Luminous Red Galaxy Population in Clusters at 0.2≤z≤0.6
Shirley Ho, Yen-Ting Lin, David Spergel, Christopher M. Hirata
aa r X i v : . [ a s t r o - ph ] J un Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 29 October 2018 (MN L A TEX style file v2.2)
Luminous Red Galaxy Population in Clusters at . ≤ z ≤ . Shirley Ho ⋆ , Yen-Ting Lin , , David Spergel and Christopher M. Hirata Department of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ 08544, USA. Departamento de Astronom´ıa y Astrof´ısica, Pontificia Universidad Cat´olica de Chile, Chile. School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA.
29 October 2018
ABSTRACT
We investigate statistical properties of LRGs in a sample of X-ray selected galaxyclusters at intermediate redshift (0 . ≤ z ≤ . N ( M ) = k × ( M/ ) a where a = 0 . ± .
105 and k = 1 . ± .
285 assuming a Poissondistribution for N ( M ).2. The halo occupation distribution of LRGs ( N ( M )) and the satellite distributionof LRGs ( N − M )) are both consistent with being Poisson. To be more quantitative,we find V ar ( N ) / h N i = 1 . ± .
351 and
V ar ( N − / h N − i = 1 . ± . . +7 . − . (6 . +3 . − . ) including (excluding) BLRGs (Brightest LRGs).We also discuss the implications of these observations on the evolution of massivegalaxies in clusters. The recent advent of large-scale galaxy surveys have revolu-tionized the field of observational cosmology. Deep spectro-scopic surveys allows us to witness the young Universe whenthe building blocks of present-day galaxies are forming, andsome of the well-known properties of the local galaxy popu-lations are about to realize. The enormous amount of datagathered by wide area surveys produce galaxy samples withexquisite statistical precision, which makes it possible to sin-gle out the most fundamental properties that govern thephysics of galaxy formation from the medley of observables.Equally impressive has been the progress in the the-oretical understanding of the structure formation in theUniverse. Techniques such as direct numerical simulationsand semi-analytic models can now reproduce the observedproperties of galaxies, such as the luminosity functionand 2-point correlation function, color, mass-to-light ra-tios over large ranges of environments and cosmic epochs(Kauffmann et al. 1999a,b; Springel et al. 2001; Cole et al.2000; White & Rees 1978).Yet another approach, the so-called halo model, whichis phenomenological in nature, has enjoyed popularity overthe recent years. An essential ingredient of this method isthe halo occupation distribution (HOD), which refers to theway galaxies (or substructures of dark matter halos) “pop-ulate” dark matter halos. In general, an HOD descriptionincludes the mean number of galaxies per halo N as a func- tion of halo mass, the probability distribution that a haloof mass M contains N galaxies P ( N | M ), and the relativedistribution (both spatial and velocity) of galaxies and darkmatter within halos (Berlind & Weinberg 2002).The halo model formalism allows fast exploration of awide range of HODs; an HOD that reproduces the observedclustering properties and luminosity function of galaxiescan be further studied to reveal the physical processes thatlead to galaxy formation and understanding of cosmolog-ical parameters. Examples of using halo model formalismto reproduce observables in order to reveal parameters incosmology, galaxy evolution and formation includes (e.g.Abazajian et al. 2005; White et al. 2007; Yoo et al. 2006;Zheng & Weinberg 2007; Kulkarni et al. 2007).Despite the success in both observational and theoret-ical sides, there remains some unsolved problems regardingthe formation of the massive, (usually) early type, galaxies.These galaxies appear ” red and dead”, with the majority ofthe stars forming at high redshift ( z > ∼
2) and evolving pas-sively since. Within the cold dark matter (CDM) paradigm,in which massive galaxies are built by smaller galaxies viamergers in the late times, mergers between gas poor systems(“dry” mergers) seem to be a promising route to form giantgalaxies. Observationally, however, the overall importanceof dry mergers is still under heated debate.Luminous Red Galaxies (LRGs) are massive galaxiescomposed mainly of old stars, with little or no on-going starformation. They demonstrate very consistent spectral en- c (cid:13) Ho et al. ergy distribution (SED). Their SEDs mainly consist of oldstar spectrum, most notably for the 4000˚A break. This al-lows one to photometrically determine their redshifts fairlyaccurately (see Padmanabhan et al. 2005). With the accu-rate photometric redshifts of LRGs, one can probe a largervolume of the universe, thus giving better constraints on theformation of massive galaxies. By studying the HOD of theLRGs, we aim to provide a simple quantitative descriptionof these galaxies in massive dark matter halos, which will en-able direct comparison with predictions of galaxy formationmodels.Here we aim to provide observational constraintson the HOD of the LRGs based on a sample of 47intermediate-redshift clusters from the ROSAT 400d sur-vey (Burenin et al. 2006), with photometric data from theSloan Digital Sky Survey (SDSS; Stoughton et al. 2002). Us-ing X-ray properties of these clusters to define the clustercenter and estimate the cluster binding mass, we determinethe mean halo occupation number N as a function of massfrom ∼ × M ⊙ to ∼ × M ⊙ and also investigatethe LRG distribution and luminosity distribution within theclusters.In §
2, we briefly describe the X-ray cluster catalog thatwe utilize and the construction of SDSS LRG sample. In §
3, we present our method and findings on the LRG distri-butions within the clusters and the mean halo occupationnumber. We discuss what is a good mass tracers and evolu-tion of massive galaxies in §
5. Possible systematics that mayaffect our results are discussed in § m h = 0 . h = 0 .
732 and the Hubble parameter H = 73 h km s − Mpc − . Our cluster sample is drawn from the 400 square degreeROSAT PSPC Galaxy Cluster Survey (Burenin et al. 2006)(hereafter the 400d survey), which is an extension of the 160square degree survey (Vikhlinin et al. 1998). The survey de-tects extended X-ray sources in archival ROSAT PSPC im-ages down to a flux limit of 1 . × − erg s − cm − , withextensive optical spectroscopic follow up. Out of the 266clusters detected in the survey, 47 lie within the redshiftrange 0 . ≤ z ≤ . L X ,which is used to estimate the cluster mass. Some of the basicinformation of the clusters in our sample is given in Table 1. The Sloan Digital Sky Survey has taken ugriz
CCDimages of 10 deg of the high-latitude sky. A ded-icated 2.5m telescope at Apache Point Observatoryimages the sky in 5 bands between 3000˚A and 10000˚A (Fukugita et al. 1996) using a drift-scanning, mosaicCCD camera (Gunn et al 1998; Gunn et al. 2006),detecting objects to a flux limit of r ∼ . targets for spectroscopy, mostof them galaxies with r < .
