An improved measurement of baryon acoustic oscillations from the correlation function of galaxy clusters at z∼0.3
A. Veropalumbo, F. Marulli, L. Moscardini, M. Moresco, A. Cimatti
aa r X i v : . [ a s t r o - ph . C O ] M a y Mon. Not. R. Astron. Soc. , 1–10 (2013) Printed 11 September 2018 (MN L A TEX style file v2.2)
An improved measurement of baryon acoustic oscillationsfrom the correlation function of galaxy clusters at z ∼ . A. Veropalumbo ⋆ , F. Marulli , , , L. Moscardini , , , M. Moresco and A. Cimatti Dipartimento di Fisica e Astronomia, Universit`a di Bologna, viale Berti Pichat 6/2, I-40127 Bologna, Italy INAF - Osservatorio Astronomico di Bologna, via Ranzani 1, I-40127 Bologna, Italy INFN - Sezione di Bologna, viale Berti Pichat 6/2, I-40127 Bologna, Italy
11 September 2018
ABSTRACT
We detect the peak of baryon acoustic oscillations (BAO) in the two-point cor-relation function of a spectroscopic sample of 25226 clusters selected from the SloanDigital Sky Survey. Galaxy clusters, as tracers of massive dark matter haloes, arehighly biased structures. The linear bias b of the sample considered in this work, thatwe estimate from the projected correlation function, is b σ = 1 . ± .
03. Thanks tothe high signal in the cluster correlation function and to the accurate spectroscopicredshift measurements, we can clearly detect the BAO peak and determine its position, s p , with high accuracy, despite the relative paucity of the sample. Our measurement, s p = 104 ± h − , is in good agreement with previous estimates from large galaxysurveys, and has a similar uncertainty. The BAO measurement presented in this workthus provides a new strong confirmation of the concordance cosmological model anddemonstrates the power and promise of galaxy clusters as key probes for cosmologicalapplications based on large scale structures. Key words: cosmology: observations – galaxy clustering – large-scale structure ofthe Universe
The clustering of cosmic structures is one of the most pow-erful tools to constrain cosmology. In particular, the sig-nal of the baryon acoustic oscillations (BAO) in the two-point correlation function acts as a standard ruler, provid-ing geometric cosmological constraints. The accuracy in thedetermination of the position of the BAO peak dependsmainly on statistical uncertainties. By now the most ac-curate measurements have been obtained with large spec-troscopic samples of galaxies (e.g. Eisenstein et al. 2005;Cole et al. 2005; Percival et al. 2007, 2010; S´anchez et al.2009; Kazin et al. 2010; Beutler et al. 2011; Blake et al.2011; Padmanabhan et al. 2012; Anderson et al. 2012, 2014)at low redshifts, z <
1, and with Ly α forest in quasar spec-tra at higher redshifts (e.g. Slosar et al. 2013; Delubac et al.2014).In recent analyses also galaxy clusters have been con-sidered as probes for the large scale matter distribution(Angulo et al. 2005). As tracers of the biggest collapsedstructures, they are more strongly clustered than galax-ies. Measurements of the two-point correlation function of ⋆ E-mail: [email protected] galaxy clusters have provided the first weak detections ofthe BAO peak. Estrada et al. (2009) and H¨utsi (2010) mea-sured, respectively, the two-point correlation function andthe power spectrum of the MaxBCG photometric catalogue,consisting of ∼ . < σ <
2. Using a similar number of objects andin an more extended redshift range, Hong et al. (2012) de-tected the BAO peak in the two-point correlation function ofthe spectroscopic cluster catalogue provided by Wen et al.(2009), with a confidence of 1 . σ .In this paper we present new measurements of theclustering of galaxy clusters, up to the BAO scale, usingthe largest spectroscopic sample currently available. As wewill show, the BAO peak is clearly detected at a scale s p ≈
105 Mpc h − .The paper is organized as follows. In § §
3. In § § c (cid:13) Veropalumbo, Marulli, Moscardini, Moresco & Cimatti
Figure 1.
Left panel : the angular distribution of galaxy clusters from the spectroscopic (blue dots) and photometric (red dots) samplesanalysed in this work.
Right panel : zoomed region of 5 × . < z < .
