The MICE Grand Challenge Lightcone Simulation II: Halo and Galaxy catalogues
M. Crocce, F. J. Castander, E. Gaztanaga, P. Fosalba, J. Carretero
MMon. Not. R. Astron. Soc. , 000–000 (0000) Printed 16 September 2015 (MN L A TEX style file v2.2)
The MICE Grand Challenge Lightcone Simulation II:Halo and Galaxy catalogues
M. Crocce, F. J. Castander, E. Gazta˜naga, P. Fosalba & J. Carretero
Institut de Ci`encies de l’Espai, IEEC-CSIC, Campus UAB, Facultat de Ci`encies, Torre C5 par-2, Barcelona 08193, Spain
16 September 2015
ABSTRACT
This is the second in a series of three papers in which we present an end-to-end sim-ulation from the MICE collaboration, the MICE Grand Challenge (MICE-GC) run.The N-body contains about 70 billion dark-matter particles in a (3 h − Gpc) comov-ing volume spanning 5 orders of magnitude in dynamical range. Here we introducethe halo and galaxy catalogues built upon it, both in a wide (5000 deg ) and deep( z < .
4) lightcone and in several comoving snapshots. Halos were resolved downto few 10 h − M (cid:12) . This allowed us to model galaxies down to absolute magnitudeM r < − .
9. We used a new hybrid Halo Occupation Distribution and AbundanceMatching technique for galaxy assignment. The catalogue includes the Spectral En-ergy Distributions of all galaxies. We describe a variety of halo and galaxy clusteringapplications. We discuss how mass resolution effects can bias the large scale 2-pt clus-tering amplitude of poorly resolved halos at the (cid:46)
5% level, and their 3-pt correlationfunction. We find a characteristic scale dependent bias of (cid:46)
6% across the BAO featurefor halos well above M (cid:63) ∼ h − M (cid:12) and for LRG like galaxies. For halos well below M (cid:63) the scale dependence at 100 h − Mpc is (cid:46)
MICECATv1.0 )publicly available through a dedicated web portal, http://cosmohub.pic.es , to helpdevelop and exploit the new generation of astronomical surveys.
Key words: (cosmology:) observations, large-scale structure of Universe, dark energy,distance scale
Over the past two decades our understanding of the Universehas improved dramatically, in good part thanks to ground-breaking observational campaigns (Riess et al. 1998; Perl-mutter et al. 1999; Bennett et al. 2003; Cole et al. 2005;Tegmark et al. 2004). Although very successful this efforthas opened the window to yet larger challenges that remainunresolved. For instance deciphering the reason for the latetime acceleration of the Universe, what can result in totallynew forms of energy or in the need to re-formulate Einstein’stheory of gravity. There is also a need to shed light into thenature of dark-matter and the neutrino sector, and of a bet-ter understanding of the galaxy formation process.The community has responded to these challenges witha multi-probe approach consisting of several observational tests carried on independently or combined. From clusterabundance and weak lensing studies to large scale galaxyclustering including the baryon acoustic oscillations and red-shift space distortions (WiggleZ , BOSS , CFHTLenS , DES ,Euclid , DESI ) in addition to state-of-the-art supernovaeand CMB experiments (Planck Collaboration et al. 2014; wigglez.swin.edu.au/ desi.lbl.govc (cid:13) a r X i v : . [ a s t r o - ph . C O ] S e p Crocce et al.
Run N part L box / h − Mpc
P MGrid m p / (10 h − M (cid:12) ) l soft / h − Kpc z i Max.T imeStep
MICE-GC 4096 .
93 50 100 0 . .
42 50 50 0 . . Table 1.
Description of the MICE N-body simulations. N part denotes number of particles, L box is the box-size, P MGrid gives the sizeof the Particle-Mesh grid used for the large-scale forces computed with FFTs, m p gives the particle mass, l soft is the softening length,and z in is the initial redshift of the simulation. All simulations had initial conditions generated using the Zeldovich Approximation. Max.Timestep is the initial global time-stepping used, which is of order 1% of the Hubble time (i.e, d log a = 0 .
01, being a the scale factor).The number of global timesteps to complete the runs were N steps > ∼ (cid:104) M allh (cid:105) (cid:104) M cenh (cid:105) M cenh, min Red Cen. [%] Red Sat. [%] M r < −
19 1 . × .
23 1 . × . × . ×
35 77 M r < −
20 8 . × .
25 2 . × . × . ×
46 80 M r < −
21 1 . × .
24 8 . × . × . ×
62 87 M r < −
22 2 . × .
13 5 . × . × . ×
85 98
Table 2.
Some basic properties of the MICE lightcone galaxy catalogue that we make publicly available with this series of papers. Thecatalogue subtends one octant of the full sky and reaches z = 1 . M r > − . M h > × h − M (cid:12) ). The table lists the number of galaxies, satellite fraction, and meanhost halo mass for all galaxies (“central+satellites”) and “centrals only” above some magnitude limits. Also listed is the minimum hosthalo mass for centrals and the fraction of red centrals and red satellites (w.r.t. all centrals and all satellites above the given M r cut). Astier et al. 2006) and deep surveys such as GAMMA orPAU , among many others.The task ahead is nonetheless very hard because thesedatasets will have an unprecedented level of precision, andthus require ourselves to match it with well suited analysistools. In this regard, large and complex simulations are be-coming a fundamental ingredient to develop the science andto properly interpret the results (e.g. see Fig. 2 in Fosalbaet al. (2013)).This paper is the second in a series of three in which wepresent a state-of-the-art end-to-end simulation composedof several steps, with a strong focus in matching observa-tional constrains and a galaxy catalogue in the lightcone asan end-product. This was built upon a new N-body sim-ulation developed by the MICE collaboration, the MICE Grand Challenge run (MICE-GC), that includes about 70billion dark-matter particles in a box of about 3 h − Gpcaside. Details of the N-body run are given in Table 1 and inthe companion Paper I (Fosalba et al. 2013). The N-bodyset up was a compromise between sampling the largest vol-ume possible without repetition, e.g. the one of the ongoingDES survey (The Dark Energy Survey Collaboration 2005), while maintaining a high mass resolution, of ∼ h − M (cid:12) (necessary to reach the observed magnitude limits of cur-rent and some future observations). The MICE-GC N-bodyrun is introduced in Paper I, with an elaborated discussionof the resulting dark-matter clustering properties and thecomparison with lower resolution runs.Next we built halo and galaxy catalogues both in co-moving and lightcone outputs. By construction the galaxycatalogue matches observed luminosity functions, color dis-tributions and clustering as a function of luminosity andcolor at low- z (Blanton et al. 2003, 2005; Zehavi et al. 2011).Galaxy properties are then evolved into the past-lightconeusing evolutionary models. Some properties of the resultinglightcone galaxy catalogue are given in Table 2. Note thatwe also built galaxy catalogues for the comoving outputs,which are very useful for some concrete studies. The dis-cussion of the halo and galaxy catalogue construction, theirproperties and their potential in terms of clustering studiesare the subject of this paper (Paper II).Lastly we used the dark-matter distribution in the light-cone discussed in Paper I to build all sky lensing potentialsand hence add lensing properties to the galaxies such asshear and kappa values, magnified luminosities and posi-tions, and ongoing work with intrinsic alignments. The de-tails of this procedure, its validation and applications are thesubject of the companion Paper III (Fosalba et al. 2015). c (cid:13)000
Some basic properties of the MICE lightcone galaxy catalogue that we make publicly available with this series of papers. Thecatalogue subtends one octant of the full sky and reaches z = 1 . M r > − . M h > × h − M (cid:12) ). The table lists the number of galaxies, satellite fraction, and meanhost halo mass for all galaxies (“central+satellites”) and “centrals only” above some magnitude limits. Also listed is the minimum hosthalo mass for centrals and the fraction of red centrals and red satellites (w.r.t. all centrals and all satellites above the given M r cut). Astier et al. 2006) and deep surveys such as GAMMA orPAU , among many others.The task ahead is nonetheless very hard because thesedatasets will have an unprecedented level of precision, andthus require ourselves to match it with well suited analysistools. In this regard, large and complex simulations are be-coming a fundamental ingredient to develop the science andto properly interpret the results (e.g. see Fig. 2 in Fosalbaet al. (2013)).This paper is the second in a series of three in which wepresent a state-of-the-art end-to-end simulation composedof several steps, with a strong focus in matching observa-tional constrains and a galaxy catalogue in the lightcone asan end-product. This was built upon a new N-body sim-ulation developed by the MICE collaboration, the MICE Grand Challenge run (MICE-GC), that includes about 70billion dark-matter particles in a box of about 3 h − Gpcaside. Details of the N-body run are given in Table 1 and inthe companion Paper I (Fosalba et al. 2013). The N-bodyset up was a compromise between sampling the largest vol-ume possible without repetition, e.g. the one of the ongoingDES survey (The Dark Energy Survey Collaboration 2005), while maintaining a high mass resolution, of ∼ h − M (cid:12) (necessary to reach the observed magnitude limits of cur-rent and some future observations). The MICE-GC N-bodyrun is introduced in Paper I, with an elaborated discussionof the resulting dark-matter clustering properties and thecomparison with lower resolution runs.Next we built halo and galaxy catalogues both in co-moving and lightcone outputs. By construction the galaxycatalogue matches observed luminosity functions, color dis-tributions and clustering as a function of luminosity andcolor at low- z (Blanton et al. 2003, 2005; Zehavi et al. 2011).Galaxy properties are then evolved into the past-lightconeusing evolutionary models. Some properties of the resultinglightcone galaxy catalogue are given in Table 2. Note thatwe also built galaxy catalogues for the comoving outputs,which are very useful for some concrete studies. The dis-cussion of the halo and galaxy catalogue construction, theirproperties and their potential in terms of clustering studiesare the subject of this paper (Paper II).Lastly we used the dark-matter distribution in the light-cone discussed in Paper I to build all sky lensing potentialsand hence add lensing properties to the galaxies such asshear and kappa values, magnified luminosities and posi-tions, and ongoing work with intrinsic alignments. The de-tails of this procedure, its validation and applications are thesubject of the companion Paper III (Fosalba et al. 2015). c (cid:13)000 , 000–000 he MICE Grand Challenge: Halos and Galaxies We make the first version of the MICE-GC lightconegalaxy and halo catalogue (
MICECAT v1.0 ) publicly availableat the dedicated web portal http://cosmohub.pic.es , withthe hope that can be of value to help develop the science, thedesign and the exploitation of new wide-area cosmologicalsurveys.In this paper, besides the catalogue validation, we studythree concrete issues: (1) how the halo clustering on large-scales depends on the mass resolution of the underlying N-body simulation for fixed halo mass samples (2) the haloand galaxy clustering from small scales to very large oneswith a focus on scale dependent bias and cross-correlationcoefficients (3) limitations of the Kaiser limit in RedshiftSpace and in particular the impact of satellite galaxies inthe multipole moments of the anisotropic clustering.This paper is organized as follows: Section 2 presentsthe MICE-GC catalogue of friend-of-friends halos, the massfunction determined from the comoving and lightcone out-puts, as well as some clustering properties. Section 3 de-scribes the galaxy mocks built upon the MICE-GC halos andtheir properties. In particular, we show predictions from thegalaxy mock for the clustering and color distribution com-pared to observations at high- z where these properties werenot matched by construction. In Sec. 4 we discuss mass res-olution effects in 2 and 3 point halo clustering statistics. InSec. 5 we investigate the scale dependence of bias for sev-eral halo and galaxy samples. In Sec. 6 we turn into redshiftspace and study the applicability of the large-scale Kaiserlimit in the lightcone, and the generation of non-trivial mul-tipole moments due to satellite motions within halos. Fi-nally, Sec. 7 describes our public catalogue release and Sec. 8summarizes our main results and conclusions. One of the fundamental data products of the MICE-GC sim-ulation are halo catalogues, which we have built out of both,comoving and lightcone dark-matter outputs subtending thefull sky.We identified halos using a Friend-of-Friend (FOF) al-gorithm with a standard linking length of 0.2 (in units ofthe mean inter-particle distance) both for the comoving out-puts and the lightcone (as the mean matter density is inde-pendent of redshift). We used a FoF code built upon theone publicly available at ,with some concrete improvements needed to handle largeamount of data in due time (each MICE-GC output is about1TB of data).The resulting halo catalogues contain basic halo infor-mation as well as positions and velocities of all the particlesforming each halo. This allowed us to measure also halo3D shapes and angular momentum. This is a key ingredi-ent for a number of further applications, but in particular it (cid:159)(cid:159)(cid:159)(cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159)(cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:64) M (cid:144)(cid:72) M (cid:159) h (cid:45) (cid:76)(cid:68) n (cid:144) n S T MICE (cid:45)
GrandChallenge
MICE (cid:45)
Intermediate z (cid:61) Crocce et. al (cid:72) (cid:76) fit
Figure 1.