77 mag (Gunn et al1998; York, Adelman, Anderson, Anderson, et al. 2000;Stoughton et al. 2002). This spectroscopic follow-up usestwo digital spectrographs on the same telescope as theimaging camera. Details of the galaxy survey are describedin the galaxy target selection papers (Eisenstein et al.2001; Strauss et al. 2002); other aspects of the surveyare mainly described in the Early Data Release paper(Stoughton et al. 2002). All the data processing, includingastrometry (Pier et al. 2003), source identification and pho-tometry (Lupton et al. 2001; Hogg et al. 2001; Ivezic et al.2004), calibration (Fukugita et al. 1996; Smith et al.2002), spectroscopic target selection (Eisenstein et al.2001; Strauss et al. 2002; Richards et al. 2002), and spec-troscopic fiber placement (Blanton et al. 2003) are doneautomatically via SDSS software ( Tucker, et al. 2006).The SDSS is well-underway, and has had six major releases(Adelman-McCarthy et al. 2007).We utilize the photometric LRGs from SDSS con-structed as described in Padmanabhan et al. (2005, here-after P05). The LRGs have been very useful as a cosmologi-cal probe since they are typically the most luminous galaxiesin the universe, thus they probe a larger volume than mostother tracers. On top of this, they also have very regularspectral energy distributions and a prominent 4000˚A break,making photometric redshift estimation much easier thanthe other galaxies. We plot the color magnitude diagram forone of the cluster and show that the LRGs in the clusterare the bright red galaxies that follow nicely along the redsequence (see Fig 1).Our selection criteria are based on the spectroscopicselection of LRGs described in Eisenstein et al. (2001), ex-tended to lower apparent luminosities (P05). We selectLRGs by choosing galaxies that both have colors consistentwith an old stellar population, as well as absolute luminosi-ties greater than a chosen threshold. The first criterion issimple to implement since the uniform SEDs of LRGs im-ply that they lie on an extremely tight locus in the spaceof galaxy colors; we simply select all galaxies that lie closeto that locus. More specifically, we can define three (notindependent) colors that describe this locus, c ⊥ ≡ ( r − i ) − . g − r ) − . ,d ⊥ ≡ ( r − i ) − . g − r ) ,c || ≡ . g − r ) + 1 . r − i − . , (1)where g , r , and i are the SDSS model magnitudes in thesebands respectively. We now make the following color selec-tions,Cut I : | c ⊥ | < . d ⊥ > . , g − r > . , (2)as well as the magnitude cutsCut I : r Petro < . c || / . r Petro < . i < . d ⊥ ,i < . (3) c (cid:13) , 000–000 uminous Red Galaxy Population in Clusters at . ≤ z ≤ . Figure 1.
Color-magnitude diagram of objects toward the fieldof Cluster 142. Those satisfying cuts I and II (see Eqn. 2 and 3)are shown as crosses and squares respectively. The red circles areselected LRGs and the blue dots are objects detected in SDSSphotometrically. As shown, the selected LRGs lie very systemat-ically along the red sequence of the cluster.
Making two cuts (Cut I and Cut II) is convenient sincethe LRG color locus changes direction sharply as the 4000˚Abreak redshifts from the g to the r band; this division dividesthe sample into low redshift (Cut I, z < .
4) and high red-shift (Cut II, z > .
4) samples. More details of these colorselection criteria are thoroughly described in P05.We do however apply slightly different cuts than thoseadopted in P05: we limit our samples to sky regions where E ( B − V ) ≤ .
08 (4)and data taken under seeing condition of
F W HM < . ′′ . (5)These cuts in extinction and seeing are applied simply byexcluding areas at which the galaxy overdensity drops sig-nificantly. Furthermore, there are a few regions in SDSS thathave 60% more red objects and less blue objects; we decideto throw away these regions.We slice our LRG sample into two redshift bins: 0 . ≤ z photo ≤ . . ≤ z photo ≤ .
6. We also regularizedour redshift distribution as described in P05. For our sam-ple, we have 855534 galaxies, covering 2,025,731 resolution10 HEALpix pixels, each with area of 11 . , giving0.422334 gal/pix.We then estimate the photometric redshift of theseLRGs with the algorithm developed by P05. The typicaluncertainty of the photo- z ’s is δ z = σ z / (1 + z ) ≈ .
03 (seeP05).
We estimate the cluster virial mass M ≡ (4 π/ r × ρ c from the X-ray luminosity using the mass–luminosityrelation given by Reiprich & B¨ohringer (2002)log (cid:20) . L X (0 . − . h − erg s − (cid:21) = A + α log (cid:18) . M h − M ⊙ (cid:19) , (6)where A = − .
055 and α = 1 . r is de-fined such that the enclosed mean overdensity is 200 timesthe critical density ρ c . The corresponding angular extent is θ . The mass–luminosity scaling relation provides a massestimate accurate to <
50% (Reiprich & B¨ohringer 2002)and a virial radius r estimate accurate to 15%.As we now have the redshifts and positions of theseclusters, we locate the LRGs as described in § θ and length of ∆ z = 0 .
06 from the cluster centerin both position and redshift space (i.e. z LRG = z c ± . δ z = 0 .
03 since that is the typical 1 σ error onthe LRG photometric redshift (P05) and z c is the clusterredshift. More discussion on the choice of cluster radius and δ z will be described in § ∼
1% (Padmanabhan et al. 2006). This is therate of which a LRG (photometrically chosen) is actually astar or a quasar after we get the spectra of the object.ii. Interlopers: There is a finite probability of findingLRGs inside the cluster purely by chance, we call these in-terlopers. We access the expected number of interlopers ineach cluster by looking at the average number of LRGs insky (2D projected) in the solid angle of radius = θ of thecluster and the average probability of finding a LRG in red-shift range of z c ± δ z where δ z = 0 .