25, as indicated by the colour map. Blue and red circlesrepresent the angular projection of the cluster radii r , from the spectroscopic (blue) and photometric (red) samples, estimated usingour fiducial cosmology. tigate the impact of photometric redshift errors. Finally, in § We consider the spectroscopic cluster sample provided byWen, Han, & Liu (2012) (WHL12), that has been extractedfrom the Sloan Digital Sky Survey (SDSS) III (Aihara et al.2011). The cluster candidates are identified by deproject-ing the transversal overdensities, using the information onphotometric redshifts. Clusters are included in the sampleif they satisfy two conditions: i) N >
8, where N isthe number of galaxy members inside the radius r , atwhich the average density is 200 times the background den-sity, and ii) R L ∗ >
12, where R L ∗ is the ratio between L ,the r -band luminosity inside r , and L ∗ , the characteristic r -band luminosity of galaxies (see Blanton et al. 2003). Thecluster centre is determined by the position of the brightestcluster galaxy (BCG), while its photometric redshift is themedian value of the photometric redshifts of its galaxy mem-bers. A spectroscopic redshift is then assigned to a clusterif it has been measured for its BCG.The total number of detected clusters in the whole pho-tometric sample is 132683, in the redshift range 0 . < z < .
8. The detection rate increases with the cluster mass. Us-ing X-ray and weak-lensing measurements available for asubsample of clusters, WHL12 showed that the sample iscomplete for M & · M ⊙ in the redshift range 0 . 42, while the detection rate decreases down to ∼ M = 6 · M ⊙ (see WHL12 for more details on the detection algorithmadopted). For this work, we use a subsample of clusters ex-tracted from the WHL12 spectroscopic sample. Specifically,we consider the complete spectroscopic cluster sample fromthe Northern Galactic Cap, with measured redshifts in therange 0 . < z < . 42. Moreover, we use only the SDSSstripes with at least 50% of the clusters with spectroscopic redshift assigned. This is to obtain the largest contiguousarea and to minimize possible selection effects. The finalnumber of objects in our selected sample is 25226. The leftpanel of Fig. 1 shows the angular distribution of the spec-troscopic cluster sample (blue dots) analysed in this work,compared to the entire photometric sample (red dots), whilethe right panel shows a zoomed 5 × r , from the spectroscopic(blue) and photometric (red) samples, estimated using ourfiducial cosmology.The main properties of the spectroscopic sample usedfor this work are summarized in Table 1. The photometricsample is used to compute the sampling rate, as describedin § z ∼ . 12 (Strauss et al.2002), and the LRG sample that covers the redshift range0 . < z < . We estimate the redshift-space two-point correlation func-tion, ξ ( s ), using the Landy & Szalay (1993) estimator: ξ ( s ) = 1 RR ( s ) × (cid:20) DD ( s ) n r n d − DR ( s ) n r n d + RR ( s ) (cid:21) , (1)where DD ( s ), DR ( s ) and RR ( s ) are the numbers ofweighted data-data, data-random and random-random pairswithin a separation s ± ∆ s/ 2, where ∆ s is the bin size, and n r and n d are the weighted number density of random andcluster sample, respectively. To compute comoving distances c (cid:13) , 1–10 AO in the clustering of galaxy clusters Figure 2. The redshift distribution of the selected galaxy clusters(histogram). The solid line shows the smoothed redshift distribu-tion obtained adopting a Gaussian filter. See WHL12 for furtherdetails on the redshift distribution of the galaxy samples used todetect the clusters. and bias (see § M = 0 . b = 0 . H = 70 km s − Mpc − , the primordial perturbationspectral index n s = 1, and the linear power spectrum am-plitude σ = 0 . w p ( r p ) = Z π max d π ′ ξ ( r p , π ′ ) , (2)where ξ ( r p , π ) is the measured two-point correlation func-tion in the directions perpendicular, r p , and parallel, π , tothe line-of-sight. The real-space two-point correlation func-tion, ξ ( r ), is then obtained from w p assuming a power-lawmodel, ξ ( r ) = ( r/R ) − γ , where R and γ are the correla-tion length and the power-law index, respectively. With theabove assumption, the relation