Mass and Concentration estimates from Weak and Strong Gravitational Lensing: a Systematic Study
Carlo Giocoli, Massimo Meneghetti, R. Benton Metcalf, Stefano Ettori, Lauro Moscardini
MMon. Not. R. Astron. Soc. , 1–18 (2014) Printed 6 November 2018 (MN L A TEX style file v2.2)
Mass and Concentration estimates from Weak and StrongGravitational Lensing: a Systematic Study
Carlo Giocoli , , (cid:63) , Massimo Meneghetti , , , R. Benton Metcalf , Stefano Ettori , ,Lauro Moscardini , , Dipartimento di Fisica e Astronomia, Alma Mater Studiorum Università di Bologna, viale Berti Pichat, 6/2, 40127 Bologna, Italy INAF - Osservatorio Astronomico di Bologna, via Ranzani 1, 40127, Bologna, Italy INFN - Sezione di Bologna, viale Berti Pichat 6/2, 40127, Bologna, Italy Jet Propulsion Laboratory, 4800 Oak Grove Dr. Pasadena, CA 91109, USA
ABSTRACT
We study how well halo properties of galaxy clusters, like mass and concentration,are recovered using lensing data. In order to generate a large sample of systems atdifferent redshifts we use the code MOKA. We measure halo mass and concentrationusing weak lensing data alone (WL), fitting to an NFW profile the reduced tangentialshear profile, or by combining weak and strong lensing data, by adding informationabout the size of the Einstein radius (WL+SL). For different redshifts, we measure themass and the concentration biases and find that these are mainly caused by the randomorientation of the halo ellipsoid with respect to the line-of-sight. Since our simulationsaccount for the presence of a bright central galaxy, we perform mass and concentrationmeasurements using a generalized NFW profile which allows for a free inner slope. Thisreduces both the mass and the concentration biases. We discuss how the mass functionand the concentration mass relation change when using WL and WL+SL estimates.We investigate how selection effects impact the measured concentration-mass relationshowing that strong lens clusters may have a concentration − higher than theaverage, at fixed mass, considering also the particular case of strong lensing selectedsamples of relaxed clusters. Finally, we notice that selecting a sample of relaxed galaxyclusters, as is done in some cluster surveys, explain the concentration-mass relationbiases. Key words: galaxies: halos - cosmology: theory - dark matter - methods: analytical- gravitational lensing: weak and strong
Galaxy clusters represent a very important cosmological lab-oratory. Their abundance and evolution is related to im-portant cosmological parameters. To first order, the clustercounts as a function of redshift mainly depend on the mat-ter density of the universe Ω m , the dark energy equation ofstate parameter w and the normalization of the initial powerspectrum of density fluctuations, σ . In the era of precisioncosmology, and with the advent of future wide field surveys,new and independent cosmological probes are necessary fordisentangling degeneracies between some cosmological pa-rameters.In particular, in order to use the cluster mass func-tion as a cosmological probe it is necessary to be able to (cid:63) E-mail: [email protected] estimate their mass with very high accuracy. Observationsin different wavelengths provide different, indirect measure-ments of cluster mass that can be compared or combinedwith each other. Many clusters have been observed in bothX-ray (CHANDRA and XMM) (Ettori et al. 2009, 2010)and optical and near-IR (HST, SUBRAU and VLT) (New-man et al. 2009, 2011). Using X-ray data it is possible toextract the cluster mass from the observed brightness andgas temperature profiles by assuming spherical symmetryand hydrostatic equilibrium (Ettori et al. 2013). However,some biases might affect the estimated mass due to unre-solved non-thermal contribution to the total gas pressureRasia et al. (2006, 2012). In a recent paper (Planck Collabo-ration et al. 2013) the Planck collaboration has presented theconstraints on cosmological parameters ( Ω m and σ ) usingnumber counts as a function of redshift for a sample of galaxy clusters selected thanks to their Sunyaev-Zeldovichsignal. The authors point out that the measured cosmologi- c (cid:13) a r X i v : . [ a s t r o - ph . C O ] F e b Giocoli et al. 2013 cal parameters are degenerate with the hydrostatic mass biasand that only a bias of about can reconcile the mea-sured σ from cluster counts with the one measured fromthe primary CMB anisotropies (although see Hajian et al.(2013)).Optical and near-IR data allow one to measure the(weak and strong) lensing signal of background galaxies. Themass and the critical lines can be recovered from elongatedarcs and multiple images. Conversely, estimating the clustermass is also very important for predicting the redshift of thearcs and to identify highly magnified high redshift galaxies(Coe et al. 2013) – recall that the size of the critical curvedepends on the redshift of the source (Zieser & Bartelmann2012; Zitrin et al. 2013b).These estimates can be turned into indirect measure-ments of the projected mass distribution of the cluster alongthe line of sight, and then into an estimate of the virialmass of the system (Hoekstra 2003; Meneghetti et al. 2010b;Hoekstra et al. 2013). Many studies have shown inconsisten-cies between the mass estimated from X -ray and lensing ob-servables (Meneghetti et al. 2010b; Rasia et al. 2012), mainlybecause clusters are not spherical or relaxed system and con-tain substructures – cluster members. It is also reasonableto suspect that clusters with strong lensing features prefer-entially have their major axis oriented along the line of sight(Sereno & Zitrin 2012) or very elongated in the plane of thesky (Zitrin et al. 2013a,b) – also because in merging or postmerging phase, which breaks down the usual assumption ofspherical symmetry when deprojecting the mass model fromtwo to three dimensions (Limousin et al. 2013).By combining the estimated mass and concentration ofobserved clusters we can measure the mass-concentration re-lation and compare it with the predictions extracted fromnumerical simulations. However, it is worth noting thatmany studies are finding a mass-concentration relation thattends to be higher than what is expected from dark mat-ter only N -body simulations (Rasia et al. 2013). However,it should be emphasized that the cooling of baryons is ex-pected to make haloes more concentrated through adiabaticcontraction while at the same time an uncertain amount offeedback from the central AGN can partially counteract thiscontraction (Killedar et al. 2012; Fedeli 2012).Several systematic errors contribute to a scatter andbias in the estimated cluster masses from lensing. The pres-ence of such systematics can be seen directly in the data.For example, the mass obtained for a single cluster usingdifferent source galaxy samples differs by ∼ . The esti-mated masses also depend on the radial range over whichthe fit is performed. Applegate et al. (2012) suggest that arange at least out to × R should be used. What massmodel is used in the fit can also affect the result. Applegateet al. (2012) fix the concentration while Okabe et al. (2013)consider it as a parameter to be estimated from the fit. In ad-dition, there are systematics arising from the assumed massmodel, shear calibration and background galaxy redshift dis-tribution. Mahdavi et al. (2013), estimating the hydrostaticand the weak lensing mass of a sample of clusters, noticedthat hydrostatic masses underestimate weak lensing massesby on average, within R .The mass estimates of strong lensing selected clustersare typically uncertain to within (Bartelmann & Stein-metz 1996b) because of substructures and projection ef- fects. More recent analyses by Kneib & Natarajan (2011);Meneghetti et al. (2010b) show that the estimates of clustercore masses from strong lensing are accurate to within .For comparison, mass estimates from X-ray obsrvations arebiased low with respect to lensing masses by around because they assume hydrostatic equilibrium (Meneghettiet al. 2010b; Rasia et al. 2012). However it is importantto note that X-ray and lensing (weak plus strong) analysescan be combined in such a way as to resolve the degeneracybetween mass and elongation (Morandi et al. 2010).In this work we will address where systematic effectsarise when trying to estimate the mass and the concentra-tion from lensing data, considering the contributions comingfrom the halo triaxiality and orientation, and from the pres-ence of substructures and/or the bright central galaxy.Another complication when estimating the mass andconcentration of halos from lensing comes from the presenceof objects along the line of sight that are unrelated to thehalo being considered. This is a relatively unexplored sourceof systematic error. In this paper we will consider only sub-structures in the main halo. In a subsequent paper we willaddress the effects of line of sight structures.The paper is organized as follows. In Section 2 we de-scribe our lens model and explain how we extract from eachsimulated cluster weak and strong lensing information. Sec-tion 3 is dedicated to present how well mass and concen-tration are recovered using as a reference a Navarro et al.(1996) (NFW) model to fit the lensing data, while in Sec-tion 4 we investigate the case of a generalized NFW model.In Section 5 we present the results for a cosmological sampleof clusters and discuss how mass and concentration uncer-tainties are reflected in the halo mass function and in theconcentration mass-relation. In Section 6 we summarize ourfinding. In order to create a large sample of triaxial and substruc-tured convergence maps of galaxy-cluster size haloes wemake use of the code MOKA (Giocoli et al. 2012a). MOKAbuilds up the convergence map of haloes in an analyticalway, treating them as made up of three components: (i)the main halo – assumed to be smooth, triaxial, with anNFW profile (in the code has been also implemented thepossibility to use a generalized-NFW profile), (ii) the clus-ter members – subhaloes, distributed to follow the main haloand to have a truncated Singular Isothermal Sphere profile(Metcalf & Madau 2001) – and (iii) the brightest clustergalaxy (BCG) modelled with a Hernquist (1990) profile (inMOKA a Jaffe (1983) model for the BCG has also beenimplemented). The axial ratios, a/b and a/c , of the mainhalo ellipsoid are randomly drawn from the Jing & Suto(2002) distributions requiring abc = 1 . The halo ellipsoid israndomly oriented choosing a point on a sphere identifiedby its azimuthal and elevation angles. We assign the sameprojected ellipticity to the smooth component, to the stel-lar density and to the subhalo spatial distribution. This ismotivated by the hierarchical clustering scenario where the http://cgiocoli.wordpress.com/research-interests/mokac (cid:13) , 1–18 alo properties from weak and strong lensing Figure 1.
Convergence map of a cluster with virial mass of M vir = 5 × M (cid:12) /h and concentration c vir = 6 located at redshift z = 0 . and sources placed at redshift z s = 2 . In the first panel on the left we show the spherical halo without substructures, whilethey are included in the second panel. In the third panel we assign a 3D ellipticity to the main halo ellipsoid and randomly rotate itwith respect to the line of sight; in the forth and fifth panel we consider also the presence of a BCG located at the centre of the clusterwith the difference that in the latter we account also for adiabatic contraction of the dark matter. In each panel the green curve in thecentral region defines the location of the critical curves, where the magnification µ is infinite. BCG and the substructures are related to the cluster as awhole and retain memory of the directions of the accretionof repeated merging events (Kazantzidis et al. 2004, 2008,2009; Fasano et al. 2010). In our simulations, we also ac-count for the adiabatic contraction of the dark matter causedby the BCG. We have implemented the adiabatic contrac-tion as described by Keeton (2001) both for the Hernquist(1990) and the Jaffe (1983) profiles. For more details aboutthe MOKA code we refer to Giocoli et al. (2012a,c). Thecode also takes into account the correlation between assem-bly history and different halo properties: (i) less massivehaloes typically tend to be more concentrated than the moremassive ones, and (ii) at fixed mass, earlier forming haloesare more concentrated and less substructured. These recipeshave been implemented considering the recent results fromnumerical simulations. In particular, we assume the Zhaoet al. (2009) relation to link the concentration to mass andthe Giocoli et al. (2010a) relation for the subhalo abundance.When substructures are included we define smooth mass as M smooth = M vir − (cid:80) i m sub ,i and its concentration c s is setin such a way that the total (smooth+clumps) mass densityprofile has a concentration c vir , equal to the original one.Throughout the paper we will denote with M vir (or M D ) the cluster mass and with c vir (or c D ) the halo con-centration. For these definitions we assume the one adoptedfor the spherical collapse model: M vir = 4 π R vir ∆ vir Ω m ( z ) Ω ρ c , (1)where ρ c = 2 . × h M (cid:12) / Mpc represents the criticaldensity of the Universe, Ω = Ω m (0) is the matter densityparameter at the present time, ∆ vir is the virial overdensity(Eke et al. 1996; Bryan & Norman 1998) and R vir symbolizesthe virial radius of the halo, i.e. the distance from the halocentre that encloses the desired density contrast; and: c vir ( M vir , z ) ≡ R vir r s = 4 (cid:40) (cid:20) t ( z )3 . t (cid:21) . (cid:41) / , (2)with r s the radius at which the NFW profile approaches alogarithmic slope of − , t ( z ) is the cosmic time correspond-ing at redshift z and t the one at which the main haloprogenitor assembles of its mass (Zhao et al. 2009; Gio-coli et al. 2012b).The characterization of galaxy cluster properties done in this way is simplified, but on average resembles – in thebest way – the properties measured from numerical simula-tions. Since we are interested in discussing only the averagelensing properties of a large sample of systems, we do notinclude in this work more particular asymmetries present ingalaxy clusters.The whole halo catalogue is made up by galaxy clustersabove M (cid:12) /h at six different redshifts ( z = 0 . , . , . , . , . and . ) with sources located at z = 2 .To make our results easily comparable to recent observa-tional data, these redshifts have been chosen to match thoseof some clusters observed during the CLASH program (Post-man et al. 2012): Abell 383, Abell 611, MACS1115.9+0129,RXJ1347.5-1145, MACS0717.5+3745 and CLJ1226.9+3332.For each redshift, we generate in total cluster mapsfrom to . × M (cid:12) /h , with a constant bin size of d log( M ) = 0 . , creating maps for each consideredbin. Because of their mass density distribution, triaxiality andbaryon content, galaxy clusters represent interesting grav-itational lenses, deflecting the light rays from backgroundgalaxies. Observing the source images, we can typicallydistinguish two different lensing regimes: one in the outerregions, where the images of background galaxies appearslightly distorted and