Mass Function in MICE at z = 0. Black symbols showthe halo abundance in the Grand Challenge run while red symbolscorrespond to the intermediate simulation with the same cosmol-ogy but 8 times worse mass resolution ( m p = 2 . × h − M (cid:12) vs. m p = 2 . × h − M (cid:12) ). The dashed line results from afit to a series of MICE simulations (Crocce et al. 2010). Abun-dances are depicted relative to the Sheth and Tormen model anderror-bars were estimated using jack-knife resampling. will permit to incorporate intrinsic alignments in the lensingcatalogues discussed in the companion Paper III.In what follows we validate the abundance and cluster-ing of the MICE-GC halo catalogues. In Sec. 4 we go onestep further and compare the clustering results with those ofprevious runs to investigate mass resolution effects in halobias. Let us begin by looking at the halo abundance. The halofinder in MICE-GC yields about 172 million FoF halos with20 or more particles in the comoving output at z = 0, andabout 74 millions in each octant of the full sky lightcone upto z = 1 . z = 0 in MICE-GC compared to both the onemeasured in MICE-IR and the numerical fit to a large setof MICE simulations from Crocce et al. (2010) (we depictthe ratio to the well known Sheth & Tormen (1999) massfunction to highlight details). For each simulation, we onlyshow a mass-range in which halos are well sampled contain-ing ∼
150 or more particles. At the high-mass end, werehalos are very well sampled in both MICE-GC and MICE-IR, both mass functions agree very well. Towards the regimeof M/ ( h − M (cid:12) ) ∼ . to 10 MICE-GC has a slightlylarger halo abundance, by ∼
2% (a trend that continues tolower mass, not shown in Fig. 1). Within 2% the fit from c (cid:13) , 000–000 Crocce et al.
Crocce et al. (2010) reproduces the shape of the MICE-GCmass function.We note that in defining our halo masses we have ac-counted for the Warren correction for discrete halo sam-pling , unless otherwise stated (Warren et al. (2006); Crocceet al. (2010); Bhattacharya et al. (2011) and ref. therein).This means that M h = n h m p (1 − n − . h ), with m p beingthe particle mass and n h the corresponding number of par-ticles in halo. This brings the shape of both mass func-tions into a much better agreement across the mass rangeshown in Fig. 1. In addition MICE-IR have been correctedfor transients as described in Crocce et al. (2010) (this cor-rection is negligible for M/ ( h − M (cid:12) ) ∼ . and ∼ M/ ( h − M (cid:12) ) ∼ ). Lastly we have also accounted forthe fact that the initial transfer function in MICE-IR wasEH instead of CAMB, see Fig. 5 in Paper I (this introducesa (cid:46)
1% correction, depending on halo mass) .Later we will argue that one possible way of reach-ing fainter magnitudes when building galaxy mock cata-logues is by using poorly resolved structures (halos of (cid:38) M h = 2 . × h − M (cid:12) ) compared to the model predic-tion using the Crocce et al. (2010) fit . The cumulativeabundance is 10% −
15% lower than the prediction at thislimit, but it goes to within 5% for halos with ∼
40 particlesalready.In turn the redshift evolution of the MICE-GC haloabundance is shown in Fig. 3, with the halo mass functionmeasured in the lightcone for several consecutive redshiftsbins, as detailed in the inset labels. The evolution is in goodagreement with the fit from Crocce et al. (2010), which doesnot assume universality. Notice how M (cid:63) (roughly the massbeyond which the abundance is exponentially suppressed)decreases with redshift, as expected in hierarchical clustering(e.g. Cooray & Sheth (2002)). z = 0We now discuss some basic characterization and validationof the halo clustering in the comoving snapshot at z = 0,with a more in depth analysis postponed to Secs. 5 and 6.The combination of large volume and good mass res-olution of MICE-GC allows to study with high precision a As previously remarked this is mostly an empirical correction,neglecting details on other quantities such as halo concentration(Luki´c et al. 2009). This was done multiplying the MICE-IR measurements by themass function model prediction for the CAMB MICE-GC powerspectrum over the one for EH used in MICE-IR. In Crocce et al. (2010) it is shown that their fit works at thepercentage level on this mass range. ••••••••••••••••••••••••••••••••••••••••••• (cid:45) (cid:45) (cid:45) (cid:45) M h (cid:64) h (cid:45) M (cid:159) (cid:68) n (cid:72) (cid:62) M h (cid:76) •••••••••••••••••••••••••••••••••••••••10 Grand Challenge (cid:72) z (cid:61) (cid:76) Figure 2.
Cumulative mass function measured in MICE-GCdown to the extreme regime of poorly resolved halos with 10 ormore particles. The inset panel show the ratio to the predictionfor this quantity using the fit from Crocce et al. (2010), which isdepicted by a solid green line in the main panel. (cid:232)(cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232) (cid:232)(cid:232) (cid:248)(cid:248) (cid:248)(cid:248)(cid:248)(cid:248)(cid:248)(cid:248)(cid:248)(cid:248)(cid:248)(cid:248)(cid:248)(cid:248)(cid:248)(cid:248)(cid:248)(cid:248)(cid:248)(cid:248)(cid:248)(cid:248)(cid:248)(cid:248)(cid:248) (cid:248) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224)
12 13 14 15 (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Log (cid:64) M (cid:144)(cid:72) M (cid:159) h (cid:45) (cid:76)(cid:68) L og (cid:64) n (cid:144) (cid:72) M p c h (cid:45) (cid:76) (cid:45) (cid:68) MICE (cid:45) GC Lightcone (cid:60) z (cid:60) (cid:60) z (cid:60) (cid:60) z (cid:60) (cid:60) z (cid:60) Figure 3.
Halo abundance as a function of redshift in the MICE-GC lightcone for several consecutive redshift bins. The theorymodel shown, from Crocce et al. 2010, does not assume univer-sality. range of quite different halo samples in terms of clustering,from anti-biased and un-biased to highly biased ones. Thered symbols in Fig. 4 show the z = 0 halo-halo power spec-trum for 3 such samples in MICE-GC ( b h = 0 . , . c (cid:13)000
Halo abundance as a function of redshift in the MICE-GC lightcone for several consecutive redshift bins. The theorymodel shown, from Crocce et al. 2010, does not assume univer-sality. range of quite different halo samples in terms of clustering,from anti-biased and un-biased to highly biased ones. Thered symbols in Fig. 4 show the z = 0 halo-halo power spec-trum for 3 such samples in MICE-GC ( b h = 0 . , . c (cid:13)000 , 000–000 he MICE Grand Challenge: Halos and Galaxies k (cid:64) h (cid:45) Mpc (cid:68) P t r ace r (cid:72) k (cid:76) (cid:144) P s m oo t h Halos M h (cid:62) (cid:180) M (cid:159) (cid:144) h Only Central GalsAll Gals M r (cid:60) (cid:45) (cid:72) b (cid:61) (cid:76) P dark (cid:45) matter k (cid:64) h (cid:45) Mpc (cid:68)
Halos M h (cid:62) M (cid:159) (cid:144) h Only Central GalsAll Gals M r (cid:60) (cid:45) (cid:72) b (cid:61) (cid:76) P dark (cid:45) matter k (cid:64) h (cid:45) Mpc (cid:68) z (cid:61) Halos M h (cid:62) M (cid:159) (cid:144) h Only Central GalsAll Gals M r (cid:60) (cid:45) (cid:72) b (cid:61) (cid:76) P dark (cid:45) matter Figure 4.
Large-scale halo and galaxy (auto) power spectrum in the MICE-GC comoving output at z = 0 (over a smooth broad-bandpower, without shot-noise correction). We display three self corresponding magnitude and mass threshold samples. For a given halo massthreshold we select the corresponding magnitude limited galaxy sample from the mean “halo mass - central galaxy luminosity” relation.We then consider both “centrals only” or “central+satellites” in each sample. The dashed black line corresponds to the clustering ofdark matter. The figure shows that the large volume and good mass-resolution in MICE-GC allows to study large-scale clustering fromanti-biased or un-biased samples to highly biased ones, with percent level error-bars at BAO scales. In addition notice how in generalsatellite galaxies increase the clustering amplitude above the “centrals only” without introducing noticeable extra scale-dependence. corresponds to the measured dark-matter clustering, whichon these scales agrees with perturbation theory predictions(RPT, Crocce & Scoccimarro (2008)) and numerical fits(Takahashi et al. 2012; Heitmann et al. 2014) to 2% or bet-ter, see Paper I. All measured spectra in Fig. 4 have beendivided by a smooth broad-band power to highlight narrowband features, and are not corrected by shot-noise. Reportederror bars assume the FKP approximation (Feldman et al.1994): σ P k = (cid:112) N m ( k ) / P k + 1 / ¯ n ), where N m is the num-ber of Fourier modes used to measure the band-power P k and ¯ n the tracer comoving number density.One basic validation is the comparison of the bias mea-sured from P hh /P mm to the prediction using the peak back-ground split argument (PBS) (Bardeen et al. 1986; Cole &Kaiser 1989), employing the mass function fit to MICE runsfrom Crocce et al. (2010) as input to the PBS formulas (fol-lowing Eq. (23) in Manera et al. (2010)). For the samples inFig. 4 we find,Halo Sample b measured bias PBS M h / ( h − M (cid:12) ) > × . ± .
002 0 . M h / ( h − M (cid:12) ) > . ± .
002 0 . M h / ( h − M (cid:12) ) > . ± .
