an d lum in o s­

ity com plete. This z-dependent lum inosity cu t is very popular and has been adopted in o ther papers (see e.g. K 11). H ow ­ ever, o ther w orks used different types o f cuts, either ignor­

ing any dependence on red sh ift (such as in M 05; C oil e t al.

2008) o r assum ing a different functional form for the red- shift evolution (e.g. A rnalte-M ur et al. 2014) . A dopting an incorrect lum inosity evolution w ould g enerate a spurious r a ­ dial gradient in the m ean density o f the objects and a w rong z-dependence in th e galaxy bias. To m inim ise th e im p act o f this p otential bias, w e carry out our analysis in relatively n ar­

row red sh ift bins, so th at adopting any o f the aforem entioned lum inosity cuts w ould pro d u ce sim ilar results, as w e verified.

T he robustness o f our re su lt to the choice o f the m agnitude cut can b e tested a posteriori. F igure 16 show s th at the d if­

ference betw een estim ates obtained w ith a z-dependent cut (filled re d dot) and w ith a z-independent cu t (open re d dot) are sm aller than the total random errors.

Selection effects induced by the colour p reselection strat­

egy w ere determ ined from the com parison betw een the spectroscopic and p hotom etric sam ples (G uzzo e t a l. 2 0 1 4 ; de la Torre e t al. 2 0 1 3 ; F ritz et al. 20 1 4 ) an d are accounted for b y assigning to each galaxy an appropriate statistical w eight d ubbed colour sam pling rate (CSR).

ii) T he surveyed area presents reg u lar gaps due to the specific footprint o f the V IM O S spectrograph th at creates a pattern o f rectan g u lar regions, called pointings, separated b y gaps w here n o spectra are taken. S uperim posed on this pattern are unobserved areas resulting from b rig h t stars and technical and m echanical problem s during observations. W e discuss o ur strategy to take into account this effect in ou r counts in cells analysis in the follow ing (see C ucciati et al. 20 1 4 , for a m o re detailed study).

Fig. 1. Luminosity selection as a function of redshift. The black dots show the W1 and W4 VIPERS galaxies (with spectroscopic redshift flag between 2 and 9.5). Yellow lines represent the principal magnitude cuts applied in every redshift bin. The green line represents the cut M 0 = -19.7 - z made to compare our results to those of K11.

iii) In each pointing, slits are assigned to a n um ber o f p otential targets th at m e et the survey selection criteria (B ottini et al.

2005). G iven the surface density o f the targeted population, the m ultiplex capability o f V IM O S , and th e survey strategy, a fraction o f about 45% o f the paren t p h otom etric sam ple can be assigned to slits. W e define the fraction o f targets th at have a m easured spectrum as the target sam pling rate (T S R ) and the fraction o f observed spectra w ith reliab le red - shift m easurem ent as the spectroscopic sam pling sate (SSR).

B oth functions are ro ughly independent o f galaxy m agnitude except the SSR, w hich decreases for 1AB > 21.0, as show n in Fig. 12 of G uzzo et al. (2014) .

A ll these selection effects are thoroughly discussed and q uan tita­

tively assessed b y de la Torre e t al. (2013) . W e m ake no attem pt to explicitly correct for these effects individually. Instead, w e a s­

sess their im pact on the estim ate o f g alaxy bias in Sect. 4 using the m ock galaxy catalogues described below.

F or the scope o f ou r analysis, the m ain advantages o f V IPER S are the relatively dense sam pling o f tracers, w hich a l­

lows us to p robe density fluctuations dow n to scales com parable to those affected by galaxy evolution processes, and the large volum e that, as discussed in the previous section, allow s us to red u ce the im p act o f cosm ic variance considerably w ith resp ect to previous estim ates o f galaxy bias a t z ~ 1.

T he paren t PDR -1 V IPER S sam ple contains 45871 galaxies w ith reliable red sh ift m easurem ents. H ere w e restrict our an aly ­ sis in the red sh ift ran g e z = [0.5,1.1] since the nu m b er density o f objects at larger distances is too sm all to p erm it a ro b u st estim ate o f galaxy bias. To investigate the possible dependence o f galaxy bias on lum inosity and redshift, w e p artitioned the catalogue into subsam ples b y applying a series o f cuts in both m agnitude and redshift.