between ξ and w p can bederived analytically: w p ( r p ) = r p (cid:18) R r p (cid:19) γ Γ( )Γ( γ − )Γ( γ ) , (3)where Γ is the Euler’s gamma function. To measure the two-point correlation function of oursources, we have to construct a sample of randomly dis-tributed objects (see Eq. 1), taking into account the selec-tion function of the sample. As a fair approximation, we canfactorise the random sample distributions into the angularand redshift components separately.The angular mask is reconstructed with the software MANGLE (Swanson et al. 2008). Using the SDSS coordi-nates system ( λ, η ), we decompose the angular distribution Table 1. Main properties of the spectroscopic cluster sample usedfor this work. See § ] ∼ z range 0 . < z < . z . R minL ∗ M min · M ⊙ of clusters in rectangular elements of equal area, that arethen randomly filled. We do not apply any weights to takeinto account sector completeness when creating the randomsample.We assign redshifts to the random objects sampling themean redshift distribution of the catalogue. The latter hasbeen obtained grouping the data in 100 redshift bins andsmoothing the distribution with a Gaussian kernel threetimes larger than the bin size. Reducing the value of thisparameter has the effect to lower the clustering signal in theradial direction. The impact of this effect is however negligi-ble, considering the estimated uncertainties in our measure-ments. To minimize the effect of shot noise, we construct arandom sample ten times denser than the cluster sample.Fig. 2 shows the redshift distribution of the cluster sample(histogram) and the smoothed distribution (solid line) usedfor the construction of the random sample. In this analysis, we apply three different weights to correctfor i) the effects of a mass-dependent detection rate in thecluster selection algorithm, W M (see WHL12), and for ii)the spectroscopic sampling rate, as a function of the clusterrichness, W N , and of the stripe location, W S , separately.We derive the above quantities directly from the data, com-paring the photometric and spectroscopic cluster samples.For each cluster, the total weight assigned is: w i ( R L ∗ , N , stripe ) = W − M · W − N · W − S . (4)The net effect is to increase the number of low mass struc-tures, whose sampling rate is lower with respect to the moremassive structures, in the spectroscopic sample. This slightlyreduces the clustering normalization, up to ∼ The errors on the clustering measurements are estimatedwith the jackknife method (see e.g. Norberg et al. 2009).The covariance matrix for the jackknife estimator is: C ij = N sub − N sub N X k =1 ( ξ ki − ¯ ξ i )( ξ kj − ¯ ξ j ) , (5)where ξ ki is the value of the correlation function at the i -thbin for the k -th subsample, and ¯ ξ i is the mean value of thesubsamples. c (cid:13) , 1–10 Veropalumbo, Marulli, Moscardini, Moresco & Cimatti We construct N sub = 140 resamplings of our cluster cat-alogue by dividing the original sample in N sub regions (i.e. 5subvolumes for each of the 28 SDSS stripes considered) andexcluding recursively one of them. Increasing the number ofsubregions provides a less scattered estimate of the covari-ance matrix. As verified directly, the value of N sub adoptedhere is large enough to assure the convergence of the results.We extensively test the jackknife algorithm exploited in thiswork using the LasDamas mock catalogues (McBride et al.2009), finding that the quoted errors are conservative esti-mates. In the following sections, we describe the models used toderive clustering parameters and cosmological constraintsfrom the projected correlation function and the BAO peak.The analysis is performed applying a Monte Carlo MarkovChain (MCMC) technique, using the full covariance matrix.We adopt a standard likelihood, L ∝ exp( − χ / χ is defined as follows: χ = i = n X i =0 j = n X j =0 ( ξ i − ˆ ξ i ) C − ij ( ξ j − ˆ ξ j ) , (6)where ξ i is the correlation function measured in the i -th bin,ˆ ξ i is the model and C − ij is the inverted covariance matrix. To measure the bias factor, b , we model the projected cor-relation function assuming a linear biasing model, w p ( r p ) = b w DMp ( r p ) , (7)where w DMp is the DM projected