Cross-correlation of galaxies and galaxy clusters in the Sloan Digital Sky Survey and the importance of non-Poissonian shot noise
Kerstin Paech, Nico Hamaus, Ben Hoyle, Matteo Costanzi, Tommaso Giannantonio, Steffen Hagstotz, Georg Sauerwein, Jochen Weller
MMon. Not. R. Astron. Soc. , 1–12 (2016) Printed 8 December 2016 (MN L A TEX style file v2.2)
Cross-correlation of galaxies and galaxy clusters in the Sloan DigitalSky Survey and the importance of non-Poissonian shot noise
Kerstin Paech , (cid:63) , Nico Hamaus , Ben Hoyle , , , Matteo Costanzi ,Tommaso Giannantonio , , , Steffen Hagstotz , Georg Sauerwein , Jochen Weller , , Universitäts-Sternwarte München, Fakultät für Physik, Ludwig-Maximilians-Universität München, Scheinerstrasse 1, 81679 München, Germany Excellence Cluster Universe, Boltzmannstr. 2, D-85748 Garching, Germany Max Planck Institute for Extraterrestrial Physics, Giessenbachstr. 1, D-85748 Garching, Germany. Kavli Institute for Cosmology Cambridge, Institute of Astronomy, Madingley Road, Cambridge, CB3 0HA, United Kingdom Centre for Theoretical Cosmology, DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
ABSTRACT
We present measurements of angular cross power spectra between galaxies and optically-selected galaxy clusters in the final photometric sample of the Sloan Digital Sky Survey(SDSS). We measure the auto- and cross-correlations between galaxy and cluster samples,from which we extract the effective biases and study the shot noise properties. We model thenon-Poissonian shot noise by introducing an effective number density of tracers and fit forthis quantity. We find that we can only describe the cross-correlation of galaxies and galaxyclusters, as well as the auto-correlation of galaxy clusters, on the relevant scales using a non-Poissonian shot noise contribution.The values of effective bias we finally measure for a volume-limited sample are b cc =4 . ± . for the cluster auto-correlation and b gc = 2 . ± . for the galaxy-clustercross-correlation. We find that these results are consistent with expectations from the auto-correlations of galaxies and clusters and are in good agreement with previous studies. Themain result is two-fold: firstly we provide a measurement of the cross-correlation of galaxiesand clusters, which can be used for further cosmological analysis, and secondly we describean effective treatment of the shot noise. Key words:
Cosmology: observations – large-scale structure of Universe – galaxies: clusters:general
The cosmological distributions of density and temperature per-turbations are well approximated over sufficiently large scales byGaussian random fields, completely described by their two-pointstatistics. One of the most powerful tools of modern cosmologyis therefore the analysis of two-point correlation functions, whichcan be measured as auto-correlations on one data set or as cross-correlations between two data sets. The strongest current con-straints on the cosmological model are indeed derived from themeasurement of the auto-correlation of the temperature anisotropyof the cosmic microwave background. Correlations can also bemeasured from the distribution of tracers of the matter in the Uni-verse: in the last decades multiple surveys have produced largegalaxy catalogues, which allowed high-precision measurements ofthe galaxy auto-correlation, such as the two degree field (2dF) (cid:63)
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Galaxy Redshift Survey (Cole et al. 2005; Percival et al. 2001)and the SDSS (York et al. 2000; Tegmark et al. 2004; Hayes et al.2011; Ho et al. 2012; Beutler et al. 2014; Grieb et al. 2016). Like-wise, the availability of large optically-selected galaxy cluster cata-logues has led to the measurement of the auto-correlation of galaxyclusters, e.g. from the SDSS catalogue (Huetsi 2009; Estrada et al.2009; Miyatake et al. 2016; Baxter et al. 2016; Veropalumbo et al.2016), and from the REFLEX X-ray survey (Collins et al. 2000;Balaguera-Antolínez et al. 2011). These measurements have alsobeen used to obtain cosmological constraints, for both the REFLEXcatalogue (Schuecker et al. 2003) and several cluster samples fromthe SDSS, such as maxBCG (Mana et al. 2013).Given the success of auto-correlation measurements and theabundance of different cosmological probes of the density field, itis increasingly interesting to combine probes via cross-correlations.Cross-correlations, such as for example between galaxy surveysand the cosmic microwave background (CMB) temperature andlensing (Giannantonio & Percival 2014; Giannantonio et al. 2016), c (cid:13) a r X i v : . [ a s t r o - ph . C O ] D ec Paech et al. or between galaxies and cosmic voids (Hamaus et al. 2014, 2016),provide new information without requiring new observations, andcan thus lead to improved and complementary cosmological con-straints.Some measurements of cross-correlation between galaxy clus-ters and galaxies were attempted in the 1970s and 1980s (Peebles1974; Seldner & Peebles 1977a,b; Lilje & Efstathiou 1988). Thesestudies were performed on relatively small and non-independentcatalogues: the cluster catalogues used by all groups were drawnfrom Abell (1958) and the galaxy catalogues were either the galaxycounts by Shane & Wirtanen (1967) or by Seldner et al. (1977).The better of these two galaxy catalogues had a resolution of 10 ar-cmin ×
10 arcmin on about
19 deg . These early cross-correlationanalyses were therefore limited in their possible applications. Somemore recent works measuring galaxy and galaxy-cluster cross-correlations are Croft et al. (1999); Sánchez et al. (2005); Zu &Weinberg (2013).Hütsi & Lahav (2008) proposed the measurement of the cor-relation between galaxy clusters and galaxies as an additional cos-mological probe, which was later extended by Fedeli et al. (2011).They showed that the cross-correlation of clusters and galaxiescould lead to better constraints on cosmological parameters, as wellas a better determination of the halo model parameters (Cooray &Sheth 2002).In this paper, we measure the cross-correlation between galax-ies and clusters derived from the final photometric data release ofSDSS (Data Release 8, DR8) (Aihara et al. 2011). When using lin-ear theory and cluster bias, as well as Poissonian shot noise, we finda discrepancy between the theoretical expectations and the mea-sured angular power spectra. We show that this tension can be re-solved by adopting a modified treatment of the shot noise.The outline of this paper is as follows: we describe in Sec-tion 2 the theoretical modelling of the angular power spectra, theshot noise, and the cluster bias. In Section 3 we introduce the cata-logues and mask used in the analysis, and in Section 4 we presentthe details of the angular power spectra C l estimation. Section 5presents the results for the auto- and cross-correlations of galaxiesand galaxy clusters. Finally, our summary and outlook are given inSection 6. In order to extract cosmological parameters from the measuredgalaxy and cluster angular power spectra C data l , we need theo-retical model predictions C model l that account for systematics andmeasurement effects affecting the observed correlation functions. We define the density field of the mass density fluctuations at co-moving coordinate r and at any redshift z as δ m ( r ) = ρ m ( r )¯ ρ m − , (1)where ρ m ( r ) is the spatially varying matter density in the Universewith a mean of ¯ ρ m . The matter overdensity δ m can be related tothe galaxy (or cluster) overdensity δ a (where a denotes a galaxy orcluster sample) via the local bias model (Fry & Gaztanaga 1993), δ a ( r ) (cid:39) b ,a δ m ( r ) + b ,a δ m ( r ) + O ( δ m ) + ε a , (2) with linear and non-linear bias parameters b ,a , b ,a , and a shotnoise term ε a .In Fourier space, we can define the matter, galaxy, or clusterpower spectra between any pair of samples ( a, b ) as: (2 π ) δ D ( k − k (cid:48) ) P ab ( k ) ≡ (cid:104) δ a ( k ) δ (cid:63)b ( k (cid:48) ) (cid:105) , (3)where k denotes a wave vector of amplitude k and angled bracketsindicate an average over all Fourier modes within a given spheri-cal shell and δ D is the Dirac delta function. Up to linear order andassuming Poissonian shot noise, the galaxy (or cluster) power spec-trum can be directly related to the matter power spectrum P ( k ) , P ab ( k ) (cid:39) b ,a b ,b P ( k ) + δ abK V /N a (4)where δ K is the Kronecker delta, and the shot noise contributionis given by the inverse number density of galaxies (or clusters), V /N a .In this analysis