magnified, called weak gravitationallensing (WL) (Bartelmann & Schneider 2001), and one inthe central part, where background sources are highly mag-nified, distorted and multiply imaged, called strong gravi-tational lensing (SL) (Kneib & Natarajan 2011; Meneghettiet al. 2013).In Fig. 1, we show, as an example, the projected den-sity map of one of our clusters with virial mass M vir =5 × M (cid:12) /h and concentration c vir = 6 located at red-shift z = 0 . . In the first panel on the left we present thecase in which the halo is perfectly smooth and spherical. Inthe second panel the halo contain substructures whose massfunction resembles the one obtained by Giocoli et al. (2010a)from a cosmological numerical simulation. In the third panel,we introduce triaxiality to the smooth component and to thesatellite spatial distribution, and randomly orient the major c (cid:13) , 1–18 Giocoli et al. 2013 axis with respect to the line of sight. The forth panel showsthe projected density map of the cluster where the BCG isincluded, finally in the fifth panel the adiabatic contractionof the dark matter is also considered. The median distanceof the tangential critical points θ E – that we will introducelater in the text, increases from left to right and assumesthe following values: θ E = 8 . , . , . , . , . . (3)The small changes of the Einstein radius when substructuresare included is due to the redistribution of the virial massbetween the smooth and the clump components.From the projected density Σ( r ) = (cid:82) ρ ( r, z )d z , we candefine the convergence as: κ ( r ) = Σ( r )Σ crit , with Σ crit = c πG D l D s D ls . where c represents the speed of light and G the universalgravitational constant; D l D s and D ls are the angular diam-eter distances observer-lens, oberver-source and source-lens,respectively. Using the convergence, we can define the ef-fective potential Φ( x, y ) , the scaled deflection angle α ( x, y ) and introduce the pseudo-vector field of the shear using thecomplex notation, γ = γ + iγ by its components: γ ( x, y ) = 12 (Φ − Φ ) (4)and γ ( x, y ) = Φ = Φ ; (5)from which we can define the tangential and the cross com-ponents of the shear γ t = − [ γ cos(2 φ ) + γ sin(2 φ )] (6) γ × = − γ sin(2 φ ) + γ cos(2 φ ) , (7)respectively. These components are respectively perpendic-ular and parallel to the radius vector; φ specifies the angleswith respect to the centre of the coordinate frame; in whatfollows we will denote with g the reduced shear as: g ≡ γ − κ . (8)The reduced shear, and not the actual shear, is the observ-able quantity from image ellipticities (Bartelmann & Schnei-der 2001; Viola et al. 2011). We recall that in the weaklensing regime the tangential shear is related to the massdensity through the relation: γ t ( θ ) = ¯ κ ( < θ ) − κ ( θ ) . Fromthe definitions of the tangential and cross components ofthe shear we can compute the corresponding reduced shearquantities g t and g × that are related to the correspondingcomponents of the image ellipticity (cid:15) t and (cid:15) × of backgroundsources (Bartelmann & Schneider 2001). In the absence ofhigher order effects, the azimuthal average of the cross com-ponent ( γ × ) is expected to vanish. In practice, the presenceof cross modes can be used to check for systematic errors.The differential deflection of light bundles propagatingfrom the source to the observer are given by to the Jacobianmatrix: A = ( δ ij − Φ ij ) = (cid:18) − κ − γ − γ − γ − γ − γ (cid:19) , (9) Figure 2.
Correlation between the median and the equivalentEinstein radius definitions for the whole sample of haloes above M (cid:12) /h at the six considered redshifts. The dashed line showsthe least-squares fit to the data, and is almost equivalent withthe bisector. The magenta points show the relation for clusterswith a convergence ellipticity within R (cid:15) κ, > . , whilethe green points for clusters with (cid:15) κ, < . . with eigenvalues λ t = 1 − κ − γ (10)and λ r = 1 − κ + γ . (11)The cases λ r = 0 and λ t = 0 define the location of radial andtangential critical lines in the lens plane; where the magnifi-cation µ is infinite (green lines in Fig. 1). In order to definethe Einstein radius of the lens, we consider all the points inthe lens plane with λ t = 0 and connect the ones that enclosethe cluster centre. As adopted by Meneghetti et al. (2010a,2011) we will define the Einstein radius θ E,med as the mediandistance from the centre of these points. To be compatiblewith other definitions (Redlich et al. 2012), for each systemwe also compute the effective Einstein radius θ E,eqv , definedas the radius of the circle enclosing the same area as centralcritical points. In Fig. 2 we show the correlation between thetwo different definitions of the Einstein radius for the totalsample of , clusters with mass between M (cid:12) /h and . × M (cid:12) /h at six different redshifts. From the figurewe notice that the median Einstein radius definition bettercaptures the presence asymmetry of the matter distributiontowards the cluster centre since many points lie above theleast-squares fit to the data: θ E , med = θ E , eqv − . . To em-phasize this point we have colored the points referring tomore asymmetric clusters (presenting a convergence ellip-ticity within R – the radius which encloses 2500 timesthe critical density of the universe – (cid:15) κ, > . ) ma-genta while those referring to the more spherical ones (with (cid:15) κ, < . ) are green. We also note that the sample byMeneghetti et al. (2011) possesses a larger scatter in the re-lation mainly because of the large number of asymmetricalobjects presented in their sample of strong lensing selectedclusters.While the halo mass and the concentration are two de-rived quantities from the weak and strong lensing signals, c (cid:13) , 1–18 alo properties from weak and strong lensing the size of the Einstein radius is more directly estimatedfrom the position of multiple images of background sources.In the literature there are many clusters in which tens ofmultiple images have been identified, allowing a very gooddetermination of the size of the strong lensing region (Broad-hurst et al. 2005a,b; Zitrin et al. 2011).In the MOKA code, all γ and α are computed in Fourierspace where derivatives are easily and efficiently calculated.The Fourier transform of the convergence ˆ κ ( l ) , is: ˆ κ ( l ) = (cid:90) R d θκ ( θ ) exp ( i l · θ ) . (12)This is computed on a map of × pixels with a zeropadding region of pixels to avoid artificial boundary ef-fects. For each system, from the potential and the conver-gence maps, we compute the ellipticity (cid:15) Φ , of the potentialand of the convergence (cid:15) κ, at R , at the distance wherethe enclosed density drops below times the critical value.We are interested in the potential ellipticity since (cid:15) Φ , is aquantity that can be directly compared with the X-ray mor-phology of observed galaxy clusters: systems that possess aregular and smooth X-ray map tend to have a small valueof (cid:15) Φ , .Combining the convergence and shear maps, for eachMOKA cluster, we compute the reduced tangential shearprofile and the error associated to each radial bin as fol-lows. We assume a background density of sources of n g = 30 gal/arcmin (which is a reasonable number for current andfuture space-based observations; see also the ESA missionEUCLID (Laureijs et al. 2011)), and locate on the map N g = n g × A map random points, where A map represents thearea of the map. We measure the reduced tangential shearat each of these N g points and build a logarithmically az-imuthally averaged profile from .