correlation function (e.g.Marulli et al. 2013). When assessing the bias through Eq. 7,the upper limit of the integration in Eq. 2, π max , has tobe fixed. The impact of this parameter choice is not signifi-cant, considering the estimated uncertainties. Nevertheless,a finite value of π max introduces unavoidable systematic er-rors, as the effect of redshift-space distortions (RSD) cannot be entirely washed out by the integration. The net ef-fect is a spurious scale-dependence in the estimated bias. Tominimize the impact of such a systematics, instead of usingEq.7 we model directly the projected correlation function asfollows: w p ( r p ) = b Z π max d π ′ ξ DM ( r p , π ′ ) , (8)where the value of π max is the same as the one used to mea-sure w p ( r p ) and ξ DM ( r p , π ) is the redshift-space DM corre-lation function in the directions perpendicular and parallelto the line of sight. RSD are introduced with the disper-sion model (Kaiser 1987; Hamilton 1992; Davis & Peebles1983), following Marulli et al. (2012). The linear DM cor-relation function is obtained by Fourier transforming thematter power spectrum computed with the software CAMB (Lewis & Bridle 2002). The linear RSD parameter is esti-mated assuming a ΛCDM cosmology, i.e. β = Ω M ( z ) γ /b ,with γ = 0 . π max , providing an approximately scale-independent biasin the range of scales considered. In this section, we descibe two different methods to detectthe BAO peak and extract cosmological information. Resultsobtained with both the methods are presented in § Empirical model We consider an empirical model similar to the one proposedby S´anchez et al. (2012), which is used to interpolate thefunction ξ ( s ) at the BAO scales: ξ ( s ) = B + (cid:18) ss (cid:19) − γ + N √ πσ exp (cid:18) − ( s − s m ) σ (cid:19) , (9)where the parameters s and γ model the shape of the cor-relation at small scales, B takes into account a possible neg-ative correlation at large scales, and s m , σ , and N are theparameters of the Gaussian function used to model the BAOfeature. We note that the true BAO peak position, s p , isshifted to smaller scales with respect to the Gaussian me-dian value s m .The empirical model given by Eq. 9 can be used to ac-curately detect the BAO peak position. To directly compareour measurements with previous studies, we compute alsothe dimensionless variable: y s = r s D V , (10)that results to be independent of the fiducial cosmol-ogy assumed to derive comoving coordinates (see e.g.S´anchez et al. 2012). The distance D V is defined as: D V = (cid:20) (1 + z ) D A ( z ) czH ( z ) (cid:21) , (11)where D A is the angular diameter distance and H ( z ) is theHubble function. Physical model To extract the full cosmological information embedded inthe position of the BAO peak, we consider a theoreticalmodel that includes the cluster bias, the effects of RSD andgeometric distortions due to a possible incorrect assumptionof the fiducial cosmology. The adopted model is the follow-ing: ξ cl ( s ) = b (cid:18) β + 15 β (cid:19) ξ DM ( αs ) , (12)where b is the linear bias factor, α is the ratio betweenthe test and fiducial values of D V and it is used to modelgeometric distortions, and β is the linear distortion pa-rameter described in § ξ DM , is computed using the software MPT-breeze (Crocce & Scoccimarro 2008), based on the renor-malized perturbation theory (Crocce & Scoccimarro 2006).This method has already been used in previous works aimedat extracting cosmological information from the position ofthe BAO peak (see e.g. Eisenstein et al. 2005; Beutler et al.2011; Blake et al. 2011). c (cid:13)000 To extract the full cosmological information embedded inthe position of the BAO peak, we consider a theoreticalmodel that includes the cluster bias, the effects of RSD andgeometric distortions due to a possible incorrect assumptionof the fiducial cosmology. The adopted model is the follow-ing: ξ cl ( s ) = b (cid:18) β + 15 β (cid:19) ξ DM ( αs ) , (12)where b is the linear bias factor, α is the ratio betweenthe test and fiducial values of D V and it is used to modelgeometric distortions, and β is the linear distortion pa-rameter described in § ξ DM , is computed using the software MPT-breeze (Crocce & Scoccimarro 