T he com plete list o f subsam ples considered in this w ork is p resented in Table 1. W e considered three red sh ift bins (z = [0.5,0.7], [0.7 ,0 .9 ], [0 .9 ,1 .1 ]) and applied different lum inosity

cuts th at w e obtained b y com prom ising betw een the n ee d o f m axim ising both com pleteness and nu m b er o f objects. D iffer­

en t lum inosity cuts w ithin each red sh ift bin allow us to study the lum inosity dependence o f g alaxy bias a t different redshifts. The m agnitude cuts, M B = - 1 9 . 5 - z - 5 l o g ( h ) a n d - 1 9 . 9 - z - 5 l o g ( h ) , th at ru n across the w hole red sh ift ran g e are used to investigate a p ossible evolution o f galaxy bias. In Table 1 the subsam ples are listed in groups. T he first three groups indicate subsam ples in the three red sh ift bins. T he la st group indicates subsam ples that are designed to m atch th e lum inosity cuts p erform ed by K11 (Mb = - 2 0 .5 - z - 5 log(h = 0.7) = - 1 9 .7 2 - z - 5 lo g (h )) and b y M 05 (M B = - 2 0 .0 - 5 log(h). T he m o st conservative cut Mb = - 1 9 .5 - z - 5 lo g (h ) guarantees 90% com pleteness out to z = 1 for th e w hole galaxy sam ple and h igher for late type objects (see Fig. 1) .

S ince the analysis p resented in this w ork is b ased on cell count statistics, a useful figure o f m erit is rep resen ted b y the n u m b er o f independent spheres th at can b e accom m odated w ithin the volum e o f the survey. C onsidering in term ediate cells w ith a radius o f 6 h -1 M pc, the nu m b er o f such independent cells is N = 3869, 5527, 6964 in the three red sh ift intervals z = [0.5 ,0 .7 ], [0.7 ,0 .9 ], [0.9 ,1 .1 ], respectively.

2.2. M o ck d a ta s e ts

W e considered a suite o f m o ck galaxy catalogues m im icking the real PD R -1 V IPER S catalogue to assess our ability to m easure the m ean biasing function and evaluate ran d o m an d system atic errors.

W e used tw o different types o f m ock galaxy catalogues. W e b ased the bulk o f ou r error analysis on the first m ock galaxy ca ta­

logue, w hich is d escribed in detail in de la Torre e t al. (2013) . In this set o f m ocks, synthetic galaxies are obtained by applying the H O D technique to the dark m a tte r halos extracted from th e M u l­

tiD ark N -b o d y sim ulation (P rada e t al. 2012) o f a flat A C D M universe w ith (Q m, O a , Q b, h, n, ^ 8)= (0.27; 0.73; 0.0469; 0.7;

0.95; 0.82). S ince th e resolution o f the p are n t sim ulation was too p o o r to sim ulate galaxies in the m agnitude ran g e sam pled by V IPE R S, de la Torre & P eacock (2013) applied an original tech ­ n iq u e to resa m p le the halo field to generate sub-resolution halos dow n to a m ass o f M = 1010 h -1 M 0 . T hese halos w ere H O D p opulated w ith m o ck galaxies b y tuning the free param eters to m atch the spatial 2-point correlation function o f V IPER S g alax ­ ies (de la Torre e t al. 2013) . O nce p opulated w ith H O D galaxies, the various outputs w ere rearran g ed to obtain 26 and 31 indepen­

den t light cones m im icking the W 1 and W 4 fields o f V IPER S and their geom etry, respectively. In ou r analysis w e considered 26 W 1 + W 4 m ock sam ples. T hey constitute ou r set o f Parent m o ck catalogues, as opposed to the R ea listic m ock catalogues th at w e obtain from the Parent set b y applying the various se­

lection effects (V IPER S footprint m ask besides TSR , SSR, and C SR ) an d by adding G aussian errors to the redshifts to m im ic the ran d o m error in the m easu red spectroscopic redshifts. The m o ck catalogues w ere built assum ing a constant SSR w hereas, as w e poin ted out, this is a declining function o f the apparent m agnitude. H owever, the dependence is w eak and only affects faint objects, i.e. preferentially objects a t large redshifts. F o r this reason w e d ecided to explicitly include this dependence b y se­

lectively rem oving objects, starting from the faintest and m oving tow ards brig h ter objects until w e m atch the observed SSR (m ) (G uzzo e t al. 2014) .