we consider the angular power spectrum C abl ,a projection of P ab ( k ) on the sky. We use the publically availablecode CLASS (Blas et al. 2011) to generate theoretical predictionsfor the angular cluster power spectrum. CLASS is a differentialequation solver for the hierarchy of Boltzmann equations governingthe perturbations in the density of dark matter, baryons, photonsand any other relevant particle species. The CLASSgal extension(Di Dio et al. 2013) calculates the angular power spectrum, C abl ,for any matter tracer as C abl = 4 π (cid:90) dk k P ini ( k )∆ al ( k )∆ bl ( k ) , (5)where P ini denotes the (dimensionless) primordial power spectrumand the transfer function for the matter component ∆ al ( k ) is givenby ∆ al ( k ) = (cid:90) d z b ,a d N a d z j l ( kr ( z )) D ( k, z ) . (6)Here r ( z ) is the comoving distance, D ( k, z ) the total comovingdensity fluctuation and we use the galaxy and cluster redshift dis-tributions d N/ d z for the observed sample, which are shown in Fig-ure 2 and introduced in Section 3.The main goal of this analysis is to measure the auto-and cross-correlation of galaxies and clusters, and to deter-mine the effective bias of these tracers. Therefore we fix thecosmological parameters to their best-fit values as obtainedby the Planck collaboration (Planck Collaboration et al. 2014)(Planck2013+WP+highL+BAO), derived by combining their ownCMB data with the Wilkinson Microwave Anisotropy Probe(WMAP) polarization data (Bennett et al. 2013), the small-scaleCMB measurements from the Atacama Cosmology Telescope(ACT) (Das et al. 2014) and the South Pole Telescope (SPT)(Reichardt et al. 2012), as well as baryonic accoustic oscillations(BAO) data from SDSS (Percival et al. 2010; Padmanabhan et al.2012; Blake et al. 2011; Anderson et al. 2012; Beutler et al. 2011).The cosmological parameters we use are: h = 0 . , Ω b = 0 . c = 0 . , σ = 0 . , z re = 11 . and n s = 0 . (we checkedthat assuming a Planck 2015 cosmology has no significant impacton the results in our analysis).For the analysis presented in this paper, we adopt a constantbias model, i.e. we define an effective bias b eff , such that we canassume for each sample b ,a ( z ) = b eff , a . The full redshift evolution http://class-code.net/ D ( k, z ) ≈ D + ( z ) T ( k ) for cold dark matter universes, where D + ( z ) isthe density growth function and T ( k ) is the matter transfer function.c (cid:13)000
19 deg . These early cross-correlationanalyses were therefore limited in their possible applications. Somemore recent works measuring galaxy and galaxy-cluster cross-correlations are Croft et al. (1999); Sánchez et al. (2005); Zu &Weinberg (2013).Hütsi & Lahav (2008) proposed the measurement of the cor-relation between galaxy clusters and galaxies as an additional cos-mological probe, which was later extended by Fedeli et al. (2011).They showed that the cross-correlation of clusters and galaxiescould lead to better constraints on cosmological parameters, as wellas a better determination of the halo model parameters (Cooray &Sheth 2002).In this paper, we measure the cross-correlation between galax-ies and clusters derived from the final photometric data release ofSDSS (Data Release 8, DR8) (Aihara et al. 2011). When using lin-ear theory and cluster bias, as well as Poissonian shot noise, we finda discrepancy between the theoretical expectations and the mea-sured angular power spectra. We show that this tension can be re-solved by adopting a modified treatment of the shot noise.The outline of this paper is as follows: we describe in Sec-tion 2 the theoretical modelling of the angular power spectra, theshot noise, and the cluster bias. In Section 3 we introduce the cata-logues and mask used in the analysis, and in Section 4 we presentthe details of the angular power spectra C l estimation. Section 5presents the results for the auto- and cross-correlations of galaxiesand galaxy clusters. Finally, our summary and outlook are given inSection 6. In order to extract cosmological parameters from the measuredgalaxy and cluster angular power spectra C data l , we need theo-retical model predictions C model l that account for systematics andmeasurement effects affecting the observed correlation functions. We define the density field of the mass density fluctuations at co-moving coordinate r and at any redshift z as δ m ( r ) = ρ m ( r )¯ ρ m − , (1)where ρ m ( r ) is the spatially varying matter density in the Universewith a mean of ¯ ρ m . The matter overdensity δ m can be related tothe galaxy (or cluster) overdensity δ a (where a denotes a galaxy orcluster sample) via the local bias model (Fry & Gaztanaga 1993), δ a ( r ) (cid:39) b ,a δ m ( r ) + b ,a δ m ( r ) + O ( δ m ) + ε a , (2) with linear and non-linear bias parameters b ,a , b ,a , and a shotnoise term ε a .In Fourier space, we can define the matter, galaxy, or clusterpower spectra between any pair of samples ( a, b ) as: (2 π ) δ D ( k − k (cid:48) ) P ab ( k ) ≡ (cid:104) δ a ( k ) δ (cid:63)b ( k (cid:48) ) (cid:105) , (3)where k denotes a wave vector of amplitude k and angled bracketsindicate an average over all Fourier modes within a given spheri-cal shell and δ D is the Dirac delta function. Up to linear order andassuming Poissonian shot noise, the galaxy (or cluster) power spec-trum can be directly related to the matter power spectrum P ( k ) , P ab ( k ) (cid:39) b ,a b ,b P ( k ) + δ abK V /N a (4)where δ K is the Kronecker delta, and the shot noise contributionis given by the inverse number density of galaxies (or clusters), V /N a .In this analysis we consider the angular power spectrum C abl ,a projection of P ab ( k ) on the sky. We use the publically availablecode CLASS (Blas et al. 2011) to generate theoretical predictionsfor the angular cluster power spectrum. CLASS is a differentialequation solver for the hierarchy of Boltzmann equations governingthe perturbations in the density of dark matter, baryons, photonsand any other relevant particle species. The CLASSgal extension(Di Dio et al. 2013) calculates the angular power spectrum, C abl ,for any matter tracer as C abl = 4 π (cid:90) dk k P ini ( k )∆ al ( k )∆ bl ( k ) , (5)where P ini denotes the (dimensionless) primordial power spectrumand the transfer function for the matter component ∆ al ( k ) is givenby ∆ al ( k ) = (cid:90) d z b ,a d N a d z j l ( kr ( z )) D ( k, z ) . (6)Here r ( z ) is the comoving distance, D ( k, z ) the total comovingdensity fluctuation and we use the galaxy and cluster redshift dis-tributions d N/ d z for the observed sample, which are shown in Fig-ure 2 and introduced in Section 3.The main goal of this analysis is to measure the auto-and cross-correlation of galaxies and clusters, and to deter-mine the effective bias of these tracers. Therefore we fix thecosmological parameters to their best-fit values as obtainedby the Planck collaboration (Planck Collaboration et al. 2014)(Planck2013+WP+highL+BAO), derived by combining their ownCMB data with the Wilkinson Microwave Anisotropy Probe(WMAP) polarization data (Bennett et al. 2013), the small-scaleCMB measurements from the Atacama Cosmology Telescope(ACT) (Das et al. 2014) and the South Pole Telescope (SPT)(Reichardt et al. 2012), as well as baryonic accoustic oscillations(BAO) data from SDSS (Percival et al. 2010; Padmanabhan et al.2012; Blake et al. 2011; Anderson et al. 2012; Beutler et al. 2011).The cosmological parameters we use are: h = 0 . , Ω b = 0 . c = 0 . , σ = 0 . , z re = 11 . and n s = 0 . (we checkedthat assuming a Planck 2015 cosmology has no significant impacton the results in our analysis).For the analysis presented in this paper, we adopt a constantbias model, i.e. we define an effective bias b eff , such that we canassume for each sample b ,a ( z ) = b eff , a . The full redshift evolution http://class-code.net/ D ( k, z ) ≈ D + ( z ) T ( k ) for cold dark matter universes, where D + ( z ) isthe density growth function and T ( k ) is the matter transfer function.c (cid:13)000 , 1–12 ross-correlation of SDSS galaxies and clusters of the galaxy and cluster bias could in principle be obtained bysubdividing our samples in multiple redshift bins, but this is beyondthe scope of the present analysis and the data available.Note that the CLASS C abl do not account for a contributiondue shot noise. As we demonstrate below, the theoretical powerspectra C abl defined by Equation (5) need a more advanced mod-elling of the shot noise contribution, which we present in the nextSection. Estimating the underlying, continuous