2008), based on the renor-malized perturbation theory (Crocce & Scoccimarro 2006).This method has already been used in previous works aimedat extracting cosmological information from the position ofthe BAO peak (see e.g. Eisenstein et al. 2005; Beutler et al.2011; Blake et al. 2011). c (cid:13)000 , 1–10 AO in the clustering of galaxy clusters Figure 3. The projected correlation function of galaxy clusters(black dots). The dashed grey line shows the best-fit linear modeldefined by Eq. 3, while the other three lines are the best-fit mod-els obtained through Eq. 8 for three different values of the massdensity parameter, Ω M = 0 . , . , . To compare with previous studies, we exploit thismethod to derive also other parameters such as y s (see also § A ( z ), defined as follows: A ( z ) ≡ D V ( z ) √ Ω M h cz . (13)This parameter results to be independent of H , since D V ∝ H − (see e.g. Eisenstein et al. 2005; Blake et al. 2011). In this section, we present the main results of our analysis.We start focusing on the small scale clustering, estimatingthe linear bias from the projected correlation function at r p < 30 Mpc h − . Then, we move to larger scales, detectingthe BAO peak and extracting cosmological information. Fi-nally, we compare our measurements with previous studies. Fig. 3 shows the projected correlation function, w p ( r p ), es-timated through Eq. 2. The error bars are the square rootof the diagonal elements of the covariance matrix given byEq. 5, i.e. σ i = √ C ii . We derive the correlation length, R ,and the power-law index, γ , assuming a power-law modelfor the real-space clustering, thus fitting the projected cor-relation function using Eq. 3. The result of the fit, obtainedin the range of scales 10 < r p [Mpc h − ] < 30, is shown bythe dashed grey line.In Eq. 2 we set the upper limit of the integration tothe value π max = 40 Mpc h − . We investigated the im-pact of this assumption and of the scale range used forthe fit, repeating the procedure for different values of π max and of the scale limits. We find that our results are onlymarginally affected by these parameters. The maximum variation in R and γ is of the order of 7%, when π max and the scale limits are changed inside reasonable ranges(i.e. 20 < π max [ Mpc h − ] < 60, 5 < r p [ Mpc h − ] < b , using themethod described in § σ uncertainties, is b σ = 1 . ± . 03, correspondingto a minimum χ value of 6 . 7, with 10 degrees of freedom.The best-fit values of R , γ and b σ are reported in Table 2.To investigate the impact of the mass density param-eter, we repeat the same measurement for Ω M = 0 . 25 andΩ M = 0 . 35. The three best-fit models corresponding to thethree assumed values of Ω M are shown in Fig. 3 with differ-ent lines, as indicated by the labels. The measured w p ( r p )results to be only marginally affected by geometric distor-tions when changing Ω M , while this is not the case for themodel. Therefore, the best-fit value of the bias does de-pend on Ω M (e.g. Marulli et al. 2012). The best-fit valueswe obtain are the following: b σ (Ω M = 0 . 25) = 1 . ± . b σ (Ω M = 0 . 35) = 1 . ± . 04. In particular, wefind that the best-fit bias values derived for different Ω M are the ones that keep the value of β approximately con-stant. The correspondent χ are: χ (Ω M = 0 . 25) = 8 . χ (Ω M = 0 . 35) = 9 . 9, respectively. The minimum of χ is obtained for Ω M = 0 . 3, thus favouring the fiducial cos-mology assumed in this work. In the next section we willperform a more detailed analysis, modelling the large scaleclustering and constraining directly Ω M , with a full MCMCmethod, finding consistent results.We notice that our analysis shows a lower clusteringwith respect to the estimates by Estrada et al. (2009) andHong et al. (2012). This is due to the lower mass limit inour cluster sample, that results in a lower bias. The left panel of Fig. 4 shows the redshift-space two-pointcorrelation function, ξ ( s ), multiplied by s , in order to mag-nify the BAO peak. We start fitting the clustering datawith the empirical model given by Eq. 9, in the scale range20 < s [ Mpc h − ] < s p = 104 ± Mpc h − , af-ter marginalizing over the other 5 parameters of the model.When using linear instead