The VIMOS VLT Deep Survey: Testing the gravitational instability paradigm at z ~ 1
C. Marinoni, L. Guzzo, A. Cappi, O. Le Fevre, A. Mazure, B. Meneux, A. Pollo, VVDS team
aa r X i v : . [ a s t r o - ph ] F e b Astronomy & Astrophysics manuscript no. ms˙marinoni November 27, 2018(DOI: will be inserted by hand later)
The VIMOS VLT Deep Survey
Testing the gravitational instability paradigm at z ∼ C. Marinoni , L. Guzzo , , A. Cappi , O. Le F`evre , A. Mazure , B. Meneux , A. Pollo , A. Iovino , H.J.McCracken , R. Scaramella , S. de la Torre , J. M. Virey , D. Bottini , B. Garilli , V. Le Brun , D.Maccagni , J.P. Picat , M. Scodeggio , L. Tresse , G. Vettolani , A. Zanichelli , C. Adami , S.Arnouts , S. Bardelli , M. Bolzonella , S. Charlot , P. Ciliegi , T. Contini , S. Foucaud , P. Franzetti ,I. Gavignaud , O. Ilbert , F. Lamareille , B. Marano , G. Mathez , R. Merighi , S. Paltani , , R.Pell`o , L. Pozzetti , M. Radovich , D. Vergani , G. Zamorani , E. Zucca , U. Abbas , M. Bondi , A.Bongiorno , J. Brinchmann , A. Buzzi , O. Cucciati , , L. de Ravel , L. Gregorini , Y. Mellier , P.Merluzzi , E. Perez-Montero , P. Taxil , S. Temporin , C.J. Walcher Centre de Physique Th´eorique, UMR 6207 CNRS-Universit´e de Provence, Case 907, F-13288 Marseille, France INAF - Osservatorio Astronomico di Brera, via Brera 28, I-20121 Milano, Italy Max Planck Institut f¨ur extraterrestrische Physik, D-85741 Garching, Germany INAF - Osservatorio Astronomico di Bologna, via Ranzani 1, I-40127 Bologna, Italy Laboratoire d’Astrophysique de Marseille, UMR 6110, BP8 Traverse du Siphon, F-13012 Marseille, France Astronomical Observatory of the Jagiellonian University, ul Orla 171, 30-244 Krakow, Poland Institut d’Astrophysique de Paris, UMR 7095, 98 bis Bvd. Arago, F-75014 Paris, France INAF - Osservatorio Astronomico di Roma, via Osservatorio 2, I-00040 Monteporzio Catone (Roma), Italy INAF - IASF, Via Bassini 15, I-20133 Milano, Italy LAT - Observatoire Midi-Pyr´en´ees, UMR5572, 14 av. E. Belin, F-31400 Toulouse, France INAF - Istituto di Radio-Astronomia, Via Gobetti 101, I-40129 Bologna, Italy School of Physics & Astronomy, University of Nottingham, University Park, Nottingham, NG72RD, UK Astrophysical Institute Potsdam, An der Sternwarte 16, D-14482, Potsdam, Germany Institute for Astronomy, 2680 Woodlawn Dr., University of Hawaii, Honolulu, Hawaii, 96822, USA Universit`a di Bologna, Dipartimento di Astronomia, via Ranzani 1, I-40127 Bologna, Italy Integral Science Data Centre, ch. d’´Ecogia 16, CH-1290, Versoix, Switzerland Geneva Observatory, ch. des Maillettes 51, CH-1290, Sauverny, Switzerland INAF - Osservatorio Astronomico di Capodimonte, via Moiariello 16, I-80131 Napoli, Italy Centro de Astrof´ısica da Universidade do Porto, Rua das Estrelas, P-4150-762, Porto, Portugal Universit`a di Milano-Bicocca, Dipartimento di Fisica, Piazza della scienza 3, I-20126 Milano, Italythe date of receipt and acceptance should be inserted later
Abstract.
We have reconstructed the three-dimensional density fluctuation maps to z ∼ . < z < Key words. cosmology:large scale structure of the Universe— cosmology:theory—galaxies:statistics—galaxies:high-redshift— galaxies:evolution
1. Introduction
According to Thomas Wright mapping the Cosmos onthe very largest scales is about gaining “a partial View
Marinoni et al.: Testing Gravity at z ∼1
Marinoni et al.: Testing Gravity at z ∼1 of Immensity, or without much Impropriety perhaps, a fi-nite View of Infinity” . Unfortunately, charting the cos-mic territory beyond our local volume into the distantUniverse is observationally challenging. Until recently, ourunderstanding of the large-scale organisation of galaxies at z ∼ > . < z < . δ g . In this way, we trace how the amplitude and spatialarrangement of galaxy fluctuations changes with cosmictime. We explore the mechanisms governing this growthby comparing the time evolution of the low-order momentsof the galaxy PDF, ( i.e. the variance amplitude < δ g > and the normalised skewness S = < δ g > c / < δ g > )with the corresponding quantity theoretically predictedfor matter fluctuations in the linear and semi-linear per-turbative regime. (Note that in the following we shall of-ten speak equivalently of the variance or of its square root,i.e. the root mean square amplitude < δ g > / when refer-ring to the second-order moment). This provides a test ofGIP-specific predictions at as-yet unexplored epochs thatare intermediate between the present era and the time ofdecoupling. Knowledge of the precise growth history ofdensity inhomogeneities provides also a way to test thetheory of gravitation (e.g., Linder, 2005). An Original Theory of the Universe (1750, 9th letter, PlateXXXI).
In addition to the statistical approach presented inthis paper, we have recently addressed this same issuealso from a dynamical point of view. We have used lin-ear redshift-space distortions in the VVDS-
Wide datato measure the growth rate of matter fluctuations at z ∼ . § § z ∼ .
5; we then characterise the evolutionof galaxy fluctuations with cosmic epoch by computingtheir PDF in two redshift slices. In § § m = 0 . Λ = 0 .