dark matter density field viathe discrete number density of observed galaxies and clusters, in-troduces a shot noise contribution which will leave a systematic im-print on the measured angular power spectrum C data l . In real space,the Poisson sampling from the true underlying density distributionintroduces a contribution to the auto-correlation at zero separation,which translates into the constant contribution in harmonic spaceshown in Equation (4). Due to the large number of galaxies ob-served in the SDSS DR8, this contribution to the measured C data l is negligible on the relevant scales for galaxies, but is the leadingcontribution for the cluster auto-correlation function.The situation is more complicated for the galaxy-cluster cross-correlation. Galaxies that are part of a cluster contribute to the shotnoise, while those that are not part of a cluster do not. Since themajority of the galaxies in our sample are not part of a galaxy clus-ter, we set the shot noise contribution for the galaxy-cluster cross-correlation to zero for now, but we will revisit this issue in Sec-tion 5.2.Additionally, we have to consider a similar, although smaller,effect for the cross-correlation of clusters in different richness bins.Assuming these clusters occupy halos of different mass, self-pairsare not taken into account in their cross-correlation, resulting in avanishing Poisson shot noise contribution. We will come back tothis issue in Section 5.2 as well.The Poisson noise contribution to the model power spectra C ab, model l can be approximated by N abl = δ abK f sky πN a , (7)where f sky is the fraction of the sky covered and N a is the numberof objects observed.While Equation (7) holds for regular masks, in the case ofirregular masks (as the one used in this analysis) a more accurateestimation of the shot noise component is required. In this case, inall generality the shot noise contribution (cid:101) N l can be determined byPoisson sampling different random realisations of a sky map witha constant matter density.Each random realisation i has a power spectrum C rand ,il , fromwhich an estimate of the shot noise contribution can be obtained byaveraging: (cid:101) N l = (cid:104) C rand ,il (cid:105) , (8)where the angular bracket (cid:104)·(cid:105) denotes the average over all randommaps i . The covariance between different angular wave numbers l and m is given by Cov[ C rand l , C rand m ] = N s N s − (cid:104) ( C rand ,il − (cid:101) N l )( C rand ,im − (cid:101) N m ) (cid:105) (9)where N s = 100 is the number of samples used. From this we can determine the covariance of the shot noise, (cid:101) N , as Cov[ (cid:101) N l , (cid:101) N m ] = N − / s Cov[ C rand ,il , C rand ,im ] . (10)We discuss how Cov[ (cid:101) N l , (cid:101) N m ] enters the analysis in Section 5.1.For full sky coverage, we recover the shot noise contributionas given by Equation (7), which remains a good approximation aslong as the shape of the mask is regular enough; we expect howeverto observe signifiant deviations for increasingly irregular masks.The amplitude of the shot noise contribution (cid:101) N l depends onthe number of objects N a distributed over the area of the mask.However, the shape of (cid:101) N l for different l is independent of N a , i.e.for a given mask and pixel size, we can determine the shot noisecontribution just once then rescale the result according to the actualnumber of objects observed.Since we are working with pixellated maps as described inSection 3.2 we will be using the average object per pixel density ¯ n when determining the shot noise contribution. The shot noisecontribution then is determined as (cid:101) N l (¯ n ) = (cid:101) N l (1) / ¯ n . (11)This means that, for a given mask and pixel size, the shot noisecontribution can be determined once for a fixed object per pixeldensity ¯ n = 1 and then rescale the result according to the actualobject per pixel density of our sample ¯ n . We discuss the actualshot noise contribution for the sky mask used in this analysis inSection 3.2. We expect deviations from a purely Poissonian shot noise contri-bution for the power spectra when measured on the galaxy clus-ter data. N -body simulations have provided significant evidencefor such deviations in the clustering statistics of dark matter ha-los (Hamaus et al. 2010). In particular, these deviations have beenshown to depend on halo mass: on large scales the shot noise con-tribution to the power spectrum of low-mass halos exceeds thefiducial value of V /N a , while it is suppressed compared to thatvalue at high masses. These effects are commonly referred to as sub- and super -Poissonian shot noise, respectively. In addition,the Poisson expectation is not only found to be violated in auto-correlations of a single tracer, but also in cross-correlations amongdifferent tracers. While Poissonian shot noise only affects the auto-correlation of self-pairs, simulations have revealed non-vanishingshot noise contributions in cross-correlations between halos of dif-ferent mass (Hamaus et al. 2010). These can be either positive ornegative, depending on the considered mass ranges.This phenomenology can be explained with two competingeffects: exclusion and non-linear clustering (Baldauf et al. 2013).The former simply specifies the fact that any two tracers, be it ha-los or galaxies, can never be closer to one another than the sumof their own extents. This violates the Poisson assumption, whichstates that tracers are randomly sampled at any given point withinsome volume. Each tracer contributes an exclusion region that ef-fectively diminishes the available sampling volume, and thereforethe shot noise contribution. As the exclusion region of halos in-creases with their mass, high-mass halos are most affected by this.Moreover, the exclusion mechanism also applies for halos of dif-ferent mass, i.e. it influences cross-correlations of tracers as well.The first-order contribution from non-linear clustering of ha-los beyond linear theory is described by the second-order bias pa-rameter b . Besides modifying the scale-dependent linear cluster-ing power spectrum of halos on small scales, it also contributes a c (cid:13) , 1–12 Paech et al. scale-independent term that cannot be distinguished from Poissonshot noise (McDonald 2006). Hence, non-linear clustering effec-tively increases the Poisson shot noise, and this effect is most im-portant for low-mass halos, where the value of b is non-zero andexclusion effects are small.While the above effects mainly apply to dark matter halos,they can be translated to galaxies and clusters by means of the halomodel (Seljak 2000; Smith et al. 2003). Given a Halo OccupationDistribution (HOD), one can assign central and satellite galaxiesto each halo of a given mass. While centrals and cluster centersclosely obey the effects outlined above, satellites add more com-plexity as they do not obey halo exclusion. In this case the satellitefraction determines the shot noise as well: a low value ( ∼ )results in sub-Poissonian, and a high value ( ∼ . ) in super-Poissonian shot noise (Baldauf et al. 2013). We should therefore expect deviations from the Poisson shotnoise predictions on all scales, and we expect this correction tobe most important for the cluster auto- and cluster-galaxy cross-correlations.As we have discussed in the previous section, on large scaleswe expect the shot noise correction to be independent of l ; we canthen model this by introducing an inverse effective (average) num-ber density ¯ n − free parameter, which we will fit from the data.In this case, we replace ¯ n in Equation (11) with ¯ n eff and use thefollowing relation for the effective shot noise: (cid:101) N eff l (¯ n eff ) = (cid:101) N l (1) · ¯ n − . (12)in the case of Poissonian shot noise we should recover ¯ n eff (cid:39) ¯ n .We determine n eff for each of the auto- and cross-correlations andmarginalise over it to determine the effective bias of each sample. We compare the effective bias we extract from the data to the-oretical expectations for a volume-limited sample, whose detailsare described below in Section 5.3. We assume here the halo massfunction n ( M, z ) and halo bias b ( M, z ) to be given by the fits to N -body simulations by Tinker et al. (2008, 2010). In order to calcu-late the expected effective bias of a volume-limited cluster sample,we average over the redshift range considered b eff = (cid:20)(cid:90) ∆ z d z d V d z dΩ (cid:90) d Mn ( M, z ) b ( M, z ) (13) (cid:90) λ min d λP ( λ | M, z ) (cid:21)(cid:30) (cid:20)(cid:90) ∆ z d z d V d z dΩ (cid:90) d Mn ( M, z ) (cid:90) λ min d λP ( λ | M, z ) (cid:21) , where d V / (d z dΩ) is the comoving volume element per unit red-shift and solid angle, and P ( λ | M, z ) is the probability that a haloof mass M at redshift z is observed with a richness λ . We modelthis probability with a log-normal distribution: P ( λ | M, z ) = 1 λ (cid:112) πσ λ exp (cid:20) − ln λ − (cid:104) ln λ ( M, z ) (cid:105) σ λ (cid:21) , (14)where we use the parameterisation and parameter values of the scal-ing relation between mass and richness (cid:104) ln λ ( M, z ) (cid:105) as determined by Farahi et al. (2016). The scatter σ ln λ is defined as σ λ = exp [ (cid:104) ln λ ( M, z ) (cid:105) ] − (cid:104) ln λ ( M, z ) (cid:105) ]) + σ λ | M , (15)where the first term accounts for the richness-dependent Poissonnoise and σ ln λ | M is the intrinsic scatter in the richness-mass rela-tion. As Farahi et al. (2016) do not specify the value for σ ln λ | M or σ ln M | λ , we determine σ ln λ | M using the value for σ ln M | λ = 0 . from Simet et al. (2016). Using Equations (13) to (15) from Simetet al. (2016), we determine σ ln λ | M via the following relation σ ln λ | M = σ ln M | λ α , (16)where α = 1 . denotes the power-law slope of mass given therichness, as defined by Simet et al. (2016). We use galaxy and galaxy cluster data drawn from the Sloan DigitalSky Survey (SDSS) (York et al. 2000). The SDSS is conductedwith a dedicated 2.5m telescope at the Apache Point Observatoryin Southern New Mexico in the United States. This telescope hasa wide field of view of , a large mosaic CCD camera and apair of double spectrographs (Aihara et al. 2011; Eisenstein et al.2011). We use the final photometric SDSS data from the eighthdata release (DR8) that combines data from the two project phasesSDSS-I and SDSS-II.The full area of SDSS DR8 is ,
555 deg and includes pho-tometric measurements of 208,478,448 galaxies. For the analysisin this paper we use the same galaxy catalogue and selection cri-teria as in Giannantonio et al. (2012). The catalogue only containsobjects with redshift between 0.1 and 0.9 that have a photo- z uncer-tainty of σ z ( z ) < . z . A completeness cut is applied by only us-ing objects with extinction-corrected r -band magnitudes between18 and 21. After these cuts, the catalogue contains 41,853,880galaxies.We use the cluster catalogue constructed from SDSS DR8with the red -sequence Ma tched-filter P robabilistic Per colation(redMaPPer) cluster finding algorithm version 5.10 (Rykoff et al.2014). It contains ∼ , galaxy clusters covering a redshiftrange between 0.1 and 0.6 and contains only clusters with richness λ > . Note that we use the richness as defined by redMaPPerthroughout this paper. As the richness λ is a measure of the numberof galaxies within the cluster, this means smaller clusters, whichare more strongly affected by systematic errors, are excluded fromthe analysis. We create pixelized maps for the galaxy and the galaxy cluster cat-alogues using the pixelization scheme HEALPix (Gorski et al.2005), in which the resolution is expressed by the parameter N side .The pixellation effectively smoothes information on scales smallerthan the pixel size, and it can be described by a multiplicative win-dow function w l given by the pixel window function provided byHEALPix. http://risa.stanford.edu/redmapper/ http://healpix.jpl.nasa.gov/ c (cid:13)000
555 deg and includes pho-tometric measurements of 208,478,448 galaxies. For the analysisin this paper we use the same galaxy catalogue and selection cri-teria as in Giannantonio et al. (2012). The catalogue only containsobjects with redshift between 0.1 and 0.9 that have a photo- z uncer-tainty of σ z ( z ) < . z . A completeness cut is applied by only us-ing objects with extinction-corrected r -band magnitudes between18 and 21. After these cuts, the catalogue contains 41,853,880galaxies.We use the cluster catalogue constructed from SDSS DR8with the red -sequence Ma tched-filter P robabilistic Per colation(redMaPPer) cluster finding algorithm version 5.10 (Rykoff et al.2014). It contains ∼ , galaxy clusters covering a redshiftrange between 0.1 and 0.6 and contains only clusters with richness λ > . Note that we use the richness as defined by redMaPPerthroughout this paper. As the richness λ is a measure of the numberof galaxies within the cluster, this means smaller clusters, whichare more strongly affected by systematic errors, are excluded fromthe analysis. We create pixelized maps for the galaxy and the galaxy cluster cat-alogues using the pixelization scheme HEALPix (Gorski et al.2005), in which the resolution is expressed by the parameter N side .The pixellation effectively smoothes information on scales smallerthan the pixel size, and it can be described by a multiplicative win-dow function w l given by the pixel window function provided byHEALPix. http://risa.stanford.edu/redmapper/ http://healpix.jpl.nasa.gov/ c (cid:13)000 , 1–12 ross-correlation of SDSS galaxies and clusters To construct the (binary) survey mask we follow the methodby Giannantonio et al. (2006) to estimate the coverage of pixelsthat straddle the survey boundaries. If we have a distribution P ( n ) of the number of galaxies per pixel, this effect causes a deviationfrom a Poisson distribution P Poiss ( n ) for low n , i.e. we will find anexcess of pixels with a small n compared to P Poiss ( n ) . In practice,we create the mask by discarding pixels where P ( n ) (cid:54)≈ P Poiss ( n ) ,i.e. pixels with n < n min where n min is a cut-off threshold wechoose.The pixel size we choose is constrained by two factors. On theone hand pixels should be large enough to ensure that the mean ofthe distribution P ( n ) is far from zero, i.e. there is only a very smalland negligible number of pixels with a small number of galaxies.On the other hand pixels should be small enough so that we canmeasure the angular power spectra at the scales of interest.We choose the HEALPix resolution N side = 512 , correspond-ing to a pixel side of (cid:48) , as this produces an average number ofgalaxies per pixel of ¯ n (cid:39) for the SDSS galaxy catalogue, andit also allows access to the scales of interest in this analysis. Allcatalogues we use here are pixelized at this resolution.Therefore, to create the mask we first determine the numberof galaxies n i in each of the pixels i and discard pixels with zerogalaxies. Then we determine the mean and variance of the Poissondistribution by calculating the average number of galaxies per pixel ¯ n and identify the value of n below which we observe a deviationfrom the Poisson distribution. We find that n min = ¯ n − √ ¯ n (i.e.the equivalent of two standard deviations below the mean numberof pixels) is a good value for the cut-off and we mask all pixelswhere n i < n min .We generate the cluster mask by repeating the above pro-cess using the available random redMaPPer cluster catalogues. Wechoose to only work with the intersection of the galaxy and galaxycluster masks. We further combine this mask with the dust extinc-tion maps by Schlegel et al. (1998), retaining only pixels with red-dening values E ( B − V ) < . , and with the SDSS seeing masksby Ross et al. (2011), retaining pixels with seeing values below . (cid:48)(cid:48) . The final mask footprint covers 6,983 deg . After the ap-plication of the complete mask, the data covers a fraction of sky f sky = 0 . and contains 671,533 unmasked pixels.In Section 2.2 we described how we determine the shot noisecontribution for a given mask. Figure 1 shows both the analytic shotnoise approximation N l from Equation (7) and the shot noise (cid:101) N l from Poisson sampling for ¯ n = 1 from Equation (8) for the maskused in our analysis. We see that taking into account the effect ofthe mask increases the shot noise by approximately 5%. Also, thereis a mild l -dependence of the shot noise for the Poisson sampledshot noise compared to the analytic approximate shot noise causedby the shape of the mask. In addition to the full sample described in Section 3.1 above, forour analysis we use different subsets of the cluster catalogue. In thefollowing, we will consider: • c all : the full cluster sample, which is richness-selected andthus not volume-limited; • c vlim : a volume-limited sample that is constructed by usingonly clusters with z < . ; • c λ low and c λ high : a low- and high-richness sample, con-structed from the full cluster sample (all redshifts) that is split atthe median richness of λ med = 33 . .