of logarithmic binning, the BAOpeak results slightly shifted to higher values. However, theeffect is of the order of 2%, well below the estimated ac-curacy on the BAO peak position, that is of the order of7%. We also fit the data with the same empirical model butwithout the Gaussian part. The ∆ χ gives a confidence levelfor the full model between 2 and 3 σ .Our measurement is in good agreement with the pre-vious detection by Hong et al. (2012). Moreover, thanks tothe higher cluster density in our sample, that is larger bya factor of two, the uncertainty in the position of the BAOpeak is significantly lower. c (cid:13) , 1–10 Veropalumbo, Marulli, Moscardini, Moresco & Cimatti Figure 4. Left panel : the redshift-space two-point correlation function of galaxy clusters (black dots), multiplied by s to magnify theBAO peak; error bars are the square root of the diagonal elements of the covariance matrix, multiplied by s . The dashed green line isthe best-fit empirical model obtained through Eq. 9. The blue line is the best-fit physical model given by Eq. 12, while the dot-dashedred line shows the no-BAO prediction, obtained with the fitting formula by Eisenstein & Hu (1999). Right panel : ∆ χ as a function of D V , for the physical (blue line) and the no-BAO (red line) models. The BAO peak is detected with a ∼ . σ confidence level. We now fit the measured correlation function with the phys-ical model described in § M h , in the linear bias b , and in the shift pa-rameter, α , that traces geometrical distortions. All the othercosmological parameters are kept fixed to the Planck values: H = 67 . − Mpc − , Ω b = 0 . h , n s = 0 . 96 and σ = 0 . 83 (Planck Collaboration 2013).The best-fit parameters are summarized in Table 2.The reported values are the medians of the MCMC pa-rameter distributions, while the 1 σ errors span from the16 th to the 86 th percentiles. The solid blue line in theleft panel of Fig. 4 shows the result of the fit obtained us-ing the MPTbreeze software to estimate ξ DM ( r ), while thered one has been obtained using the fitting formula givenby Eisenstein & Hu (1999) with no BAO. As shown in theright panel of Fig. 4, the BAO feature is detected with a ∼ . σ confidence level, in agreement with what obtainedwith the empirical model. We achieve a distance measure of D V = 1031 ± . Constraints on the distortion parameters y s are of the order of 7%, in good agreement with the valueobtained with the empirical model in § < s [ Mpc h − ] < M h = 0 . ± . . , aftermarginalizing over the other two model parameters α and b (see § σ marginalized probability con-tours in the Ω M h − D V plane. The dotted line indicatesthe points with constant y s , i.e. it represents the degener-acy direction between parameters that would occur if thefit was driven by the BAO feature only. The dashed lineshows the opposite case in which the fit is driven only by Table 2. Best-fit parameters and 1 σ uncertainties obtained fromthe projected and redshift-space two-point correlation functionsof the selected spectroscopic cluster sample. For more details see § § σ uncertainties R [ Mpc h − ] 11 . ± . w p ( r p ) γ . ± . b σ . ± . s p [ Mpc h − ] 104 ± D V (˜ z ) [Mpc] 1031 ± ξ ( s ) y s . ± . . A (˜ z ) 0 . ± . . Ω M h . ± . . b σ . ± . . the shape of the two-point correlation function. As it can beseen, the orientation of the parameter degeneracy obtainedin this work lies approximately in the middle between thesetwo extremes, closely following the solid line of costant A (Eq. 13). In Table 2 we report the best-fit values of the cos-mological parameters, as well as the estimated uncertaintiesderived from the MCMC analysis after marginalizing overall the free parameters of the fit. As it can be seen, the esti-mated value of b is consistent with the one derived in § c (cid:13) , 1–10 AO in the clustering of galaxy clusters Figure 5. Marginalized probability contours at 1 and 2 σ forΩ M h − D V , obtained fitting Eq. 12 in the scale range 20 04, where the central value is set to thePlanck value Ω M h = 0 . The cluster centres of the spectroscopic sample analysed inthis work are determined by the positions of the BCGs (see § full LRGsample. Thus, it is worth wondering if there is any advan-tage of using a sparse cluster sample, instead of a largergalaxy sample, for BAO analyses. We address this questionin § § For any clustering analysis, the WHL12 spectroscopic clus-ter sample can be considered just as a particularly selectedsubsample of LRGs. To investigate the impact of such a se-lection on the detection of the BAO peak, we consider herethe large LRG sample by Kazin et al. (2010). The catalogueconsists of ∼ . < z < . 