7. Throughout, the Hubbleconstant is parameterised via h = H /
2. The First-Epoch VVDS-Deep Redshift Sample
The primary observational goal of the VIMOS-VLTRedshift Survey as well as the survey strategy and first-epoch observations in the VVDS-0226-04 field (from nowon simply VVDS-02h) are presented by Le F`evre et al.(2005). Here it is enough to stress that, in order to min-imise selection biases, the VVDS-Deep survey has beenconceived as a purely flux-limited (17 . ≤ I ≤
24) survey,i.e., no target pre-selection according to colors or com-pactness is used. Stars and QSOs have been a-posteriori removed from the final redshift sample. Photometric datain this field are complete and free from surface bright-ness selection effects, down to the limiting magnitude I AB =24 (Mc Cracken et al., 2003). Spectroscopic observa-tions were carried out using the VIMOS multi-object spec-trograph using one arcsecond wide slits and the LRRedgrism which covers the spectral range 5500 < λ (˚ A ) < R ∼
227 at λ = 7500˚ A . The rms accuracy in the redshift measure-ments is ∼
275 km/s. Details on the observations and arinoni et al.: Testing Gravity at z ∼ data reduction are given in Le F`evre et al. (2004) and inLe F`evre et al. (2005).The VVDS-02h data sample extends over an area of0.7 × z ∼ ≥ z = 1 .
5) of nearly 1 . · h − Mpc . This volume has transverse dimensions ∼ × h − Mpc at z = 1 . h − Mpc in the radial direction.For the statistical analysis presented in this paper, wefirst define a sub-sample (VVDS-02h-4) including galax-ies with redshift z < × ) that was repeatedly covered by four independentVIMOS observations in each point. Even if measured red-shifts in the VVDS reach up to z ∼ § . < z < . ∼
30% (i.e. on average about oneover three galaxies with magnitude I AB ≤
24 has a mea-sured redshift). This high spatial sampling rate is a crit-ical factor for minimising biases in the reconstruction ofthe 3D density field of galaxies. To optimise the analy-sis of the associated probability density function, we fur-ther select only galaxies with absolute blue magnitude M B < −
20 + log h . With this selection, we define twonearly volume-limited sub-samples in the redshift ranges0 . < z < . . < z < . §
3. The galaxy density field at high redshift
The first large redshift surveys of the local Universe(e.g. Davis & Huchra, 1981; Geller & Huchra, 1991;Giovanelli & Haynes, 1991; Strauss et al., 1992a;da Costa et al., 1994) showed that galaxies have ahighly non-random spatial distribution and cluster in ahierarchical fashion. The corresponding three-dimensionalmaps reveal a complex web-like network of thin, filamen-tary structures connecting centrally condensed clusters ofgalaxies, punctuated by large, quasi-spherical, low-densityvoids. These structures are the outcome of more than 13billion years of evolution of small-amplitude fluctuationsthat we see reflected in the temperature anisotropy ofthe Cosmic Microwave Background (CMB) at z ≃ z ∼ z ∼ The continuous galaxy density fluctuation field δ g ( r , R ) = ρ ( r , R ) − ¯ ρ ¯ ρ (1)represents the adimensional excess/deficit of galaxies on ascale R, at any given comoving position r with respect tothe mean density ¯ ρ . As suggested by Strauss and Willick(1995) we estimate the smoothed number density of galax-ies brighter than M c on a scale R , ρ ( r, R, < M c ), bysumming over an appropriately weighted convolution ofDirac-delta functions with a normalised Gaussian filter F ρ ( r , R, < M c ) = X i R ∞ δ D ( u − | ∆r i | /R ) F ( u ) duS ( r i , M c )Φ( m ) ζ ( r i , m )Ψ( α, δ ) (2) F ( u ) = (cid:0) πR (cid:1) − / exp h − u i . (3)Here ∆ r = ( r i − r ) is the separation between galaxy po-sitions and the location r where the density field is evalu-ated. We compute the characteristic mean density at po-sition r using equation (2) by simply averaging the galaxydistribution in survey slices r ± R s , with R s = 400 h − Mpc.The four functions in the denominator of equation 2 cor-rect for various observational characteristics:- S ( r i , M c ) is the distance-dependent selection func-tion of the sample. This function is identically one when avolume-limited sample is used. When the full magnitude-limited survey (17 . < I <
24 in our case) is used, how-ever, this function corrects for the progressive radial in-completeness due to the fact at any given redshift wecan only observe galaxies in a varying absolute magni-tude range. While the PDF of galaxy fluctuations will bederived from volume-limited samples, in the next sectionwe shall make use of this function when reconstructing aminimum-variance 3D density map from the full VVDSsurvey.The actual values of S ( r, M c ) are derived using theVVDS galaxy luminosity function (Ilbert et al. 2005), as-suming a minimum absolute magnitude M c = −
15 +5 log h and accounting for its evolution as measured fromthe VVDS itself. A more detailed discussion of the deriva-tion of the selection function can be found in Paper I Marinoni et al.: Testing Gravity at z ∼1