T he average galaxy n um ber densities in the m ocks are listed in Col. 4 o f Table 1. F or z < 0.9 the n um ber density in th e m ocks is sim ilar o r som ew hat sm aller than in the re a l catalogue. This A62, page 4 of 22

C. Di Porto et al.: The VIMOS Public Extragalactic Redshift Survey (VIPERS). Measuring non-linear galaxy bias at z - 0.8

Table 1. VIPERS subsamples. 3. Theoretical background

z -range Mb- cut _{nVIPERS n m ock}

Mb - 5 log(h) 10-3 h3 M pc-3

10-3 h3 Mpc-3

0 .5 -0 .7 - 1 8 .6 - z 4.78 4.36

0 .5 -0 .7 - 1 9 .1 - z 3.16 2.43

0 .5 -0 .7 - 1 9 .5 - z 2.10 1.37

0 .5 -0 .7 - 1 9 .9 - z 1.24

0.68

0 .7 -0 .9 - 1 9 .1 - z 2.71 2.55

0 .7 -0 .9 - 1 9 .5 - z 1.86 ^1.47

0 .7 -0 .9 - 1 9 .9 - z 1.07 0.72

0 .9 -1 .1 - 1 9 .5 - z 0.62 0.63

0 .9 -1 .1 - 1 9 .9 - z 0.42 0.43

0 .5 -0 .7 - 1 9 .7 - z 1.64 1.36

0 .7 -0 .9 - 1 9 .7 - z 1.13 1.05

0 .9 -1 .1 - 1 9 .7 - z 0.53 0.53

0 .7 -0 .9 - 20.0 1.42 1.49

Notes. Column 1: redshift range. Column 2: B-band magnitude cut (computed for h = 1). Column 3: galaxy number density in the real VIPERS sub-catalogues. Column 4: galaxy number density in the HOD-mock VIPERS sub-catalogues. In the Parent mock catalogue the number density is a factor ~3.7 larger. Cells fully contained in the surveyed volume (i.e. not overlapping with gaps or empty areas) contain

~40% more objects on average.

discrepancy increases with the lum inosity and probably origi­

nates from the uncertainty in the procedure to HOD-populate halos with bright m ock galaxies. The consequence for our analy­

sis is an overestimation o f the random errors in the measurement o f the bias o f VIPERS galaxies. A t higher redshift the trend is reversed; the number density o f objects in the m ocks is system ­ atically larger than in the real catalogue. In this case, to avoid underestimating errors, w e randomly diluted the galaxies in the m ocks. Hence the perfect match o f number densities in the red- shift bin z = [0.9, 1. 1], as shown in the table.

On the sm allest scale investigated in this paper, R = 4 h-1 Mpc, the second-order statistics o f simulated galaxies and the variance o f the galaxy density field are underestimated by -1 0 % (B e le t a l. 2014) . Therefore, to check the robustness o f our bias estimate to the galaxy m odel used to generate the m ock catalogues and to the underlying cosm ological m odel, w e con­

sidered a second set o f m ocks. These were obtained from the M illennium N -body simulation (Springel et al. 2005) o f a flat A C DM universe with (Qm, QA, Qb, h, n , o 8) = (0.25; 0.75;

0.045; 0.73; 1.00; 0.9) and using the semi-analytic technique o f D e Lucia & Blaizot (2007), an alternative to the HOD. As a re­

sult o f the lim ited size o f the computational box, it was p ossi­

ble to create light cones with an angular size o f 7 x 1 deg2, i.e.

smaller than the individual W1 and W 4 fields. Overall, w e con­

sidered 26 + 26 reduced versions o f the W 1+ W 4 fields. From these light cones w e created a corresponding number o f R ea lis­

tic m ock catalogues.

Robustness tests that involve both types o f m ock catalogues were restricted to a lim ited number o f samples (one for each redshift bin). In these tests w e sim ply compared the errors in the bias estimates after accounting for the larger cosm ic variance in the M illennium m ocks. Since these robustness tests turned out to be successful in the sense that errors estimated with the two sets o f m ocks turned out to be consistent with each other, w e do not m ention these m ocks again and, for the rest o f the paper, fully rely on the error estimates obtained from the HOD m ocks.

In this section w e briefly describe the formalism proposed by D ekel & Lahav ( 1999) and the method that w e use to estimate bias from galaxy counts. The key step is the procedure to esti­

mate the galaxy PDF, P (6g), from the measured probability o f galaxy counts in cells, P (N g). We review som e o f the techniques proposed to perform this crucial step and describe in detail the technique used in this work.

