Galaxy Cluster Correlation Function to z ~ 1.5 in the IRAC Shallow Cluster Survey
M. Brodwin, A. H. Gonzalez, L. A. Moustakas, P. R. Eisenhardt, S. A. Stanford, D. Stern, M. J. I. Brown
aa r X i v : . [ a s t r o - ph ] N ov A P J L
ETTERS , IN PRESS
Preprint typeset using L A TEX style emulateapj v. 5/14/03
GALAXY CLUSTER CORRELATION FUNCTION TO Z ∼ . M. B
RODWIN , A. H. G
ONZALEZ , L. A. M OUSTAKAS , P. R. E ISENHARDT , S. A. S TANFORD , D. S
TERN , AND
M. J. I. B
ROWN ApJ Letters, in press
ABSTRACTWe present the galaxy cluster autocorrelation function of 277 galaxy cluster candidates with 0 . ≤ z ≤ . area of the IRAC Shallow Cluster Survey. We find strong clustering throughout our galaxy clustersample, as expected for these massive structures. Specifically, at h z i = 0 . r = 17 . + . - . h - Mpc, in excellent agreement with the Las Campanas Distant Cluster Survey, the only othernon–local measurement. At higher redshift, h z i = 1, we find that strong clustering persists, with a correlationlength of r = 19 . + . - . h - Mpc. A comparison with high resolution cosmological simulations indicates theseare clusters with halo masses of ∼ M ⊙ , a result supported by estimates of dynamical mass for a subsetof the sample. In a stable clustering picture, these clusters will evolve into massive (10 M ⊙ ) clusters by thepresent day. Subject headings: galaxies: clusters: general — cosmology: observations — large–scale structure of the uni-verse INTRODUCTION
The clustering amplitude of massive galaxy clusters, andin particular its dependence on richness or cluster mass, is astrong function of the underlying cosmology. Recent theo-retical work (e.g., Majumdar and Mohr 2003; Younger et al.2006) has demonstrated how cluster surveys can be “self–calibrated,” providing precise simultaneous constraints onboth the cosmology and cluster evolution models. Given red-shift and approximate mass information for each cluster, re-liable cosmological parameter estimation is feasible even inthe presence of significant, and potentially unknown, evolu-tion in cluster physical parameters (e.g., Gladders et al. 2007).The ultimate goal of these cosmological studies, a measure-ment of the equation of state of dark energy with an accuracycompetitive with SNe Ia methods, also requires knowledge ofthe cluster power spectrum or autocorrelation function (Ma-jumdar and Mohr 2004; Wang et al. 2004). Such extensivesample characterization naturally emerges from mid–infraredselected photometric redshift cluster surveys like the IRACShallow Cluster Survey (ISCS; Eisenhardt et al. 2007, here-after E07).Previous studies (e.g., Bahcall et al. 2003, and referencestherein) have produced largely local ( z . .
3) galaxy clusterclustering measurements with limited baselines for evolution-ary studies. The ISCS provides the opportunity to addressthese issues in one of the largest statistical samples of highredshift clusters to date. In this Letter we present the firstmeasurement of the galaxy cluster autocorrelation functionextending over more than half the cosmic age of the Uni-verse. Such measurements also allow evolutionary connec-tions between galaxy clusters and high redshift, highly clus-tered galaxy populations to be explored.We use a concordance cosmology throughout, with Ω M =0 . Ω Λ = 0 .