009 1 . P hh and P mm into Eqs. (16) and (17) of Smith et al. (2007) for the first 10 k -bins at the largestscales (0 . < k/ h Mpc − < . . For b ∼ ∼ low biasing regime has not been explored before. We next explore how the clustering evolves with redshiftusing the lightcone halo catalogue (see also Sec. 6.1).Figure 5 shows the effect of halo bias on the 2-pointcorrelation at BAO scales, for a sample of FoF halos withmasses 6 × < M h / ( h − M (cid:12) ) < . × (more than20 particles and less than 50 particles). We can see how themain effect of halo biasing with redshift (for constant halomass) is to change the amplitude of the correlations in away that is roughly degenerate with the linear growth in thedark-matter (DM) distribution. The linear bias b , definedas the square root of the ratio of the halo correlation to thecorresponding DM correlation is about b (cid:39) . b (cid:39) . Notice however that we refer to mass threshold samples, andbias from halo-halo power spectra.c (cid:13) , 000–000
Crocce et al.
Figure 5.
Symbols show the 2-point correlation function ξ ( r )(scaled by r ) in FoF halos (with 20-50 particles) for three redshiftbins in the lightcone (in real space). Dashed lines are the corre-sponding DM smoothed linear theory predictions (which resemblenon-linear predictions). Notice how halo biasing for constant massis roughly degenerate with growth, yielding a constant clusteringamplitude. redshifts (i.e. b ∝ /D ( z )), in contrast to the correspondingDM results shown in solid lines. Note that the BAO peakcan be well detected despite the bias. We leave for Sec. 5 amore detailed study of scale dependent bias across the BAOscales.Lastly we note that the binning used in Fig. 5 (and laterin Fig. 12) is rather broad to make the figure less crowdedand reduce the covariance between data-points, but we havechecked that a narrower binning does not change the results. We have built galaxy mock catalogues from both lightconeand comoving outputs, starting from the corresponding halocatalogues discussed in the previous section. These mocksare generated to provide a tool to design, understand andanalyze cosmological surveys such as PAU, DES and Euclid.We have used a new hybrid technique that combineshalo occupation distribution (HOD; e.g. Jing et al. (1998);Scoccimarro et al. (2001); Berlind & Weinberg (2002)) andhalo abundance matching (HAM; e.g. Vale & Ostriker (2004);Tasitsiomi et al. (2004); Conroy et al. (2006)) prescriptions.We do not intend to reproduce the details of the cataloguegeneration here, but just to present some validation resultsto give an idea of how the mocks compare to observationsand therefore provide a glimpse of their possible use. Thegalaxy assignment method is described in full detail in Car- retero et al. (2015). Further details regarding redshift evo-lution of galaxy properties will be given in Castander et al.(2014), in preparation.We stress however that our galaxy catalogue is not in-tended to reproduce one given galaxy population, as typ-ically needed for spectroscopic surveys (e.g. the CMASSsample for BOSS in Manera et al. (2013) and White et al.(2014)). In that sense our scope is larger and more complexas we aim to reproduce the abundance and clustering acrossluminosity and color space, and its evolution with redshift(i.e. luminosity functions, color-magnitude diagrams, clus-tering as a function of color and magnitude cuts, and more).In order to generate the galaxy catalogue, we assumethat halos are populated by central and satellite galaxies.We assume that all halos have one central galaxy and anumber of satellites given by an HOD prescription, whichdetermines the mean number of satellite galaxies as a func-tion of the halo mass. We assign luminosities to the centralgalaxies with a halo mass-luminosity relation computed withHAM techniques, matching the abundances of the galaxy lu-minosity function and the halo mass function. We need tointroduce scatter in this relation in order to fit the galaxyclustering at bright luminosities. We then populate the haloswith the number of satellites given by the HOD parametersthat fit the resulting 2-point projected correlation functionof galaxies to observations. We assign luminosities to thesatellites to preserve the observed galaxy luminosity func-tion and the dependence of clustering with luminosity, im-posing the additional constraint that the luminosity of satel-lites in a given halo cannot exceed the luminosity of thecentral galaxy by more than 5% (Carretero et al. 2015).Central galaxies are placed at the halo center of mass.Satellite galaxies are positioned into the halos following atriaxial NFW profile with fixed axis ratios (e.g. Jing & Suto(2002)). In order to fit clustering observations as a functionof luminosity on small scales (one-halo regime) we need toconcentrate satellite galaxies more than the standard dark-matter distribution relation given in Bullock et al. (2001).Hence we employ a concentration parameter in each axisgiven by the relation in Bullock et al. (2001) with slightchanges depending on galaxy luminosities (Carretero et al.2015). Similar conclusions are found in Watson et al. (2012).Central galaxies are assigned the center of mass velocityof the host halo. In turn, we assume that satellite galaxieshave in addition a virial motion on top of the bulk halovelocity (Sheth & Diaferio 2001). We assume that the halosare virialized and that the satellites velocities within the halohave a velocity dispersion given by the halo mass (Bryan &Norman 1998). We assign velocities to the satellite galaxiesdrawing each component of the velocity vector from threeindependent Gaussian distribution of dispersion 1 / √ c (cid:13)000
Symbols show the 2-point correlation function ξ ( r )(scaled by r ) in FoF halos (with 20-50 particles) for three redshiftbins in the lightcone (in real space). Dashed lines are the corre-sponding DM smoothed linear theory predictions (which resemblenon-linear predictions). Notice how halo biasing for constant massis roughly degenerate with growth, yielding a constant clusteringamplitude. redshifts (i.e. b ∝ /D ( z )), in contrast to the correspondingDM results shown in solid lines. Note that the BAO peakcan be well detected despite the bias. We leave for Sec. 5 amore detailed study of scale dependent bias across the BAOscales.Lastly we note that the binning used in Fig. 5 (and laterin Fig. 12) is rather broad to make the figure less crowdedand reduce the covariance between data-points, but we havechecked that a narrower binning does not change the results. We have built galaxy mock catalogues from both lightconeand comoving outputs, starting from the corresponding halocatalogues discussed in the previous section. These mocksare generated to provide a tool to design, understand andanalyze cosmological surveys such as PAU, DES and Euclid.We have used a new hybrid technique that combineshalo occupation distribution (HOD; e.g. Jing et al. (1998);Scoccimarro et al. (2001); Berlind & Weinberg (2002)) andhalo abundance matching (HAM; e.g. Vale & Ostriker (2004);Tasitsiomi et al. (2004); Conroy et al. (2006)) prescriptions.We do not intend to reproduce the details of the cataloguegeneration here, but just to present some validation resultsto give an idea of how the mocks compare to observationsand therefore provide a glimpse of their possible use. Thegalaxy assignment method is described in full detail in Car- retero et al. (2015). Further details regarding redshift evo-lution of galaxy properties will be given in Castander et al.(2014), in preparation.We stress however that our galaxy catalogue is not in-tended to reproduce one given galaxy population, as typ-ically needed for spectroscopic surveys (e.g. the CMASSsample for BOSS in Manera et al. (2013) and White et al.(2014)). In that sense our scope is larger and more complexas we aim to reproduce the abundance and clustering acrossluminosity and color space, and its evolution with redshift(i.e. luminosity functions, color-magnitude diagrams, clus-tering as a function of color and magnitude cuts, and more).In order to generate the galaxy catalogue, we assumethat halos are populated by central and satellite galaxies.We assume that all halos have one central galaxy and anumber of satellites given by an HOD prescription, whichdetermines the mean number of satellite galaxies as a func-tion of the halo mass. We assign luminosities to the centralgalaxies with a halo mass-luminosity relation computed withHAM techniques, matching the abundances of the galaxy lu-minosity function and the halo mass function. We need tointroduce scatter in this relation in order to fit the galaxyclustering at bright luminosities. We then populate the haloswith the number of satellites given by the HOD parametersthat fit the resulting 2-point projected correlation functionof galaxies to observations. We assign luminosities to thesatellites to preserve the observed galaxy luminosity func-tion and the dependence of clustering with luminosity, im-posing the additional constraint that the luminosity of satel-lites in a given halo cannot exceed the luminosity of thecentral galaxy by more than 5% (Carretero et al. 2015).Central galaxies are placed at the halo center of mass.Satellite galaxies are positioned into the halos following atriaxial NFW profile with fixed axis ratios (e.g. Jing & Suto(2002)). In order to fit clustering observations as a functionof luminosity on small scales (one-halo regime) we need toconcentrate satellite galaxies more than the standard dark-matter distribution relation given in Bullock et al. (2001).Hence we employ a concentration parameter in each axisgiven by the relation in Bullock et al. (2001) with slightchanges depending on galaxy luminosities (Carretero et al.2015). Similar conclusions are found in Watson et al. (2012).Central galaxies are assigned the center of mass velocityof the host halo. In turn, we assume that satellite galaxieshave in addition a virial motion on top of the bulk halovelocity (Sheth & Diaferio 2001). We assume that the halosare virialized and that the satellites velocities within the halohave a velocity dispersion given by the halo mass (Bryan &Norman 1998). We assign velocities to the satellite galaxiesdrawing each component of the velocity vector from threeindependent Gaussian distribution of dispersion 1 / √ c (cid:13)000 , 000–000 he MICE Grand Challenge: Halos and Galaxies Figure 6.
Scatter plot of apparent magnitudes, i AB , versus red-shifts for a random set of galaxies in our mock catalog. Thereare missing faint galaxies at low redshift because the sample islimited in absolute magnitude. At z (cid:39) i AB ∼
24, while at z (cid:39) . i AB ∼ Lastly we assign spectral energy distributions to thegalaxies with a recipe that preserves the observed color-magnitude diagram and the clustering as a function of color.The method has been tuned to match observational con-straints from SDSS at low redshift where they are morestringent (detailed further below). We nevertheless applythe same method throughout the lightcone at all redshiftswith slight modifications. In order to reproduce the observedgalaxy properties at higher redshifts, we impose evolution-ary corrections to the galaxy colors and obtain a final spec-tral energy distribution (SED) resampling from the COS-MOS catalogue of Ilbert et al. (2009) galaxies with compat-ible luminosity and (g-r) color at the given redshift. Onceeach galaxy has a SED assigned, we can compute any desiredmagnitude to compare to observations.One feature of the current version of the galaxy mockcatalogue presented in this paper is that it is absolute mag-nitude limited. This is inherited from the fact that is gen-erated from a parent halo catalogue that is mass limited.Current cosmological imaging surveys are normally apparentmagnitude limited down to faint magnitudes. The version ofthe catalogue that is now made public is complete down to i <
24 for redshifts z (cid:38) .
9. For lower redshifts the catalogis complete only to brighter magnitudes as illustrated byFig. 6.In order to overcome this incompleteness at low z wehave started populating sub-resolved halos (thanks to thefact that at the resolution limit of MICE-GC the halo biashas a weak dependence with mass) yielding catalogues which
Figure 7.
Contour plots for the (g-r) vs. (r-i) color distributionin the COSMOS sample (red) and the MICE galaxy mock (blue),for galaxies at z < < z < are complete to observed magnitudes i ∼
24 (for all z < . c (cid:13) , 000–000 Crocce et al.
Figure 8.
