Density Jumps Near the Virial Radius of Galaxy Clusters
aa r X i v : . [ a s t r o - ph . C O ] S e p Draft version September 28, 2015
Preprint typeset using L A TEX style emulateapj v. 5/2/11
DENSITY JUMPS NEAR THE VIRIAL RADIUS OF GALAXY CLUSTERS
Anna Patej and Abraham Loeb Draft version September 28, 2015
ABSTRACTRecent simulations have indicated that the dark matter halos of galaxy clusters should featuresteep density jumps near the virial radius. Since the member galaxies are expected to follow similarcollisionless dynamics as the dark matter, the galaxy density profile should show such a feature aswell. We examine the potential of current datasets to test this prediction by selecting cluster membersfor a sample of 56 low-redshift (0 . < z < .
3) galaxy clusters, constructing their projected numberdensity profiles, and fitting them with two profiles, one with a steep density jump and one without.Additionally, we investigate the presence of a jump using a non-parametric spline approach. We findthat some of these clusters show strong evidence for a model with a density jump. We discuss avenuesfor further analysis of the density jump with future datasets. INTRODUCTION
Galaxy clusters contain a representative sample of thematter in the universe: the dominant constituent is darkmatter, while the baryonic components include hot gasand the galaxies themselves (for a review, see Voit 2005).A longstanding analytical prediction for cosmologicalstructure formation is the existence of a shock bound-ing the hot, gaseous intracluster medium and a corre-sponding jump in the dark matter profile, coincident withthe virial radius of the galaxy cluster (e.g., Bertschinger1985). More recently, simulations by Diemer & Kravtsov(2014) (hereafter DK14) have reinforced the expecta-tion that the dark matter halo itself should exhibit asharp density steepening near the virial radius (see alsoAdhikari, Dalal, & Chamberlain 2014).The member galaxies of a cluster are expected to tracethe cluster’s dark matter profile in the cluster outskirts,as they are subject to similar collisionless dynamics; ac-cordingly, we may expect to see such a feature not justin the dark matter profile but also in the galaxy den-sity profile. Tully (2010) examined the distributions ofthe galaxies in the Coma and Virgo clusters and de-tected sharp density cut-offs at radii of 3 and 2 Mpc,respectively, from the cluster centers, which he identifiedwith the caustics of second turnaround. These corre-spond roughly to the virial radii of these clusters (e.g.,Kubo et al. 2007; Karachentsev et al. 2014). The analy-sis of Trentham & Tully (2009) identified a similar fea-ture in the galaxy group NGC 1023.More, Diemer, & Kravtsov (2015) mentioned thathints of such a jump may have been seen in otherdata sets. For instance, Tully (2015) measured thesecond turnaround radius, which is associated withthe aforementioned density jumps, of several groupsand clusters. Additionally, Rines et al. (2013) (here-after R13) presented spectroscopic velocities for clus-ter members in 58 galaxy clusters, from which they re-constructed the density profiles, which appear to show
Electronic Addresses: [email protected], [email protected] Department of Physics, Harvard University, 17 Oxford St.,Cambridge, MA 02138 Harvard-Smithsonian Center for Astrophysics, 60 GardenSt., Cambridge, MA 02138 a deficit of galaxies with respect to an NFW profile(Navarro, Frenk, & White 1997). However, they notedthat their results at large radii from the cluster centermay be affected by having few spectroscopically observedcluster members in the exterior regions.In this paper, we focus on the R13 sample of clus-ters and use public data from the Sloan Digital Sky Sur-vey (SDSS) (York et al. 2000) to select cluster membersphotometrically. Rather than use spectroscopic veloci-ties, we construct the radial number density profiles ofmember galaxies and fit them with two functional formsto examine whether we can detect a feature consistentwith the predicted density jump in any of the clustersusing available data sets. Where applicable, we assumethe standard ΛCDM cosmology with Ω Λ = 0 .
73 andΩ m = 0 .