7, and H = 70 km s - Mpc - . For consistencywith previous studies we report distances, including correla- JPL/Caltech, 4800 Oak Grove Dr., Pasadena, CA 91109 NOAO, 950 N. Cherry Ave., Tucson, AZ 85719 Dept. of Astronomy, Univ. of Florida, Gainesville, FL 32611 University of California, Davis, CA 95616 IGPP, LLNL, Livermore, CA 94550 School of Physics, Monash Univ., Clayton 3800, Victoria, Australia tion lengths, in units of comoving h - Mpc, with H = 100 h km s - Mpc - . IRAC SHALLOW CLUSTER SURVEY
The ISCS is a sample of 335 galaxy clusters spanning0 . < z < µ Jy at 4.5 µ min a 7.25 deg region (Brodwin et al. 2006, hereafter B06;E07). The galaxy photometric redshifts, which are key to thecluster–finding algorithm, are derived from the joint ISS, ND-WFS (DR3) and FLAMEX (Elston et al. 2006) data sets.The AGES survey (Kochanek et al. in prep) in Boötes pro-vides spectroscopic confirmation for dozens of clusters at z ≤ . z > and contains 320 of the 335 galaxy clusters in the full ISCS,of which 277 are within the redshift range (0 . ≤ z ≤ . GALAXY CLUSTER AUTOCORRELATION MEASUREMENTS
Angular Correlation Function in Redshift Slices
The redshift distribution of galaxy clusters is presented inFigure 1. A fit of the form N ( z ) ∝ ( z / z ) α exp ( - ( z / z ) β ), with α = 1 . β = 1 . z = 0 .
7, is overplotted. Cosmologicalstudies with this observed redshift distribution must await adetailed characterization of the survey selection function, tobe presented in a forthcoming paper. Galaxy clusters are splitinto two similarly populated redshift bins, 0 . ≤ z ≤ . . < z ≤ . σ z = 0 . + z ), and reli-ability of the galaxy photometric redshifts have been demon-strated (B06; see also Brown et al. 2007). N
320 Clusters F IG . 1.— Observed redshift distribution of galaxy clusters in the ISCS. Thedashed lines illustrate the two redshift bins considered here, 0 . ≤ z ≤ . . < z ≤ .
5, containing 277 clusters between them. The curve is a fitto the distribution and is used in the Limber deprojection. This distributionshould not be used for quantifying number density evolution without carefulinclusion of selection biases.
The angular correlation function (ACF) is parametrizedhere as a simple power law, ω ( θ ) = A ω θ - δ . (1)This can be deprojected (Limber 1954) to yield a measure-ment of the real–space correlation length, r ( z ), over the red-shift range spanned by the 2–D sample, r γ ( z eff ) = A ω " H H γ c R z z N ( z ) [ x ( z )] - γ E ( z ) dz [ R z z N ( z ) dz ] - , (2)where z eff = R z z z N ( z ) [ x ( z )] - γ E ( z ) dz R z z N ( z ) [ x ( z )] - γ E ( z ) dz , (3) γ ≡ + δ , H γ = Γ (1 / Γ [( γ - / / Γ ( γ/ N ( z ) is the red-shift distribution, and E ( z ) and x ( z ) describe the evolution ofthe Hubble parameter and the comoving radial distance, re-spectively (e.g., Hogg 1999). Since the cluster redshift uncer-tainties are much smaller than the width of our redshift bins(E07), they have little impact on the redshift distribution andthe modeling of the spatial correlation function. Results
We calculate the ACF using both the Landy and Szalay(1993) and Hamilton (1993) estimators using 500,000 ran-doms to ensure a robust Monte Carlo integration. The re-sults are nearly identical with both estimators and we reportthe results obtained with the latter. Two independent fittingtechniques are applied to the data. The standard frequentist(or classical) approach is used to simultaneously fit the slope,and, through the use of the relativistic Limber equation, the correlation length, r . In calculating these correlation lengthswe have adopted the N ( z ) parametrization shown in Figure 1.A Bayesian technique is also employed to directly determinethe correlation lengths, marginalizing over the slope, δ , sub-ject to the weak prior that it is in the range 0 . ≤ δ ≤ . r . q (deg)0.010.101.0010.00 w ( q ) (h −1 Mpc) 0.51.0 L i k e li hood £ q (deg)0.010.101.00 w ( q ) (h −1 Mpc) 0.51.0 L i k e li hood £ z £ F IG . 2.