Angular two point correlation function of the COS-MOS (black dots and error bars) and MICE-GC catalogues (redsolid line) at redshift z = 1 . < i AB <
24. We also plot the MICE dark matter angular cor-relation function (blue dashed line) for comparison. In this plot,the MICE correlation has been computed in an area sixty timesthat of the COSMOS catalogue using the same mask. Given thelarger area of the MICE catalogue and to avoid overcrowding theplot, we do not show the MICE error bars. alogue. The COSMOS data has been cut in absolute mag-nitude, M V < − .
0, and redshift, z < .
4, to expand thesame ranges as the MICE catalogue. On the top panel wepresent the galaxies at z < .
8, where the overall color-colordistribution of our mock is a reasonable fit to observations.On the bottom panel we show the distribution of galaxies at0 . < z < .
4. In this case our catalogue is also an accept-able fit to observations albeit slightly over-producing bluegalaxies.Turning to clustering properties, in Fig. 8 we comparethe two point angular correlation function of the COSMOScatalogue (black dots and error bars) to our mock catalogue(red solid line) at redshift z = 1 .
0. In both catalogues, wehave selected all galaxies in the redshift range 0 . < z < . . < i <
24. The value of the corre-lation amplitude is very similar, except at scales larger than5 arc-minutes, where the COSMOS amplitude is larger thanthe catalogue (although at the 1- σ level). The COSMOSfield presents an overdensity at z ∼ . b ∼ . . < z <
75. Here we restrictboth samples to brighter galaxies 17 . < i AB < .
5, so thatboth are complete (see Fig. 6). The COSMOS photo-z er-rors are also much smaller for these brighter galaxies, so thecomparison is more direct. We also include the DM linearprediction (dotted), the non-linear DM prediction (contin-
Figure 9.
Top Panel: Angular two point correlation function ofthe COSMOS (circles with jack-knife errorbars) at photometricredshift 0 . < z <
75 and for a galaxy sample with 17 .
5. Triangles with errorbars correspond to the meanand dispersion of 50 COSMOS like MICE-GC catalogues. We usethe same magnitude and photo-z limits for MICE as in COSMOS.The redshift distribution of galaxies is shown in the inset for bothCOSMOS (continuous) and MICE (dashed), with (cid:104) z (cid:105) = 0 . uous) and the prediction including the finite area correc-tion (dashed line) resulting from the integral constrain inthe COSMOS area (of about 1 . ). We have simulatedthe COSMOS sample with 50 separate MICE-GC regionsof equal size, same magnitude limit (17 . < i < .
5) andsame redshifts distribution (shown in the inset). The COS-MOS data and the MICE-GC catalog agree remarkably well.In the bottom panel we also show the effective galaxy bias b estimated as b ( θ ) = (cid:112) w GAL ( θ ) /w DM ( θ ), the ratio of thegalaxy to non-linear DM prediction. The results are quiteconsistent between MICE-GC and COSMOS, despite thefact that MICE galaxy catalog was not built to match clus-tering by construction at these redshifts.As a further validation we have compared the cluster-ing in our catalogue to the one in the CMASS sample of theSDSS-III BOSS survey, on BAO scales. Note that this testis quite more challenging than before as it involves botha magnitude and a color selection of galaxies evolved inthe past light-cone. We built a MICE-CMASS sample byapplying the same selection for luminous red galaxies de-scribed in Eqs.(1-6) in Anderson et al. (2012). In the cur-rent version of the MICE-GC catalogue there is no evolutionof the luminosity function beyond the one of the underly-ing mass function. Before doing a combined color and lu- c (cid:13)000
5) andsame redshifts distribution (shown in the inset). The COS-MOS data and the MICE-GC catalog agree remarkably well.In the bottom panel we also show the effective galaxy bias b estimated as b ( θ ) = (cid:112) w GAL ( θ ) /w DM ( θ ), the ratio of thegalaxy to non-linear DM prediction. The results are quiteconsistent between MICE-GC and COSMOS, despite thefact that MICE galaxy catalog was not built to match clus-tering by construction at these redshifts.As a further validation we have compared the cluster-ing in our catalogue to the one in the CMASS sample of theSDSS-III BOSS survey, on BAO scales. Note that this testis quite more challenging than before as it involves botha magnitude and a color selection of galaxies evolved inthe past light-cone. We built a MICE-CMASS sample byapplying the same selection for luminous red galaxies de-scribed in Eqs.(1-6) in Anderson et al. (2012). In the cur-rent version of the MICE-GC catalogue there is no evolutionof the luminosity function beyond the one of the underly-ing mass function. Before doing a combined color and lu- c (cid:13)000 , 000–000 he MICE Grand Challenge: Halos and Galaxies Figure 10.
A MICE-GC luminous red galaxy (LRG) sample.Solid black line (labeled MICE-CMASS) is the redshift distri-bution for an LRG sample in the MICE-GC catalogue follow-ing the color and magnitude selection criteria as the SDSS-IIIBOSS CMASS sample in Anderson et al. (2012). Filled dots isthe actual distribution from CMASS-DR10. After the color se-lection the redshift distribution of MICE-CMASS matches verywell CMASS-DR10. Dashed line corresponds to a sample selectedonly in absolute luminosity such as to yield the same clusteringas DR10 (see Fig. 11) and the same total number of objects. minosity cut is important to account for this evolution .Hence we first correct the absolute magnitudes according toa functional fit derived by abundance matching between theevolving galaxy luminosity function and the halo mass func-tion, which was not included in the MICECAT v1.0 release, M evolvedr = M r + 0 . ∗ ( atan (1 . ∗ z ) − . . < i < .
9. Next we impose thecolor selection d ⊥ > .
46 and i < .
13 + 1 . d ⊥ − . d ⊥ = r − i − ( g − r ) / . ∼ of MICE-GC) and a very similar redshift distri-bution as the BOSS-CMASS, starting at z ∼ . z ∼ .
7. This is shown in Fig. 10. We then apply theredshift selection 0 . < z < .
7, as done in Anderson et al.(2012).We next measured the monopole and quadrupole corre-lation functions, focusing on large BAO scales. This is shownin Fig. 11 that compares our clustering estimate in the BOSSCMASS-DR10 sample with the one over the CMASS se-lection of MICE galaxies. Both the shape and BAO scale are We have tested that the clustering validation against COS-MOS discussed before is not affected by applying the evolution inluminosity prior to sample selection. Mainly because the samplesare not as bright as CMASS, neither they involve a color cut. We have checked that our results are in good agreement withthose in S´anchez et al. (2014)
Figure 11.
Quadrupole and Monopole correlation functions forthe CMASS samples built from the MICE-GC lightcone cata-logue, see Fig. 10. MICE-CMASS is shown in solid black whilethe actual CMASS DR10 measurements with filled symbols. Theoverall shape and BAO position is well traced by MICE-CMASS,while the bias is ∼
10% higher (see text for details). Dashed linecorresponds to a M r < − . quite well reproduced by our catalogue. While the relativelinear bias of MICE-CMASS is ∼
10% higher than the onein CMASS-DR10, both in monopole and quadrupole. Thereare several factors that can contribute to this small discrep-ancy. First, the MICE cosmology is different from the one inBOSS. Second, we have fixed HOD parameters to clusteringat z ∼
0, so we do not expect a perfect match at higher red-shift. This probably needs some HOD evolution other thanthe evolution in halo properties (for example in the mass-luminosity scatter which we apply always after the sameluminosity threshold across redshift). Third, MICE-CMASShas 27% satellites which is a factor of 3 times larger thanBOSS predictions (Tojeiro et al. 2012). As we fixed the to-tal galaxy abundance, a lower satellite fraction requires in-cluding more central galaxies, which by construction havesmaller halo mass and therefore smaller bias. The mean halomass in MICE-CMASS is M h = 3 . × h − M (cid:12) which is afactor 2 times higher than that expected in BOSS-CMASS.Such increase of a factor of 2 in halo mass can yield a ∼ M r < − .
2, which turnsout very close to the actual minimum luminosity of BOSS-CMASS galaxies. After the luminosity selection we dilute c (cid:13) , 000–000 Crocce et al.
MICE-GC to have the same number of objects as BOSS-CMASS. The redshift distribution is shown by a dashed linein Fig. 10, while the clustering in Fig. 11. The later agreesvery well with CMASS-DR10 although there is no color se-lection imposed.Although we have shown some concrete examples, theoverall comparison between photometric and clustering prop-erties of our catalogue to observations is good. This vali-dates our approach in constructing the galaxy mock cata-logue where we have applied stellar evolutionary correctionsto the colors of the galaxies to construct the mock cataloguein the lightcone extrapolating the other low redshift recipesto higher redshifts. We have also discussed a simple imple-mentation for evolution in galaxy luminosity, particularlyrelevant for narrow magnitude range selections.
The HOD prescription described above and used to populatethe MICE-GC simulation with galaxies is based on matchingthe observed luminosity function and the small scale galaxyclustering ( r (cid:46) h − Mpc). In this section we investigatewhat it implies for the clustering of galaxies on large-scales,in particular how this compares to the halo clustering al-ready discussed.Figure 4 has three panels corresponding to the powerspectrum of anti-biased, unbiased and highly biased halosamples discussed in Sec. 2.2.1. In each panel we now in-clude the clustering of galaxies brighter than the luminosityset by the corresponding mass-luminosity relation from theHOD+HAM prescription. We divide the (magnitude lim-ited) galaxy sample into centrals only (orange symbols) and central+satellites (blue symbols). The left-most panel showsthat faint central galaxies have almost the same clustering astheir host halos. This is because the mass-luminosity relationat this regime is one-to-one (and the clustering is dominatedby the most abundant galaxies). The addition of satellitesboost the clustering signal because faint satellites can live inmassive halos. On the other end bright central galaxies (rightmost panel) have less clustering strength than their corre-sponding halos from the mean halo mass - central luminosityrelation. This is due to the scatter in L cen = L ( M h ), for fixedmass. Hence, a sample of centrals with L > L (cid:63) = L ( M (cid:63) ) hasgalaxies residing in halos with M < M (cid:63) , what determinesthe (smaller) galaxy bias. Again adding the satellites boostthe signal, in this particular case to match that of the halos(right panel of Fig. 4).Lastly we turn to the evolution of clustering in the light-cone (see also Sec. 6.1). Figure 12 shows the monopole 3Dcorrelation function measured at BAO scales in 3 redshiftbins, for a magnitude limited galaxy sample ( r <
24) ex-tracted from one octant of the MICE lightcone catalogue inredshift space. The dashed lines are the linear theory predic-tions for the corresponding dark-matter clustering in real-
Figure 12.