100 200 300 400 500 l . . . . . s h o t n o i s e × − e N l (from mocks) N l (analytic) Figure 1.
Shot noise predictions for an approximate analytic shot noiseestimate N l (Equation 7) and the Poisson sampled shot noise ( mocks ) (cid:101) N l (Equation 8) for an average number of objects per pixel density ¯ n = 1 . . . . . . . . . . z d N / d z gc all c v lim c λ low c λ high Figure 2.
Normalized redshift distributions d N/ d z of galaxies and galaxyclusters samples as discussed in Section 3.3. The different samples are summarised in Table 1. The sample c all containing all the clusters of the redMaPPer catalogue is the start-ing point of our analysis as it makes use of all the objects avail-able and allows us to investigate the shot noise properties of ourmeasurements, especially for the galaxy and galaxy cluster cross-correlation (Sections 5.1 and 5.2). In a second step we use the c vlim sample in order to be able to compare our best-fit results to the the-oretical expectation for the value of the effective bias (Section 5.3).Finally, we use the samples split into two richness bins c λ low and c λ high to investigate the shot noise properties of cluster clustering(Section 5.4).In Figure 2 we show the redshift distribution d N/ d z of galax-ies and galaxy clusters, after masking has been applied, for the dif-ferent samples described above. In order to account for the uncer-tainties in cluster redshifts when performing the theoretical predic-tions in Sections 2.1, we randomly sample from the redshifts errorsfor the respective object type. For galaxies we assume an overall c (cid:13) , 1–12 Paech et al.
Table 1.
Summary of all samples used in this analysis. N is the total num-ber of objects left after masking and cuts, ¯ n is the average number of objectsper pixel ( N side = 512 ), z median is the median redshift of the sample andthe selection column indicates the additional cuts done beyond masking.Sample N object type z median ¯ n selectiong 25,959,346 galaxies 0.31 39 see text c all c vlim z < . c λ low λ < . c λ high λ > .
5% photometric redshift error as the individual redshift errors inthe catalogues underestimate the true redshift error. In the case ofthe galaxy clusters we re-sample the provided redshift accordingto the error provided by redMaPPer. Therefore the redshift distri-bution of galaxies does not have a sharp boundary at low redshiftsand the distribution of clusters for the c vlim sample does not have asharp boundary at z = 0 . . We measure the angular power spectrum C l for all the data productsdescribed in Section 3 using the Spatially Inhomogeneous Correla-tion Estimator for Temperature and Polarisation (PolSpice) (Chonet al. 2004; Szapudi et al. 2000; Challinor & Chon 2005). The ad-vantage of using PolSpice is that the algorithm corrects for distor-tions of the measured power spectrum caused by masking and thepixel window function w l . The partial sky coverage has more com-plex effects (see eg Efstathiou 2004), which we assume here to becorrected by PolSpice.PolSpice is used to estimate the C data l from pixellated densitycontrast maps δ i derived from both the galaxy and galaxy clusterdensity per pixel n i δ i = n i ¯ n − , (17)where ¯ n denotes the average over all pixels i .For our analysis we consider multipoles l in the range
20 and N jk =
100 to determine the covariance matrices of the C l measure-ments. This yields an uncertainty in the error bars of the extractedparameters of 16%.Note that when calculating the inverse covariance, we needto multiply it by the de-biasing factor introduced by Hartlap et al.(2007) and Taylor et al. (2013): f corr = N jk − N jk − N bin − , (19)where N bin is the size of the data vector, i.e. the number of C l bins N bin . In the following section we will present the results of ouranalysis. In Figure 3 we present the measured angular power spectra for thegalaxy ( gg ), cluster ( c all c all ) and galaxy-cluster ( gc all ) cases usingthe full cluster sample c all and the analysis method described inSection 4. The best fitting models to these measurements are de-scribed in Sections 5.1 and 5.2 below and are shown as lines inFigure 3.In order to determine the effective bias for the different trac-ers, we use two different models to account for the shot noise. Wefirst analyse the data using Poissonian shot noise as described inSection 2.2 and discuss the results of those fits. In a second step weaccount for non-Poissonian shot noise as discussed in 2.4. We first fit to the C data l the model given by: C model l ( b ) = C th l ( b ) + (cid:101) N l (¯ n ) , (20)where C th l is the theoretical angular power spectrum as given inEquation (5) for a given value of the effective bias b and (cid:101) N l (¯ n ) isthe Poisson noise term given in Equation (7).For the covariance, there are two contributions: the covariancefrom the power spectrum measurement given in Equation (18), aswell as the covariance from the Poisson noise contribution (cid:101) N l givenin Equation (10). The covariance of the shot noise contributionarises from the fact that we determine the shot noise from the Pol-Spice measurements of random maps as discussed in Section 2.2. c (cid:13)000
100 to determine the covariance matrices of the C l measure-ments. This yields an uncertainty in the error bars of the extractedparameters of 16%.Note that when calculating the inverse covariance, we needto multiply it by the de-biasing factor introduced by Hartlap et al.(2007) and Taylor et al. (2013): f corr = N jk − N jk − N bin − , (19)where N bin is the size of the data vector, i.e. the number of C l bins N bin . In the following section we will present the results of ouranalysis. In Figure 3 we present the measured angular power spectra for thegalaxy ( gg ), cluster ( c all c all ) and galaxy-cluster ( gc all ) cases usingthe full cluster sample c all and the analysis method described inSection 4. The best fitting models to these measurements are de-scribed in Sections 5.1 and 5.2 below and are shown as lines inFigure 3.In order to determine the effective bias for the different trac-ers, we use two different models to account for the shot noise. Wefirst analyse the data using Poissonian shot noise as described inSection 2.2 and discuss the results of those fits. In a second step weaccount for non-Poissonian shot noise as discussed in 2.4. We first fit to the C data l the model given by: C model l ( b ) = C th l ( b ) + (cid:101) N l (¯ n ) , (20)where C th l is the theoretical angular power spectrum as given inEquation (5) for a given value of the effective bias b and (cid:101) N l (¯ n ) isthe Poisson noise term given in Equation (7).For the covariance, there are two contributions: the covariancefrom the power spectrum measurement given in Equation (18), aswell as the covariance from the Poisson noise contribution (cid:101) N l givenin Equation (10). The covariance of the shot noise contributionarises from the fact that we determine the shot noise from the Pol-Spice measurements of random maps as discussed in Section 2.2. c (cid:13)000 , 1–12 ross-correlation of SDSS galaxies and clusters However,