27, consistent with the parameters selected byDK14 to enable direct comparison. To compare resultsacross clusters we also use the measure R ∆ , which is theradius within which the mean mass density ¯ ρ = ∆ ρ b ( z ),where ρ b is a specified cosmological background densityand ∆ is the density contrast with respect to ρ b ( z ). DATA
SDSS Catalogs
To probe the existence of the density jump predictedby DK14, we focus our attention on the sample of 56 low-redshift (0 . < z < .
3) galaxy clusters from R13 (of theiroriginal 58 clusters, we do not include MS0906/A750 inour samples, since these clusters are nearly coincidentand hence it is difficult to make a clean selection of mem-ber galaxies for each). Since their cluster sample was se-lected from regions of sky covered by SDSS, we use pub-licly available data from the most recent Data Release12 (Alam, et al. 2015) to select cluster members.We obtain photometric catalogs from the SDSS Sky-Server SQL server . For each cluster, we query galax-ies within 1.5 degrees of the cluster center (which,from R13, is the X-ray center) in both right ascension(RA) and declination. We further restrict our queryto sources in the ‘Galaxy’ view with observed r -bandmagnitude between 14 and 22 and which have ‘clean’photometry as determined by the SDSS pipeline. For http://skyserver.sdss.org/dr12/en/tools/search/sql.aspx each galaxy, we obtain the dereddened ‘model’ mag-nitudes (which we will use throughout the remainderof this work), corrected for Galactic extinction accord-ing to Schlegel, Finkbeiner, & Davis (1998) by the SDSSpipeline, and we also select photometric redshift esti-mates and associated redshift quality estimates from the‘Photo-z’ table. Cluster Member Selection
We will test two cluster member selections. The first,which we will refer to as Selection A, is based solely onphotometry, primarily comprising red cluster galaxies se-lected from their location on a color-magnitude diagrambut also including some galaxies with sufficiently securephotometric redshift estimates that are blueward of thered sequence. The second, which we call Selection B,folds in the spectroscopy of R13 to include additionalcluster members and to reject some of the photomet-rically selected galaxies whose redshift estimates placethem beyond the cluster. These selection procedures areoutlined in more detail in the following sections.
Red Sequence Cluster Member Selection
Both of our methods of cluster member selection arebased upon the target selection process of R13, whichrelied on the red sequence method of Gladders & Yee(2000). The red sequence refers to a linear feature inthe color-magnitude diagram of galaxies in a cluster fieldthat arises from the population of red, early-type galax-ies that have been observed to comprise the majority ofcluster members. By selecting galaxies within a rangearound this line, we can obtain a sample of cluster mem-bers. As in R13, we construct a diagram of g − r vs r and fix the slope of the red sequence line to − . ± . Contribution of Non-Red Sequence Galaxies
Since the red sequence method selects a specific pop-ulation of galaxies, it is worth examining whether theexclusion of other types of galaxies in the cluster affectsour result. The analysis of Dressler (1980), which exam-ined the galactic populations of over 50 galaxy clusters,found that blue, late-type galaxies comprise a small frac-tion of the populations of rich galaxy clusters, although this fraction increases as the density of galaxies decreases.This is supported by the spectroscopic data provided byR13, who find that within the virial radius of the clus-ter the fraction of blue galaxies is . . . h − Mpc, thereare on average only about 60 galaxies with photometricredshifts satisfying | z c − z g | < .
03 and subject to theabove quality cuts, many of which were already pickedup via the red sequence method. Additionally, basing aselection on these redshifts would exclude most clustermembers with r &
19. Accordingly, as noted above, wemust combine them with the galaxies selected via the redsequence method. This final selection, denoted SelectionA, for an example cluster is shown in the lefthand panelof Figure 2.
Additional Tests of the Cluster Member Selection
While neither the SDSS photometric redshifts nor thespectroscopy of R13 identify enough cluster members forour analysis, we can use this additional information torefine the red sequence selection in a second selection.As before, after selecting red sequence cluster membersusing the procedure outlined in Section 2.2.1, we add inbluer cluster members based on SDSS photometric red-shifts as described in Section 2.2.2. However, in thiscase, we can further refine the selection by also reject-ing cluster members if the SDSS photometric redshiftsare sufficiently secure and the members in question haveredshifts in the range | z c − z g | < .