03 (as defined above). Wecan write down the expected number of interlopers ( h N int i )as: h N int i = ¯ nπθ Z z c + δ z z c − δ z P ( z p ) dz p (7)where P ( z p ) is the normalized (photometric) redshift distri-bution of LRGs, ¯ n is the 2D average LRG density.iii. Missing galaxies due to errors in photometric red-shift: As an LRG can be scattered out of the cluster (due tophotoz error), we need to account for this process by look-ing at the probability of LRG having been photometricallydetermined to be outside of the cluster, but in fact has spec-troscopic redshift that falls within the range of the cluster: P ( | z p − z c | > δ z , || z s − z c | < δ z ) = Z z max z min ( F ( z s )+ B ( z s )) dz s , (8) z min = max (0 . , z c − δ z c ,in ) ,z max = min (0 . , z c + δ z s ,in ) ,F ( z ) = Z z c − δ z − z −∞ P ( δ, z ) dδ , c (cid:13) , 000–000 Ho et al.
Figure 2.
Spatial distribution of LRGs in Cluster 142. The bluepoints represent all objects detected in the SDSS photometricsurvey. Those satisfying cuts I and II are shown as crosses andsquares, respectively. The LRGs that have photo- z consistent withthe cluster redshift ( z = 0 . B ( z ) = Z + ∞ z c + δ z − z P ( δ, z ) dδ , (9)where P ( δ, z ) is the probability of finding δ (= z s − z p ) at z s , given by Padmanabhan et al. (2005) and these are onlycharacterized within the spectroscopic redshift range from z = 0 .
05 to z = 0 . δ z c ,in is the redshift range we allow aLRG to be a cluster member when we have its spectroscopicredshift, and this is set to be 0 . h N corr i = ( h N obs i − h N int i ) /f ( z p , z c , z s ) , (10) f ( z p , z c , z s ) = (1 − P ( | z p − z c | > δ z || z s − z c | < δ z )) × (1+ F )) , (11)and F is the LRG identification failure rate.We list these corrected LRG counts in Table 1.To convert the observed magnitudes of the LRGs intothe rest-frame luminosity at z = 0, we follow the evolutionof a simple stellar population formed in a burst at z = 5,with solar metallicity and Salpeter initial mass function, us-ing the model of Bruzual & Charlot (2003). The LRGs areselected so that their present-day magnitude lies in the range − . ≤ M g ≤ −
21 (roughly corresponding to 1–7 L ∗ , where L ∗ is the characteristic luminosity).For each cluster, we visually inspect the spatial andcolor distributions of LRGs with respect to all objects de-tected by SDSS. An example is shown for cluster 142. Per-haps not surprisingly, the spatial distribution of the LRGsseems concentrated towards cluster center (Fig. 2).A general scenario that has been painted about LRGsand clusters is that there is a massive red galaxy sitting rightin the middle of the cluster. Then some other process may Figure 3.
The distribution of LRGs in the clusters. The numberof LRGs in each bin are normalized by dividing the number ofLRGs in each bin by the total number of LRGs in all bins. sometimes bring in other massive red galaxies, but they willprobably sink into the center over several dynamical times.Here, we actually have the ability to see if this scenariois true: we have the number of LRGs inside each of theseclusters and we know where they are. Below we present re-sults on the spatial distribution of LRGs in clusters ( § § § We show the spatial distribution of LRGs within the clus-ters in Fig. 3. Previous studies (e.g. Jones & Forman 1984;Lin & Mohr 2004) have shown that brightest galaxies tendto lie at the center of the clusters. Here we test if this is truefor the LRGs. We plot the distribution of brightest LRGs ineach of the cluster alongside with their companions in eachof the cluster (see Fig. 4). One realizes that most ( ∼ ∼
20% of the “central” LRGswhich are not central at all. This may suggest a few scenar-ios, one being that the cluster is not relaxed enough for thecentral LRG to sit at the center of the gravitational potential(which is supposedly traced by the intracluster light). Thecentroiding of the clusters in X-ray is called into question,and we will address this in § c (cid:13) , 000–000 uminous Red Galaxy Population in Clusters at . ≤ z ≤ . Figure 4. (Above) The distribution of brightest LRGs (BLRGs)and the non-brightest LRGs in the clusters. As shown above, theBLRGs tend to lie at centers of the clusters, while the ones thatare not BLRGs have a shallower radial distribution. (Below) Thedistribution of most central LRGs and the non-central LRGs inthe clusters. The distribution of the central LRGs are very similarto the BLRGs. the halo model formalism. One would like to understandhow statistically LRGs populate the clusters they are resid-ing. We try to fit the NFW profile here to the LRG sur-face density of stacked clusters in our sample and find thatthe concentration of the surface density to be 17 . +7 . − . with χ = 4 .
29. We also plot the fitted profile in Fig. 5. We also fitthe NFW profile to LRG surface density of stacked clusterswithout the BLRG (Brightest LRG in the cluster), and thisgives a concentration of 6 . +3 . − . with χ = 6 .
6. Both profileshave very similar concentration as the K-band cluster pro-file discussed in Lin & Mohr (2007) (see Figure 6). Errorsin r determination do not affect the fit in any significantfashion as demonstrated in Lin & Mohr (2007) appendix . As the halo occupation number of the clusters is a key ingre-dient to the halo model formalism, we investigate the num-ber of LRGs in these clusters as a function of their masses.As the size of our sample is not large and the mass estimateo the clusters are accurate to 30% −
50% only, one will haveto be extra cautious in finding a fit for the average numberof LRGs in the mass range of these clusters. We take thefollowing approach, assuming two different models: N ( M t ) = a × M t + kN ( M t ) = k × M at (12)where M t is true value of cluster virial mass in 10 h − M ⊙ ,a Poisson distribution of N ( M t ) and two distributions forthe probability finding M t given M i where M i is the mea-sured mass of the i-th cluster (in same units as in M t ), one Figure 5.
The distribution of LRGs in the clusters with the fit tothe NFW profile. Blue line: we fit the surface density of the LRGs(including BLRGs) to a NFW profile and get a concentration of17 . +7 . − . with χ = 4 .