Cov[ C rand ,il , C rand ,im ] is about one order of magni-tude smaller than Cov data [ C l , C m ] and N − / s = 10 , and there-fore the covariance originating from the Poisson noise correction Cov[ (cid:101) N l , (cid:101) N m ] is about two orders of magnitude smaller than thecovariance contribution from the data Cov data [ C l , C m ] . Hence weneglect the covariance contribution of the shot noise error and use Cov fit [ C l , C m ] = Cov data [ C l , C m ] (21)as the covariance in the Gaussian likelihood L of the effective biasparameters b .We then use this covariance to calculate the Gaussian like-lihoods of the effective bias parameters from all spectra we con-sider, i.e. from galaxy and cluster auto-spectra and from the cross-spectrum; we label these likelihoods L g , L gc all , and L c all respec-tively.Additionally, we can estimate the effective bias likelihoodfrom the cross-correlation given the results from the two corre-sponding auto-correlations. For example, from the likelihoods ofthe galaxy and cluster auto-correlations L g and L c , we can con-struct the following likelihood L√ b g b c ( b ) = (cid:90) (cid:90) d˜ b g d˜ b c δ D (cid:18) b − (cid:113) ˜ b g ˜ b c (cid:19) L g (˜ b g ) L c (˜ b c ) . (22)This likelihood serves as a cross-check for the biases obtained byour analysis and is equivalent to drawing values for b g and b c from the respective auto-correlation distributions and determininga new distribution from the corresponding (cid:112) b g b c . We determinethese cross-check likelihoods and bias values for all the cross-correlations we determine in our analysis.We start by analysing the auto- and cross-correlations for thelargest samples available, i.e. the galaxy sample g and the full clus-ter sample c all , as we expect to obtain the most accurate mea-surements from these samples. The best-fit models as defined inEquation (20) are shown as dashed lines in Figure 3. While thegalaxy auto-correlation ( gg ) is accurately described by the model,the galaxy-cluster cross-correlation ( gc all ) as well as the clusterauto-correlation ( cc all ) are poorly described by the fit. Best-fit pa-rameters, as well as χ , are listed in rows 1 – 3 of Table 2. Note thatthe χ values are too large for the gc all and c all c all correlations,indicating that the model is not a good description of the measure-ments. The likelihoods for the fits and cross-check are shown inFigure 4 and indicate that also L√ b g b c does not agree well withthe results for L g and L c , i.e. the results of the auto- and cross-correlations are inconsistent with each other.Concentrating on the mismatch between the data and modelfor the galaxy-cluster ( gc all ) and cluster ( c all c all ) correlation, weremove the shot noise contribution, which is dominant for the c all c all correlation, from both the data and model. In addition, webring the C l to the same scale by renormalizing them by the best fitbias; the resulting C l are shown in Figure 5.For the galaxy-cluster cross-correlation, the model (orangesymbols and line) clearly overestimates the C l for low l and un-derestimates them for high l . This is not unexpected, because ne-glecting the shot noise contribution for the galaxies that are partof clusters (as discussed in Section 2.2) is expected to be a poorapproximation.In the case of the cluster auto-correlation, we can now see themismatch between the data and model more clearly because theextracted signal is dominated by the shot noise. Also in this case,the figure shows a severe mismatch, as the model overestimatesthe C l on all scales. Above l ≈ the noise-corrected C l even l − − l ( l + ) π C l gggc all c all c all Figure 3.
Galaxy ( gg – red lines and symbols), cluster ( c all c all – greenlines and symbols) and galaxy-cluster ( gc all – orange lines and symbols)angular power spectra for the data described in Section 3. Lines indicatethe best-fit models described in Sections 5.1 and 5.2: dashed lines show thebest-fit model specified in Equation (20) using Poissonian shot noise, whilesolid lines use the best-fit model specified in Equation (23) where ¯ n eff isadded as a fit parameter to adjust the shot noise contribution. b . . . . . . L / L m a x gggc all c all c all p b g b c all Figure 4.
Likelihood functions for the effective bias of the samples weconsider, obtained using a fixed shot noise contribution, for galaxies ( gg – red lines), clusters ( c all c all – green lines) and galaxy-clusters ( gc all –brown lines), as well as for the cross-check case L√ b g b c ( (cid:112) b g b c - dashedgrey line) described in Section 5.1. turn negative, indicating the shot noise is overcorrected using theaverage object per pixel density ¯ n for clusters. Following the reasoning of Section 2.4, we can effectively accountfor a modification of the shot noise contribution if we limit our-selves to a regime where the correction is (to a good approximation)independent of l . We can then introduce an effective number den-sity of objects per pixel ¯ n eff as a nuisance parameter. This allows c (cid:13) , 1–12 Paech et al.
Table 2.
Results for the different parameter fits from the angular power spectra we use. Column 2 indicates the correlator used, column 3 indicates themaximum likelihood value for bias b and the statistical error for the fit, column 4 the χ and number of degrees of freedom. If the effective noise contributionis also determined in the fit, the inverse of the maximum likelihood effective pixel density ¯ n eff , ML is listed in column 5, while column 6 lists the actual inversepixel density of objects ¯ n − (only available for auto-correlations). In case the cross-check can be performed, the corresponding value for bias b cross − check (including error) is listed in column 7.Row correlator b ± σ stat χ /dof / ¯ n eff ± σ stat / ¯ n b cross − check gg ± gc all ± ± c all c all ± gg ± ± gc all ± ± ± c all c all ± ± gc vlim ± ± ± c vlim c vlim ± ± gc λ low ± ± ± gc λ high ± ± ± c λ low c λ low ± ± c λ high c λ high ± ± c λ low c λ high ± ± ± l − . . . . . l ( l + ) π ( C l − f N l ( ¯ n )) b − gc all c all c all Figure 5.