03, where z c is thecluster redshift and z g is the photometric redshift esti-mate.We can use the spectroscopic data of R13 similarly. Af-ter matching their tables to the SDSS catalogs, we addin cluster members as identified by the spectroscopy andreject galaxies that were originally selected by the redsequence method or photometric redshifts but are iden-tified as non-members by R13. This method of clustermember selection, which we term ‘Selection B,’ is sum-marized in the righthand panel of Figure 2. This selec-tion is more observationally expensive than Selection A,so it is worth testing both methods to see whether theadditional information makes a difference in the results.On average, the numbers of cluster members selected bythese two methods differ by about 5% within 1 . h − Mpcof the cluster center. We will test the effect of this selec-tion on our analysis in Section 4.As an additional step in verifying our cluster mem-ber selection, we construct a projected radial densityplot consisting of the total radial density profiles of all
14 15 16 17 18 19 20 r − . . . . . . . g − r A1068 ( g − r ) i =1.15All Galaxies θ <
14 15 16 17 18 19 20 r − . . . . . . . g − r A1068 ( g − r ) i =1.15All GalaxiesRed Sequence
14 15 16 17 18 19 20 r − . . . . . . . g − r A1068 ( g − r ) i =1.15All GalaxiesSpec. Cluster Members Fig. 1.—
The red sequence cluster member selection for example cluster A1068.
Left : in the color-magnitude diagram, we initially plotthe location of galaxies within 5 arcminutes of the cluster center, which should be dominated by cluster members, to provide an indicationof the red sequence.
Middle : The galaxies selected by the red sequence method using the limits described in Section 2.2.1; we plot thegalaxies thus selected that are within 2.5 h − Mpc of the cluster center. The labelled intercept ( g − r ) i is selected to be the value at r = 16. Right:
The cluster members as selected via the spectroscopy of R13, which can be used to verify the red sequence selection. sources in our catalog (subject to r <
20) as well asthe non-cluster density profile, computed by subtractingthe counts of cluster members from the total number ofgalaxies in each bin. In the total profile, we expect tosee a steep increase in galaxies at small radial distancesthat flattens out at large R . After subtracting out thecontribution from these cluster members, the resultingnon-cluster member profile should be roughly flat. Fig-ure 3 shows these density plots for four example clusters,using Selection A. METHOD
Approach
Our fiducial analysis relies on cluster members selectedas described in Section 2, subject to a magnitude cut of r <
20. We determine the number counts of galaxies inbins of 0 . h − Mpc, from which we construct the pro-jected number density profile of member galaxies, N ( R ),with R indicating the projected radius, using the clustercenters listed in R13. We use Poissonian error bars.We employ two methods to test for evidence of thedensity jump. In the first, we fit the profiles using thefitting formula provided by DK14 to incorporate theirpredicted density steepening: n DK ( r ) = n in ( r ) " (cid:18) rr t (cid:19) β − γβ + n m " b e (cid:18) r R (cid:19) − s e + 1 . (1)As suggested by DK14, we fix β = 6 and γ = 4, a choicethat yields a dependence of r t on the mass accretion rate,Γ: r t = (cid:16) .