01 Mpc /h up to the virialradius with bins equispaced by d log( r ) = 0 . . This ensuresa good S/N in each bin both for low and high-redshift clus-ters. To each radial bin we assign an error given by the sumof two components: σ g + = σ + σ g,(cid:15) , (13)where σ int represents the rms of the measured reduced tan-gential shear from the map, and σ g,(cid:15) is the intrinsic shapeof the background galaxies, that depends on the considerednumber density of the sources and on the intrinsic scatterin the ellipticity σ (cid:15) = 0 . (Hoekstra 2003; Hoekstra et al.2013): σ g,(cid:15) = σ (cid:15) π ( θ − θ ) n g ; (14)here θ and θ the two extrema of the bin. The intrinsicscatter of the ellipticity of the background galaxies providesa noise that limits the accuracy of the shear measurements(Hoekstra 2003; Hirata & Seljak 2004). We recall that in ouranalysis we did not consider any uncertainty on the photo-metric redshift which would dilute the weak lensing signal.An example is shown in Fig. 3 where the red circle withthe error bars shows the average reduced tangential shearprofile measured for a cluster at redshift z = 0 . withsources located at redshift z s = 2 ; the vertical line representsthe median distance of the critical points from the clustercentre. The average of the reduced tangential shear in eachannulus is calculated by averaging the corresponding values. Figure 3.
The reduced tangential shear profile of a galaxy clusterat redshift z = 0 . with sources at redshift z s = 2 . The errorbars show the error associated to g t , as in equation (13). Thevertical line defines the size of the Einstein radius of the cluster.The solid curve represents the best fit NFW profile obtained bytaking into account both weak and strong lensing data, while thedashed one is the best fit when only the reduced tangential shearprofile is considered. Here we prefer not to use the weighted average, as done byUmetsu et al. (2011), because we assume the variance for theshear estimate to be zero since the value of g + is directlycomputed using the corresponding map. To compute thespherical averaged profile we take the centre of the clusterto be the position of the BCG. In fact the location of theBCG can sometime be offset from the mass centroid of thecorresponding matter density distribution (Oguri et al. 2010;Oguri & Takada 2011). However, in analyzing the stronglensing mass model of five clusters, Umetsu et al. (2011)find only a small offset (of the order of
20 kpc /h ) betweenthe BCG and the centre of mass.For each cluster, we estimate the weak-lensing (WL)mass M est and concentration c est using an NFW profile fitto the reduced tangential shear profile by minimizing thequantity: χ WL ( M, c ) = N (cid:88) i =1 [ g NFW ( θ i | M, c ) − g t ( θ i )] σ g + ,i , (15)where the index i runs on the number of bins and g NFW ( θ i ) is computed using the relations for the convergence and theshear valid for a NFW halo (Bartelmann 1996). In Fig. 3,the dashed blue curve represents the best fit to the reducedtangential shear profile.The information about the Einstein radius of the clus-ter allows also to define a strong lensing constraint (Sereno& Zitrin 2012; Zitrin et al. 2011). For a spherical NFW halowe use analytic formula for the convergence and the shear(Bartelmann 1996) and compute from equation (10) the cor-responding Einstein radius given a certain mass and concen-tration. For the strong lensing constraint on the mass andconcentration estimates we minimize the following quantity: χ SL ( M, c ) = [ θ E − θ E,NFW ( M, c )] σ E , ; (16)we assume σ E = 1 arcsec (Jullo et al. 2010; Zitrin et al. c (cid:13) , 1–18 Giocoli et al. 2013
Figure 4.
Weak lensing (left), strong lensing (centre) and weak+strong lensing (right) constraints estimating mass and concentrationfor a smooth, spherical NFW halo. The black dot represents the input halo mass and concentration. In all three cases the minimum ofthe χ corresponds with the black dot. χ = χ WL + ηχ SL . The parameter η is an integer (1 (cid:54) η < N ) equal to the bins of g + containing the valueof the Einstein radius θ E . This ensures that when combin-ing weak and strong lensing χ we are putting together thesame information, taking into account that the shear pro-file represents a differential quantity of the cluster matterdistribution, while the size of the Einstein radius is a cu-mulative quantity since it is related to the total enclosedmass (Narayan & Bartelmann 1996; Bartelmann 2010) (forthe case presented in Fig. 3 we have η = 1 ). We recall thereader that even if we are using only the Einstein radius asstrong lensing constraint we will label it as SL in all thefigures.In Fig. 4, we show the constraints obtained when re-covering mass and concentration for the reference case of asmooth, spherical NFW halo. The left, center and right pan-els show weak, strong and weak+strong lensing constraints,respectively. In all the panels the black dot represents themass and the concentration of the input halo which corre-sponds in all cases to the location of the minimum of thecorresponding χ . In this section we will present the results on the mass andthe concentration estimates obtained from the whole con- structed MOKA cluster sample. Since average halo struc-tural properties depends on mass and redshift we will studyhow the mass and concentration bias depends on these quan-tities.In Fig. 5 we show the average rescaled estimated mass asa function of the cluster mass for three of the six consideredredshifts. The results of the three redshifts not displayed areconsistent with those here presented. The rescaled mass isthe ratio between the estimated and the true (3D) clustermass (see Appendix A for the comparison between the trueand 2D cluster masses). We recall that the simulations ofthe MOKA clusters include the presence of a BCG, the adi-abatic contraction (ADC) of the dark matter component,triaxiality and subhaloes. Filled circles and triangles showthe masses estimated considering SL+WL and WL alone,respectively; the shaded region encloses the σ scatter ofthe distribution. From the figure we notice that for groupsand small clusters the mass is typically underestimated byabout , for massive clusters the mass has a bias rang-ing from down to a few precent, consistently with whathas been found by Becker & Kravtsov (2011); Meneghettiet al. (2010b); Rasia et al. (2012). The higher bias in themass estimate for the smallest systems is due to the triaxi-ality model by Jing & Suto (2002) implemented in MOKAand extended down to these masses. In this model, typi-cally smaller systems tend to be more prolate than the moremassive ones in agreement with the fact that they are morestretched by the gravitational field of the surrounding mat-ter density distribution during their collapse (Sheth et al.2001). The small trend of the normalization of the relation M est /M vir − M vir as a function of redshift reflects the factthat MOKA clusters at higher redshifts tend on average topossess more substructures (van den Bosch et al. 2005; Gio-coli et al. 2010a) and to have a larger 3D ellipticity (cid:15) (Shawet al. 2006, 2007; Giocoli et al. 2010a; Despali et al. 2013;Limousin et al. 2013). From the figure we notice that whenthe constraint on the size of the Einstein radius is includedin the mass estimate the corresponding mass bias tends to bereduced: the modeling of the Einstein radius size is done us-ing a spherical model and so on average we tend to measure ahigher mass with the SL constrain (Bartelmann & Steinmetz1996b). The absence of difference in the rescaled estimated c (cid:13) , 1–18 alo properties from weak and strong lensing Figure 5.