15 +5 log h and accounting for its evolution as measured fromthe VVDS itself. A more detailed discussion of the deriva-tion of the selection function can be found in Paper I Marinoni et al.: Testing Gravity at z ∼1 - Φ( m ) corrects for the slight bias against brightobjects introduced by the slit positioning toolVMMPS/SPOC (Bottini et al. 2005).- ζ ( r i , m ) is the correction for the varying spectroscopicsuccess rate as a function of the apparent I AB magnitudeand of the distance of the object (see Ilbert et al. 2005).- Ψ( α, δ ) is the angular selection function correctingfor the uneven spectroscopic sampling of the VVDS onthe sky (see Fig 1. of Cucciati et al. 2006). Its purpose isto make allowance for the different number of passes doneby the VIMOS spectrograph in different sky regions (afactor which is anyway maximised in the 4-pass sub-areaof the sample).The analytical form of these selection functions is dis-cussed in Cucciati et al. (2006). The underlying assump-tion in this reconstruction scheme is that the subset ofobserved galaxies (e.g. in the case of a flux-limited sam-ple, those luminous enough to enter the sample at a givenredshift) is representative of the full population. This as-sumption clearly neglects any dependence of clustering onluminosity and could bias the density field reconstructedfrom the pure flux-limited sample at different redshifts; forthis reason, the quantitative measurements presented inthis paper will all be based on quasi-volume-limited sam-ples, limited to an absolute magnitude M B = − h .Finally, it should also be mentioned that in adopting a uni-versal luminosity function we do not take into account thepossible dependence of the luminosity function on mor-phological type and environment; this is, however, a sec-ond order effect in this work.The shot-noise error affecting the reconstructed fieldat different r is estimated by computing the square rootof the variance ǫ ( r ) = 1 ρ g " X i F (cid:0) | ∆r i | R (cid:1) S ( r i , M c )Φ( m i ) , ζ ( z, m ) , Ψ( α, δ ) ! / . (4)The amplitude of the shot noise increases as a functionof redshift in a purely flux-limited survey. We deconvolvethe signature of this noise from the density maps by apply-ing the Wiener filter (cf. Press et al., 1992; Strauss et al.,1992b) which provides the minimum variance reconstruc-tion of the smoothed density field, given the map of thenoise and the a priori knowledge of the underlying powerspectrum (e.g., Lahav et al., 1994). For this we assumethat the observed galaxy density field δ g ( r ), and the true(i.e. including all galaxies) underlying field δ T ( r ), bothsmoothed on the same scale, are related via δ g ( r ) = δ T ( r ) + ǫ ( r ) , (5)where ǫ ( r ) is the local contribution from shot noise (seeEq. (4)). The Wiener filtered density field, in Fourierspace, is˜ δ F ( k ) = F ( k )˜ δ g ( k ) , (6)where F ( k ) = h ˜ δ T ( k ) ih ˜ δ T ( k ) i + (2 π ) P ǫ ( k ) . (7) where brackets denote statistical averages and where P ǫ ( k ) = (2 π ) − h| ˜ ǫ ( k ) |i is the power spectrum of thenoise. Assuming ergodic conditions, this last quantity canbe computed as P ǫ ( k ) = (2 π ) − | ˜ ǫ ( k ) | . The calculationof h ˜ δ T ( k ) i taking into account the form of the windowfunction F and the peculiar VVDS survey geometry ispresented in paper I. z = 1 . We have first applied our reconstruction technique to theglobal flux-limited VVDS sample to build a visual three-dimensional map of galaxy density fluctuations to z = 1 . I ≤
24 sample is characterised by an effective meaninter-particle separation of ( h r i ∼ . h − Mpc ) in the red-shift range [0,1.5]. For comparison, this sampling is better(denser) than the early CfA1 survey ( h r i ∼ . h − Mpc)used by Davis & Huchra (1981) to reconstruct the 3D den-sity field of the local Universe (i.e., out to ∼ h − Mpc ).Also, at the median depth of our survey, i.e., in the redshiftinterval 0 . < z < .
8, the mean inter-particle separationis 4.4 h − Mpc, a value nearly equal to the 2dFGRS at itsmedian depth.The recovered galaxy density field is presented in Fig.1. Fluctuations have been smoothed on a scale R =2 h − Mpc. Only density contrasts with signal-to-noise ra-tio
S/N > geographical” explorationof the Universe at early cosmic epochs is the abundance oflarge-scale structures similar in density contrast and size(at least in one direction) to those observed by local sur-veys. In particular, it is tempting to identify qualitativelya few filament-like density enhancements bridging morecondensed structures along the line of sight, although thesurvey transverse size is still too small to fully sample theirextent. Nevertheless, it is interesting to notice that theseapparently one-dimensional structures remain coherentover scales ∼ h − Mpc, separating low-density regionsof similar size. Figs. one and two visually confirm that thefamiliar web pattern observed in the local Universe is nota present-day transient phase of the galaxy spatial organ-isation but it is already well-defined at ∼ . ∼
30% its present age (e.g., Le F`evre et al.,1996; Gerke et al., 2005; Scoville et al., 2007). This im-plies that large-scale features of the galaxy distributionessentially reflects the long-wavelength modes of the initialpower spectrum, in agreement with theoretical predictionsof the CDM hierarchical scenario. Numerical simulationsof large scale structure formation in fact show that thepresent-day web of filaments and walls is actually presentwhen the universe was in embryonic form in the overden-sity pattern of the initial fluctuations, with subsequentlinear and non-linear gravitational dynamics just sharp-ening its features (e.g., Bond et al., 1996; Springel et al.,2005). arinoni et al.: Testing Gravity at z ∼ z = . z = . z = . z = . z = . z = . z = . z = . z = . z = . z = . z = . z = . z = . z = . z = . z = . z = . z = . z = . M p c M p c − δ Fig. 1.
The reconstructed density field for 0 . < z < .
4, as traced by the galaxy distribution in the VVDS-Deepredshift survey to I ≤
24. This figure preserves the correct aspect ratio between transverse and radial dimensions. Themean inter-galaxy separation of this sample at the typical depth of the VVDS ( z = 0 .
75) is 4 . h − Mpc, comparableto local redshift surveys as the 2dFGRS. The galaxy density distribution has been smoothed using a 3D Gaussianwindow of radius R = 2 h − Mpc and noise has been filtered away using a Wiener filtering technique (see Strauss &Willick 1995, Marinoni et al. 2005). Only fluctuations above a signal-to-noise threshold of 2 are shown. The accuracyand robustness of the reconstruction methods have been tested using realistic mock catalogues (Pollo et al., 2005;Marinoni et al., 2005).