29. Black line: we fit the surface densityof the LRGs (excluding BLRGs) to a NFW profile and get aconcentration of 6 . +3 . − . with χ = 6 . being Gaussian, the other Log-Normal. In short, we have thefollowing: L tot = N c Y i Z P ( N i , M t,i | a, k ) P ( M t,i | M i ) dM t,i log P ( N, M | a, k ) = N × log( µ ) + const − µ log P g ( M t | M i ) = − ( M t − M i ) / (2 σ M ) + const log P ln ( M t | M i ) = − ( M t,l − M i,l ) / (2( σ M l ) ) + const (13)where µ = a × M + k or µ = k × M a , M t,i stands for the M t for the i-th cluster and M x,l stands for log ( M x ) for theabove mentioned form of fit. We then maximize the totalLikelihood within a grid of resolution 100 , , a , k and we also vary the size of dM t,i to ensure that ourresults are robust with respect to varying grid size. We alsotry a variety of ranges for both a and k (such as 0 < a < − < a <
10 and 0 < a <
3; and 0 < k < − < k < < k < P g ( M t | M i ) gives a = 0 . ± .
215 and k = 1 . ± .
705 for 68 .
3% confidence intervals. This power-law fit with P g ( M t | M i ) gives a = 0 . ± .
245 and k =1 . ± .
540 for 68 .
3% confidence intervals. See Fig. 7 forthe data and the fit using P ln ( M t | M i ). N ( M ) distribituion: Poisson or not? Since we assume a Poisson distribution for N ( M t ) (hereafter N for simplicity in this section), we test if this is a goodassumption by looking at ( h N i − h N i ) / h N i . We define γ N = ( h N i − h N i ) / h N i , and since we only have N ( M i ),but not N (Number of LRG given the true measure of clustermass) for each cluster, therefore, we have to consider the c (cid:13) , 000–000 Ho et al.
Figure 6.
The distribution of LRGs in the clusters with the fitto the NFW profile. Blue line: we fit the surface density of theLRGs (including Brightest LRGs) to a NFW profile and get aconcentration of 17 . +7 . − . with χ = 4 .
29. Black line shows thefit of the profile of bright galaxies (normalized to this plot) from(Lin et al. 2006).
Figure 7.
The number of LRG per halo N as a function of binnedhalo mass (10 h − M ⊙ ), the fit (red line) is calculated by max-imizing the likelihood given a model of N ( M t ) = a ∗ M t + k where M t is the true measure of M in 10 h − M ⊙ , assumingPoisson distribution of N ( M t ), where M t is the true mass of thecluster. We also assume a log-normal distribution for probabilityof P ln ( M t | M i ), where M i is the measured mass. This gives a =0 . ± .
205 and k = 1 . ± .
72. The blue line fit is calculatedby maximizing the likelihood given a model of N ( M t ) = k ∗ M at and it gives a = 0 . ± .
250 and k = 1 . ± . Figure 8.
To test whether our assumption of a Poisson distri-bution for N ( M t ) is valid, we compute h N ( N − i / h N i for thecombined sample of 400 d + Y x . It does not deviate drasticallyfrom being Poisson. contribution of scatter from the various systematic effectswe mentioned in § h N i i ( M i ) = h N int i + h N int i + Y + Z + WY = 2 f h N int ih N i ( M i ) Z = Z dM t [ P ( M t | M i )( h N i M t ) + V ) V = γ h N i ( M t ) W = f (1 − f ) h N i ( M i ) (14)where f = f ( z p , z c , z s ) and is defined in § N int is thenumber of interloper as discussed in § X ( M i ) ( X ( M t ) )means the quantity X conditioned on M i ( M t ).Subtracting h N ( M i ) i M i ) from the equation will reduceto: h N i i ( M i ) − h N ( M i ) i M i ) = P Q + R + S + TP = f γ + f (1 − f ) Q = h N i ( M i ) R = h N int i S = f ( Z dM t ([ P ( M t | M i ) h N i M t ) ) T = − f h N i M i ) (15)Note that h N i ( M i ) = R dM t [ P ( M t | M i ) h N i ( M t ) ].A Poisson distribution is completely characterized by itsfirst moment, γ would be 1 if the distribution is completelyPoisson. We calculate the γ from the combined sample of400 d and Y x (please refer to § Y x sample) sample and use h N i from the fit of N ( M ) = k × ( M/ ) a . We bin the cluster such that there are equalnumber of clusters in each mass bin (See Figure 8). We findthat γ = 1 . ± .
351 and thus the N ( M t ) distribution isconsistent with being Poisson.Furthermore, one important ingredient of halo occupa- c (cid:13) , 000–000 uminous Red Galaxy Population in Clusters at . ≤ z ≤ . Figure 9.
To test if the satellite LRG distribution is Poisson, wecalculate ( h ( N − i − h ( N − i ) / h ( N − i for the combinedsample of 400 d + Y x . It is consistent with being Poisson. tion distribution is the assumption of Poisson distribution ofthe satellite galaxies. We test the assumption here by com-puting ( h ( N − i − h ( N − i ) / h ( N − i for the h N − i distribution (see Fig. 9) in a similar way as we compute γ forthe h N i distribution. We find that γ N − = 1 . ± .
496 andso the satellite LRG distribution in clusters is also consis-tent with being Poisson. However, one should note that it ismathematically impossible for both N and N − σ ( X ) = p (( X − ¯ X ) ) /N c . Finally we study the multiplicity of LRGs in clusters(Fig. 10).We calculate the multiplicity function by counting the1 /V max weighted number of cluster in each bin. We compute V max (the comoving search volume of the cluster) by:i.We find the flux of the cluster via the followingBurenin et al. (2006): f = L πd L ( z ) K ( z ) (16)where L is the luminosity of the cluster, d L ( z ) is is the cos-mological luminosity distance, K ( z ) is the K-correction fac-tor for X-ray clusters (for more details see Burenin et al.(2006)).ii. We find the comoving search volume that each clusterwith luminosity L can be detected given by the following: V max ( L ) = Z z = z c z =0 P sel ( f, z ) dVdz dz (17)where P sel ( f, z ) is the selection efficiency of the 400 squaredegrees ROSAT PSPC Galaxy Cluster Survey providedby A. Vikhlinin and R. Burenin (private communication), dV /dz is the cosmological comoving volume per redshift in-terval (see Burenin et al. (2006) for more details). Figure 10.
The volume weighted multiplicity function of LRGs inthese clusters. Blue (Black) line: the volume weighted multiplicityfunction for clusters with X-ray luminosities > = ( < ) 10 ergs − .The variance is calculated by taking the sum of 1 / ( V max ) for each N LRG bin.