Average of the rescaled estimated mass as a function of the true cluster mass. We show the case in which the mass is estimatedusing WL+SL (circles) and only WL (triangles) information for systems at three different redshifts. The upper and lower curves show the σ scatter of the corresponding distributions, red for WL+SL and blue for WL only. Each mass bin contains cluster realizations. mass between WL and WL+SL for the smallest mass binsis due to the fact that most of those clusters are not stronglenses or have small Einstein radii: in those cases χ SL hasa negligible contribution on the total χ . Meneghetti et al.(2010b) perform a similar analysis. They study the lensingsignals of three projections of three clusters extracted from anumerical simulation. In particular, in the left panel of theirFig. 16 they show the ratio between the estimated and true3D mass obtained best fitting the reduced tangential shearprofile of each cluster using, as we have done, an NFW func-tional. For the case M , the authors find a negative biasof about − , consistent with what we have found inthis work. However, the flexibility and the speed of our algo-rithm MOKA allow us to generate and analyze a sample ofclusters which is more than five orders of magnitude largerthan the one studied by Meneghetti et al. (2010b).We have also investigated what effect the cluster ellip-ticity on the plane of the sky have on the estimated mass.For each mass bin we have computed the median potentialellipticity (cid:15) Φ , – measured within R – and divided thehalo sample into haloes with smaller or higher ellipticity.In Fig. 6 we present the average estimated mass as a func-tion of the cluster mass for these two samples, consideringonly the clusters at redshift z = 0 . . The results for theother redshifts are quantitatively consistent. From the fig-ure we see that the orientation of the main halo ellipsoid isan important source of bias in the measured halo mass, asalready discussed by Meneghetti et al. (2010b). We considerin this case the situation in which mass and concentrationhave been estimated from the WL+SL constraints. Typi-cally we see that for clusters whose major axis is orientedalong the line of sight the mass tends to be overestimated– that are more spherical in the plane of the sky (Morandiet al. 2010), while the opposite occurs for clusters elongatedin perpendicular direction: most spherical clusters are lessbiased than the most elliptical ones. An analogous result ispresented in Meneghetti et al. (2010b) (see their fig. 17),where the cluster masses tend to be under (over)-estimatedwhen large (small) angles between the line-of-sight and themajor axis of the halo ellipsoid are present.In Fig. 7, we show the average ratio between the esti- Figure 6.
Average of the rescaled estimated mass as a function ofthe true cluster mass for haloes with higher or smaller ellipticitythe plane of the sky at redshift z = 0 . . Masses and concen-trations have been evaluated using both weak and strong lensingconstraints. mated concentration and the true one as a function of thecluster mass. The different panels refer to different redshiftsand the data points have the same meaning as in Fig. 5. Theresults show that adding to the fit the constraint on the sizeof the Einstein radius does not change significantly the av-erage value of the distribution, while it reduces the scatterfor the most massive systems. The different panels show thepresence of a positive bias for the smallest systems by lessthan ∼ , which decreases with the lens redshift.Our cluster lensing simulations include the presence ofa BCG and a recipe for the adiabatic contraction. To under-stand how much these ingredients affect our mass and con-centration measurements, we present in the left (right) panel c (cid:13) , 1–18 Giocoli et al. 2013
Figure 7.