Galaxy 3D monopole correlation function in thelightcone (open circles) for three redshift bins and a magnitudelimited sample r <
24 (to compare with Fig. 5). Dashed lines arethe corresponding linear theory predictions for dark-matter inreal-space, while solid lines include linear bias and redshift spacedistortions. The modeling, where bias has been obtained fromreal space measurements, agrees well with the galaxy monopole.In this case the bias evolves stronger than the growth such thatthe galaxy clustering amplitude increases with z , contrary to thecase of halos. space at the given redshift. In turn the solid lines are thelinear modeling for biasing and redshift space distortions(i.e. the Kaiser effect, Kaiser (1984), discussed in Sec. 6.1 inmore detailed), angle averaged and evaluated at the meanof each redshift bin, see Eq. (10). The bias used in the mod-eling, and shown in the inset top-right labels, was obtainedfrom the ratio of the two point correlation of galaxies to DMin real space. As we can see, the bias evolves quite stronglywith redshift such that the clustering amplitude is largestfor higher z (where dark-matter clustering is weaker). Thisis because we have selected a magnitude limited sample,hosted by halos of increasing mass as we increase the red-shift. This can be compared with Fig. 5 that has the corre-sponding study with halos of fixed mass showing a clusteringamplitude that is roughly independent of redshift, meaning b ( z ) ∼ D ( z ) − . Overall the linear modeling and the clus-tering measurements agree quite well on large scales in thelightcone provided with the larger statistical error bars (al-though more realistic from an observational point of view)compared to a comoving output, as we investigate in Sec. 5.The largest differences, still at the 1 − σ level, are foundat 0 . < z < .
44 where sampling variance is largest andnonlinear effects strongest. c (cid:13)000
44 where sampling variance is largest andnonlinear effects strongest. c (cid:13)000 , 000–000 he MICE Grand Challenge: Halos and Galaxies k (cid:64) h Mpc (cid:45) (cid:68) b X (cid:61) P h m (cid:144) P mm (cid:180) (cid:60) M h (cid:144)(cid:72) M (cid:159) h (cid:45) (cid:76) (cid:60) (cid:180) z (cid:61) Grand ChallengeIntermediate run
Idem with no Warren mass correction k (cid:64) h Mpc (cid:45) (cid:68) b X (cid:61) P h m (cid:144) P mm (cid:180) (cid:60) M h (cid:144)(cid:72) M (cid:159) h (cid:45) (cid:76) (cid:60) z (cid:61) k (cid:64) h Mpc (cid:45) (cid:68) b X (cid:61) P h m (cid:144) P mm (cid:60) M h (cid:144)(cid:72) M (cid:159) h (cid:45) (cid:76) (cid:60) (cid:180) z (cid:61) k (cid:64) h Mpc (cid:45) (cid:68) b X (cid:61) P h m (cid:144) P mm (cid:180) (cid:60) M h (cid:144)(cid:72) M (cid:159) h (cid:45) (cid:76) (cid:60) z (cid:61) Figure 13.
Mass resolution effects in large-scale halo bias.Solid magenta lines correspond to the cross-correlation bias b X = P hm /P mm measured in the MICE-GC run ( m p = 2 . × h − M (cid:12) ), while solid blue to the one measured in MICE-IR( m p = 2 . × h − M (cid:12) ) at z = 0 .
5. By default masses werecorrected for finite halo sampling (Warren et al. 2006). The cor-responding results for halo samples with the naive definition ofmass M h = m p n p is depicted by the dashed lines in each panel. In Paper I we studied how the matter distribution, in par-ticular the clustering, depends on the simulation mass reso-lution. In this section we now extend that study to halo biasderived from both 2-pt and 3-pt clustering.
Let us start with the 2-pt clustering in Fourier Space. Inorder to avoid noise due to low halo densities we look at thebias through the cross-correlation with the matter field, i.e. b X ≡ P hm /P mm (1)where P hm and P mm are the halo-matter and matter-matterpower spectra respectively.Figure 13 shows the large scale bias for different halosamples at z = 0 . . × < M h / ( h − M (cid:12) ) < × ,5 × < M h / ( h − M (cid:12) ) < , 10 < M h / ( h − M (cid:12) ) < × and M h / ( h − M (cid:12) ) > , from top to bottom.In each panel MICE-GC measurements are depicted withsolid magenta lines for samples in which halo masses havebeen corrected for finite sampling prior to selection (follow-ing the discussion in Sec 2.1), or with magenta dashed linesotherwise (here halo masses are defined simply as m p n p ).Similarly for MICE-IR we use solid blue lines for “Warren”corrected masses, and dashed otherwise. In all cases thebias asymptotically approaches a scale independent value(although at progressively large scales with increasing halomass, as expected) but with some slight differences depend-ing on the simulation and halo mass range.The top panel shows an extreme case of very poorly re-solved halos (or “groups”), formed by 10 or more particles(up to 20). At this mass scale, MICE-GC halos have 80 to170 particles. Even such extreme scenario yields quite rea-sonable clustering, with bias miss-estimated by ∼ −
10% level (see Fig. 5 in Carreteroet al. (2015)).As pointed in the introduction, the next generation sur-veys will reach very faint magnitudes, challenging the res-olution limit of current state-of-the-art simulations. Hencedifferent approaches are being proposed to improve on massresolution in approximated ways (Angulo et al. 2014). This c (cid:13) , 000–000 Crocce et al. panel intends to highlight one such approach which is sim-ply to consider samples of very poorly resolved halos aslong as one is interested in a halo mass scale M h < few × h − M (cid:12) where b (cid:46)
1. At this scale halo bias becomesvery weakly dependent on mass (e.g. Fig 5 in Carretero et al.(2015)). Thus for clustering measurements we can make alarge error in the halo mass and still obtain accurate re-sults. We should stress that we make this comment with theconcrete goal of producing galaxy mock catalogues for dataanalysis. And is mainly relevant for completeness at low zbecause a galaxy catalog is typically limited in apparentmagnitude therefore at high redshifts the galaxies are quiteluminous and reside only in high mass halos. In turn error inthe mass function (because of possible errors in halo mass)are automatically corrected by the calibration to low red-shift galaxy luminosity with the SAM . The actual HODparameters will be different than in a high resolution run,but the galaxy distribution will be quite similar. This is whywe believe that halos with small number of particles can giveresults which are similar to those in higher resolution simu-lationsFor the next mass bin in Fig. 13 MICE-IR halos have ∼ −
50 particles (as opposed to 170 to 350 in MICE-GC). The large-scale bias in MICE-IR is higher by ∼
5% ifmasses have been corrected or 3% otherwise (see top panelin Fig. 13). The effect diminishes in the intermediate massrange at the middle panel (with 50 to 200 particles in MICE-IR halos) to 4% and 2% roughly. For well sampled halos( M h / ( h − M (cid:12) ) > × , bottom panel) the bias recov-ered from MICE-IR is compatible within ∼
1% to the onefrom MICE-GC. For low number of particles per halo, the“Warren” correction seems to introduce a mass shift toolarge, that translates into artificial bias differences. A slightmodification such as the one proposed in Bhattacharya et al.(2011) might alleviate this issue.In summary the bias differences found between the dif-ferent resolutions is within 3% or better for well resolvedhalo masses, more standard in the literature, and up to10%, for poorly resolved ones formed by as low as 10 par-ticles. Similar conclusions were reached studying other red-shifts. Hence this kind of effect can then be of importancefor studies of accuracy in halo bias modeling, for instancethe stated accuracy of peak-background split approach is (cid:46)
10% (Manera et al. 2010; Tinker et al. 2010; Manera &Gazta˜naga 2011), not far from the effects purely dependenton simulation parameters discussed above. Note that SAM will correct the abundance and large scaleclustering (two-halo term) by a suitable choice of the number ofsatellite and central galaxies at a given luminosity. This in princi-ple modifies also the one-halo term. But our satellite assignmentalgorithm has freedom to control the distribution of the satellitesaway from an NFW (and their velocities) in such a way that onecan simultaneously match the small-scale one-halo clustering toobservations (Carretero et al. 2015).
A further quantity of interest on top of the linear large-scalebias discussed above is the second order bias b that natu-rally appears at the leading order in higher order correlations(Fry & Gaztanaga 1993).On large enough scales, where the fluctuations in thedensity field are smoothed so that the matter density con-trast is of order unity or smaller, one can assume a generalnon-linear (but local and deterministic) relation between thedensity contrast in the distribution of halos δ h and dark mat-ter δ m that can be expanded in a Taylor series δ h = ∞ (cid:88) k =0 b k k ! δ km = b + b δ m + b δ m + · · · , (2)where the k = 0 term comes from the requirement that < δ h > = 0. Within this local bias model, at scales where ξ ( r ) ≡ (cid:104) δ m ( r ) (cid:105) <
1, we can write the biased (halo or galaxy)two and three point functions to the leading order in ξ (Fry& Gaztanaga 1993; Frieman & Gaztanaga 1994) ξ h ( r ) (cid:39) b ξ ( r ) ζ h ( r , r , r ) (cid:39) b ζ ( r , r , r ) ++ b b ( ξ ( r ) ξ ( r ) + cyc) (3)where ζ is the matter 3-pt function, which is O ( ξ ) for Gaus-sian initial conditions. From the above we obtain the reduced3-point function Q (Groth & Peebles 1977) defined as: Q ≡ ζ ( r , r , r ) ξ ( r ) ξ ( r ) + ξ ( r ) ξ ( r ) + ξ ( r ) ξ ( r ) (4)such that Q h (cid:39) b ( Q + c ) (5)where c ≡ b /b , and the (cid:39) sign indicates that this is theleading order contribution in the expansion given by Eq. (2)above. In the local bias model we can use Q as measuredin the DM field to fit Q h from halos, and obtain an estimateof b and c , that could be used to break the full degeneracyof b and growth in ξ ( r ).Figure 14 shows a comparison of Q h in halos of mass M h > . × h − M (cid:12) (without Warren correction) fromsimulations with different mass resolution, as detailed in Ta-ble 1: MICE-GC in black squares, MICE-IR with red tri-angles and MICE-SHV with blue crosses. The simulationwith intermediate resolution has 8 times less particles perhalo than the one with higher resolution, while the lowestresolution has 125 fewer particles. Notice that in the latercase we are using as few as 5 particles per halo as thresh-old and yet Q has a very reasonable shape. In additionto the measured Q h we include Q for the dark-matter inMICE-GC (solid black) and a linearly biased version Q /b (dashed black), see Eq. (5). The dotted magenta is a similarestimate but assuming a theory Q derived from a powerspectrum with no wiggles. The figure focuses on large scales( r = 2 r = 96 h − Mpc) and the BAO peak is clearlydetected at r ≈ h − Mpc. c (cid:13)000
1, we can write the biased (halo or galaxy)two and three point functions to the leading order in ξ (Fry& Gaztanaga 1993; Frieman & Gaztanaga 1994) ξ h ( r ) (cid:39) b ξ ( r ) ζ h ( r , r , r ) (cid:39) b ζ ( r , r , r ) ++ b b ( ξ ( r ) ξ ( r ) + cyc) (3)where ζ is the matter 3-pt function, which is O ( ξ ) for Gaus-sian initial conditions. From the above we obtain the reduced3-point function Q (Groth & Peebles 1977) defined as: Q ≡ ζ ( r , r , r ) ξ ( r ) ξ ( r ) + ξ ( r ) ξ ( r ) + ξ ( r ) ξ ( r ) (4)such that Q h (cid:39) b ( Q + c ) (5)where c ≡ b /b , and the (cid:39) sign indicates that this is theleading order contribution in the expansion given by Eq. (2)above. In the local bias model we can use Q as measuredin the DM field to fit Q h from halos, and obtain an estimateof b and c , that could be used to break the full degeneracyof b and growth in ξ ( r ).Figure 14 shows a comparison of Q h in halos of mass M h > . × h − M (cid:12) (without Warren correction) fromsimulations with different mass resolution, as detailed in Ta-ble 1: MICE-GC in black squares, MICE-IR with red tri-angles and MICE-SHV with blue crosses. The simulationwith intermediate resolution has 8 times less particles perhalo than the one with higher resolution, while the lowestresolution has 125 fewer particles. Notice that in the latercase we are using as few as 5 particles per halo as thresh-old and yet Q has a very reasonable shape. In additionto the measured Q h we include Q for the dark-matter inMICE-GC (solid black) and a linearly biased version Q /b (dashed black), see Eq. (5). The dotted magenta is a similarestimate but assuming a theory Q derived from a powerspectrum with no wiggles. The figure focuses on large scales( r = 2 r = 96 h − Mpc) and the BAO peak is clearlydetected at r ≈ h − Mpc. c (cid:13)000 , 000–000 he MICE Grand Challenge: Halos and Galaxies Figure 14.