Marinoni et al.: Testing Gravity at z ∼1
Marinoni et al.: Testing Gravity at z ∼1 The limited angular size of the survey is exemplifiedby a dense “wall” at z = 0 .
97 that stretches across thewhole survey solid angle (0 . × . ∼ h − Mpc (comoving) in the transverse direction, isonly ∼ h − Mpc thick along the line of sight, and hasa mean overdensity δ g = 2 . ± .
3. This makes it sim-ilar to the largest and rarest structures observed in thelocal Universe, such as the Shapley concentration (e.g.Scaramella et al., 1989; Bardelli et al., 2000). By applyinga Voronoy-Delaunay cluster finding code (Marinoni et al.,2002), we find 10 distinct groups in this structure, withbetween 5 and 12 galaxy members each (down to the lim-iting magnitude I=24), for a total of 164 galaxies. If oneconsiders the evolution of mass fluctuations in the stan-dard ΛCDM model, the probability of finding a struc-ture with similar mass overdensity at such early times(0 . < z <
1) would be nearly 4 times smaller than to-day: one such mass fluctuation would be expected in avolume of ∼ · h − Mpc , i.e., nearly 5 times largerthan our surveyed volume up to z ∼
1. In fact, as we shalldescribe in section 3.3, finding such a galaxy overdensityis not so unusual: it is clear evidence that the biasing be-tween galaxies and matter at these epochs is higher thantoday. This makes fluctuations in the galaxy distributionto be highly enhanced with respect to those in the mass.
Several approaches may be used to characterise in a quan-titative way the distribution of galaxy fluctuations δ g shown in Fig 1. A complete specification of the overdensityfield may be given by the full set of galaxy N-point correla-tion functions (Davis & Peebles, 1977). This approach hasbeen explored and routinely applied over the past decadeas better and deeper redshift surveys have become avail-able. An alternative description may instead be given interms of the probability distribution function of a ran-dom field. By definition, the PDF of cosmological densityfluctuations describes the probability of having a fluctu-ation in the range ( δ, δ + dδ ), within a spherical regionof characteristic radius R randomly located in the surveyvolume. In principle, it encodes all the information con-tained within the full hierarchy of correlation functions,and provides insights about the time evolution of densityfluctuations. This definition can be applied either to thedistribution of galaxies, characterizing their number den-sity fluctuations, or to the dark-matter dominated massdistribution. For the latter case, the expected shape ofthe PDF can be predicted as a function of redshift givena cosmological model, at least for large-scale fluctuations;this can be done either analytically (see below) or usingnumerical simulations.On the observational side, in the case of surveys of thelocal Universe this fundamental statistics has been oftenoverlooked (but see Marinoni & Hudson 2002, Ostriker et al. 2003). On the other hand, only recently deep red-shift surveys have reached sufficient volumes to allow thesemeasurements to be extended back in time. In paper I, wehave discussed and tested in detail the methodology toestimate the PDF from this kind of samples. In particu-lar we have checked the robustness of the reconstructionagainst the specific VVDS-02h survey selection function,shot-noise errors and other observational biases. We usedfully realistic mock samples of the VVDS-02h survey dataand showed that, once the smoothing scale R is largerthan the mean inter-galaxy separation, the overall shapeof the reconstructed PDF is an unbiased realisation ofthe complete parent galaxy population. In particular, weshowed that for redshifts up to z = 1 . R ≥ h − Mpc. Clearly, thedegree with which the PDF measured from this sample isa fair representation of the “universal” PDF up to z =1.5is a separate, yet critical question. A difference is natu-rally expected due to fluctuations on scales larger thanthe volume probed (“cosmic variance”). An estimate ofthis effect is actually included in our error bars, as thesewere drawn from the scatter among our set of VVDS mocksamples.We have therefore applied the estimator of Eq. 2 andthe full de-noising technique described in § §
2, reconstructing the PDF of galaxy fluctuations in top-hat spheres of radius R = 10 h − Mpc at two differentepochs (0 . < z < . . ≤ z < . M B ≤ −
20 + 5 log h ) corresponds to a median luminosity L B ≃ L ∗ B at z ∼ δ -values; secondly, the low-density tail is en-hanced, with more low-density regions appearing at lowerredshifts. Quantitatively, this implies in particular thatthe probability of having an under-dense ( δ g <
0) regionof radius R = 10 h − Mpc at 0 . ≤ z < . . ≤ z < . The shape of the PDF of the galaxy overdensities isstrongly dependent on the non-linear effects implicitboth in the gravitational growth and in the physi-cal mechanisms responsible for galaxy formation (e.g.Watts & Taylor, 2000). Initial density fluctuations are arinoni et al.: Testing Gravity at z ∼
40 Mpc15 Mpc125 Mpc z=0.97
Fig. 2.
Density distribution and properties of a large-scale planar structure at z = 0 .