Average of the rescaled estimated concentration as a function of the true cluster mass. We show the case in which theconcentration is estimated using WL+SL (red circles) and only WL blue (triangles) information for systems at three different redshifts.For each considered case, the corresponding upper and lower curves enclose the σ scatter of the distribution at fixed mass. of Fig. 8 the ratio between the mass (concentration) derivedin simulations including BCG and ADC and that estimatedin simulations without the central galaxy. Blue triangles andred circles refer to the case where only WL or WL+SL con-strains are considered, respectively. We recall that in thefirst case holds M vir = M smooth + M in subs + M BCG while forthe second M vir = M (cid:48) smooth + M in subs . We notice that fittingthe whole profile with an NFW function we have an under-estimate of the mass by only few percent with respect tothe case in which the BCG is not present. The trend is dif-ferent for the concentration which for small systems, wherethe cold baryon contribution is more important since starformation is fractionally more efficient in low-mass objectsit is overestimated by − , while for the more massiveclusters it is only few percent. The two data points referagain to the cases in which the fit is performed consideringSL+WL (circles) and WL only (triangles). The shaded re-gion encloses the σ scatter of the distribution for a fixedvalue of the true cluster mass. Many studies analyzing haloes from DM-only numerical sim-ulations have shown that the halo density profile (Neto et al.2007; Gao et al. 2012) is well described by the NFW rela-tion. However, in our case, we need to take into account thefact that real galaxy clusters are not only made up by darkmatter – that accounts for more than of the total mass– but also by baryons, divided into cold and hot compo-nents. While the hot component is more evenly spread inthe potential well of the cluster – possessing a scale radiusof the density profile r s,h of the order of hundreds kpc/ h –the cold component, that turns into the presence of a brightcentral galaxy, is more concentrated toward the center witha scale radius r s,c much smaller than r s,h . This translatesinto a total density profile which is different from an NFWrelation and typically has an inner slope larger than unity.In order to better model the increase of the density distri-bution towards the cluster centre we can introduce a freeparameter β in the NFW equation which allows the central slope to freely vary (Zhao 1996; Jing 2000): ρ gNFW ( r, β | M vir , c vir ) = ρ s ( r/r s ) β (1 + r/r s ) − β , (17)where ρ s represents the density within the scale radius r s .In order to define the concentration it is useful to introducethe quantity r − as the radius at which the logarithmic den-sity profile is − . This allows us to write r − = (2 − β ) r s and the concentration c − ≡ R vir /r − = c vir / (2 − β ) . Theprofile and the corresponding definitions match the NFWfunction when β = 1 . The generalized NFW convergencecan be obtained by integrating the profile in equation (17)along the line of sight: κ gNFW ( r (cid:48) ) = (cid:90) ∞−∞ ρ gNFW ( r, ζ )d ζ , (18)with r → r (cid:48) + ζ and r (cid:48) the radius vector on the plane ofthe sky. We can now define the dimensionless mass m ( x ) = (cid:82) x xκ ( x )d x and the shear: γ gNFW ( x ) = m gNFW ( x ) x − κ gNFW ( x ) , (19)with x = r/r s . By setting λ t = 0 in equation (10), we canfind the Einstein radius for the generalized NFW profile,that we use as reference model in constructing χ SL . The χ is in this case minimized in order to obtain three parameters:the virial mass, the concentration and the inner slope β ofthe total density profile.In Fig. 9, we show as a function of the halo mass the av-erage rescaled mass and concentration, and the inner slopeestimated by fitting the tangential shear profile and the sizeof the Einstein radius using a generalized NFW profile (therelations at the other redshifts are consistent with what pre-sented here at z = 0 . ). The shaded region encloses the σ scatter of the distributions at fixed halo mass, while thepoints represent the average value. From the figure we noticethat using the gNFW profile the bias in the mass is reducedby about for the smallest systems, but the scatter re-mains as large as in the NFW case. The average measureof the concentration is almost unbiased – of the order offew percent. From the right panel we notice that on averagewe tend to measure an inner slope larger than unity for the c (cid:13) , 1–18 alo properties from weak and strong lensing Figure 8.
Ratio between different quantities estimated in simulations with BCG+ADC and without BCG as a function of the clustermass. Left and right panels refer to mass and concentration, respectively. Blue triangles and red circles refer to the case where mass andconcentration are estimated using only WL or WL+SL constrains, respectively.
Figure 9.
Average mass (left panel), concentration (center panel) and inner slope (right panel) estimated best fitting with the generalizedNFW profile the reduced shear profile and the size of the Einstein radius, as a function of the cluster mass. The shaded region enclosesthe σ scatter of the distribution. smallest systems. This behavior is related to the fact that weare fitting the total 2D matter density distribution includ-ing the contribution both from DM and the bright centralgalaxy. Since the BCG steepens the profile and affects moresignificantly the core of the smallest systems, we tend tomeasure on average an inner slope that is even largerthan one. Even when the BCG is removed, a bias of − still remains for the smallest clusters due to the prolatenessof their ellipsoids. Until now many numerical simulations and analytical pre-dictions have been developed to interpret the number of collapsed objects and their concentration-mass relation ata given redshift. With the advent of the era of precision cos-mology it is possible to use the halo mass function and theconcentration-mass relation as an additional cosmologicalprobe. Currently the relatively small number of availabledata combined with the bias and the scatter in estimatedcluster properties limits the constraints on cosmological pa-rameters. However, recently the Planck team (Planck Col-laboration et al. 2013), using a sample of 189 galaxy clustersfrom the Planck SZ catalogue, was able to put good con-straints on Ω m and σ the also emphasize the fact that thevalues of the cosmological parameters are degenerate withthe hydrostatic mass bias and that the agreement betweenthe cluster counts and the primordial CMB anisotropies can c (cid:13) , 1–18 Giocoli et al. 2013 be reached only assuming a mass bias of about . Howeverit is worth mentioning that the tension between the cosmo-logical parameters ( Ω m and σ ) derived from cluster countsand the ones derived from the Planck CMB temperaturemaps is alleviated when the SZ clusters are cross-correlatedwith the X-ray cluster maps from ROSAT (Hajian et al.2013). At the same time the CLASH collaboration (Postmanet al. 2012), reconstructing the mass distribution of a sampleof galaxy clusters using weak and strong lensing measure-ments, is exploring the possibility of eventually measuringdeviations of the concentration-mass relation from the onemeasured in Λ CDM numerical simulations. However theseworks require precise knowledge of bias and scatter whencomparing estimated and true cluster properties, like massand concentration.In this section, we will discuss how the mass and theconcentration measured by fitting the tangential shear pro-file and constraining the size Einstein radius tend to modifythe intrinsic concentration-mass relation for galaxy clusters.We will also discuss how selection effects on the cluster sam-ple could modify the slope and the zero point of the relation.As done previously, we consider the case of six differentredshifts but with the halo sample extracted from the ana-lytical Sheth & Tormen (1999) mass function. We considerhaloes with a mass larger than M (cid:12) /h . At each redshift,the number of haloes is created to match the number of col-lapsed objects present on the whole sky between z − ∆ z/ and z + ∆ z/ (with ∆ z = 0 . ). To increase the statisticalsample, for each redshift we perform different realizations.We investigate the ellipticity distribution of these cos-mological samples. In the left panel of Fig 10, we show thedistribution of the 2D ellipticities, for each of the six red-shifts considered. The parameters a , b and c are the small-est, the intermediate and largest axes of the halo ellipsoiddescribing the dark matter halo, obtained from the Jing &Suto (2002) model. The corresponding 2D ellipticity distri-butions (on the plane of the sky with a (cid:48) and b (cid:48) representingthe smallest and the longest axis of the ellipse) measuredfrom the cluster convergence maps within R are shownin the right panel. It is interesting to notice the agreementwith the results obtained by Meneghetti et al. (2010a) an-alyzing the clusters extracted from the MARENOSTRUMUNIVERSE simulation (Gottloeber et al. 2006; Gottlöber &Yepes 2007): our distributions lie in between with respect tothe their measurements of the convergence ellipticity mea-sured at R vir and at . × R vir . As expected from numeri-cal simulations (Jing & Suto 2002; Despali et al. 2013) andalso from analytic predictions of collapsing ellipsoids (Rossiet al. 2011), high-redshift clusters tend to be more ellipti-cal, since more ongoing merging events make them typicallyunrelaxed. Notice that the higher ellipticities of high red-shift clusters tend to enhance their strong lensing efficiencyby stretching and increasing the critical area Zitrin et al.(2013a,b). In Appendix A we discuss the correlation betweenconvergence and potential ellipticities comparing our findingwith the model proposed by Golse & Kneib (2002). As described previously in the paper, for each cluster weestimate the mass and the concentration in two ways. In thefirst case we use only the tangential shear profile (WL), while
Table 1.