3.1. S to c h a s tic n o n -lin e a r bias

D ekel & Lahav ( 1999) proposed a probabilistic approach to galaxy bias in which non-linearity and stochasticity are treated independently. In this framework, galaxy bias is described by the conditional probability o f galaxy over-density, 6g, given the m ass over-density 6: P (6g|6). Both quantities are smoothed on the same scale and treated as random fields. If biasing is a local process then P (6g|6) fully characterises galaxy bias. Key quanti­

ties formed from the conditional probability are the mean biasing function

(1⁾

(2⁾

where o

2

2

b

g

eralisation o f the linear bias parameter. The ratio

b

b

It measures the non-linearity o f the mean biasing relation and, in realistic cases, is close to unity. In the lim it o f linear and deter­

m inistic bias, the two moments b and

b

stant) mean biasing function

b

b LiN

b LIN

miliar linear bias parameter. We note that

b

1

m ents’ ratio is very insensitive to it, b /b <x o

015

2000) . These scaling relations are used in Sect. 5 to compare re­

sults obtained assuming different values o f o

8

b var

g

2

g

gression o f 6 over 6

g

b inv

g

g

In this paper w e focus on the

b

lows us to compare our results with those o f K11 (but not with M 05, in which the focus is instead on

b

If bias is deterministic, then it is fully characterised by the mean biasing function b (6)6. However, w e do not expect this to be the case since galaxy formation and evolution are regulated by com plex physical processes that are not solely determined by the local m ass density. Therefore, for a given value o f 6 there is a w hole distribution o f 6

g

and its non-trivial second-order m om ents f (b(S)ó2) r2 { (b (ó )ó f)

b = — b = 2 ,

referred to as bias stochasticity, is contributed by two sources:

shot n oise due to the discrete sampling o f a continuous underly­

ing density field and those astrophysical processes relevant to the formation and evolution o f galaxies that do not depend (solely) on the local mass density.

Previous studies (Branchini 2 0 0 1 ; Marinoni et al. 2 0 0 5 ; V iel et al. 2 0 0 5 ; Kovac et al. 2011) that, like this one, used the galaxy 1-point PDF to recover the biasing function ignored the impact o f stochasticity and assumed a deterministic bias. We aim to improve the accuracy o f the bias estimator by taking bias stochasticity into account and w e do this by assuming that shot noise is the only source o f stochasticity. This sim plify­

ing assumption can be justified theoretically by both numerical and analytic arguments. Numerical experiments in which semi- analytic galaxies are used to probe the m ass density field in sam­

ples m imicking SDSS (S zap u d i& P an 20 0 4 , see Figs. 11 and 16) and 2MRS (N usser et al. 2 0 1 4 , see Fig. 1), i.e. two surveys with galaxy number densities similar to that o f VIPERS, do in­

deed show that shot noise is the dominant source o f scatter. More specifically, Poisson noise accounts for the scatter in the 6g ver­

sus 6 relation except at large over-density where the relation is over-dispersed. A nalytic arguments in the framework o f the halo m odel also confirm that the main source o f stochasticity is shot n oise with the halo-halo scatter providing a significant contri­

bution for faint objects alone (Cacciato et al. 2012). A ssessing the impact o f this shot noise only assumption is not simple, but som e arguments can be made to quantify the system atic effect o f underestimating stochasticity.

An upper lim it can be obtained when stochasticity is ignored altogether. In the case o f linear and stochastic bias, for exam ­ ple, b and b w ould be equal whereas binv w ould be systemat­

ically larger by about 10% (Som erville et al. 2000) . The more realistic case o f a non-linear and stochastic bias was consid­

ered by Sigad et al. (2 000) using numerical simulations again.

In this case, the effect o f ignoring stochasticity is that o f over­

estimating both b and b . The amplitude o f the effect depends on both the cosm ological m odel assumed and the scale consid­

ered. To obtain estimates relevant to our analysis w e repeated the Sigad et al. (2000) test in Sect. 4 .1 . The results, which w e antic­

ipate here, indicate that b and b are overestimated by 8(4)% on a scale o f 4(8) h -1 Mpc. As for the ratio, b /b w e also confirm that it is remarkably insensitive to stochasticity and, as expected, to the m odel adopted (Sigad et al. 2000) .

A nalyses o f the datasets may also constrain the size o f the effect. Galaxy clustering, higher order statistics, or gravitational lensing generally indicate that galaxy bias cannot be linear and deterministic. However, as w e anticipated in the introduction, it is not possible to disentangle the effects induced by non-linearity and stochasticity, except for the case o f relative bias between two types o f tracers. With respect to this, the largest stochastic­

ity Ob/b = 0.4 4 so far was measured by W ild e ta l. (2005) . If ignored, this w ould induce a system atic error o f ~20% on the relative b moments.

Overall, the variety o f evidenceindicates that if stochasticity is ignored then <r b and b are overestimated by 10-20% , whereas their ratio is unaffected. However, w e stress that in our work stochasticity is, at least in large part, taken into account. There­

fore, w e expect that our assumption that shot noise is the only source o f bias stochasticity generates system atic errors w ell b e­

low the 10% level.