36. To avoidany possible systematic effect, we restrict our analysis to theNorthern Galactic Cap, reducing the number of objects to ∼ z = 0 . ∼ b BCG /b LRG ∼ . 16. Moreover,while the 1 σ error bars are smaller for the LRGs, due totheir higher number density, the BAO peak is significantlybetter determined for the BCG sample. Following the sameanalysis performed in § . σ level, and the error on the BAO peak results more thantwo times larger relative to the one obtained with the BCGsample. To investigate the robustness of our data reductionand clustering measurements, we compare our results withthe literature (Kazin et al. 2010), finding good agreementand confirming that our jackknife method slightly overpre-dicts the uncertainties relative to external methods basedon mock catalogues, thus providing conservative estimatesfor the errors.To further investigate the differences between BCG andLRG clustering, we extract two subsamples of LRGs withthe same number of objects, one totally random and theother reproducing the BCG absolute magnitude distribu-tion. Then we measure the two-point correlation functionfor both the samples, and we repeat the BAO analysis. Inboth cases, we find that the BAO peak is less accurately de-termined with respect to the BCG case. We conclude thatBCGs, or equivalently galaxy clusters, are optimal tracersto detect the BAO peak. This is due to the dynamical stateof these objects, that have significantly lower peculiar veloc-ities with respect to other galaxies. Indeed, as we verifieddirectly, the Fingers of God feature is almost absent in theBCG sample analysed here. This is the crucial property that c (cid:13) , 1–10 Veropalumbo, Marulli, Moscardini, Moresco & Cimatti Figure 6. D V ( left panel ) and y s ( right panel ) as a function of redshift. Comparison of our measurements (red square) with previousestimates from large galaxy samples (grey/black symbols): 6dF Galaxy Survey ( z = 0 . 106 by Beutler et al. 2011), SDSS ( z = 0 . , . 35 byPercival et al. 2010; z = 0 . 278 by Kazin et al. 2010), WiggleZ ( z = 0 . , . . 73 by Blake et al. 2011) and BOSS ( z = 0 . , z = 0 . Lyα ( z = 2 . 34 by Delubac et al. 2014). Solid and dashed lines show the ΛCDM predictions obtainedadopting the Planck and WMAP9 parameters, respectively. The y s ( z ) values are normalized to the Planck values. The shaded area isobtained changing the mass density parameter in the range { Ω M − . , Ω M + 0 . } , where the central value is set to the Planck valueΩ M h = 0 . can reduce the width of the BAO peak, thus improving thesignificance of the detection. Actually, such a small scale ef-fect can directly impact the large scale clustering, as clearlyshown in Fig. 7. The effect of small scale dynamics on the two-point corre-lation function appears quite similar to the one of redshifterrors (Marulli et al. 2012). Thus, for what we have seen in § ∼ . . < z < . . In this paper we presented new measurements of the two-point correlation function of a spectroscopic sample ofgalaxy clusters, selected from the SDSS (WHL12) in theredshift range 0 . < z < . 42. From the projected corre-lation function, we derive the correlation length and thepower-law index of the real-space clustering, and the lin-ear bias factor. As shown in Fig. 4, we could clearly de-tect the BAO peak. Fitting the measured ξ ( s ) with an em-pirical model with a Gaussian function at the BAO scale,we find s p = 104 ± Mpc h − , y s = 0 . ± . . and D V = 1031 ± Mpc h − . We test two different methods toanalyse the BAO feature, both providing compatible con-straints. We estimate a confidence level in the BAO detec-tion of ∼ . σ , despite the sparseness of the spectroscopiccluster sample considered. This is comparable to what ob-tained from many large galaxy surveys, though