We choose to use θ since it is closest to the virial radiusof the clusters. We also look at how the uncertainties of r will affect our results. r is accurate up to ∼ r : Z r ρ ( r ) dr/ Z . r ρ ( r ) dr = 0 .
95 (18)and Z r ρ ( r ) dr/ Z . r ρ ( r ) dr = 1 .
06 (19)We set the density profile ρ ( r ) as a NFW profile withconcentration of 8 (which is approximately what we getwhen we fit the surface density of the cluster when we ex-clude the BCG). This shows that the uncertainties in θ ,thus r only affect our estimation of N ( M ) at the level of ∼ Cluster mass estimation is crucial in our analysis, as it de-fines the cluster virial region to search for member LRGs,and provides a fundamental radius to scale the distance ofLRGs to cluster center. We infer cluster mass through the X-ray luminosity–mass scaling relation (Reiprich & B¨ohringer2002), which has been shown as a unbiased estimator(Reiprich 2006). Compared to other X-ray–based clusterproxies such as temperature and Y X (the product of gasmass and temperature, which is proportional to the thermal c (cid:13) , 000–000 Ho et al.
Figure 11.
Distribution of the clusters on the mass–redshiftplane. The squares, red points, and blue points denote the whole400d survey sample, the subsample used in this study, and the Y X sample, respectively. energy of the cluster Kravtsov et al. 2006), L X – M correla-tion shows higher degree of scatter. We therefore seek foranother cluster sample with better measured mass (despitewithout well-defined selection criteria).Recently, Maughan et al. (2007) have presented a largecluster sample selected from the Chandra archive, for whichthe cluster mass is inferred from Y X , and the cluster centeris inferred from the Chandra images. 26 of these clusters liewithin our SDSS DR5 masks and the redshift range 0 . ≤ z ≤ .
6. 16 of these 26 clusters do not overlap with our 400dsample and we use them to examine the results presented in §§ Y X sample).Because of the flux-limited nature of the 400d survey,low mass ( ∼ M ⊙ ) clusters will be only detected at lowerredshifts. In Fig. 11 we show the mass distribution of thewhole 400d sample (open squares) and the subsample usedin our analysis (red points) within 0 . ≤ z ≤ .
9. It showsthat our sample is a random subsample of the whole 400dsample Interestingly, at z ∼ . z ≤ . Y X sample.Very curiously, the distribution of this sample on the mass–redshift space seems to be roughly orthogonal to that of our400d sample. Since our results derived from the Y X sample isconsistent with those based on the 400d sample, we combinethe two samples to enhance the mass coverage (especially forclusters at z ≥ .
4) and the statistical signal. We calculatedthe N ( M ) for the combined sample and assuming power-law model, we have N ( M ) = k ∗ ( M/ ) a , where a =0 . ± .
105 and k = 1 . ± . Figure 12.
Combining the Y X sample (green triangles), we havea larger mass coverage, thus giving stronger constraints on theslope. We have fairly similar fits between the two different models(linear (red) and power-law (blue)). The fits are also consistentwith the respective fits using only clusters from 400d survey. As clusters are becoming more important cosmological tools,we need to characterize the masses of clusters more thanever. “What is a good mass tracer?” has been a very well-motivated question.Here we try to investigate a few options that peoplehave suggested before as possible solutions:Fist, as we see earlier in § N LRG − M relation by the following quantities in a 3 mass bins: σ ( ln ( N LRG )) = p ( γ ) p ( N LRG ) σ ( ln ( M )) = 1 a σ ( ln ( N LRG )) (20)where a is as defined in N ( M ) = k × ( M/ ) a . We foundthat the scatter in ln ( N LRG ) ( ln ( M )) in low, middle andhigh mass bins are 0 .
332 (0 . M = 2 . × h − M ⊙ ), 0 .
281 (0 . M = 3 . × h − M ⊙ )and 0 .
21 (0 . M = 9 . × h − M ⊙ ) respec-tively.Second, we look at the luminosities of the central LRG.As previous studies suggested in some bands, the brightestcluster galaxies traces the mass of the cluster (Lin & Mohr2004) and that the brightest cluster galaxies tend to be thecentral galaxies of the cluster, we look at the relation be-tween the luminosities of the brightest LRG in clusters andtheir X-ray masses. However, the correlation between thedistribution of luminosities of central LRG and the massesof the clusters in our sample does not look promising(seeFig. 13). c (cid:13) , 000–000 uminous Red Galaxy Population in Clusters at . ≤ z ≤ . Figure 13. (Above) Luminosities of the central LRGs for eachcluster. (Below) Luminosities of the brightest LRGs for each clus-ter.
We then look at the correlation between the luminositiesof the brightest LRG and their cluster X-ray masses. How-ever, it does not seem to be promising either (see Fig. 13).This is also seen in Lin & Mohr (2004) when we look at thesame mass range and when one looks at the correlation be-tween the brightest LRGs and the richness of the maxBCGcatalog (Koester et al. 2007), there is not a strong correla-tion for 14,000 clusters (R. Reyes 2007, private communica-tion). However, there are several caveats that would requirefurther investigations, such as the possibility of photo-z fail-ure for the CLRG or BLRG in the clusters and possiblephotometry problem that could destroy the correlation. Welook into the available spectroscopic data in SDSS and foundno extra LRGs that are targeted by the SDSS spectroscopy.This rules out the possible missing LRGs that have M r ofrange ∼ − . z = 0 .
2) and ∼ − . z = 0 . ∼
70% of BLRGs lie in the central ∼
20% of the virial radius, thus, most clusters do have aLRG at their centers. If we are missing Brightest LRGs incenters of clusters, we need to expect the scenario of hav-ing more than 1 LRG at the central ∼
20% of cluster virialradius to be prevalent. This scenario is not supported bythe distribution of LRGs as shown in Figure 4. Given thecaveats and findings here, we conclude that further workwill be needed to make this more quantitative, especially toquantify the effect of photometry errors on the correlation.
Insights into the evolution of massive galaxies in clustersmay be gained by comparing some of the results presentedin § M ∗ − z < .