62 + 1 . e − / (cid:17) × R (2)We further select the NFW profile with two parameters, n s and r s , as our inner density profile n in : n NFW ( r ) = n s r/r s (1 + r/r s ) . (3) We note that DK14 used the Einasto function (Einasto1965) instead of the NFW for the inner profile, but inthe regime of interest, the distinction between the two isnegligible. We will refer to the model given by Equation 1with n in ( r ) = n NFW ( r ) as the ‘Density Jump’ model, or‘DJ’ model in abbreviation.Since the density jump feature occurs around the virialradius, we use a fitting range of R < R , which isbeyond the range that is typically fitted well by an NFWprofile (we also exclude the inner R < . R vir in the fit,consistent with DK14). Accordingly, the two models thatwe will compare via fitting will be the DJ profile and aprofile given by an NFW profile with an outer term: n ( r ) = n NFW ( r ) + n m " b e (cid:18) r R (cid:19) − s e + 1 . (4)These profiles are projected numerically and fitted toour galaxy density data. The free parameters in ourfit are n s , r s , b e , and s e for both profiles; the full DJformula has one additional free parameter, Γ. We re-strict the fits to the range 0 < Γ < n s and r s , the latter of which is re-stricted by the value of the NFW concentration pa-rameter, c = R c /r s . Observations have suggestedthat this value is lower for galaxy profiles than for darkmatter (e.g., Lin, Mohr, & Stanford 2004; Hansen et al.2005; Budzynski et al. 2012), and we set the range as2 . < c < . s e (0 . − .
0) and b e (0 . − . n m , which will be dis-cussed further in Section 3.2, and we select as our upperlimit of integration R = 10 R vir , same as used by DK14.We then examine the results of the fits using theAkaike Information Criterion (AIC; Akaike 1974) andthe Bayesian Information Criterion (BIC; Schwarz 1978),which provide a means of comparing models fitted todata. As the use of these methods in astrophysics andcosmology has been discussed in a number of papers(e.g., Takeuchi 2000; Liddle 2007; Broderick et al. 2011;Tan & Biswas 2012), we simply mention the most salientqualities here and refer the reader to these works for fur-
14 15 16 17 18 19 20 r − . . . . . . . g − r A1068 ( g − r ) i =1.15All GalaxiesCluster Members, A
14 15 16 17 18 19 20 r − . . . . . . . g − r A1068 ( g − r ) i =1.15All GalaxiesCluster Members, B Fig. 2.—
Top:
The cluster member selection for example clusterA1068 using Selection A, which comprises galaxies selected usingthe red sequence method and galaxies with SDSS photometric red-shifts, which add in some galaxies blueward of the red sequence.
Bottom:
The cluster member selection for this same cluster usingSelection B, which combines the red sequence, photometric red-shifts, and spectroscopic redshifts. ther details. To apply these criteria, we compute thefollowing statistics for each fit:AIC = χ + 2 p + 2 p ( p + 1) N − p − , (5)BIC = χ + p ln( N ) , (6)where p is the number of parameters in the fit, N isthe number of data points being fitted, and χ is thestandard minimized goodness-of-fit parameter. In thecase of the AIC, we have also included a correction termof 2 p ( p + 1) / ( N − p −
1) which is recommended for smallvalues of N (Burnham & Anderson 2002, 2004). The model that is preferred by these criteria is the one withthe lower IC = AIC, BIC value. If we compute ∆IC =IC high − IC low , then, roughly, values of ∆IC = 1 − > y i and errors σ i at a setof points r i , the smoothing spline f ( r i ) is constructed tosatisfy the condition N X i =1 (cid:18) y i − f ( r i ) σ i (cid:19) ≤ S, (7)where S is a constant that interpolates between smooth-ing and fitting: that is, when S = 0, the spline isforced to pass through every data point, so that thereis no smoothing, whereas as S is increased, the curve be-comes smoother at the expense of the fit (de Boor 2001).Reinsch (1967) argues that the smoothing parameter S should be chosen in the range N −√ N ≤ S ≤ N + √ N ,where N is the number of data points over which weconstruct the spline, if the σ i are estimates of the stan-dard