— Angular correlation functions in redshift bins at 0 . ≤ z ≤ . top ) and 0 . < z ≤ . bottom ). The error bars are estimated via bootstrapresampling. The red solid lines show the best–fit ACFs, with correspondingslopes listed in Table 1. The blue dashed lines show the fits with a fixed slope( δ = 1 . r presented in the insets. The 68% and 90% confidence intervals are shown. In this highly clustered population, sample (or cosmic) vari-ance dominates over simple Poisson errors. We therefore cal-culate the error bars using 100 bootstrap resamplings with re-placement. To test for possible systematics across the fieldwe divided the field into halves, once north–south and onceeast–west, and for each subfield we computed the ACF. Theresults for all half–fields agree within 1 σ , indicating that weare not adversely affected by an unidentified bias in the spatialselection function.Bootstrap simulations of the cluster detection process indi-cate that the spurious fraction is less than 10% at all redshifts(E07), and spectroscopic observations indicate it is likelymuch lower. Conservatively assuming 5–10% of the sample isindeed spurious, and that these are uncorrelated, then at mostwe are underestimating the clustering by ≈ r values from the Bayesian fits,alaxy Cluster Correlation Function to z ∼ . z ≈ . z ≈
1, is the first to probe struc-ture on the largest scales in the first half of Universe.The space densities for these samples and the mean inter-cluster distances, d c , are also presented in Table 1. The re-lationship between clustering amplitude and d c predicted ina concordance cosmology, and observed in practice (Bahcallet al. 2003, and references therein), is only weakly dependenton redshift. As shown in Figure 3, the ISCS samples are quiteconsistent with the LCDM predictions between 0 < z < .
20 40 60 80 100d c (h -1 Mpc)10203040 r ( h - M p c ) ISCSLCDCSSDSSAbellAPMREFLEXXBACSLCDCS F IG . 3.— Plot of r vs. d c for the present sample, along with severalmeasurements taken from the compilation of Bahcall et al. (2003). Ourresults (filled blue circles) are consistent with LCDM predictions between0 < z < . z ∼ . DISCUSSION
A key theoretically predictable cluster observable is the cor-relation function as a function of halo mass. In simulationsthe halo mass, M , is defined as the mass inside the radiusat which the mean overdensity is 200 times the critical den-sity. We compare our clustering results with the Younger et al.(2005) analysis of the Hopkins et al. (2005) high–resolutioncosmological simulation, which had a 1500 h - Mpc boxlength, an individual particle mass of 1 . × M ⊙ , and apower spectrum normalization of σ = 0 .
84. We infer thatthe ISCS cluster sample has average log[ M / M ⊙ ] masses of ∼ . + . - . and ∼ . + . - . at z eff = 0 .
53 and 0 .
97, respectively.Direct dynamical masses for the 10 z > r for massive galaxy clustersout to z = 1 is a robust confirmation of a key prediction fromnumerical simulations, reflecting the relative constancy of themass hierarchy of clusters with redshift (Younger et al. 2005).That is, the N most massive clusters at one epoch roughly r ( h - M p c ) ISCSLCDCSSDSSAbellAPMREFLEXXBACS EROULIRGSMGFRII F IG . 4.— Comoving correlation lengths for the ISCS clusters, along withother cluster samples at lower redshift (offset slightly in z for clarity) andhighly clustered galaxy samples at higher redshifts (see the text for a fulldescription and references). Two evolutionary models are overplotted, in-cluding the Fry (1996) biased structure formation model (shaded region anddotted lines for z eff = 0 .
53 and 0 .
97, respectively), and a simple stable clus-tering model (dashed lines). correspond to the N most massive clusters at a later epoch,and therefore have similar clustering.In Figure 4 we plot a compilation of recent clustering am-plitudes for various cluster surveys, as well as for highlyclustered galaxy populations, including FRII radio galaxies(Overzier et al. 2003), EROs (Brown et al. 2005; Daddiet al. 2001, 2004), ULIRGs (Farrah et al. 2006; Maglioc-chetti et al. 2007), and SMGs (Blain et al. 2004). Sinceoptically–selected QSOs are more modestly clustered (Coilet al. 2007; Croom et al. 2005; Myers et al. 2006; Porcianiet al. 2004), and therefore reside in considerably less massivehalos ( ∼ - M ⊙ ) than the ISCS clusters, they are notincluded in Figure 4.Following Moustakas and Somerville (2002), we overplotthe halo conserving model of Fry (1996) normalized to ourtwo measurements in order to explore possible evolutionaryconnections with structures at other redshifts. The shadedarea (dotted lines) shows the 1 σ region for the z eff = 0 . .