2, find that luminous clustergalaxies ( M K ≤ −
25) follow an NFW profile with concen-tration of 18 . . § M K ≤ −
25 with the z < . § N thus obtained as a upper limit. We find thatthe nearby ¯ N – M relation is similar to that shown in § z ∼ . z ≈
0. The occupa-tion number comparison basically suggests that the shapeof the luminosity function is similar in clusters at thesetwo epochs, after the passive evolution has been taken intoaccount. This is consistent with several previous studies,both for cluster galaxies (Lin et al. 2006; Andreon 2006;De Propris et al. 2007; Muzzin et al. 2007) and the fieldpopulation e.g. (Wake et al. 2006; Brown et al. 2007).However, evidence for mergers that produce massivegalaxies has been found (van Dokkum 2005; Bell et al. 2006;some other references). In the ΛCDM model, formation ofmassive objects through mergers of less massive ones is ageneric feature. To reconcile the apparent no-evolution of theaforementioned bulk properties with this picture, we suggesttwo considerations. (1) Irrespective of the role of mergers inthe formation and evolution of the LRGs, their spatial dis-tribution seems to be similar out to z ∼ .
5. This is similarto the “attractor” hypothesis of Gao et al. (2004). (2) Thedegree of evolution, be it an increase in the number of LRGsdue to mergers of the host halo with less massive halos, ora decrease due to dynamical processes (e.g. tidal disruption,mergers), would be seen more clearly through (Monte Carlo)simulations where the merger history of the halos is fully fol-lowed. In a companion paper such an approach is adopted toinfer the merger rate of LRGs (Conroy, Ho & White 2007).
We investigate statistical properties of LRGs in a sampleof X-ray selected galaxy clusters at intermediate redshift(0 . ≤ z ≤ . . +7 . − . with χ = 4 .
29 when we includethe brightest LRG. When we do not include the brightestLRG, we find concentration of 6 . +3 . − . with χ = 6 .
6. Consid-ering the sample size and mass errors on our sample, we use c (cid:13)000
6. Consid-ering the sample size and mass errors on our sample, we use c (cid:13)000 , 000–000 Ho et al. the maximum likelihood method to find the best fit param-eters for halo occupation distribution ( N ( M )). The resultdepends on what kind of models we adopt, but are fairlyinsensitive to what model we use, results are shown in Ta-ble 2.Uncertainties in photometric redshifts are taken intoaccount by including different possible effects such as in-terlopers and missing LRGs due to errors in photometricredshifts (see § N ( M ) atthe level of ∼ Y X sample)to test the mass estimation of our sample. However, we doimplicitly assume that the scatter of M − L X relation doesnot correlate with N ( M ) during the analysis. The result wederive from a combined analysis of both sample on N ( M )is consistent with using our sample alone (see Table 2). Wealso find that there are no obvious good mass tracer as welook at different correlations between various quantities ofclusters and their galaxies. Last, we discuss the evolutionof massive galaxies from different perspectives. We concludethat it would be important to study low-z LRG populationto better constrain the evolution of the population (Ho et al.2007). c (cid:13) , 000–000 uminous Red Galaxy Population in Clusters at . ≤ z ≤ . Name RA DEC redshift M θ LRG LRG count(deg) (deg) (10 M ⊙ h − ) (arcmin) count corrected20 29.8258 0.5025 0.386 3.815 4.0608 2 2.56436 46.7695 -6.4808 0.347 4.879 4.8054 2 2.33680 122.4208 28.1994 0.399 5.357 4.428 4 5.00386 132.2975 37.5230 0.240 1.419 4.326 2 2.19188 133.3058 57.9955 0.475 3.536 3.3564 5 7.02799 149.0116 41.1188 0.587 3.731 2.895 1 1.243100 149.5541 55.2683 0.214 2.516 5.7744 3 4.031101 149.5804 47.0380 0.390 3.552 3.933 0 -0.0116103 150.7687 32.8933 0.416 4.334 3.9906 4 5.659107 152.8558 54.8350 0.294 2.142 4.185 1 1.112108 153.3658 -1.6116 0.276 2.157 4.4214 1 1.052110 154.5037 21.9097 0.240 1.867 4.74 1 1.093111 156.7945 39.1350 0.338 4.079 4.6248 1 1.083121 169.3754 17.7458 0.547 3.343 2.949 3 3.389123 170.2429 23.4427 0.562 4.277 3.1344 1 1.281124 170.7941 14.1611 0.340 2.375 3.843 2 2.388134 178.1487 37.5461 0.230 2.260 5.238 1 1.279136 180.0320 68.1519 0.265 2.726 4.9458 3 3.334137 180.2062 -3.4583 0.396 2.818 3.5964 1 1.265142 183.0800 27.5538 0.353 7.677 5.5116 7 8.071144 183.3933 2.8991 0.409 2.518 3.3756 0 -0.008145 184.0825 26.5558 0.428 2.742 3.3492 2 2.517146 184.4320 47.4872 0.270 3.743 5.412 2 2.164150 185.5079 27.1552 0.472 3.384 3.324 7 10.194163 193.2695 62.8027 0.235 1.810 4.7766 3 3.714166 197.1370 53.7041 0.330 2.208 3.8436 0 -0.007167 197.8029 32.4827 0.245 2.876 5.379 1 1.089168 198.0808 39.0161 0.404 3.500 3.8046 5 6.069172 202.8791 62.6400 0.219 1.619 4.8876 7 8.992175 204.7091 38.8550 0.246 3.342 5.6358 1 1.087181 208.5695 -2.3627 0.546 3.360 2.958 4 4.525184 212.5558 59.7105 0.316 3.140 4.479 4 4.879185 212.5662 59.6408 0.319 2.356 4.0386 4 4.775188 214.6300 25.1797 0.290 4.599 5.4612 1 1.127198 231.1679 9.9597 0.516 4.881 3.5016 3 3.662202 243.5479 34.4236 0.269 2.110 4.4844 0 -0.006208 250.4679 40.0247 0.464 4.316 3.654 3 4.946209 254.6412 34.5022 0.330 3.293 4.3914 2 2.237210 255.1779 64.2161 0.225 2.576 5.5746 1 1.209211 255.3441 64.2358 0.453 4.940 3.8952 4 5.670212 260.7245 41.0916 0.309 2.799 4.3908 3 3.375s9 209.89166 62.3169 0.332 4.162 4.725 2 2.217s11 221.0266 63.7483 0.298 2.006 4.0488 5 5.595s12 225.0108 22.5680 0.230 1.353 4.4142 2 2.564s13 228.5916 36.6061 0.372 4.795 4.5156 1 1.448s14 234.1470 1.5556 0.309 4.793 5.253 2 2.244s17 236.8350 20.9502 0.266 2.201 4.5912 4 4.426 Table 1.