deviation in y i . We accordingly choose three val-ues within this range to compare to an NFW fit: S = N − √ N , N , N + √ N . The NFW model is expectedto be a good fit to the inner parts of the profile, so we usethe analytical expression for the projected NFW density(e.g., Wright & Brainerd 2000) to fit the cluster galaxydensity profiles within R , establishing the values of n s and r s . We then calculate the logarithmic derivative d log( N ) /d log( R ) of the splines for each cluster to testfor the presence of the density jump feature, comparingit to the logarithmic derivative of the NFW fit. Fixed Parameters
As noted in Section 1, we define measures of clus-ter size such that the mean mass density inside the ra-dius R ∆ is ¯ ρ = ∆ ρ b ( z ); commonly used values are R and R . The other quantity that needs to be spec-ified is the background density ρ b ( z ); one choice oftenused in observational work is the critical density ρ c ( z ) ≡ H ( z ) / πG , where H ( z ) is the Hubble constant at red-shift z , which is given by H ( z ) = H (cid:2) Ω m (1 + z ) + Ω Λ (cid:3) with H = 100 h km / s / Mpc. Another choice is to usethe mean matter density ρ m ( z ) = ρ c ( z )Ω m ( z ), whereΩ m ( z ) = Ω m (1 + z ) / (cid:2) Ω m (1 + z ) + Ω Λ (cid:3) , which is usedwith ∆ = 200.As noted above, DK14 use the mean matter density ρ m ( z ) to define the radius R = R m used in thefitting formula given by Equation (1). However, R13measures R = R c for their sample of clusters usingthe critical density as reference, so we need to converttheir measure to that of DK14. To do so, we note that agiven mean density ¯ ρ may be written in two ways:¯ ρ = ∆ c ρ c ( z ) = ∆ m ρ m ( z ) . (8)If we specify that ρ = ρ NFW , then in the outskirts of thecluster (i.e., including near R ), we have:¯ ρ ( R ∆ ) ∝ R . (9) θ [arcmin] . . . . . . N [ a r c m i n − ] A1068 All GalaxiesNon-cluster GalaxiesCluster Galaxies θ [arcmin] . . . . . . . N [ a r c m i n − ] A1302 All GalaxiesNon-cluster GalaxiesCluster Galaxies θ [arcmin] . . . . . . N [ a r c m i n − ] A646 All GalaxiesNon-cluster GalaxiesCluster Galaxies θ [arcmin] . . . . . . . N [ a r c m i n − ] A1132 All GalaxiesNon-cluster GalaxiesCluster Galaxies
Fig. 3.—
Plots of the projected densities of galaxies with r <
20 in four cluster fields: A1068, A1302, A646, and A1132. Both thetotal galaxy density (black points) and the cluster density (red points) are expected to rise steeply towards the cluster center. The bluepoints show the density after removing cluster members from the total count, with a line drawn to indicate the average value beyond 20arcminutes for comparison. The non-cluster galaxy distribution should be roughly flat if we have adequately selected cluster members, asfor A1068 and A1302. A646 and A1132, on the other hand, show some residual overdensity.
Combining this relation with Equation (8) yields R m R c = (cid:18) m ( z ) (cid:19) / . (10)For z = 0 . − .
3, this implies that R m ≈ . × R c . (11)Accordingly, for simplicity, we will henceforth refer to R m as R , and the values from R13 will be convertedto this measure using Equation (11).Lastly, we need to establish an appropriate value for n m . We note that the original prescription of DK14 de-fines Equation (1) in terms of mass densities ρ ratherthan number densities n , and their fitting results assumea fixed value of ρ m ( z ) = ρ c ( z )Ω m ( z ). The translationinto a number density needs to take into account the im-pact of the primary selection function by which we obtaincluster members, the red sequence method. In the ab-sence of a cluster, this method would select a population of galaxies that lies in the appropriate region of color-magnitude space; in this case, the projected density ofthese galaxies is expected to be roughly constant acrossthe field of view.Recalling the outer profile term of Equation (1), n out ( r ) = n m " b e (cid:18) r R (cid:19) − s e + 1 , (12)we can analytically determine the contribution to thesurface density of the last, constant term. The surfacedensity is the line-of-sight