3.2. D irect e s tim a te o f b(6)6

Under the hypothesis that bias is deterministic and m onotonic the mean biasing function, b (6)6, can be estimated by compar­

ing the PDFs o f the mass and o f the galaxy over-density. We let C(6) = P (> 6) and Cg(6g) = P (> 6g) be the cumulative probabil­

ity distribution functions [C D Fs] obtained by integrating the two PDFs. M onotonicity guarantees that the ranking o f the fluctua­

tions 6 and 6g is preserved and b(6)6 can be obtained by equating the two CDFs at the same percentile,

b(6)6 = C-1 (C(6)), (3)

where C-1 indicates the inverse function o f Cg.

Equation (3) provides a practical recipe to estimate galaxy bias from observed counts in cells o f a given size. It requires three ingredients: the galaxy over-density 6g, its PDF, and that o f 6. 6g can be estimated from galaxy counts in cell, Ng as

(4) where (Ng) represents mean over all counts. From Eq. (4 ) one can form the galaxy PDF, P (6g) and the count probability P (Ng).

The biasing function can then be obtained by comparing Cg(6g) with a m odel C (6).

This sim ple bias estimator has been used by several au­

thors (Sigad e ta l. 2 0 0 0 ; Branchini 2 0 0 1 ; Marinoni et al. 2 0 0 5 ; Viel et al. 2 0 0 5 ; Kovac et al. 2011) . It is potentially affected by several error sources that should be system atically investigated.

The first error source is shot noise that affects the estimate o f 6g

from Ng. Shot noise induces stochasticity in the bias relation in contrast with the hypothesis o f deterministic bias. Stochasticity affects the estimate o f b(6)6 from Eq. (3) , especially at large val­

ues o f 6g, where the CDF flattens and the evaluation o f the in­

verse function C-1 becom es noisy. A second issue is the mass PDF for which no simple theoretical m odel is available. The last error source is redshift distortions. Galaxy over-densities are computed using the redshift o f the objects rather than distances.

This induces systematic differences between densities evaluated in real and redshift space (Kaiser 1987) .

A ll these issues potentially affect the estimate o f galaxy bias and should be properly quantified and accounted for. In the next section, w e review som e existing estimators designed to m in­

im ise the impact o f the shot noise and propose a new estimator that w e apply in this paper. We investigate the performance o f this new strategy in Sect. 4 .

3.3. From P(Ng) to P(6g)...

The probability o f galaxy counts, P (Ng), can be expressed as

X +ro

P (6g)P (Ng |6g)d6g, (5)

where the conditional probability function P (Ng|6g) specifies the way in which discrete galaxies sample the underlying, continu­

ous field. The com mon assumption that galaxies are a local Pois- son process im plies that

(6)

A62, page 6 of 22

[<Ng)(1 + ó g )]^ e -<N*>(1+5*)

= 1---Ng!

---

C. Di Porto et al.: The VIMOS Public Extragalactic Redshift Survey (VIPERS). Measuring non-linear galaxy bias at z ~ 0.8

The Poisson m odel provides a good match to numerical experiments except at large densities where a negative binomial distribution seem s to provide a better fit (Sheth 1995; Somerville et al. 2 0 0 1 ; Casas-Miranda et al. 2002) . In this work w e adopt the Poisson m odel. However, di

ff

The follow ing strategies have been proposed to estimate P (6g) from P (Ng) using Eq. (5) :

- Richardson-Lucy deconvolution. Szapudi & Pan (2004) pro­

posed this iterative, non-parametric method to reconstruct P (6g) by comparing the observed P (Ng) to that computed from Eq. (5) at each step o f the iteration, starting from an initial guess for P (6g).

- Skewed lognorm al m odel fit. This parametric method was also implemented by Szapudi & Pan (2004) . In this approach one assumes a skewed lognormal form for P (6g) and then de­

termines the four free parameters o f the m odel by m inim ising the di

ff

- Gamma expansion [

T

sion co e

ffi

served counts. Because o f this, the full shape o f the galaxy PDF can be recovered directly from the observed P (Ng) with no need to integrate Eq. (5) .

Szapudi & Pan (2004) have tested the ability o f the first two methods in reconstructing the PDF o f halos and m ock galax­

ies obtained from N -body simulations. They showed that a suc­

cessful reconstruction can be obtained when the sampling is (Ng) > 0.1; safely a factor 3 smaller than the sm allest mean galaxy density in our VIPERS subsamples. B el et al. (2016) ex ­ tensively tested the

r

perior to that o f the other methods. This com es at the price o f discarding counts in cells that overlap the observed areas by less than 60%, which is a constraint that further reduces deviations from the Poisson sampling hypothesis.