Reduced 3-point function in Eq. (4), for halos above1 . × h − M (cid:12) measured in simulations with different parti-cle mass resolutions (as labeled). We include Q measured in thedark-matter and Q /b corresponding to a local bias model (with b estimated from 2-pt functions as in Sec. 5). The dotted linecorresponds to the no-wiggle EH power spectrum. The imprint ofthe BAO feature in Q is clearly significant at r ≈ h − Mpc.
The differences between MICE-GC and MICE-IR aremarginal, with derived linear bias values that agree at thepercent level as found in Sec. 4.1 for well resolved halos.In turn MICE-SHV yield larger differences showing thatsuch a low resolution is inappropriate for percent level accu-racy studies. One subtlety is that even at the level of dark-matter there are some differences among these simulations,as discussed in Paper I. Figure 15 shows the effect of resolu-tion on nonlinear bias by plotting Q h − Q dm /b where both Q h and Q dm ≡ Q are measured in each given run. Thuswe subtract the linear bias which also has some resolutioncoming effects coming from the DM. Moreover we focus onsmaller scales, r = 2 r = 48 h − Mpc. In Fig. 15 the lo-cal bias model corresponds to an horizontal line set by thenon-linear bias c while the non-local model of Chan et al.(2012) is given by the dashed black line . For isosceles tri-angles ( r = 48 h − Mpc) all three simulations agree, butfor collapsed and elongated shapes MICE-SHV exceeds con-siderably the other two runs. In fact, while MICE-GC andMICE-IR track well the non local model clearly deviatingfrom a horizontal line, MICE-SHV seems to be consistentwith it for c ∼ γ = 0). Further work re-garding higher order halo bias in MICE-GC computed withdifferent methods can be found in Hoffmann et al. (2015). Note that our best fit non-local coefficient γ is in perfectagreement with the one derived from b = 1 . γ (cid:39) − b − / − . Figure 15.
Amplitude of the reduced 3-point function relativeto the dark matter one measured in each run. Note how MICE-GC and MICE-IR agree with each other and with the non-localmodel. In turn, the MICE-SHV yield differences of few percentand seems to follow the local model (dotted line)
As recalled in the introduction, one of the most interestingaspects of the MICE-GC run and its derived products isthe combination of large volume and good mass resolution.In this section we profit from these by looking in detail atthe scale dependence of halo and galaxy bias in configura-tion space, from small scales relevant for full-shape fitting(S´anchez et al. 2009, 2013, 2014) to large scales tracing theBAO feature (Anderson et al. (2012) and references therein).In Fig. 16 we show the halo bias from two point cor-relation functions for three “mass threshold” halo samples, M h / ( h − M (cid:12) ) > × , × and 7 × . These sam-ples were selected to have low, mid and high bias (top to bot-tom panels respectively). On the one hand Fig. 16 focuseson the comparison of the bias from the halo cross-correlationsignal with dark-matter ( b = ξ hm /ξ mm , red line) versusthe one from halo auto-correlation ( b = ( ξ hh /ξ mm ) / , blueline). On the other hand the panels split the bias measure-ment into small scales (shown with logarithmic binning) andlarge-scales (shown with linear binning). This is useful todetermine on what scales the cross-correlation coefficient r cc ≡ ξ hm / √ ξ mm ξ hh departs from unity, a point that is typ-ically linked to the brake-down of a local and deterministicbiasing (Tegmark & Peebles 1998; Dekel & Lahav 1999).In each panel the filled region shows 1% of the mean lin-ear bias defined as the error weighted average over scales s (cid:62) h − Mpc. Defined in this way we find b = 0 . , . .
43 (top to bottom). Both, Figs. 16 and 18, correspond c (cid:13) , 000–000 Crocce et al. • • • • • • • • • • • • • ••••••••••••••• • • • • • • • • • • • • • • • • • • • • • • ••••••••••••••• • • • • • • • • • (cid:64)
Mpc h (cid:45) (cid:68) b i a s z (cid:61) • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • •
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Mpc h (cid:45) (cid:68) b i a s Ξ hm (cid:144) Ξ mm (cid:72) Ξ hh (cid:144) Ξ mm (cid:76) (cid:144) M h (cid:62) (cid:180) M (cid:159) (cid:144) h • • • • • • • • • • ••••••••• • • • • • • • • • •••••••• (cid:64) Mpc h (cid:45) (cid:68) b i a s z (cid:61) • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • •
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Mpc h (cid:45) (cid:68) b i a s M h (cid:62) (cid:180) M (cid:159) (cid:144) h • • • • • • • • • • •• • • • • • • • • • • (cid:64) Mpc h (cid:45) (cid:68) b i a s z (cid:61) • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • •
40 60 80 100 1202.32.42.52.6 r (cid:64)
Mpc h (cid:45) (cid:68) b i a s (cid:173) BAO M h (cid:62) (cid:180) M (cid:159) (cid:144) h Figure 16.
Halo bias in MICE-GC for 3 different mass threshold samples (as labeled). The shaded regions indicates a 1% around themean value measured at r > h − Mpc. For an M (cid:63) -like sample the bias from halo-matter correlations (red symbols) is very close toscale-independent from small to large scales. At small scales ( r (cid:46) h − Mpc) there is a clear trend of the clustering with increasinghalo mass (with b hm ≡ ξ hm /ξ mm being smaller, similar and larger than the large-scale value). Across the BAO region there is slightclustering decrement ( ∼ h − Mpc for the least massive halos and a clustering excess of ∼ −
5% at the BAO peak position(marked with a vertical arrow) for the most massive ones. For clarity error bars are only displayed for bias derived from ξ hh . to comoving catalogues at z = 0. Error bars in those figureswere obtained using jack-knife resampling with njk = 64 re-gions (measuring the bias in each region and the mean andvariance weighted by njk − − h − Mpc, with the cross correlation coefficient r cc being close to unity on this regime. On the largest scalesthere are however some residual effects worth highlighting.For the least massive halos there seems to be a decre-ment of clustering amplitude around 100 h − Mpc, althoughwith a marginal amplitude of about 2%. As we increasethe sample mass to M h > ∼ M (cid:63) (see middle panel) thebias is consistent with linear bias almost within 1%. How-ever for masses considerably above M (cid:63) (such as M h (cid:62) × h − M (cid:12) as shown in the bottom panel of Fig. 16) we find Where M (cid:63) is defined as the mass scale with a variance ν = δ c /σ ( M (cid:63) ) = 1, which for MICE yields M (cid:63) = 2 . × h − M (cid:12) at z = 0. an excess of clustering of 4% −
5% precisely at the BAO peak( r ≈ h − Mpc for our cosmology). This excess cluster-ing increases with increasing mass, for M h (cid:38) h − M (cid:12) isseparately shown in Fig. 17. This pattern has been discussedby Desjacques et al. (2010) in the context of peak biasingand attributed for the most part to first order effects in La-grangian Space, but more work is needed to characterize thisas a function of mass and redshift.At scales smaller than 20 h − Mpc the bias becomessteadily scale dependent due to nonlinear gravitational ef-fects ( b terms in the language of PT) , again with an in-teresting dependence with mass since b changes sign fromnegative to positive across the three halo samples shown (topto bottom respectively). Notice that this is the expected be-havior for b given the values of the linear bias b at largescales (Cooray & Sheth 2002).At these scales the cross-correlation coefficient also de-parts from unity by up to 10% at r = 5 h − Mpc (see alsoSato & Matsubara (2013)). This is compatible with the c (cid:13)000
5% precisely at the BAO peak( r ≈ h − Mpc for our cosmology). This excess cluster-ing increases with increasing mass, for M h (cid:38) h − M (cid:12) isseparately shown in Fig. 17. This pattern has been discussedby Desjacques et al. (2010) in the context of peak biasingand attributed for the most part to first order effects in La-grangian Space, but more work is needed to characterize thisas a function of mass and redshift.At scales smaller than 20 h − Mpc the bias becomessteadily scale dependent due to nonlinear gravitational ef-fects ( b terms in the language of PT) , again with an in-teresting dependence with mass since b changes sign fromnegative to positive across the three halo samples shown (topto bottom respectively). Notice that this is the expected be-havior for b given the values of the linear bias b at largescales (Cooray & Sheth 2002).At these scales the cross-correlation coefficient also de-parts from unity by up to 10% at r = 5 h − Mpc (see alsoSato & Matsubara (2013)). This is compatible with the c (cid:13)000 , 000–000 he MICE Grand Challenge: Halos and Galaxies • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • •
40 60 80 100 1202.52.62.72.82.93.0 r (cid:64)
Mpc h (cid:45) (cid:68) b i a s M h (cid:62) (cid:180) M (cid:159) (cid:144) h Figure 17.
Scale dependence in bias across the BAO feature fora cluster mass scale halo sample. The sample is selected from thecomoving output at z = 0. Shaded regions as in Fig. 16. Errorbars are only displayed for bias derived from halo auto-correlationfor clarity, as in Fig. 16 emergence of nonlinear bias on these scales but might alsosignal a stochastic relation between halos and mass.We next turn to investigate similar issues in the galaxycatalogue. Fig. 18 shows the galaxy bias from galaxy-masscross correlations (red line) and from galaxy-galaxy auto-correlations for two distinctive samples. The top panel fo-cuses on one faint magnitude limited sample ( M r < − . M h ∼ h − M (cid:12) . Bottom panel corresponds to a Lumi-nous Red Galaxy sample defined through a bright absoluteluminosity cut ( M r < −
21) and one color cut ( g − r ) > . r ∼ h − Mpc, where nonlinear effects increase the clus-tering above the linear value. Notably the cross-correlationcoefficients remains tightly close to unity all the way to r ∼ h − Mpc, a clear and remarkable difference with re-spect to the halos in Fig. 16. In a follow up work we willexplore to what extent this depends on the satellite pro-files or the halo exclusion. But for instance notice that oursatellite galaxies do not necessarily follow the distributionof matter as we place them using a pre-determined profile.Turning to the LRG sample in the bottom panel ofFig. 18 we find a clear scale dependent bias across the BAOfeature, with an excess power at the BAO peak of about 5%and a small 2% deep at 80 < r/ ( h − Mpc) < In this section we will discuss the properties of our galaxycatalogue in redshift space, which is a measure of how galaxyvelocities are assigned.