integral, N ( R ) = 2 Z ∞ R n ( r ) r √ r − R dr ; (13)this integral diverges for a constant n ( r ). However, inpractice we must truncate this integral at some maxi-mum radius R max . As noted above, DK14 use R max =10 R vir ≈ R . In that case, N m ( R ) = 2 n m Z R R r √ r − R dr, (14) N m ( R ) = 2 n m R s − (cid:18) RR (cid:19) . (15)For the scales of interest in our fits, R . R (and evena bit beyond), this value is roughly constant, N m ≈ . n m R . (16)The redshift dependence of N m can be determined byapplying the red sequence method in test fields that arenot centered on low redshift clusters. If we select a pop-ulation of galaxies with this method, then we can con-struct the projected density in a given radial bin i as N m,i = N i π (cid:16) R i, max − R i, min (cid:17) , (17)= 1 D A ( z ) N i π (cid:16) θ i, max − θ i, min (cid:17) , (18)where N i is the number of galaxies in the i th bin and D A ( z ) denotes the angular diameter distance. We expectthat the angular projected density (the term in brackets)over the radial range is a roughly constant value, whichwe denote by η . Then N m ( z ) = ηD A ( z ) = η (1 + z ) D c ( z ) , (19)where D c is the comoving distance, which at the smallredshifts considered here, is given by D c ( z ) ≈ ( c/H ) z .Upon absorbing the factor of c/H into η , we have: N m ( z ) = η (1 + z ) z . (20)To obtain the value of η , we apply our red sequence cutsin random test fields from SDSS (also subject to our ini-tial magnitude cut of r <
20) and fit N with the abovefunction. This yields η ≈ . h Mpc − .Accordingly, combining Equation (16) with (20), wefind that: n m ( z, R ) = 4 . × − R (1 + z ) z . (21)We fix this value individually for each cluster using itsmeasured redshift and R (the latter converted as dis-cussed above) from R13. RESULTS
Fitting
The results of fitting the projected number density pro-files with Equations (1) and (4) are summarized in Ta-ble 1 for both the fiducial method and variations, thelatter of which will be discussed in the next section.However, as the DJ model has one additional parame-ter than the NFW+Outer model, we then compare thetwo fits using the AIC and BIC. We compute, in eachcase, ∆IC = IC
NFW+O − IC DJ . Since lower values of IC are favored, this quantity will be positive if the criteriaindicate evidence in favor of the DJ model. In Table 1, wethus first list in the column, ‘% χ / ndf < χ / ndf,’the number of galaxy clusters for which the reduced χ is lower for the DJ model and second the num-ber of galaxy clusters whose fits pass a general qual-ity cut, including requiring that χ / ndf < χ / ndf < NFW+O − IC DJ > >
5, indicating strong evidence in favor ofthis model.Our fiducial method uses a binning of ∆ R =0 . h − Mpc and employs a magnitude cut of r < > > θ < ′ of thecluster center after the cluster member selection (c.f. Fig-ure 3), which could be indicative either of having missedsome population of galaxies in our selection or that thereare some other agglomerations of galaxies along the lineof sight that contribute to the overdensity. High-quality,dense redshift estimates would help resolve this ambigu-ity.We additionally test the fits by making the radial binstwice as large (∆ R = 0 . h − Mpc). In this case, thereduced χ values are a bit worse for Selection A, asindicated in Table 1. We find that the larger binningyields the greatest drop in the number of clusters withevidence favorable towards the DJ model using the AIC;the BIC results, however, are virtually unchanged fromthe earlier case. The discrepancy is likely caused by thesmaller number of data points over which we fit relativeto the number of parameters, which makes affects thecorrection term for the AIC. . . . . . . l og ( N [ h M p c − ] ) A655
NFWNFW+OuterDJ
A1033 A1246 A1437 . . R/R . . . . . . l og ( N [ h M p c − ] ) A1689 . . R/R
A1914 . . R/R
A2034 . . R/R
A1835
Fig. 4.—