To illustrate the performance o f the

r

r

=

- 1 9 .1 - z - 5lo g (h ) in the range z

=

=

The reconstruction is compared with the “reference” PDF (solid, red line) obtained by averaging over the PDFs recon­

structed, with the same

r

logues. We regard this as the “reference” PDF since, as shown by Szapudi & Pan (2004) and checked by us, when the sampling is dense, all the above reconstruction methods recover the PDF o f the m ass, P (Ng) and the mean biasing function very accurately.

In the plot w e show P(1

+

+

ing. The reconstructed PDF underestimates the reference PDF in the low - and high-density tails and overestimates it at 6 ~ 0.

Systematic deviations in the low- and high-density tails are to be expected since the probability o f finding halos, and therefore m ock galaxies, in these regim es significantly deviates from the

Fig. 2. Reconstructed PDF of the mock VIPERS galaxies measured in cells of R = 4 h-1 Mpc (top) and R = 8 h-1 Mpc (bottom). The blue solid curve represents the reference galaxy PDF obtained by averag­

ing over the PDFs reconstructed from the Parent mocks using the r E method. The blue dashed curve shows the average PDF reconstructed from the Realistic mocks using the same method. The blue shaded re­

gion represents the 1ix scatter among the 26 Realistic mocks. We plot P(1 + <5g)(1 + Sg) to highlight the performance of the reconstruction at high and low over-densities. We note the different Y-ranges in the two panels. The bottom panels in each plot show the difference Ap between the reconstructed and reference PDFs in units of the random error ixp.

Horizontal, dashed lines indicate systematic errors equal to 1ixp random uncertainties.

probability expected for a Poisson distribution. However, these differences are w ell within the 1^ uncertainty strip as shown in the bottom panels o f each plot.

The rE method used to reconstruct the galaxy PDF from dis­

crete counts is implemented as follows:

- We consider as the input dataset one o f the volume-lim ited, lum inosity com plete subsamples listed in Table 1. The p o­

sition o f each object in the catalogue is specified in redshift space, i.e. by its angular position and measured spectroscopic redshift.

- Spherical cells are thrown at random positions within the surveyed region. We consider cells with radii R = 4, 6, and 8 h-1 Mpc. The sm allest radius is set to guarantee (Ng) > 0.3.

The largest radius is set to have enough cell statistics to sam­

ple P (Ng) at large Ng. We only consider cells that overlap by more than 60% with the observed areas. This constraint reduces deviations from Poisson statistics (B el & Marinoni 2014) . Counts in the partially overlapping cells are weighted by the fraction f o f the surveyed volum e in the cell: Ng/ f . The probability function P (Ng) is then computed from the counts frequency distribution.

- We use the measured P (Ng) and its moments to m odel the galaxy PDF with the rE m ethod that w e compute using all factorial moments up to the sixth order.

3 .4 a n d from P(6g) to b (6)6.

To estimate the mean biasing function from the galaxy PDF, w e solve Eq. (3) . To do so, w e assume that shot noise is the main source o f stochasticity and that a reliable m odel for the mass PDF is available. D espite its conceptual simplicity, this proce­

dure requires several non-trivial steps that w e describe below.

The uncertainties introduced in each step are estimated in the 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 m odel PDF for the mass density field in redshift space. Rather than adopting som e approximated, analytic m odel, w e measure the mass PDF directly from a dark mat­

ter only N -body simulation with the same characteristics and cosm ological m odel as the M illennium run (Springel et al.

2005) , thatis not based on the same m odel used to build the HO D-m ock VIPERS catalogues. The use o f an incor­

rect mass PDF is yet another possible source o f systematic errors that w e quantify in Sect. 4 . However, this error is ex ­ pected to be small since b and b are m ainly sensitive to a and their ratio is largely independent o f the underlying cosm ol­

ogy (Sigad et al. 2000) .

- After computing the cumulative distribution function from the mass and galaxy PDFs, w e use Eq. (3) to estimate the mean biasing function.

- We determine the maximum over-density 6MAX at which the reconstructed mean biasing function can be considered re­

liable. To estimate 6MAX w e compare the measured P (N g) with the estimated P (N g) follow ing the procedure described in Sect. 4 .4 .4 .