In the large-scale linear regime and in the plane-parallelapproximation (where galaxies are taken to be sufficientlyfar away from the observer that the displacements inducedby peculiar velocities are effectively parallel), the distortioncaused by coherent infall velocities takes a particularly sim-ple form in Fourier space (Kaiser 1984): δ ( s ) ( k, µ ) = (1 + fµ ) δ m ( k ) (6)where µ is the cosine of the angle between k and the line-of-sight, the superscript s indicates redshift space, and f ( z ) isgiven by, f ( z ) ≡ d ln Dd ln a . (7)The second term in Eq. (6) is caused by radial peculiar ve-locities. If we assume that galaxy fluctuations are linearlybiased by a factor b relative to the underlying matter density δ m (i.e. δ G = b δ m ) but velocities are unbiased, then δ ( s ) G ( k, µ ) = ( b + fµ ) δ m ( k ) (8)where δ ( s ) G are the measured galaxy fluctuations in redshiftspace. We then have an anisotropic power spectrum: P ( s ) gg ( k, µ ) = < ( δ ( s ) G ( k )) > = ( b + fµ ) P mm ( k ) (9)where P mm ( k ) = < δ m ( k ) > is the real space matter powerspectrum. This can be Fourier transformed and averagedover angles to obtain the monopole correlation function: ξ gg ≡ ξ (cid:96) =0 ,gg = K (cid:96) =0 ( z ) ξ mm K (cid:96) =0 ( z ) ≡ b ( z ) + 23 b ( z ) f ( z ) + 15 f ( z ) b ( z ) (10)where ξ mm is the matter correlation function at redshift z (i.e. in linear theory ξ mm = D ( z ) ξ L ( r, z = 0)) and we havedefined K ( z ) to be the monopole “linear Kaiser” factor.Figure 19 shows the ratio ξ gg /ξ mm measured in theMICE-GC galaxy lightcone catalogue (error weighted aver-aged on scales r > h Mpc − ) for an apparent magnitudelimited sample ( r < K , where we use b ( z ) as measured in real space and f ( z ) given by the MICE cosmology. Note how both b ( z )and f ( z ) change with redshift and that the predictions de-pend strongly on both ( b or f alone cannot account for theobserved variations, as indicated by red and dotted lines).There is an excellent agreement with the linear Kaiser model(in blue) for all redshifts and for the concrete bias evolution c (cid:13) , 000–000 Crocce et al. • • • • • • • • • • • •• • • • • • • • • • • • (cid:64)
Mpc h (cid:45) (cid:68) b i a s z (cid:61) • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • •
40 60 80 100 1201.101.151.201.251.30 r (cid:64)
Mpc h (cid:45) (cid:68) b i a s Ξ gm (cid:144) Ξ mm (cid:72) Ξ gg (cid:144) Ξ mm (cid:76) (cid:144) All Galaxies, M r (cid:60) (cid:45) • • • • • • • • • • • • • • ••••••••••••••• • • • • • • •• • • • • • • • • • • • • • •••••••••••• •••• • • • • • • • (cid:64) Mpc h (cid:45) (cid:68) b i a s • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • •
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Mpc h (cid:45) (cid:68) b i a s LRGs
Figure 18.
Scale dependence in galaxy bias for two samples in the MICE-GC comoving catalogue at z = 0. Top panel correspond toan absolute magnitude limited sample ( M r < − . M r < −
21 and g − r > . r > h − Mpc. The panels show trends resembling those present for halo clustering.The M r < − .
16 galaxies show a remarkably flat bias for r > h − Mpc while the LRGs have a scale dependent feature across theBAO region of order 4 − r < h − Mpc) but, contrary to the case of halosor centrals only shown in Fig. 16, the cross correlation coefficient remains close to unity down to ∼ few h − Mpc scales that results from cutting galaxies to r <
24. On the onehand this serves as an excellent validation of the large-scalebulk galaxy velocities in the catalogue (which in turn arebased on halo velocities). On the other hand, it means thatboth bias b ( z ) and f ( z ) can be constrained from observa-tions using this simple modeling. In the previous section we showed that the Kaiser limit isa good model to describe the large-scale ( s (cid:62) h − Mpc)clustering amplitude of the monopole correlation functionand its lightcone evolution, provided with the bias as a func-tion of redshift. In this section we investigate the break downof this large-scale limit due to the departure from purelybulk motions, in particular the impact of satellite galaxies.We will focus on the multipole moments of the anisotropicgalaxy power spectrum in redshift space, P ( s ) gg,(cid:96) ( k ) = 2 (cid:96) + 12 (cid:90) − P ( s ) gg ( k, µ ) L (cid:96) ( µ ) dµ (11)with L (cid:96) being the Legendre polynomials. On large scales wecan assume the “linear” relation in Eq. (9) and obtain thecorresponding Kaiser limits : P ( s ) gg,(cid:96) ( k ) = K (cid:96) ( b, f ) P mm ( k ),where K is given in Eq. (10) and, K ( b, f ) ≡ bf + 47 f K ( b, f ) ≡ f (12) Figure 20 shows the first three moments, monopole ( (cid:96) =0), quadrupole ( (cid:96) = 2) and hexadecapole ( (cid:96) = 4) for themagnitude limited sample of galaxies discussed in Sec. 3.2,i.e. M r < − .
16, in the comoving catalogue at z = 0. Inthe mean, this luminosity corresponds to halos more massivethan 10 h − M (cid:12) . In order to understand what is the impactin the anisotropy of large-scale fluctuations from the motionof satellite galaxies inside halos we split the sample into“centrals only” (i.e. bulk motion of halos only), shown by redfilled dots, and “all galaxies” (central plus satellites) shownby blue empty triangles. For this sample the satellite fractionis 24%. The corresponding multipole spectra for dark-matteris shown by dashed black lines.On the largest scales the Kaiser limit (shown in short-dashed) is reached for both the “centrals only” sample andthe “central+satellites”, although in a more limited range ofscales for the later. Notice that the large-scale bias of thesetwo samples is slightly different ( b cen = 0 .
98 and b cen + sat =1 .
2) because of the scatter in the mass-luminosity relationdiscussed in Sec. 3.2. Hence the different Kaiser asymptoticsin the monopole and quadrupole panels of Fig. 20 (while K does not depend on bias). In turn, at k < . h Mpc − sampling variance dominates the hexadecapole results, de-spite the large simulation size. For reference we show thecorresponding cosmic variance error assuming the multipolemoments in redshift space to be Gaussian random fields (e.g.Taruya et al. (2009) and references therein).This is in contrast to the monopole or quadrupole, which c (cid:13)000
2) because of the scatter in the mass-luminosity relationdiscussed in Sec. 3.2. Hence the different Kaiser asymptoticsin the monopole and quadrupole panels of Fig. 20 (while K does not depend on bias). In turn, at k < . h Mpc − sampling variance dominates the hexadecapole results, de-spite the large simulation size. For reference we show thecorresponding cosmic variance error assuming the multipolemoments in redshift space to be Gaussian random fields (e.g.Taruya et al. (2009) and references therein).This is in contrast to the monopole or quadrupole, which c (cid:13)000 , 000–000 he MICE Grand Challenge: Halos and Galaxies Figure 19.
Ratio of galaxy monopole 3D correlations in redshiftspace to the matter correlation in real space (points with errors),see Eq. (10). Dashed line shows the unbiased Kaiser prediction,while dotted line shows the bias measured in real space averag-ing over scales s (cid:62) h − Mpc. The blue line corresponds to thelinear Kaiser model in Eq. (10) with this measured bias. This cor-respond to r <
24 galaxies in the MICE-GC lightcone catalogue. can be measured to much smaller k , and results from thestronger dependence in the shape ( µ ).In order to investigate departures from the Kaiser limitwe fit the following model to our monopole and quadrupolemeasurements (Scoccimarro 2004), P ( s ) ( k, µ ) = (cid:2) b P δδ + 2 bfµ P δθ + f µ P θθ (cid:3) × e − ( kµfσ v ) (13)where we take b to be the large-scale linear bias measuredin real space (i.e. with a fixed value), f = 0 .
46 for our cos-mology at z = 0 and σ v is a nuisance parameter related to(1D) velocity dispersion. In Eq. (13) P XY are the nonlineardensity ( δ ) and velocity divergence ( θ ) auto and cross powerspectra which we compute using MPTbreeze (Crocce et al.2012). We stress that Eq. (13) is not expected to give accu-rate results but it is useful to hint on departures from thesimplest linear Kaiser model discussed before.From the monopole and quadrupole in Fig. 20 we findthe best-fit to be σ v = 6 h − Mpc for dark matter (equiv-alent to 600 km / s), very close to the linear value σ v, Lin = (cid:18) π (cid:90) P Lin ( q ) dq (cid:19) / = 6 . h − Mpc , (14)in agreement with Taruya et al. (2010). The “centrals” onlysample (or halos) yields a smaller value σ cenv = 3 h − Mpccharacteristic of a more coherent bulk motion. In turn the in- We limit to scales k (cid:54) . h − Mpc where the model fits thethree multipoles, provided with one nuisance parameter. (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:243) (cid:243) (cid:243) (cid:243) (cid:243) (cid:243) (cid:243) (cid:243) (cid:243) (cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243) (cid:230) (cid:230) (cid:230) (cid:243) (cid:243) (cid:243) k (cid:64) h Mpc (cid:45) (cid:68) P gg , l (cid:61) s (cid:144) P mm r mag limit M r (cid:60) (cid:45) All GalaxiesCentrals Only
Dark MatterKaiser Limit
Monopole (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:243) (cid:243) (cid:243) (cid:243) (cid:243) (cid:243) (cid:243) (cid:243) (cid:243) (cid:243) (cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243) k (cid:64) h Mpc (cid:45) (cid:68) P gg , l (cid:61) s (cid:144) P mm r Quadrupole (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:243) (cid:243) (cid:243) (cid:243) (cid:243) (cid:243) (cid:243) (cid:243) (cid:243) (cid:243) (cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243) (cid:45) k (cid:64) h Mpc (cid:45) (cid:68) P gg , l (cid:61) s (cid:144) P mm r Hexa (cid:45) decapole z (cid:61) Figure 20.
The first 3 multipole power spectra for a magnitudelimited galaxy sample ( M r < − .
16) at z = 0. In each panelthe corresponding P (cid:96) has been divided by the measured (non-linear) matter power spectrum. The figure shows the case forcentral galaxies only, or the full sample (cen+sat) as well as thecorresponding dark-matter. Hence it stresses the significant im-pact of satellite galaxies into the anisotropic clustering, basicallyby adding velocity dispersion. Notice how in all cases the Kaiserlimit (short dashed line) is reached but only for the largest scales. clusion of satellite galaxies leads to virialized motions closerto those of dark matter, with a best-fit σ allv = 8 . h − Mpc. c (cid:13) , 000–000 Crocce et al.
As we discussed in Sec. 3.1 the spatial distribution of satel-lite galaxies is set by observational constraints from pro-jected clustering (Carretero et al. 2015). However equiva-lent observational constraints for the distribution of satel-lite velocities are not that well stablished, hence our choicearises from well known results using hydrodynamical sim-ulations (Bryan & Norman 1998). In this section we studyquantitatively how this assumption impact the anisotropicclustering.Our procedure is to give the satellite galaxies the bulkmotion of the halo plus an additional virial motion thatfollows a Gaussian distribution (in each axis) with a velocitydispersion σ vir = (cid:104) v vir (cid:105) ∝ M / h (Sheth & Diaferio 2001), so v = v h + v vir where v h is the halo center of mass velocity(also the one of the central galaxy).For the magnitude limited sample M r < − .