Least-squares fit to the estimated mass as a function ofthe true mass: log M set / M = a log M + b ( M D is in unit of M (cid:12) /h ). On the left for the case WL+SL, while on the right forthe WL alone.redshift a b | a b0.187 0.062 -0.970 | 0.045 -0.7310.288 0.077 -1.179 | 0.051 -0.8110.352 0.073 -1.121 | 0.046 -0.7270.450 0.059 -0.905 | 0.037 -0.5970.548 0.059 -0.900 | 0.040 -0.6270.890 0.055 -0.813 | 0.040 -0.608 in the second one we combine this with the measurementof the size of the Einstein radius adopting as a referencea NFW model (WL+SL); for the case WL+SL we performthe measurement also considering a generalized NFW model(gNFW WL+SL).In Fig. 11, we show the cumulative all sky cluster massfunction at three different redshifts. In each panel, the solidcurve represents the analytical mass function of Sheth &Tormen (1999), while the black filled diamonds representa catalogue extracted sampling this relation. Red filled cir-cles and blue filled triangles show the mass function fromweak and weak+strong lensing, respectively. In all panelsthe vertical line marks M (cid:12) /h : the minimum mass ofthe extracted cluster sample. We notice that while for smallmasses the behavior is dominated by the difference betweenthe true and the estimated mass, for large masses – sincethere are less haloes – it is dominated mainly by the scat-ter. We have tested this statement modifying the mass ofthe cosmological sample using the least-squares fit to theWL+SL masses performed to the data in Fig. 5, as expressedin Table 1, with a scatter σ log M = 0 . (open green squarespoints in the figure). From the bottom frame presented ineach panel, where we show the difference of the measure-ments with respect to the analytical prediction, we noticethe biased Gaussian sample matches quite well the WL+SLdistribution up to the mass bin where the S/N = 5 – wedefine the signal to noise for each bin as the ratio betweenthe number of clusters and the corresponding Poisson er-ror.u The mass function obtained using a gNFW model asreference is similar to the others even if the mass bias is re-duced in this case. This is due to the scatter still present inthe rescaled estimated mass: the discrepancy with respect tothe theoretical prediction is further reduced if σ log M = 0 . .If these deviations with respect to the theoretical mass func-tion are not reduced, they will bias the estimated cosmolog-ical parameters in favor of models with higher σ and lower Ω m . Biases and scatter of mass and concentration impact onthe concentration-mass relation. In Fig. 12, we show theconcentration-mass relation at three considered redshifts.The solid curve in each panel represents the c − M model byZhao et al. (2009) at the corresponding redshift, while theblack filled diamonds is the median intrinsic relation for theclusters obtained from this model assuming a log-normalscatter of . . The blue filled triangles and the red filledcircles show the median c − M relations obtained using WL c (cid:13) , 1–18 alo properties from weak and strong lensing Figure 10.
Left panel: intrinsic 3D ellipticity distribution of the main halo component for the samples of galaxy cluster at six differentredshifts. Right panel: measured 2D ellipticity of the convergence at the radius at which the enclosed density reaches times thecritical value. Different line styles refer to the various considered redshifts, as labelled.
Figure 11.
Cumulative all sky cluster mass functions at redshift z=0.288 (left panel), z=0.450 (central panel) and z=0.890 (right panel).In each panel panel, the solid line in the top frame represents the analytical halo mass function (Sheth & Tormen 1999), while theblack filled diamonds show the mass function of the cluster above M (cid:12) /h obtained sampling this function. The filled blue trianglesand the red filled circles represent the cluster mass function obtained using WL and WL+SL masses, respectively. For both cases, wehave considered as a reference model a NFW halo. The orange cross show the WL+SL mass function, where a generalized NFW modelhas been used as a reference. The open green squares represent the mass function obtained biasing the halo mass of each cluster usingthe best fit relations reported in Table 1, and assuming a gaussian scatter of σ log M = 0 . . For all data the error bars represent theircorresponding Poisson uncertainties. In each panel, the bottom frame represents the residuals as a function of the cluster mass of theWL, WL+SL and WL+SL using gNFW model with respect to the analytical Sheth & Tormen (1999) mass function. and WL+SL estimates. In both cases, we notice that theconcentration-mass relation for clusters tends to be overes-timated by . On average the difference with respect tothe intrinsic c − M relation is reduced using gNFW massesand concentrations (orange crosses) and the median pointsare on average well inside the two central quartiles of the in-trinsic distribution. For each data, the curves with the corre-sponding color enclose the first and the third quartiles of thedistribution at fixed halo mass. From the figure we noticethat describing the mass distribution as a triaxial ellipsoidtends to increase the normalization of the concentration-mass relation (Comerford & Natarajan 2007) . As discussed by Oguri et al. (2005) in the analysis of A1689, this mayeventually reduce the apparent discrepancy between theoryand observations.Different observational campaigns (Okabe et al. 2010b;Postman et al. 2012) are trying to use the concentration-mass relation to test the agreements of observations withthe predictions of structure formation in a Λ CDM cosmol-ogy. However it is reasonable to ask if the selection functionof the observed clusters is important in reconstructing the c − M relation and, if so, how does it reflects, in both thetrue and reconstructed samples. For example it is interest-ing to point out the work by Sereno & Zitrin (2012), where c (cid:13) , 1–18 Giocoli et al. 2013 the c − M relation of the MACS cluster sample has beenstudied using strong lensing mass model reconstructions.This work first underlines the importance of triaxiality ofthe clusters when reconstructing their properties and secondthe selection function since high-redshift, unrelaxed clustersmay form a different class of prominent strong gravitationallenses.In order to understand how different selection functionschange the c − M relation, in Fig. 13 we show the medianrelation for the cluster sample at redshift z = 0 . (theresults at other redshifts are consistent with these) whenthe objects are selected to have different projected poten-tial ellipticities, (cid:15) Φ , . Selecting clusters in ellipticity im-plies also a selection in the shape of the X-ray emission: theones with smaller ellipticity will also present a more spher-ical and relaxed X-ray morphology. Going from the left tothe right panel, we show the c − M relations for clusters with (cid:15) φ, < . , (cid:15) φ, < . and (cid:15) φ, < . , respectively;the adopted symbols are are the same as in Fig. 12. Look-ing at the filled black diamonds we notice that no particularbias appears in the intrinsic c − M relation, while the onesreconstructed using the concentrations and the masses de-rived from WL