- We estimate the second-order moments b and b and their ra­

tio by integrating over all 6 up to 6MAX

X

X

m a x

(b (6)6)2P (6) d6. (7)

and test the robustness o f the result with respect to the choice o f6m a x .

4. Error sources

In this section w e review all possible sources o f uncertainty that m ight affect the recovery o f the biasing function and assess their amplitude using m ock catalogues. In this process w e need to consider a reference biasing function to compare with the re­

sults o f the reconstruction. This could be estimated directly from the distribution o f the dark matter particles and m ock galaxies within the simulation box. However, w e use the mean biasing function obtained from the Parent m ocks as reference. We justify this choice as follow s. First, Szapudi & Pan (20 0 4 ) showed that when the sampling is dense both the Richardson-Lucy and the skewed lognormal fit methods recover the mean biasing function with high accuracy. Second, in Sect. 3.3 w e found that when the sampling is dense the r E method accurately recovers the mean biasing function in the Parent mocks.

4.1. S e n s itiv ity to the g a la x y P D F re co n stru ctio n m e th o d M ost o f the previous estimates o f the m ean biasing function did not attempt to account for shot noise directly. This choice can

Fig. 3. Mean biasing function of mock VIPERS galaxies computed from counts in cells of R = 4 h-1 Mpc (bottom panel) and R = 8 h-1 Mpc (top panel). The magnitude cut and redshift range of the mock VIPERS subsample, indicated in the plot, are the same as Fig. 2. Solid red curve:

reference biasing function obtained from the Parent mock catalogues.

Blue dashed curve and blue-shaded region: average value and l a scat­

ter of the biasing function reconstructed from the Realistic mocks using the r E method. Brown dot-dashed curve and orange-shaded band: av­

erage value and l a scatter of the biasing function reconstructed from the Realistic mocks using a “direct” estimate of the galaxy PDF. Bot­

tom sub-panels: difference Ap between the reconstructed and reference PDFs in units of the random error a p. Dashed lines indicate systematic errors equal to 1 ap random errors.

hamper the recovery o f b (6)6 when the sampling is sparse. To estimate errors induced by ignoring shot noise and quantify the benefit o f using the rE method w e compared the biasing func­

tions reconstructed using both procedures. The result o f this test is shown in Fig. 3 . The red curve represents the reference bi­

asing function obtained by averaging over the Parent m ocks. In each m ock the biasing function was estimated from the galaxy PDF using the rE method. The blue dashed curve represents the same quantity estimated from the 26 R ealistic m ocks using the rE method. The blue band represents the 2 a scatter. For negative values o f 6g the reconstructed biasing function is below the refer­

ence biasing function, but the trend is reversed for 6g > 0, reflect­

ing the m ismatch between the reconstructed and reference PDFs in Fig. 2 . The discrepancy however, is m ostly within the 2 a scat­

ter (horizontal dashed line in the bottom sub-panels). On the con­

trary, the biasing function obtained from the “direct” estimate o f 6g (brown dot-dashed curve and the corresponding 2 a scatter, orange band) is significantly different from the reference func­

tion. The discrepancy increases at lo w densities and for small spheres, i.e. when the counts per cell decrease and the shot noise is large.

4.2. S e n s itiv ity to the m a ss P D F

Another key ingredient o f the mass reconstruction is the mass PDF. In principle this quantity could be obtained from galaxy A62, page 8 of 22

C. Di Porto et al.: The VIMOS Public Extragalactic Redshift Survey (VIPERS). Measuring non-linear galaxy bias at z ~ 0.8

peculiar velocities or gravitational lensing. However, in practice, errors are large and would need to be averaged out over scales much larger than the size o f the cells considered here. For this reason w e need to rely on theoretical m odelling. Coles & Jones ( 1991) and Kofman et al. ( 1994) found that the m ass PDF can be approximated by a lognormal distribution and this m odel was in­

deed adopted in previous reconstructions o f the biasing function (e.g. M05; W ild et al. 2 0 0 5 ; K11).