97, that completely fills theVVDS-02h field-of-view.normally assumed to have a Gaussian PDF; this is thenmodified by the action of gravity and, in the case of thegalaxy field, by the way galaxies trace the underlying mass( biasing scheme). If galaxies were faithful and unbiasedtracers of the underlying mass, the peak shift and the de-velopment of a low-density tail we observe in Fig.3 couldbe naturally interpreted as the key signature of dynamicalevolution purely driven by gravity. In fact, gravitationalgrowth in an expanding Universe makes low density re-gions propagate outwards and become more common astime goes by, while at the same time the high-density tailincreases.If this interpretation is correct, we expect the PDF ofgalaxy overdensities to coincide with the PDF of mass fluc-tuations in each redshift range, once they are normalisedto the observed clustering at z ∼
0, where we know that L ∼ L ∗ galaxies trace the mass (Verde et al., 2002). Letus verify whether this is the case by first summarising themain formalism to compute the PDF of mass fluctuationsin a given cosmological scenario.In hierarchical models, it is well established from nu-merical simulations that when structure growth reachesthe nonlinear regime on a scale R, the PDF of mass den-sity contrasts in comoving space is well described by a log- normal distribution (Coles & Jones, 1991; Kofman et al.,1994; Taylor & Watts, 2000; Kayo, Taruya & Suto, 2001), f R ( δ ) = (2 πω R ) − / . δ exp n − [ln(1 + δ ) + ω R / ω R o . (8)This is fully characterised by a single parameter ω R , re-lated to the variance of the δ -field on a scale R as ω R = ln[1 + h δ i R ] (9)At high redshifts, the variance σ R ≡ h δ i R over suf-ficiently large scales R (those explored in this paper) isgiven in the linear theory approximation by σ R ( z ) = σ R (0) D ( z ) (10)where D ( z ) is the linear growth factor of density fluctua-tions (normalised to unity at z = 0), D ( z ) = exp − h Z z f(z) d ln(1 + z) i . (11)In the standard ΛCDM cosmological model, the expres-sion for the logarithmic derivative of the growth factor,f = d log D / d log a (with a = (1 + z ) − ), can be approxi-mated to excellent accuracy asf(z) ∼ Ω γ m (z) Marinoni et al.: Testing Gravity at z ∼1
0, where we know that L ∼ L ∗ galaxies trace the mass (Verde et al., 2002). Letus verify whether this is the case by first summarising themain formalism to compute the PDF of mass fluctuationsin a given cosmological scenario.In hierarchical models, it is well established from nu-merical simulations that when structure growth reachesthe nonlinear regime on a scale R, the PDF of mass den-sity contrasts in comoving space is well described by a log- normal distribution (Coles & Jones, 1991; Kofman et al.,1994; Taylor & Watts, 2000; Kayo, Taruya & Suto, 2001), f R ( δ ) = (2 πω R ) − / . δ exp n − [ln(1 + δ ) + ω R / ω R o . (8)This is fully characterised by a single parameter ω R , re-lated to the variance of the δ -field on a scale R as ω R = ln[1 + h δ i R ] (9)At high redshifts, the variance σ R ≡ h δ i R over suf-ficiently large scales R (those explored in this paper) isgiven in the linear theory approximation by σ R ( z ) = σ R (0) D ( z ) (10)where D ( z ) is the linear growth factor of density fluctua-tions (normalised to unity at z = 0), D ( z ) = exp − h Z z f(z) d ln(1 + z) i . (11)In the standard ΛCDM cosmological model, the expres-sion for the logarithmic derivative of the growth factor,f = d log D / d log a (with a = (1 + z ) − ), can be approxi-mated to excellent accuracy asf(z) ∼ Ω γ m (z) Marinoni et al.: Testing Gravity at z ∼1 where γ ≃ .
55 (Wang & Steinhardt 1998, Linder 2005)andΩ m ( z ) = Ω m (1 + z ) E ( z ) E ( z ) = Ω m (1 + z ) + Ω Λ . The lognormal approximation formally describes thedistribution of matter fluctuations computed in real co-moving coordinates. On the contrary, the PDF of galax-ies is observationally derived in redshift space, whereits shape is distorted by the effects of peculiar motions(e.g. Marinoni et al., 1998; Guzzo et al., 2008). In orderto map properly the mass overdensities into galaxy over-densities the mass and galaxy PDFs must be computedin a common reference frame. It has been shown bySigad, Branchini & Dekel (2000) that an optimal strat-egy to derive galaxy biasing is to compare both mass andgalaxy density fields directly in redshift space. Implicit inthis approach is the assumption that mass and galaxiesare statistically affected in the same way by gravitationalperturbations, i.e. that there is no velocity bias in the mo-tion of the two components.The relation between the variances measured in realand redshift comoving space is σ zR ( z ) = p ( z ) σ R ( z ) (12)where p ( z ) is a redshift-dependent correcting factor whichtakes into account the average contribution of the linearredshift distortions induced by peculiar velocities (Kaiser,1987). Its expression, in the high redshift regime, is givenby (Hamilton, 1998; Marinoni et al., 2005) p ( z ) = (cid:2) (z) (cid:3) / . (13)We have used this formalism to compute the PDF ex-pected for mass fluctuations in the redshift ranges ex-plored using our galaxy samples. This is given, for thetwo ranges, by the curves in the top panel of Fig. 3. Theevident discrepancy between the galaxy and mass PDF’sindicates that the observed evolution cannot be only theproduct of gravitational growth (in the adopted cosmo-logical model), but that a time-evolving bias between thegalaxy and mass density fields is needed: at high redshiftsand on large scales galaxy overdensities trace in a morebiased way the underlying pattern of dark matter fluctu-ations. In the following section we shall summarise ourcurrent knowledge on the properties and evolution of thebiasing function and show how the presence of a non-linearbias is a necessary ingredient to theoretically understandthe evolution of the PDF in Fig.3. This will in particularprovide us with the necessary background to interpret theevolution of the low-order moments of the PDF at differ-ent redshifts, which is our aim in this paper. Fig. 3.