Cluster ( c all c all ) and galaxy-cluster ( gc all ) power spectra withPoissonian shot noise (cid:101) N l removed according to Equation (11) and rescaledby the best-fit bias b . Dashed lines show the best-fit model specified in Equa-tion (20). us to account for any sub- and super-Poissonian shot noise contri-butions where the shape of the shot noise correction is unaffectedand only the magnitude of (cid:101) N l is adjusted, i.e. C model l ( b, ¯ n eff ) = C th l ( b ) + (cid:101) N l (¯ n eff ) . (23)To account for measuring systematics affecting the powerspectra, a similar treatment was used by Ho et al. (2012); Zhaoet al. (2013); Beutler et al. (2014); Johnson et al. (2016); Griebet al. (2016) in their analyses.The results for the fits including the effective shot noise contri-bution as a free parameter are shown as the solid lines in Figure 3.The C model l ( b, ¯ n eff ) for the fit describe the data much more accu-rately. The corresponding constant likelihood contours are shownin Figure 6 where the dashed lines indicate the actual / ¯ n for thePoissonian shot noise contribution (not applicable to the cross-correlation gc all ). Both the galaxy and cluster auto-correlations slightly favour sub-Poissonian noise contributions, though the devi-ation from the Poissonian noise case is below 1- σ for galaxies andabout 1.5- σ for the clusters. As expected, the noise contribution forthe galaxy auto-correlation is small. However, the shot noise forthe cluster auto-correlation is a dominant contribution, and there-fore we see a much better description of the data compared to theprevious fit where we only fitted for the bias. The same holds for thegalaxy-cluster cross-correlation for which the data clearly favoursa non-zero shot noise contribution.We show in Figure 7 the marginalised likelihoods for the biasparameters; we can see that in this case the cross-check likeli-hood L√ b g b c agrees well with the result obtained from the galaxy-cluster cross-correlation gc all . This means that the measurementsof auto- and cross-correlation are now in good agreement with eachother when introducing n eff as an additional model parameter.The results of these fits, including maximum likelihood valuesfor L√ b g b c , are summarised in rows 4-6 of Table 2. Note that the χ have improved significantly compared to the bias-only fits.The fact that we obtain a sub-Poissonian shot noise from theauto-correlation of galaxies argues for a relatively low satellite frac-tion in the sample, as discussed in Section 2.3. The sub-Poissonianshot noise obtained from the cluster auto-correlation is expectedbecause clusters obey halo exclusion. The measurement of the cluster auto-correlation for the full sam-ple c all yields a bias value of b c = 4 . ± . . However, the c all sample is not volume-limited and we should therefore use thevolume-limited sample c vlim when comparing the effective bias totheoretical expectations.We perform the same analysis of Section 5.2 on the volume-limited sample c vlim as discussed in Section 3 and summarized inTable 1. The fits for the volume-limited cluster sample are qualita-tively similar to the c all samples and we therefore do not show plotsof the power spectra and best fits. Results for the auto-correlation c vlim c vlim as well as the cross-correlation gc vlim are listed in rows7 and 8 of Table 2. The bias from the cross-check and the bias for c (cid:13)000
Cluster ( c all c all ) and galaxy-cluster ( gc all ) power spectra withPoissonian shot noise (cid:101) N l removed according to Equation (11) and rescaledby the best-fit bias b . Dashed lines show the best-fit model specified in Equa-tion (20). us to account for any sub- and super-Poissonian shot noise contri-butions where the shape of the shot noise correction is unaffectedand only the magnitude of (cid:101) N l is adjusted, i.e. C model l ( b, ¯ n eff ) = C th l ( b ) + (cid:101) N l (¯ n eff ) . (23)To account for measuring systematics affecting the powerspectra, a similar treatment was used by Ho et al. (2012); Zhaoet al. (2013); Beutler et al. (2014); Johnson et al. (2016); Griebet al. (2016) in their analyses.The results for the fits including the effective shot noise contri-bution as a free parameter are shown as the solid lines in Figure 3.The C model l ( b, ¯ n eff ) for the fit describe the data much more accu-rately. The corresponding constant likelihood contours are shownin Figure 6 where the dashed lines indicate the actual / ¯ n for thePoissonian shot noise contribution (not applicable to the cross-correlation gc all ). Both the galaxy and cluster auto-correlations slightly favour sub-Poissonian noise contributions, though the devi-ation from the Poissonian noise case is below 1- σ for galaxies andabout 1.5- σ for the clusters. As expected, the noise contribution forthe galaxy auto-correlation is small. However, the shot noise forthe cluster auto-correlation is a dominant contribution, and there-fore we see a much better description of the data compared to theprevious fit where we only fitted for the bias. The same holds for thegalaxy-cluster cross-correlation for which the data clearly favoursa non-zero shot noise contribution.We show in Figure 7 the marginalised likelihoods for the biasparameters; we can see that in this case the cross-check likeli-hood L√ b g b c agrees well with the result obtained from the galaxy-cluster cross-correlation gc all . This means that the measurementsof auto- and cross-correlation are now in good agreement with eachother when introducing n eff as an additional model parameter.The results of these fits, including maximum likelihood valuesfor L√ b g b c , are summarised in rows 4-6 of Table 2. Note that the χ have improved significantly compared to the bias-only fits.The fact that we obtain a sub-Poissonian shot noise from theauto-correlation of galaxies argues for a relatively low satellite frac-tion in the sample, as discussed in Section 2.3. The sub-Poissonianshot noise obtained from the cluster auto-correlation is expectedbecause clusters obey halo exclusion. The measurement of the cluster auto-correlation for the full sam-ple c all yields a bias value of b c = 4 . ± . . However, the c all sample is not volume-limited and we should therefore use thevolume-limited sample c vlim when comparing the effective bias totheoretical expectations.We perform the same analysis of Section 5.2 on the volume-limited sample c vlim as discussed in Section 3 and summarized inTable 1. The fits for the volume-limited cluster sample are qualita-tively similar to the c all samples and we therefore do not show plotsof the power spectra and best fits. Results for the auto-correlation c vlim c vlim as well as the cross-correlation gc vlim are listed in rows7 and 8 of Table 2. The bias from the cross-check and the bias for c (cid:13)000 , 1–12 ross-correlation of SDSS galaxies and clusters .
00 1 .
05 1 .
10 1 .
15 1 . b g − . − . − . . . . . . . / ¯ n e ff . . . . . . . . . . . L / L m a x . . . . . . b gc all . . . . . . . . . / ¯ n e ff . . . . . . . . . . . L / L m a x b c all / ¯ n e ff . . . . . . . . . . . L / L m a x Figure 6.
The 2d-likelihoods for the fits described in Section 5.2 for theparameters bias b and effective number of objects per pixel ¯ n eff . Solid linesindicate the , and confidence regions respectively. For theauto-correlation cases the dotted lines indicate the actual inverse number ofobjects per pixel ¯ n − . b . . . . . . L / L m a x gggc all c all c all p b g b c all Figure 7.
Likelihoods of the effective bias, marginalised over the ampli-tude of the shot noise contribution as described in Section 5.2, for the dif-ferent correlators we consider: galaxies auto- ( gg – red lines), clusters auto-( c all c all – green lines) and galaxy-cluster cross-correlation ( gc all – brownlines) as well as the consistency check bias ( (cid:112) b g b c – dashed grey line) asdescribed in Section 5.1. the cross-correlation are again in agreement. From the cluster auto-correlation we find b c vlim = 4 . ± . .From the theoretical modeling of the effective bias in Sec-tion 2.5 we expect b ≈ . for the volume-limited cluster sample,which is in tension with the value extracted from the data at the2-3 σ level. We explored – and ruled out – the following systematic effectswhich might explain this discrepancy. The redshift and richness dis-tributions of clusters we expect from the predictions in Section 2.5are in good agreement with those measured for the volume-limitedsample as shown in Figure 2 and cannot be used to explain this dis-crepancy. Neither are there extremely high mass/bias objects thatcould explain the difference. We have checked if statistical and sys-tematic uncertainty of the mass-richness relation for galaxy clus-ters could account for the tension. Even shifting the mass-richnessrelation by 30% in mass does not alleviate the tension. The ef-fect of the measurement uncertainties of the mass-richness relationparameters is negligible. The measurement error on σ from thePlanck2013+WP+highL+BAO measurement also cannot accountfor the discrepancy, i.e. shifting the value of σ by 1- σ does onlyhave a very small effect on our measurement of the effective bias.In this work we have not explored the effect of assembly bias(Sheth & Tormen 2004; Gao et al. 2005; Gao & White 2007; Wech-sler et al. 2006; More et al. 2016), i.e. the dependence of haloclustering on assembly history. However, our effective bias mea-surement is consistent with the results examining this effect as re-ported in Miyatake et al. (2016) and Baxter et al. (2016) and ref-erences therein. Miyatake et al. (2016) split the clusters into twosubsamples based on the average member galaxy separation fromthe cluster center and find significantly different values for the biasfor those subsamples: b = 2 . ± . for clusters with a low av-erage member galaxy separation and b = 3 . ± . for largeaverage separation. Baxter et al. (2016) measures the angular cor- c (cid:13) , 1–12 Paech et al. relation function w ( θ ) for clusters in different richness and redshiftbins and find high (compared to the prediction using the Tinkermass function) bias values between 3 and 5 for the λ > richnessbins. It will be very interesting to study this effect in more detailusing angular power spectra in harmonic space in a future analysis. Next, we investigate the shot noise properties in the cross-correlations of clusters in different richness bins and hence dif-ferent halo mass. Therefore, we divide the c all sample into twohalves, splitting it at the median richness λ median = 33 . with λ low < λ median and λ low > λ median with median redshifts 0.315and 0.42. The redshift distributions for these two sub-samples areshown in Figure 2.The auto-correlations of the richness-split cluster samples andtheir relative best-fit bias values are qualitatively similar to the c all sample. We therefore do not show the results for the auto-correlations and will only discuss and show the cross-correlationmeasurements below. The fit results are summarized in rows 9-13of Table 2. As expected, for the auto-correlation of the low-richnesssample c λ low we observe a smaller bias than for the c all sample,while for the high-richness sample c λ high the bias is shifted to evenhigher values (rows 11 and 12). The effective shot noise contribu-tion is larger for these samples as reflected by the smaller valuesof ¯ n eff , because there are half as many objects in the sub-samples.A similar systematic bias shift can be seen for the galaxy-clustercross-correlation, which is shown in rows 7-8, while the shot-noisecontribution is smaller for the gc λ low and higher for gc λ high cross-correlation.From the cross-correlation between the low- and high-richnesssamples, we expect a small or vanishing value of the effective ob-jects per pixel density, because there is no overlap of objects be-tween the two samples (except for systematic effects from clusterfinding/identification and line of sight effects). However, the ac-tual value of the effective pixel density is negative at the 2- σ level.Again, this argues for strong exclusion effects between clusters ofdifferent richness (and thus halo mass), as observed in N -body sim-ulations (Hamaus et al. 2010; Baldauf et al. 2013).Figure 8 shows the measured angular power spectrum as wellas the best-fit models using a vanishing shot-noise contribution(dashed line) and fitting for the shot-noise contribution (solid line).The measured C l show an unusual behavior as the correlation in-creases up to l ≈ and then decreases again. This cannotbe described by a model with a vanishing (shown by the dashedline) shot-noise contribution, as a positive shot-noise contributionwould worsen this mismatch. However, if allowing for negativenon-Poissonian shot noise, the model yields a good description ofthe data. It will be interesting to see whether this measurement per-sists for larger sample sizes and other data sets.We have summarized the bias fit results for the different cata-logue samples in Figure 9, which includes 1- and 2-sigma error barsfor all measurements. This figure illustrates that the cross-checkvalues (black symbols) and errors for bias from Equation (22) ofthe measurements of auto- and cross-correlations are consistentwith each other for all measurements, including the volume-limitedsample c vlim as well as for the two richness bin samples c λ low and c λ high . l − . − . − . − . − . . . . . . l ( l + ) π C l c λ low c λ high Figure 8.