the latestmeasurements provide even stronger constraints, e.g. SDSSDR11 LRGs and QSO Ly- α provide ∼ σ and ∼ σ BAOdetection, respectively (Anderson et al. 2014; Delubac et al.2014). Overall, our measurements appear consistent with allprevious studies and with the ΛCDM predictions. Our errorestimates are quite conservative, due to the method used tocompute the covariance matrix. The goodness of our resultsis due to the high clustering signal (i.e. high bias) of thecluster sample analysed, and to the accuracy in the spectro-scopic redshift measurements. Indeed, as we have verified c (cid:13) , 1–10 AO in the clustering of galaxy clusters Figure 7. Comparison between the redshift-space two-point correlation functions of cluster (black dots) and LRG (magenta triangles)spectroscopic samples ( left panel ), and between spectroscopic (black dots) and photometric (magenta triangles) cluster samples ( rightpanel ). The lines show the best-fit empirical models obtained through Eq. 9, for spectroscopic clusters (blue lines) and photometricclusters/LRG (dashed red lines). directly, the BAO peak is weakly constrained when usinglarger LRG or photometric cluster catalogues. This resultshows that galaxy clusters are powerful cosmological probesfor the detection of BAO, even with a fairly limited statis-tics, and highly competitive with respect to galaxies. Fu-ture massive surveys such as Euclid (Laureijs et al. 2011;Amendola et al. 2013) and eROSITA (Merloni et al. 2012)will allow this approach to be fully exploited in several openkey questions (e.g. the dark energy equation of state). Ac-curate forecasts on the cosmological constraints achievableby these future cluster surveys will be provided in a futurework. Moreover, thanks to the ongoing BOSS program, weplan to enrich the spectroscopic cluster sample analysed inthis work, providing new BAO constraints at different red-shifts, and as a function of the cluster richness. ACKNOWLEDGMENTS We would like to thank the anonymous referee for provid-ing useful comments, that significantly helped to improvethe paper. We acknowledge financial contributions by grantsASI/INAF I/023/12/0, PRIN MIUR 2010-2011 “The darkUniverse and the cosmic evolution of baryons: from currentsurveys to Euclid” and PRIN INAF 2012 “The Universe inthe box: multiscale simulations of cosmic structure”. REFERENCES Aihara H., Allende Prieto C., An D., et al., 2011, ApJS,193, 29Amendola, L., Appleby, S., Bacon, D., et al. 2013, LivingReviews in Relativity, 16, 6Anderson L., Aubourg E., Bailey S., et al., 2012, MNRAS,427, 3435Anderson, L., Aubourg, ´E., Bailey, S., et al. 2014, MNRAS,441, 24 Angulo, R. E., Baugh, C. M., Frenk, C. S., et al. 2005,MNRAS, 362, L25Beutler F., Blake C., Colless M., et al., 2011, MNRAS, 416,3017Blake C., Kazin E. A., Beutler F., et al., 2011, MNRAS,418, 1707Blanton M. R., Hogg D. W., Bahcall N. A., et al., 2003,ApJ, 592, 819Cole S., et al., 2005, MNRAS, 362, 505Crocce M., & Scoccimarro R. 2006, PhRvD, 73, 063519Crocce M., Scoccimarro R., 2008, PhRvD, 77, 023533Davis, M., & Peebles, P. J. E. 1983, ApJ, 267, 465Delubac, T., Bautista, J. E., Busca, N. G., et al. 2014,arXiv:1404.1801Eisenstein, D. J., & Hu, W. 1999, ApJ, 511, 5Eisenstein, D. J., Annis, J., Gunn, J. E., et al. 2001, AJ,122, 2267Eisenstein D. J., et al., 2005, ApJ, 633, 560Estrada J., Sefusatti E., Frieman J. A., 2009, ApJ, 692, 265Hamilton, A. J. S. 1992, ApJL, 385, L5Hong T., Han J. L., Wen Z. L., et al., 2012, ApJ, 749, 81H¨utsi G., 2010, MNRAS, 401, 2477Kaiser, N. 1987, MNRAS, 227, 1Kazin E. A., Blanton M. R., Scoccimarro R., et al., 2010,ApJ, 710, 1444Koester, B. P., McKay, T. A., Annis, J., et al. 2007, ApJ,660, 239Komatsu, E., Smith, K. M., Dunkley, J., et al. 2011, ApJS,192, 18Landy S. D., Szalay A. S., 1993, ApJ, 412, 64Laureijs R., Amiaux J., Arduini S., et al., 2011, ArXiv e-printsLewis A., Bridle S., 2002, PhRvD, 66, 103511Marulli, F., Bianchi, D., Branchini, E., et al. 2012, MN-RAS, 426, 2566Marulli, F., Bolzonella, M., Branchini, E., et al. 2013, A&A,557, A17McBride, C., Berlind, A., Scoccimarro, R., et al. 2009, Bul- c (cid:13) , 1–10 Veropalumbo, Marulli, Moscardini, Moresco & Cimatti letin of the American Astronomical Society, 41, c (cid:13)000