The PDF of galaxy fluctuations (in units y = 1+ δ )for VVDS galaxies with M B ≤ −
20 + 5 log h from theVVDS within two independent volumes, corresponding todifferent cosmic epochs: 0 . < z < . . ≤ z < . R = 10 h − Mpc.The histograms actually correspond to the distributionfunction G ( y ) = ln(10) yg ( y ) because the binning is donein log( y ). The two observed histograms have been repro-duced in both the upper and lower panels. They are com-pared to the theoretical predictions for the PDF of, respec-tively, mass fluctuations (top, from Eq. 8) and of galaxy fluctuations as inferred from Eq. 14 using the non-linearbiasing function measured from the VVDS (bottom). Theblue and red lines correspond to the higher- and lower-redshift samples respectively. Biasing lies at the heart of all interpretations of large-scalestructure models. Structure formation theories predict thedistribution of mass; thus, the role of biasing is pivotal in arinoni et al.: Testing Gravity at z ∼ mapping the observed light distribution back into the the-oretical model. In our case we need to disentangle the im-print of biasing from that of pure gravity in the evolutionof the galaxy PDF.In paper I we inferred the biasing relation δ g = δ g ( δ ) = b ( δ ) δ between mass and galaxy overdensities from their re-spective probability distribution functions f ( δ ) and g ( δ g ).Assuming a one-to-one mapping between mass and galaxyoverdensity fields, conservation of probabilities implies(e.g. Sigad, Branchini & Dekel, 2000; Wild et al., 2005) g ( δ g ) dδ g = f ( δ ) dδ (14)This approach implies the assumption of a cosmologi-cal model (the standard ΛCDM model in our case) fromwhich to compute f ( δ ), the mass PDF. The advantageover other methods is that we can explore the functionalform of the relationship δ g = b ( z, δ, R ) δ over a wide rangein mass density contrasts, redshift intervals and smoothingscales R without imposing any a-priori parametric func-tional form for the biasing function. Note that, by defi-nition, this scheme is ineffective in capturing informationabout possible stochastic properties of the biasing func-tion.The numerical solution δ g = δ g ( δ ) of Eq. 14 maps themass PDF (solid lines in the top panel of Fig. 3) intothe galaxy PDF (solid lines in the bottom panel of Fig.3) and can be analytically approximated using a Taylorexpansion (Fry & Gazta˜naga, 1993) δ g ( δ ) = n X k =0 b k ( z ) k ! δ k , (15)where the coefficients b i depend on redshift. We considerthis power series only to second order, and fit the nu-merical solution for the biasing function leaving b as afree parameter. Avoiding setting b as an integral con-straint ( < δ g > = 0) allows us to account for possible (un-modelled) contributions from higher order moments of theexpansion. This approach has the advantage of minimis-ing biases in the estimates of the lower moments of theexpansion, specifically b and b .The key result from Paper I has been to show thatgalaxy biasing is poorly described in terms of a sin-gle scalar and better characterised by a more sophis-ticated representation. Specifically, always considering ascale R = 10 h − Mpc, the ratio between the quadraticand linear bias terms has been evaluated in four differenthigh redshift intervals (see Table 2 of Paper I). When av-eraged over the full redshift baseline 0 . < z < .
5, thisratio turns out to be (cid:28) b b (cid:29) = − . ± .
04 (16)i.e. different from zero at more than 4 σ confidence level.This means that – at least over the redshift range andscales considered here – the level of biasing depends onthe underlying value of the mass density field. In otherwords, the way galaxies are distributed in space depends in a non-linear manner on the local amplitude of darkmatter fluctuations.The measurement of a non-linear term in the biasingrelation is fully consistent with a parallel analysis of thehierarchical scaling of the N-point correlation functionsin the same VVDS sample (Cappi et al. 2008). These re-sults confirm a generic prediction of hierarchical modelsof galaxy formation (Somerville et al., 2001, e.g., ). It isrelevant to compare them to estimates of the bias func-tion at the current epoch. Early works indirectly suggestedthat also at z ∼ b < z > . b /b ∼ − .
35) (Feldman et al., 2001; Gazta˜naga et al.,2005), our estimate seems to suggest that the amplitudeof the quadratic term b /b decreases (in absolute terms)as a function of redshift, a results in qualitative agreementwith indications from simulations. It seems therefore morelikely that the difference in the reconstruction methodsused (with different sensitivity to higher order terms inEq. 15) is a better explanation of the discrepant resultsat z ∼
0. We will show in the next section ( §
4) how theself-consistency of the evolution of the variance and skew-ness of galaxy counts with redshift, indeed requires thepresence of a non-linear biasing component.The information contained in the non-linear functionof Eq. 15 can be compressed into a single scalar termthat can be used to interpret the evolution of two-pointstatistics (correlation function) as well as the variance ofthe galaxy density field (see § h b ( δ ) δ i = 0, the most interesting linear bias estimatorsare associated to the second order moments of the PDFs,i.e., the variance h δ g i and the covariance h δ g δ i . Following ∼1
4) how theself-consistency of the evolution of the variance and skew-ness of galaxy counts with redshift, indeed requires thepresence of a non-linear biasing component.The information contained in the non-linear functionof Eq. 15 can be compressed into a single scalar termthat can be used to interpret the evolution of two-pointstatistics (correlation function) as well as the variance ofthe galaxy density field (see § h b ( δ ) δ i = 0, the most interesting linear bias estimatorsare associated to the second order moments of the PDFs,i.e., the variance h δ g i and the covariance h δ g δ i . Following ∼1 the prescriptions of Dekel & Lahav (1999), we character-ize the biasing function as follows b L ≡ h b ( δ ) δ ih δ i (17)where b L is an estimator of the linear biasing parameterdefined, in terms of the two-point correlation function, as ξ g = b L ξ . We evaluate Eq. 17 using Eq. 15 with parame-ters b i ( z ) estimated locally by Verde et al. (2002) and inthe redshift range 0 . < z < . L ∼ L ∗ ( z = 0) is b L ( z ) = 1 + (0 . ± . z ) . ± . While today ∼ L ∗ galaxies trace the underlyingmass distribution on large scales (Lahav et al., 2002;Verde et al., 2002; Gazta˜naga et al., 2005), in the past thetwo fields were progressively dissimilar and the relativebiasing systematically higher. In Paper I we showed howthis observed redshift trend compares to different theoret-ical models for biasing evolution, i.e. a “galaxy conserv-ing” model (Fry et al. 1996), a “halo merging” model (Mo& White 1996) and a “star forming” model (Tegmark &Peebles 1998).
4. Testing gravitational instability with thelow-order moments of the PDF
Having decoupled biasing effects from the purely gravita-tional evolution of the galaxy PDF we have now all theingredients to use this latter quantity to test the consis-tency of some general predictions of the GIP. The evo-lution of the low-order statistical moments of the galaxyPDF, specifically its second and third moments can becompared, on large scales with analytical predictions oflinear and second order perturbation theory respectively.