Basic parameters of our cluster sample. Naming schemefollows the cluster number as given in table 4 of Burenin et al.(2006) those start with s are from table 5 of Burenin et al. (2006),and are not part of the main sample of serendipitous 400d survey.c (cid:13) , 000–000 Ho et al. N ( M ) Data Poisson+Gaussian Poisson+log-Normal a ∗ M + k a = 0 . ± . a = 0 . ± . k = 1 . ± . k = 1 . ± . a ∗ M + k Y X a = 0 . ± . a = 0 . ± . k = 1 . ± . k = 1 . ± . k ∗ M a a = 0 . ± . a = 0 . ± . k = 1 . ± . k = 1 . ± . k ∗ M a Y X a = 0 . ± . a = 0 . ± . k = 1 . ± . k = 1 . ± . Table 2.
Results of maximizing likelihood assuming different pa-rameters. This table describes the model of the N ( M ) in the firstcolumn, dataset we use in the ”Data” column, results of maximiz-ing likelihood by assuming Poisson distribution for N ( M ) (bothcolumn 3 and 4), Gaussian and Log-Normal distribution for thecluster mass distribution for column 3 and 4 respectively. REFERENCES
Abazajian K., Zheng Z., Zehavi I., Weinberg D. H., Frie-man J. A., Berlind A. A., Blanton M. R., Bahcall N. A.,Brinkmann J., Schneider D. P., Tegmark M., 2005, ApJ,625, 613Adelman-McCarthy J., et al., 2007, ApJS, in pressAndreon S., 2006, A&A, 448, 447Bell E. F., Naab T., McIntosh D. H., Somerville R. S.,Caldwell J. A. R., Barden M., Wolf C., Rix H.-W., Beck-with S. V., Borch A., H¨aussler B., Heymans C., Jahnke K., Jogee S., Koposov S., Meisenheimer K., Peng C. Y.,Sanchez S. F., Wisotzki L., 2006, ApJ, 640, 241Berlind A. A., Weinberg D. H., 2002, ApJ, 575, 587Blanton M. R., Lin H., Lupton R. H., Maley F. M., YoungN., Zehavi I., Loveday J., 2003, AJ, 125, 2276Brown M. J. I., Dey A., Jannuzi B. T., Brand K., BensonA. J., Brodwin M., Croton D. J., Eisenhardt P. R., 2007,ApJ, 654, 858Bruzual G., Charlot S., 2003, MNRAS, 344, 1000Burenin R. A., Vikhlinin A., Hornstrup A., Ebeling H.,Quintana H., Mescheryakov A., 2006, ApJSConroy C., Ho S., White M., 2007, MNRAS, in pressCole S., Lacey C. G., Baugh C. M., Frenk C. S., 2000,MNRAS, 319, 168De Propris R., Stanford S. A., Eisenhardt P. R., HoldenB. P., Rosati P., 2007, AJ, 133, 2209Eisenstein D. J., Annis J., Gunn J. E., Szalay A. S., Con-nolly A. J., Nichol R. C., Bahcall N. A., Bernardi M.,Burles S., Castander F. J., Fukugita M., Hogg D. W.,Ivezi´c ˇZ., Knapp G. R., Lupton R. H., Narayanan V.,Postman M., Reichart D. E., Richmond M., SchneiderD. P., Schlegel D. J., Strauss M. A., SubbaRao M., TuckerD. L., Vanden Berk D., Vogeley M. S., Weinberg D. H.,Yanny B., 2001, AJ, 122, 2267Fukugita M., Ichikawa T., Gunn J. E., Doi M., ShimasakuK., Schneider D. P., 1996, AJ, 111, 1748Gao L., Loeb A., Peebles P. J. E., White S. D. M., JenkinsA., 2004, ApJ, 614, 17Gunn J. E., Carr M., Rockosi C., Sekiguchi M., Berry K.,Elms B., de Haas E., Ivezi´c ˇZ., Knapp G., Lupton R.,Pauls G., Simcoe R., Hirsch R., Sanford D., Wang S.,York D., Harris F., Annis J., Bartozek L., Boroski W.,Bakken J., Haldeman M., Kent S., Holm S., HolmgrenD., Petravick D., Prosapio A., Rechenmacher R., Doi M.,Fukugita M., Shimasaku K., Okada N., Hull C., SiegmundW., Mannery E., Blouke M., Heidtman D., Schneider D.,Lucinio R., Brinkman J., 1998, AJ, 116, 3040Gunn J. E., et al., 2006, AJ, 131, 2332Ho S., et al. 2007, in prepHogg D. W., Blanton M., SDSS Collaboration, 2001, inBulletin of the American Astronomical Society, Vol. 34,Bulletin of the American Astronomical Society, pp. 570–+Ivezic Z., Juric M., Lupton R. H., Tabachnik S., QuinnT., The SDSS Collaboration, 2004, NASA Planetary DataSystem, EAR-A-I0035-3-SDSSMOC-V1.0:SDSS-Z, 9, 7Jones C., Forman W., 1984, ApJ, 276, 38Kauffmann G., Colberg J. M., Diaferio A., White S. D. M.,1999a, MNRAS, 303, 188Kauffmann G., Colberg J. M., Diaferio A., White S. D. M.,1999b, MNRAS, 307, 529Koester B.P. et al. ApJ, 660, 239Kravtsov A. V., Vikhlinin A., Nagai D., 2006, ApJ, 650,128Kulkarni G. V., Nichol R. C., Sheth R. K., Seo H.-J., Eisen-stein D. J., Gray A., 2007, ArXiv Astrophysics e-printsLauer T. R., Faber S. M., Richstone D., Gebhardt K.,Tremaine S., Postman M., Dressler A., Aller M. C., Filip-penko A. V., Green R., Ho L. C., Kormendy J., MagorrianJ., Pinkney J., 2006, ArXiv Astrophysics e-printsLin Y.-T., Mohr J. J., 2004, ApJ, 617, 879Lin Y.