Cross-correlation between the low- and high-richness samplesdescribed in Section 3. Lines indicate the best-fit models described in Sec-tions 5.1 and 5.2: Dashed line shows the best-fit model specified in Equa-tion (20), using a vanishing shot-noise contribution (as there is no overlapbetween the two samples), solid line represents the best-fit model specifiedin Equation (23), where ¯ n eff is added as a fit parameter to adjust the shot-noise contribution. The best fit ¯ n eff is negative in this case. We have presented a first measurement of the cross-correlation an-gular power spectrum of galaxies and galaxy clusters using theSDSS DR8 galaxy and galaxy cluster sample. Further, we mea-sured the auto- and cross-correlations of different sub-samples ofthe full cluster catalogue: a volume-limited sample as well as twosamples of low and high richness.We argued that in order to get a good theoretical descriptionfor the cross-correlation measurements we need to add an effectiveshot-noise contribution as an additional component to our model.Because there is some overlap of galaxies and galaxy clusters, weexpect a non-vanishing shot-noise contribution. We find the mea-surements are much better described by a model containing sub-Poissonian shot noise and that using a regular Poisson shot noisecorrection results in an overcorrection. Since we also expect a de-viation from Poissonian shot noise for cluster auto-correlations dueto halo exclusion and non-linear clustering (Baldauf et al. 2013),we investigated if the cluster auto-correlation shows deviation fromPoissonian shot noise as expected from simulations (Hamaus et al.2010).We extracted the effective bias for our measurements and usedthe results for the effective bias from auto-correlation measure-ments to perform a cross-check on the effective bias from the cross-correlation measurements. These cross-checks were in very goodagreement after we allowed for non-Poissonian shot noise contri-bution. We performed the same cross-checks for measurements in-volving the subsamples of the cluster data and we find all measure-ments of effective bias to be consistent.To compare our measurement of effective bias to theoreticalexpectations, we constructed a volume-limited cluster sample andfound a relatively high value of . ± . compared to our expec-tation of b = 2 . . However, this value is consistent with previousmeasurements and supports the case for a fuller exploration of ad-ditional systematic effects, such as halo assembly bias.Finally, we constructed a low and high richness sample from c (cid:13)000
Cross-correlation between the low- and high-richness samplesdescribed in Section 3. Lines indicate the best-fit models described in Sec-tions 5.1 and 5.2: Dashed line shows the best-fit model specified in Equa-tion (20), using a vanishing shot-noise contribution (as there is no overlapbetween the two samples), solid line represents the best-fit model specifiedin Equation (23), where ¯ n eff is added as a fit parameter to adjust the shot-noise contribution. The best fit ¯ n eff is negative in this case. We have presented a first measurement of the cross-correlation an-gular power spectrum of galaxies and galaxy clusters using theSDSS DR8 galaxy and galaxy cluster sample. Further, we mea-sured the auto- and cross-correlations of different sub-samples ofthe full cluster catalogue: a volume-limited sample as well as twosamples of low and high richness.We argued that in order to get a good theoretical descriptionfor the cross-correlation measurements we need to add an effectiveshot-noise contribution as an additional component to our model.Because there is some overlap of galaxies and galaxy clusters, weexpect a non-vanishing shot-noise contribution. We find the mea-surements are much better described by a model containing sub-Poissonian shot noise and that using a regular Poisson shot noisecorrection results in an overcorrection. Since we also expect a de-viation from Poissonian shot noise for cluster auto-correlations dueto halo exclusion and non-linear clustering (Baldauf et al. 2013),we investigated if the cluster auto-correlation shows deviation fromPoissonian shot noise as expected from simulations (Hamaus et al.2010).We extracted the effective bias for our measurements and usedthe results for the effective bias from auto-correlation measure-ments to perform a cross-check on the effective bias from the cross-correlation measurements. These cross-checks were in very goodagreement after we allowed for non-Poissonian shot noise contri-bution. We performed the same cross-checks for measurements in-volving the subsamples of the cluster data and we find all measure-ments of effective bias to be consistent.To compare our measurement of effective bias to theoreticalexpectations, we constructed a volume-limited cluster sample andfound a relatively high value of . ± . compared to our expec-tation of b = 2 . . However, this value is consistent with previousmeasurements and supports the case for a fuller exploration of ad-ditional systematic effects, such as halo assembly bias.Finally, we constructed a low and high richness sample from c (cid:13)000 , 1–12 ross-correlation of SDSS galaxies and clusters b b g b gc all p b g b c all b gc vlim p b g b c vlim b gc λ low q b g b c λ low b gc λ high q b g b c λ high b c all b c vlim b c λ low b c λ low c λ high q b c λ low b c λ high b c λ high Figure 9.
The maximum likelihood galaxy and dark mater halo bias aftermarginalising over over ¯ n eff , as well as the expectations denoted by √ b a b b from the cross-check described in Section 5.2. We also show (solidlines) and (faint solid lines) error bars. the full cluster sample and measured the auto- and cross-correlationas well. Again, the values for the effective bias are consistent witheach other and are all relatively large for the cluster samples.Most notably, we found a negative shot noise contribution forthe cross-correlation at the 2- σ level. This argues for strong exclu-sion effects between clusters of different richness (and thus halomass), in agreement with N -body simulations. As larger and betterdata sets will become available, it will be interesting to see if thismeasurement of negative shot noise persists.An appropriate treatment of shot noise is important for manyother auto- and cross-correlation large scale structure analyses, i.e.different galaxy types and multi-tracer surveys. In the future, it willbe essential to account for these effects to derive unbiased cosmol-ogy constraints from correlation functions and power spectra anal-yses. ACKNOWLEDGMENTS
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