Following standard conventions, we define the second- andthird-order moments, on a scale R , of a continuous, zero-mean overdensity field as h δ g i R = Z ∞− δ g g R ( δ g ) dδ g . (18)and h δ g i R = Z ∞− δ g g R ( δ g ) dδ g . (19)Note that the moments cannot be estimated as ensem-ble averages over the reconstructed PDF. In fact, this lastquantity has been reconstructed using the Wiener filter-ing technique. This minimises the shot noise contribution( § assuming a second moment). Astandard practical way to estimate moments is to ran-domly throw spherical cells down within the galaxy distri-bution and reconstruct the count probability distributionfunction P k = n k /N (where n k is the number of cells thatcontain k galaxies out of a total number of cells N. Themoments are then estimated as h δ pg i = ¯ N − p ∞ X k =0 P k h ( k − ¯ N ) p i (20)where ¯ N = P ∞ k =0 kP k .The quantities we are interested in are the cumulants h δ p i c of the one-point density PDF. For a density fieldsmoothed with a top-hat window, the p -order cumulant h δ pg i c = 1 v pR Z ξ p ( r , r ...r p ) d r d r ...d r p (21)is the average of the N-point reduced correlation func-tion over the corresponding cell of volume v R (from nowon we will only consider the scale R = 10 h − Mpc and wewill drop the suffix R , unless we need to emphasize it).This is defined as the connected part of the N-point cor-relation function h δ g ( r ) δ g ( r ) ...δ g ( r p ) i in such a way thatfor p > ξ p = 0 for a Gaussian field. Since the galaxy dis-tribution is a discrete process (Eq. 2 is a sum over Diracdelta functions) and since, by definition, the density con-trast has a zero mean, the connection between low-ordercumulants and moments is given by h δ g i c = h δ g i − N (22) h δ g i c = h δ g i − h δ g i c ¯ N − N (23)These relations accounts for discreteness effects usingthe Poisson shot-noise model (e.g., Peebles, 1980; Fry,1985). Possible biases introduced by this technique arediscussed by Hui & Gazta˜naga 1999, while an alternativeapproach is detailed by Kim & Strauss (1998).Finally, it is necessary to devise a strategy to compen-sate for the fact that a cell will sample regions that havevarying angular and spectroscopic completeness and whichmay even span the survey boundary. For this reason thegalaxy counts are scaled up in proportion to the degreeof incompleteness in the cell. This is done by weightinggalaxy counts using the selection functions Φ( m ), ζ ( r, m ),and Ψ( α, δ ) defined in § §
2. As a conse-quence, the radial selection function is constant and anyvariations in the density of galaxies are due only to large-scale structure. arinoni et al.: Testing Gravity at z ∼ rms and skewness of galaxyfluctuations Since in perturbation theory higher order cumulants arepredicted to be a function of the variance, it is useful, inthe following, to define the normalized skewness S = h δ g i c /σ , (24)where the shot-noise corrected variance σ is given byEq. 22. Fig. 4 shows the evolution of the rms fluctuationand the normalized skewness on a scale R = 10 h − Mpc,as measured from the VVDS volume-limited sub-samples.Errors have been computed using the 50 fully-realisticmock catalogs of VVDS-Deep discussed in Pollo et al.(2005). This allows us to include an estimate of the con-tribution of cosmic variance, which represents the mostsignificant term in our error budget.The top panel of Fig. 4 shows that the square-rootof the variance, which measures the r.m.s. amplitude offluctuations in galaxy counts, is with good approxima-tion constant over the full redshift baseline investigated:in redshift space, the mean value of σ g for our volume-limited galaxy samples is 0 . ± .
09 for 0 . < z < . z ∼ .
15 from the 2dF galaxy red-shift survey (Croton et al., 2004) that is also reported insame figure. This means that over nearly 2/3 of the ageof the Universe the observed fluctuations in the galaxydistribution look almost as frozen, despite the underlyinggravitational growth of mass fluctuations. This quantifiesthe visual impression we had from Fig. 1, that the distri-bution of galaxies is as inhomogeneous at z ∼ h − Mpc ) even at these remote epochs ( ∼ σ detection).In particular we find indication for an increase of the nor-malised skewness with cosmic time, when comparing theVVDS values to the local measurement by 2dFGRS.Using the measured bias evolution, we can translatethe specific predictions of the GIP for the variance andskewness of the matter density field into the correspondingobserved quantities. Using linear perturbation theory, thescaling of the rms of number density fluctuations is σ g ( z ) ∼ b L ( z ) D ( z ) p ( z ) σ (0) , (25)In a Universe in which primordial density fluctuationswere Gaussian, the non-linear nature of gravitational dy-namics leads to the emergence of a non-trivial skewness ofthe local density PDF.Within the framework of gravitational perturbationtheory, the first non-vanishing term describing the evolu-tion of the skewness of a top-hat filtered, initially Gaussianmatter density field corresponds to second-order.According to non-linear, second-order perturbationtheory predictions, the skewness of the mass distribu- Fig. 4.
Evolution of the r.m.s deviation (top) and skew-ness (bottom) of the PDF of galaxy fluctuations on ascale R = 10 h − Mpc. The filled squares correspond totwo volume-limited samples from the VVDS with M B < −
20 + 5 log h covering the redshift intervals indicated bythe shaded regions. Triangles correspond to the 2dFGRSmeasurements at z ≃ .
15 (Croton et al., 2004), froma sample including similarly bright galaxies. Error barsgive 68% confidence errors, and, in the case of VVDSmeasurements, include the contribution from cosmic vari-ance. The dashed lines in both panels show the theo-retical predictions for the evolution of the variance (Eq.25) and skewness (Eq. 27) inferred using VVDS mea-surement of biasing. Predictions for the skewness (basedon the ( b ( z ) , b ( z )) measurements in the redshift range0 . < z < . z ∼ b /b = − .