-T., Mohr J. J., 2007, ApJS, 170, 71 c (cid:13) , 000–000 uminous Red Galaxy Population in Clusters at . ≤ z ≤ . Lin Y.-T., Mohr J. J., Gonzalez A. H., Stanford S. A., 2006,ApJ, 650, L99Lupton R., Gunn J. E., Ivezi´c Z., Knapp G. R., Kent S.,2001, in Harnden Jr. F. R., Primini F. A., Payne H. E.,eds, ASP Conf. Ser. 238: Astronomical Data AnalysisSoftware and Systems X The SDSS Imaging Pipelines.p. 269Maughan B. J., Jones C., Forman W., Van Speybroeck L.,2007, ArXiv Astrophysics e-printsMuzzin A., Yee H. K. C., Hall P. B., Lin H., 2007, ApJPadmanabhan N., Budav´ari T., Schlegel D. J., Bridges T.,Brinkmann J., Cannon R., Connolly A. J., Croom S. M.,Csabai I., Drinkwater M., Eisenstein D. J., Hewett P. C.,Loveday J., Nichol R. C., Pimbblet K. A., De ProprisR., Schneider D. P., Scranton R., Seljak U., Shanks T.,Szapudi I., Szalay A. S., Wake D., 2005, MNRAS, 359,237Padmanabhan N., Schlegel D. J., Seljak U., Makarov A.,Bahcall N. A., Blanton M. R., Brinkmann J., EisensteinD. J., Finkbeiner D. P., Gunn J. E., Hogg D. W., IvezicZ., Knapp G. R., Loveday J., Lupton R. H., Nichol R. C.,Schneider D. P., Strauss M. A., Tegmark M., York D. G.,2006, ArXiv Astrophysics e-printsPier J. R., Munn J. A., Hindsley R. B., Hennessy G. S.,Kent S. M., Lupton R. H., Ivezi´c ˇZ., 2003, AJ, 125, 1559Reiprich T. H., 2006, A&A, 453, L39Reiprich T. H., B¨ohringer H., 2002, ApJ, 567, 716Richards G. T., Fan X., Newberg H. J., Strauss M. A.,Vanden Berk D. E., Schneider D. P., Yanny B., BoucherA., Burles S., Frieman J. A., Gunn J. E., Hall P. B., Ivezi´cˇZ., Kent S., Loveday J., Lupton R. H., Rockosi C. M.,Schlegel D. J., Stoughton C., SubbaRao M., York D. G.,2002, AJ, 123, 2945Smith J. A., Tucker D. L., Kent S., Richmond M. W.,Fukugita M., Ichikawa T., Ichikawa S.-i., Jorgensen A. M.,Uomoto A., Gunn J. E., Hamabe M., Watanabe M.,Tolea A., Henden A., Annis J., Pier J. R., McKay T. A.,Brinkmann J., Chen B., Holtzman J., Shimasaku K., YorkD. G., 2002, AJ, 123, 2121Spergel D. N., Bean R., Dor´e O., Nolta M. R., BennettC. L., Dunkley J., Hinshaw G., Jarosik N., Komatsu E.,Page L., Peiris H. V., Verde L., Halpern M., Hill R. S.,Kogut A., Limon M., Meyer S. S., Odegard N., TuckerG. S., Weiland J. L., Wollack E., Wright E. L., 2006,ArXiv Astrophysics e-printsSpringel V., White S. D. M., Tormen G., Kauffmann G.,2001, MNRAS, 328, 726Stoughton C., Lupton R. H., Bernardi M., et al., 2002, AJ,123, 485Strauss M. A., Weinberg D. H., Lupton R. H., NarayananV. K., Annis J., Bernardi M., Blanton M., Burles S., Con-nolly A. J., Dalcanton J., Doi M., Eisenstein D., FriemanJ. A., Fukugita M., Gunn J. E., Ivezi´c ˇZ., Kent S., KimR. S. J., Knapp G. R., Kron R. G., Munn J. A., New-berg H. J., Nichol R. C., Okamura S., Quinn T. R., Rich-mond M. W., Schlegel D. J., Shimasaku K., SubbaRaoM., Szalay A. S., Vanden Berk D., Vogeley M. S., YannyB., Yasuda N., York D. G., Zehavi I., 2002, AJ, 124, 1810Tucker D. L., Kent S., Richmond M. W., Annis J., SmithJ. A., Allam S. S., Rodgers C. T., Stute J. L., Adelman-McCarthy J. K., Brinkmann J., Doi M., Finkbeiner D.,Fukugita M., Goldston J., Greenway B., Gunn J. E., Hendry J. S., Hogg D. W., Ichikawa S.-I., Ivezi´c ˇZ., KnappG. R., Lampeitl H., Lee B. C., Lin H., McKay T. A., Mer-relli A., Munn J. A., Neilsen Jr. E. H., Newberg H. J.,Richards G. T., Schlegel D. J., Stoughton C., Uomoto A.,Yanny B., 2006, Astronomische Nachrichten, 327, 821van Dokkum P. G., 2005, AJ, 130, 2647Vikhlinin A., McNamara B. R., Forman W., Jones C.,Quintana H., Hornstrup A., 1998, ApJ, 502, 558Wake D. A., Nichol R. C., Eisenstein D. J., Loveday J.,Edge A. C., Cannon R., Smail I., Schneider D. P., Scran-ton R., Carson D., Ross N. P., Brunner R. J., CollessM., Couch W. J., Croom S. M., Driver S. P., da ˆAngelaJ., Jester S., de Propris R., Drinkwater M. J., Bland-Hawthorn J., Pimbblet K. A., Roseboom I. G., ShanksT., Sharp R. G., Brinkmann J., 2006, MNRAS, 372, 537White M., Zheng Z., Brown M. J. I., Dey A., Jannuzi B. T.,2007, ApJ, 655, L69White S. D. M., Rees M. J., 1978, MNRAS, 183, 341Yoo J., Tinker J. L., Weinberg D. H., Zheng Z., Katz N.,Dav´e R., 2006, ApJ, 652, 26York D. G., Adelman J., Anderson Jr. J. E., AndersonS. F., et al. , 2000, AJ, 120, 1579Zheng Z., Weinberg D. H., 2007, ApJ, 659, 1 c (cid:13)000