Plots of the projected number density profiles for the 8 clusters with the highest ∆AIC and ∆BIC values in the fiducial analysisusing Selection A. Three fitted functions are shown: a base NFW, fitted interior to R , an NFW+Outer model, given by Equation (4)and fitted interior to 2 R , and the full Density Jump model given by Equation 1 with an inner NFW profile, fitted interior to 2 R . . . . . . . l og ( N [ h M p c − ] ) A655
NFWNFW+OuterDJ
A1033 A1246 A1437 . . R/R . . . . . . l og ( N [ h M p c − ] ) A1689 . . R/R
A1914 . . R/R
A2034 . . R/R
A1835
Fig. 5.—
Plots of the projected number density profiles for the 8 clusters with ∆IC > TABLE 1Information Criteria Results
Method % χ / ndf < χ / ndf ∆AIC > > > > R = 0 . h − Mpc, Sel. A 34/55 12 22 5 7 . . . . . l og ( N [ h M p c − ] ) b e = 0 . , s e = 0 . NFW z = 0 . , Γ = 4 . z = 0 . , Γ = 4 . z = 0 . , Γ = 4 . z = 0 . , Γ = 2 . z = 0 . , Γ = 1 . − . − . − . − . − . . d l og N / d l og R . . . . . l og ( N [ h M p c − ] ) b e = 0 . , s e = 1 . . . R/R − . − . − . − . − . . d l og N / d l og R b e = 1 . , s e = 0 . b e = 1 . , s e = 1 . . . R/R
Fig. 6.—
The predicted signature of the density jump is a steep-ening of slope that can be analyzed using the logarithmic derivativeof the DK14 model for four combinations of s e and b e values. Theredshift dependence enters via n m ( z, R ) as given by Equation(21), and we show the result for three values of z in the range ofthe R13 clusters at fixed Γ = 4. For the z = 0 .
15 case we alsoshow the variation of the profile with Γ. An NFW profile (dashed)is drawn for comparison. All other model parameters are fixed forall curves shown.
Smoothing Splines
In addition to fitting the projected galaxy density pro-files with Equations (1) and (4), we also fit a smooth-ing cubic spline as described in Section 3.1. This latterapproach has the benefit of being model-independent;we use the spline fits to construct a smooth logarithmicderivative of the profile to test for the existence of a den-sity jump.First, to provide context for the spline results, in Fig-ure 6, we plot the DJ model at fixed Γ = 4 for variousredshifts, which determine n m , and for an example fixedredshift z = 0 .
15, we also show the variation with Γ. Thebottom panel of each plot shows the corresponding loga-rithmic derivatives. The effect of the redshift dependenceis to make the density jump steeper at higher redshift dueto the lower background n m . At a fixed n m , a smallervalue of Γ means a shallower density jump, whose maxi-mum amplitude slope is attained at larger radii, althoughthe values of b e and s e , which are free parameters in ourfits, also contribute to the variation in the amplitude.Figure 6 can be compared to Figure 13 of DK14. Whilethe behavior of the density jump is qualitatively thesame, we note that in our analysis, the maximum slopeamplitudes are smaller than those predicted by DK14 due to the elevated background; accordingly, in our case themaximum slope amplitudes are governed not only by themass accretion rate Γ but also the redshift of the cluster.However, the density jump location also depends on Γ.Accordingly, we model our spline fits on Figure 6. Weshow the cubic spline fits for the same subset of clustersselected via the information criteria in the fiducial anal-ysis. Figure 7 shows the spline fits for Selection A, whileFigure 8 is its counterpart for Selection B. The spline fitsdo appear to pick up a modest steepening of slope, con-sistent with the low redshift of these clusters. Since atlow redshift we can expect to almost exclusively detectvery large density jumps (Γ ≈ DISCUSSION
We have tested whether cluster galaxy density pro-files show evidence for a density jump feature near thevirial radius using two methods: profile fitting and splinesmoothing. We have examined the evidence in favor ofthe presence of a steep density jump in the galaxy den-sity profiles of clusters, and also investigated the resultsvia spline smoothing.There does appear to be some dependence of our re-sults on cluster redshift. In our fiducial analysis of Sec-tion 4.1, the seven clusters that showed strong evidence(both ∆IC >
5) for the density jump using SelectionA spanned the redshift range of roughly z = 0 . − . > z = 0 . z = 0 . − .