34, dot-dashed line).tion is approximately independent of time, scale, den-sity, or geometry of the cosmological model. Assumingthat its evolution only depends on the hypothe-sis that the initial fluctuations are small and quasi-Gaussian and that they grow via gravitational clus-tering one derives that, in redshift-distorted space(Peebles, 1980; Juszkiewicz, Bouchet, & Colombi, 1993;Bernardeau, 1994; Hivon et al., 1995) S ∼ . − . n + 3) (26)where n is the effective slope of the power spectrum on thescales of interest [i.e. in our case, since R = 10 h − Mpc ,n is approximately given by -1.2 (e.g. Bernardeau et al., ∼1
34, dot-dashed line).tion is approximately independent of time, scale, den-sity, or geometry of the cosmological model. Assumingthat its evolution only depends on the hypothe-sis that the initial fluctuations are small and quasi-Gaussian and that they grow via gravitational clus-tering one derives that, in redshift-distorted space(Peebles, 1980; Juszkiewicz, Bouchet, & Colombi, 1993;Bernardeau, 1994; Hivon et al., 1995) S ∼ . − . n + 3) (26)where n is the effective slope of the power spectrum on thescales of interest [i.e. in our case, since R = 10 h − Mpc ,n is approximately given by -1.2 (e.g. Bernardeau et al., ∼1 S ,g ∼ b ( z ) − h S + 3 b ( z ) b ( z ) i . (27)The curves in both panels of Fig. 4 show that equations(25) and (27) reproduce extremely well the evolution ofvariance and skewness observed within the VVDS.The mass PDF is a one-parameter family of curvescompletely specified once the linear evolution model forthe mass variance h δ i is supplied. This implies that our non-linear biasing estimate is fully independent from pre-dictions of higher-order perturbation theory. On the con-trary, non-linear biasing at z = 0 is inferred by directlymatching 3-point galaxy statistics with the correspond-ing mass statistics derived from weakly non-linear per-turbation theory (e.g. Verde et al. 2000, Gazta˜naga etal. 2005). As a consequence, the agreement we find be-tween predicted and observed third-order moments is nota straightforward consequence of the method used to de-rive the biasing function. These results provide an indi-cation of the consistency, at z = 1, of some constitutiveelements of the standard picture of gravitational instabil-ity from Gaussian initial conditions.Concerning the local measurements from 2dFGRS, thepredicted scaling for the skewness continues to show verygood agreement if the local, non-linear measurement ofGazta˜naga et al. (2005) is considered. The value of S ,g ,however, cannot be consistent with GIP predictions if inthe local universe the simple linear biasing measurementof Verde et al. (2002) (i.e. b = 0) is adopted. In the above comparison of galaxy samples at three differ-ent epochs, we have so far neglected an important point.Galaxy luminosity evolves significantly between z ∼ z ∼
1, with a mean brightening of at least 1 magnitudefor an average spectral type (Ilbert et al. 2005). Thus,the contribution to the clustering signal at progressivelyearlier epochs may not be be due to the progenitors ofthe galaxies that are sampled at later times in the sameluminosity interval. Luminosity evolution between z = 1and z = 1 . z = 1 and z ∼
5. Conclusions
The results presented in this paper provide the first directevidence at z ∼ < z < . M B ≤ −
20 + 5 log h ) at different epochsunambiguously reveals the time-dependent effects of grav-itational evolution:a) underdense regions progressively occupy a largervolume fraction as a function of cosmic time, as expectedfrom gravitational growth in an expanding background;b) the second moment of the field traced by this“normal” population of galaxies (with median luminos-ity ∼ L ∗ ) is statistically consistent with the local ( z ∼ < z < .
5. This implies that the apparent inhomogeneity in thegalaxy distribution remains similar, i.e. galactic fluctu-ations have almost frozen over nearly 2/3 of the ageof the universe (Giavalisco et al., 1998; Coil et al., 2004;Pollo et al., 2005). We have shown that this is readily ex-plained by the combination of the gravitational growth ofmass fluctuations with the evolution of the bias betweengalaxies and mass. These two factors almost cancel eachother out;c) there are some hints that the skewness increaseswith cosmic time, its value at z ∼ . σ times lower than that measured locally by the 2dF-GRS for similarly luminous galaxies. In particular, themeasured value of the skewness at z ∼ . R = 10 h − Mpc ) indicates that galaxy fluctuations arestrongly non-Gaussian ( ∼ σ detection) even at such anearly epoch (see Cappi et al. 2008 for a different approachwhich arrives at similar conclusions);d) remarkably, once VVDS measurements of non-linearbiasing are included, both these trends are consistent withpredictions of linear and second-order perturbation the- arinoni et al.: Testing Gravity at z ∼ ory for the evolution of gravitational perturbations as de-scribed within the framework of general relativity;e) we have shown that the values of the skewnesswe measure at high redshift are difficult to reconcilewith the 2dFGRS measurements if local biasing is linear(Verde et al., 2002). A fully coherent gravitational pictureemerges, however, over the whole baseline 0 < z < . | b /b | is a decreasing function of red-shift at least up to z ∼ . Acknowledgments
We thank the referee, M. Strauss, for important sugges-tions that significantly improved the manuscript. LG ac-knowledges the hospitality of MPE and the “ExcellenceCluster Universe” in Garching, where part of thiswork was completed. This research has been developedwithin the framework of the VVDS consortium andit has been partially supported by the CNRS-INSUand its Programme National de Cosmologie (France),and by the Italian Ministry (MIUR) grants COFIN2000(MM02037133) and COFIN2003 (num.2003020150). TheVLT-VIMOS observations have been carried out on guar-anteed time (GTO) allocated by the European SouthernObservatory (ESO) to the VIRMOS consortium, under acontractual agreement between the Centre National de laRecherche Scientifique of France, heading a consortium ofFrench and Italian institutes, and ESO, to design, manu-facture and test the VIMOS instrument.
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