However, the lognormal approximation is known to perform poorly in the high- and low-density tails and for certain spec­

tra o f density fluctuations. An improvement over the lognor­

mal m odel is represented by the skewed lognormal distribution (Colom bi 1994) . This m odel proved to be an excellent approx­

imation to the PDF o f the dark matter measured from N -body experiments over a w ide range o f scales and o f over-densities (U eda & Yokoyama 1996) . The impact o f adopting either m odel for the m ass PDF can be appreciated in Fig. 4 . The solid red curves represent the same biasing functions shown in Fig. 3 ob­

tained from the galaxy PDFs o f the Parent m ocks and from a mass PDF obtained directly from an N -body simulation with the same cosm ological parameter and size as the M illennium sim u­

lation using the output corresponding to z = 0.8. As in the previ­

ous test, w e consider the red solid curve as the reference biasing function. The brown dot-dashed curve shows the mean biasing function reconstructed assuming a lognormal m odel for the mass PDF, i.e. a lognormal fit to the PDF measured from the N -body simulation. The curve represents the average among 26 mocks and the orange band is the 2 ^ scatter. For R = 8 h-1 Mpc, the b i­

asing function is system atically below the reference whereas for R = 4 h-1 M pc is above the reference at both high and low densi­

ties. The m ismatch is very large and significantly exceeds the 1^

scatter (bottom sub-panels). The skewed lognormal m odel (blue dashed curve) performs significantly better with differences w ell below 1^ except at very negative 6 values.

We conclude that, for the practical purpose o f reconstruct­

ing galaxy bias, the mass PDF measured from N -body data and a skewed lognormal fit perform equally well. The main advan­

tage o f using the latter w ould be the possibility o f determining the four parameters o f the fit experimentally. Since, however, the parameters are poorly constrained by observations, w e decided to adopt the m ass PDFs from N -body simulations. This choice introduces a dependence on the cosm ological m odel, however, that is m ostly captured by one single parameter, ^ , for which b and b exhibit a linear dependent. With respect to this, the mass PDF used to obtain the biasing functions in Fig. 4 is not the true mass PDF since it is obtained from an N -body simulation that uses a cosm ological m odel that is different from the m odel used to produce the m ock catalogues. We did this on purpose to m im ic the case o f the real analysis for which the underlying cosm olog­

ical m odel is not known.

4.3. S e n s itiv ity to re d s h ift d isto rtio n s

Galaxy positions are measured in redshift space, i.e. using the observed redshift to estimate the distance o f the objects.

The presence o f peculiar velocities induces apparent radial anisotropies in the spatial distribution o f galaxies and, as a con­

sequence, modifies the local density estimate and their PDF (Kaiser 1987) . However, our goal is to reconstruct the mean b i­

asing function in real space without redshift distortions. Con­

sidering the difficulties and uncertainties in determining the galaxy PDF in real space, one could instead consider the galaxy and mass PDFs both measured in redshift space under the

z = [ 0 . 7 , 0 . 9 ] MB< — 19.1 —z —5 l o g ( h )

Fig. 4. Solid red curve: reference mean biasing function of Fig. 3 com­

puted using the mass PDF from N-body simulations. Brown dot-dashed curve and orange band: biasing function obtained using a lognormal fit to the mass PDF and l a scatter from the mocks. Blue dashed curve and blue band: biasing function obtained using a skewed lognormal fit to the mass PDF and l a scatter from the mocks. Bottom panels: difference Ap between the reconstructed and reference PDFs in units of the random error a p. Dashed lines indicate systematic errors equals to 1 ap random errors.

assumption that peculiar velocities induce similar distortions in the spatial distribution o f both dark matter and galaxies so that they cancel out when estimating the mean biasing relation from Eq. (3) . In the lim it o f the Gaussian field, linear perturbation theory and no velocity bias, the cancelation is exact. However, non-linear effects have a different impact on the m ass and galaxy density fields and induce different distortions in their respec­

tive PDFs. To assess the impact o f these effects w e compared the mean biasing function o f m ock galaxies reconstructed from PDFs estimated in real and redshift space.

The results are shown in Fig. 5 . The solid red curve repre­

sents the mean biasing function o f galaxies in the R ealistic mock catalogues estimated using the PDFs o f galaxies and mass in real space. The blue dashed line shows the same function esti­

mated in redshift space. Both curves are obtained by averaging over the 26 m ocks and the blue band represents the 2 ^ scat­

ter in redshift space. The redshift space biasing function under­

estimates the true biasing function in low-density regions and overestimates it at high densities, i.e. in the presence o f highly non-linear flows. The difference is systematic but its amplitude is within the 2 ^ random errors estimated by adding in quadra­

ture the scatter among m ocks in real and redshift space (bottom panels in each plot). The biasing functions shown in Fig. 5 repre­

sents a demanding test in which w e consider the sm allest cells o f 4 h-1 M pc where deviations from linear m otions are larger. The discrepancy decreases if the size o f the cell increases.

These system atic differences induce errors in the estimated moments b and b. To quantify the effect w e computed the m o­

ments as a function o f 6 (i.e. by varying 6MAX in Eq. (7)) both in real and redshift space. The results are shown in Fig. 6 . The