97) measurement. In this model, representative of a classof merger-free biased structure formation models, the ISCSclusters will evolve into typical present–day massive clusters,such as those in the SDSS, APM or Abell surveys. In thestable–clustering picture, in which clustering is fixed in phys-ical coordinates (Groth and Peebles 1977, dashed lines), the z eff = 0 .
97 ISCS clusters grow into the most massive clustersin the local Universe, typically identified in X–ray surveys.Most of the plotted high redshift galaxy clustering measure-ments are rather uncertain due to both small number statis-tics and poorly known redshift distributions. Mindful of thiscaveat, we observe that FRII radio galaxies, some ERO sam-ples, and z ∼ & M ⊙ ). As a cau-tion against overinterpretation, however, we note that only Brodwin et al. T ABLE
1. A
MPLITUDE OF C ORRELATIONS FOR
ISCS C
LUSTERS ∆ z z eff N log A ω a r δ b r ¯ n d c ( h - Mpc) ( h - Mpc) (10 - h Mpc - ) ( h - Mpc)0.25 – 0.75 0.53 136 - . ± . . . ± . . . ± . . . ± . . . ± . . ± . - . ± . . . ± . . . ± . . . ± . . . ± .
76 48 . ± . a Formal fits for r and δ were computed directly from the data. The error in A ω corresponds to the error in r at the best–fit δ . b Best–fit parameters from frequentist fits. c Best–fit r from Bayesian marginalization. ULIRGs have space densities similar to the present clustersamples, a prerequisite for drawing evolutionary connectionsfrom these particular models. Thus the present work offersa measure of support for recent studies (e.g., Magliocchettiet al. 2007) indicating that ULIRGs may be associated with,or progenitors of, groups or low–mass clusters. CONCLUSIONS
By deprojecting the angular correlation function measuredin redshift bins spanning z = 0 .
25 to z = 1 .
5, we have deter-mined the real–space clustering amplitudes for ISCS clus-ters at h z i = 0 .
53 and h z i = 0 .
97 to be r = 17 . + . - . and r = 19 . + . - . h - Mpc, respectively. These measurementsare consistent with the relation between correlation amplitudeand mean intercluster distance predicted by LCDM. The ISCSclusters have total masses exceeding 10 M ⊙ and will evolve into very massive clusters by the present day.Based in part on observations made with the Spitzer SpaceTelescope , operated by the Jet Propulsion Laboratory, Cali-fornia Institute of Technology, under a contract with NASA.This paper made use of data from the NDWFS, which wassupported by NOAO, AURA, Inc., and the NSF. We thankA. Dey, B. Jannuzi and the entire NDWFS survey team. Wealso thank J. Younger and P. Hopkins for providing their sim-ulation results, C. Porciani and K. Blindert for useful discus-sions, and the anonymous referee for a very helpful report.SAS’s work was performed under the auspices of the U.S.DoE, National Nuclear Safety Administration by the Univer-sity of California, LLNL under contract No. W-7405-Eng-48., operated by the Jet Propulsion Laboratory, Cali-fornia Institute of Technology, under a contract with NASA.This paper made use of data from the NDWFS, which wassupported by NOAO, AURA, Inc., and the NSF. We thankA. Dey, B. Jannuzi and the entire NDWFS survey team. Wealso thank J. Younger and P. Hopkins for providing their sim-ulation results, C. Porciani and K. Blindert for useful discus-sions, and the anonymous referee for a very helpful report.SAS’s work was performed under the auspices of the U.S.DoE, National Nuclear Safety Administration by the Univer-sity of California, LLNL under contract No. W-7405-Eng-48.