25, effectively the entire range of R13.More massive clusters tend to show stronger evidence fora density jump: these same 8 clusters have virial massesin the range M vir = (2 − × h − M ⊙ , while a sig-nificant fraction of the clusters in the R13 sample have M vir = (0 . − × h − M ⊙ (and may also be con-sidered galaxy groups). This trend continues even whenconsidering the additional clusters that pass the crite-ria using Selection B. This behavior can be expected, ashigher mass clusters tend to have higher values of Γ andtheir profiles are likely to be better sampled than lowermass systems.We see variations based on the cluster member selec-tion. Our primary selection uses only photometric data,but redoing the analysis with the inclusion of the R13spectroscopy does shift the results; in particular, the re-fined selection yields a larger sample of clusters that showsome evidence for the jump. This suggests that addi-tional data is required to make a firm detection of thedensity jump. This additional data could be in the formof dense spectroscopy out to large radii of cluster mem-ber galaxies, building upon the R13 catalog, which wouldprovide the most secure cluster member determination.Otherwise, high-quality photometric redshifts could alsoimprove upon our estimates.Additionally, it is worth noting that in interpretingthese results it is necessary to keep in mind some ofthe other factors that can impact the jump signature.In particular, cluster asphericity and contamination byother small groups and clusters along the line of sight . . . . . l og ( N [ h M p c − ] ) A655
NFW S = N + √ N S = N S = N − √ N − . − . − . − . . d l og N / d l og R . . . . . l og ( N [ h M p c − ] ) A1689 . . R/R − . − . − . − . . d l og N / d l og R A1033A1914 . . R/R
A1246A2034 . . R/R
A1437A1835 . . R/R
Fig. 7.—
The top panel shows spline fits for various values of the smoothing parameter S for the clusters in Figure 4. The lower paneldisplays the logarithmic derivatives of the splines. An NFW fit and its logarithmic derivative (dashed line) are shown for comparison. can contribute to a diminishing of the jump signal. Thelatter of these can be addressed with redshift estimates.For the first, we note that the signature of the jump be-comes more pronounced and thus easier to detect whenΓ is large; however, clusters with large values of Γ maybe more likely to have disturbed shapes and substruc-tures, which when examined in projection could obscurethe signature of the density jump.More generally, it would be useful to compare the re-sults in Table 1 to simulations. Mock halo catalogs couldgive an indication of the conditions needed for a clusterto show a discernible jump and thus the fraction of clus-ters in which we can expect to find this signature, whichwould provide context for the fractions we find using ob- servational data. While this is a promising avenue forfuture analyses, such a comparison is beyond the scopeof this work. SUMMARY AND CONCLUSIONS
Using the cluster sample of R13 and optical data fromSDSS, we have searched for the signature of a densityjump feature near the virial radius ( ∼ R ) predictedby the simulations of DK14. Our fiducial analysis selectscluster members from SDSS photometric catalogs usingthe red sequence method and photometric redshifts, andcompares this to a selection refined by the inclusion ofthe spectroscopy of R13. After constructing the radialdensity profiles of the clusters, we fit two models, one0 . . . . . l og ( N [ h M p c − ] ) A655
NFW S = N + √ N S = N S = N − √ N − . − . − . − . . d l og N / d l og R . . . . . l og ( N [ h M p c − ] ) A1689 . . R/R − . − . − . − . . d l og N / d l og R A1033A1914 . . R/R
A1246A2034 . . R/R
A1437A1835 . . R/R
Fig. 8.—
Same as Figure 7, but for Selection B profiles. with a density jump — Equation (1) — and one without— Equation (4) — and used the Akaike and BayesianInformation Criteria (AIC and BIC) to examine the evi-dence in favor of the density jump model. These criteriaindicate that, using our fiducial methods, at least 10%showed strong evidence (∆IC >
5) for the model thatincludes a density jump. The clusters with strong evi-dence for the density jump tend to have a higher mass( M vir & × h − M ⊙