16 dis-cussed in Sec. 6 the velocity dispersion of satellites is 422 Km sec − (the distribution is narrower than a Gaussian, because itarises from a range of halo masses) while the satellite frac-tion is ∼ ±
20% keeping the bulk motion of thehalos unaltered. As expected increasing the satellite veloc-ity dispersion to 500 km / s induces more FoG effects (fromsatellite-central correlations in different halos) and a strongerscale dependent suppression of power. The monopole is sup-pressed at the 5% level on scales k ∼ . h Mpc − comparedto the fiducial case, while the quadrupole is more affected(20% at the same scale). In turn the hexa-decapole is toonoisy on these scales, but the impact is clearly stronger. Re-ducing the satellite velocity dispersion to 340 km / sec (i.e.by 20% less w.r.t the fiducial) has the opposite effects. Wehave done a more extreme case in which all satellites movewith the bulk motion of the halo (setting v vir = 0). This isshown by short-dashed lines in Fig. 21. The result is thatthe anisotropic clustering in this case is well described bythe simple linear Kaiser effect down to smaller scales.Overall we find that satellite galaxies give a signifi-cant contribution to the anisotropic clustering through non-linear redshift space distortions even on quite large-scales(see also Hikage & Yamamoto (2013); Masaki et al. (2013);Nishimichi & Oka (2014) for the case of LRG’s), yielding ve-locity dispersion effects similar (or larger) to those of dark-matter. Together with this series of papers we make a first publicdata release of the current version of the MICE-GC light-cone catalogue (
MICECAT v1.0 ). The halo and galaxy cata-logue can be obtained at http://cosmohub.pic.es , a dedi- (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:243) (cid:243) (cid:243) (cid:243) (cid:243) (cid:243) (cid:243) (cid:243) (cid:243) (cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243) (cid:230) (cid:230) (cid:230) (cid:243) (cid:243) (cid:243) k (cid:64) h Mpc (cid:45) (cid:68) P gg , l (cid:61) s (cid:144) P mm r (cid:144) K a i s e r l (cid:61) M r (cid:60) (cid:45) Σ virsat (cid:61)
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Figure 21.
Change in the first 3 multipole power spectra for amagnitude limited galaxy sample ( M r < − .
16) when the virial-ized motion of satellite galaxies within the host halos is changedby ∼ ± cated database portal hosted by Port d’Informaci´o Cient´ıfica(PIC). It corresponds to one octant of the full sky (5000 deg )from z = 0 to z = 1 .
4. In the current version (v1.0) galax-ies are limited in absolute magnitude to M r < − . M h > . × h − M (cid:12) . Among otherproperties we provide angular positions and observed red-shifts for all galaxies, flags to central/satellite distinctions,host halo masses, peculiar velocities, comoving distances, ob- c (cid:13)000
4. In the current version (v1.0) galax-ies are limited in absolute magnitude to M r < − . M h > . × h − M (cid:12) . Among otherproperties we provide angular positions and observed red-shifts for all galaxies, flags to central/satellite distinctions,host halo masses, peculiar velocities, comoving distances, ob- c (cid:13)000 , 000–000 he MICE Grand Challenge: Halos and Galaxies served magnitudes (and magnitude errors) in several bandsfor surveys such as SDSS, DES and VISTA. Besides, weprovide the SED template and dust extinction assigned toeach galaxy which rely on the template library of Ilbert et al.(2009). These SEDs were used to generate the observed mag-nitudes. We also provide lensing information for each galaxysuch as shear and convergence, as well as magnified magni-tudes and angular positions (described in Paper III). Lastlywe provide photometric redshift errors and error distribu-tions based on a photometric template code.The web-portal is set up to facilitate the download ofthe data, either the full compressed catalogue or by queryingparticular regions of the sky or data columns. We have presented the MICE-GC halo and galaxy cata-logues built upon one the largest N-body runs completedto date: the MICE Grand Challenge lightcone simulation.This N-body run contains about 70 billion particles in a 3 h − Gpc periodic box, a unique combination of large volumeand fine mass resolution sampling 5 orders of magnitude indynamical range.We identify bound structures using a Friends-of-Friendsalgorithm with linking length b = 0 .
2. Halos were resolveddown to few times 10 h − M (cid:12) with a total of about 157million identified in each octant of the full sky lightconeup to z = 1 .
4. A similar procedure was followed in severalcomoving outputs. We then populated the dark-matter haloswith galaxies following a hybrid HOD and HAM scheme,matching the luminosity, color distributions and projectedclustering properties (as a function of luminosity and g-rcolor) of SDSS galaxies at low- z . Lastly galaxy propertieswere evolved into the past lightcone using stellar evolutionmodels. In all, this resulted in a catalogue limited in absolutemagnitude to M r < − . ∼ × galaxies(considering only one octant of the full sky and z < . • Halo Catalogue:
We showed that the halo mass func-tion at z = 0 agrees at the 1% −
2% level with the Crocceet al. (2010) fit for well resolved halos (similarly for othercomoving redshifts and the lightcone, were the fit does notassume universality). The cumulative abundance of groupswith as low as 10 particles is up to 15% below the modelprediction using the Crocce et al. (2010) fit (which is a nu-merical fit calibrated to higher-resolution runs). In turn, theMICE-GC resolution and volume allow us to study halo clus-tering with good precision for samples with a broad range oflinear bias values, even b (cid:46)
1. The PBS prediction for thislow bias sample agrees at the 2% level with the bias mea-sured from P hm /P mm , a better performance than for mas-sive objects (Manera et al. 2010). We note that this regime of low-bias was not well explored previously and deservesa more detailed analysis for more robust conclusions aboutthe performance of PBS. Lastly, halos in the lightcone pre-sented an almost constant clustering amplitude, i.e. degen-erate with the growth factor evolution, for constant masssamples. Instead, galaxies selected above an apparent lumi-nosity threshold show a clustering amplitude that increaseswith redshift. • Galaxy Catalogue:
Starting from fits at low redshift andimplementing evolutionary corrections to galaxies and re-sampling SEDs from COSMOS (Ilbert et al. 2009), we wereable to predict the color distributions and clustering prop-erties of higher-redshift galaxy populations (0 . < z < . z , whereas the MICEsample is slightly bluer at higher redshifts, as depicted byFig. 7. As for the clustering, MICE mock galaxies matchvery well the shape of the angular correlation function ofCOSMOS galaxies at z = 0 . i AB < .
5. A similar match is found at z = 1 for galaxieswith i AB <
24, except for the rise in clustering strength inCOSMOS at angular scales larger than 5 arc minutes, whichwe attribute to the known excess of clustering power in theCOSMOS field (Skibba et al. 2014). Compared to the darkmatter the galaxy clustering of these samples is consistentwith a simple linear bias model with b ∼ .
16 and b ∼ . θ (cid:38) • We have studied how the large-scale halo clustering de-pends on the mass resolution of the underlying N-body simu-lation. We focused first in the halo-matter cross-power spec-trum which is a robust measure of halo clustering againstshot noise. Using this estimator we find the bias to be upto 5% larger for halos resolved with 20 −
50 particles in ourMICE-IR run than for the corresponding sample in MICE-GC (a factor of 8 more particles), and 10% for 10 −
20 particlehalos. The exact value depends on whether halo masses arecorrected for discreteness following Warren et al. (2006) ornot (for poorly resolved halos the applicability of this correc-tion is unclear and makes the effect worse). For well resolvedhalos we find no significant difference between MICE-IR andMICE-GC large scale clustering. Although we concentratedin the comoving output at z = 0 . z = 0. c (cid:13) , 000–000 Crocce et al. • We also looked into this effect in higher order statisticsby measuring the reduced 3-point function, Q , of massivehalos M h (cid:62) . × h − M (cid:12) (sampled with N p (cid:62) N p (cid:62)
78 particles in MICE-IR and only 5particles in MICE-SHV) at z = 0. Mass resolution effectsfor this halo resolution do not affect the shape of the 3-point function unless we use extremely low resolution as forMICE-SHV. Although the MICE-SHV halos yield the cor-rect shape for Q there are few percent level differences. Forsmaller scales MICE-GC and MICE-IR deviate clearly fromthe simple local model and track well the non-local predic-tion from Chan et al. (2012), see Fig. 14. • We investigated scale dependent bias from small (few h − Mpc) to large BAO scales (up to ∼ h − Mpc) inthe two-point correlation function of halos and galaxies at z = 0. We focused on three halo mass threshold samples, M h / ( h − M (cid:12) ) (cid:62) × , × and 7 × , and foundthe bias to be remarkably close to scale independent (within2%) for scales 20 (cid:46) r/ ( h − Mpc) (cid:46)
80. For the interme-diate mass scale (roughly M (cid:63) halos) the bias is flat alsoacross the BAO. However for more massive halos we findan excess of clustering at BAO scales of ∼ ∼ −
4% at 80 (cid:46) r/ ( h − Mpc) (cid:46) r < h − Mpc together with departures of thecross-correlation coefficient r cc from unity. We then investi-gated how this translates to the clustering of galaxies, whichas a non-trivial combination of the one of halos through theHOD. For a faint luminosity cut M r < .
16, correspondingto an L cen = L cen ( M h ) relation of M h ∼ h − M (cid:12) , wefind the bias to be constant with scale for r > h − Mpc.In turn, for an LRG type selection (bright M r < −
21 andred g − r > . − h − Mpc. Overall these are relevantconclusions for standard ruler tests that aims to extractdistance-redshift relations from galaxy clustering (as theyimpact the observed BAO feature) or for modeling the full-shape of the correlation function. We leave a more detailedanalysis for follow-up work. • Lastly we studied galaxy clustering in redshift space,a testing ground for galaxy peculiar velocities. Using thelightcone we find the averaged amplitude of the monopolecorrelation function on scales r > h − Mpc to be very con-sistent with the linear Kaiser model (with an input bias fromreal-space measurements). This was true across all redshiftssampled in the lightcone ( z < .
4) which is a non-trivial testof both b ( z ) and f ( z ). We next looked into departures fromthe linear Kaiser model in the multipole moments of thegalaxy anisotropic power spectrum at the z = 0 snapshot. While on large scales all multipoles agree with the Kaiserlimit there are departures already at k ∼ . h Mpc − . No-tably the satellite galaxies make the anisotropic clusteringstronger, in the sense of increasing Finger-of-God effects toreach (or surpass) those of dark-matter.In a series of three papers we introduce in detail theMICE-GC mock galaxy catalogue, the ending product ofan elaborated step-by-step process that puts together dark-matter, halos, galaxies and lensing, with a strong observa-tional angle. The success of the largest ongoing and futurecosmological surveys is based upon our ability to developsuitable simulations for their analysis and science. We makeour catalogue publicly available, with the aim of contribut-ing to the community wide effort in shaping the upcomingera of precision cosmology. ACKNOWLEDGMENTS
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