and WL+SL tends to move up for the morespherical systems. This behavior reflects the fact that, sincehaloes are in general prolate ellipsoids, the more sphericalthey are in the plane of the sky, the more elongated theymust be along the line-of-sight – or they could be intrinsi-cally spherical. So, even if the intrinsic c − M relations lieon the theoretical ones, the selection in ellipticity could in-troduce a bias in the estimated masses and concentrations.This effect is also seen in the c − M relation of clusters whentheir properties are estimated using a gNFW model as areference: going from left to right, this is shown by the or-ange crosses, which tend to move toward higher values ofconcentration.The effect is more drastic when we select clusters bytheir strong lensing features. In Fig. 14 we show the c − M relation selecting clusters to have an Einstein radius largerthan , . and arcsec, respectively. In this case, wenotice again that the intrinsic c − M relation – showed by theblack filled diamonds – tends to move up with respect to thesolid curve, representing the reference model: intrinsicallymore concentrated haloes tend to be selected (Meneghettiet al. 2010a) . Lensing reconstructed c − M relations areusually above the solid curve even by − . Selectingclusters by their strong lensing features not only picks upthe ones that are more elongated along the line of sight butalso the ones that are more elliptical in the plane of the sky(Zitrin et al. 2013b). As a first case, we analyze the situation of building up theconcentration-mass relation using a sample of relaxed clus-ters selected in a similar way to the CLASH sample. Sinceour cosmological cluster catalogues have been created forsix discrete redshifts, we group the CLASH clusters in thesebins. This corresponds to create six sub-samples, for the clusters, with a temporal bin size of approximately Gyrwhich is of the order of the cluster relaxing time. Each re-alization is constructed looping through the 25 clusters andthen randomly selecting, from the corresponding catalogue,
Table 2.
Least-squares fit to the samples WL+SL selected inpotential ellipticity: log c = a log M + b ( M is in unit of M (cid:12) /h )case a σ a | b σ b all clusters -0.06 0.03 | 0.69 0.03 θ E > -0.21 0.03 | 0.92 0.03 θ E > . -0.25 0.04 | 1.00 0.05 θ E > -0.34 0.06 | 1.16 0.09 a cluster with a true mass of at least × M (cid:12) /h and witha 2D ellipticity of the potential as in the Table 4 of Postmanet al. (2012), when available. We repeat this procedure cre-ating 10,000 different realizations of the CLASH-like sample.The ellipticity distribution of the sample nicely shows thatthe stronger the lens the larger is the elongation, and thatthe strongest lenses are very elongated either along the lineof sight or in the lens plane.In Fig. 15 we show the concentration-mass relation ofthese 10,000 realizations. For each cluster, we use the massand the concentration estimated using a NFW model asreference combining weak and strong lensing constraints.In each panel the solid black curve shows the median ofthe sample and the gray region encloses the first and thethird quartiles. The two external gray curves enclose of the data. The different dashed curves show the theoreti-cal concentration-mass relations at three redshifts. The fourpanels show: on the top left the case in which each sampledoes not contain any constrain on the minimum size of theEinstein radius, while in the other threes the clusters are ran-domly selected with the condition of possessing an Einsteinradius of at least 5, 12.5 and 20 arcsec, respectively. The c-Mrelation built from the relaxed samples when no strong lens-ing selection is present is in very good agreement with thetheoretical input models, while it increasingly steepens de-viating from the theoretical expectations when a lower limitto the Einstein radius size is imposed. In Table 2 we presentthe slope and the zero point of the least-squares fit relationto the data, for each panel of the figure.As a second case, in Fig. 16 we show the concentration-mass relation for 10,000 realizations of MOKA clustersselected to match the sample by Oguri et al. (2012) as partof the Sloan Giant Arcs Survey (SGAS). In this case wehave selected the clusters from our database to match theredshift, the ellipticity and the size of Einstein radius aslisted in Tables 2 and 3 in Oguri et al. (2012). The redcircles represent the median of the c − M relation for thoseclusters while the orange and pink lines enclose the quartilesand of the data, respectively. The black line shows theleast squares fit, that can be read as: c vir = 11 . ± . (cid:18) M vir (cid:19) − . ± . . (20)The green triangles show the location of the SGAS-SDSSclusters in the figure when mass and concentration are es-timated using both weak and strong lensing analyses, whilethe dotted line is the best fit relation (equation 26 fromOguri et al. (2012)). From the figure we notice that apply-ing a realistic selection function to the simulated sample – inwhich multiple mass components, presence of the BCG andadiabatic contraction are considered – clarifies the tension c (cid:13) , 1–18 alo properties from weak and strong lensing Figure 12.
Concentration-mass relation at three different redshifts. In each panel the solid curve represents the Zhao et al. (2009) model,which we use as a reference when we assign the concentration to each halo. The black filled diamonds represent the intrinsic measured c − M relation, while the blue filled triangles and the red filled circles correspond to the c − M relations obtained for WL and WL+SL,respectively. The orange crosses show the relation for WL+SL when the mass and the concentration of the halo are obtained using thegNFW model as a reference. For each data, the curves with the same color enclose the first and the third quartiles of the distribution atfixed halo mass. Figure 13.
Same as Fig. 12 but for clusters selected with different limits in the potential ellipticity: (cid:15) φ, < . (left panel), (cid:15) φ, < . (central panel) and (cid:15) φ, < . (right panel). The results refer to z = 0 . only. Figure 14.
Same as Fig. 12 but for clusters selected with different limits in the size of the Einstein radius: θ E > (left panel), θ E > . (central panel) and θ E > arcsec (right panel). Also in this case the results refer to z = 0 . only.c (cid:13) , 1–18 Giocoli et al. 2013
Figure 15.
Median concentration-mass relation for 10,000 realizations of clusters selected in potential ellipticity, where mass andconcentration are estimated using WL+SL constraints considering an NFW profile as reference model. The shaded dark gray regionencloses the first and the third quartiles of the distribution, while the gray curves are of the data. The solid (magenta, for thecolored version of the figure) represent the least-squares fit to the data. Figure 17.
Median concentration-mass relation for 10,000 realizations of clusters selected in redshift as the LoCuSS sample by Okabeet al. (2010b), where mass and concentration are measured using only WL data through the reduced tangential shear profile. In the left,central and right panel we show the relation obtained from our MOKA sample, considering all clusters, those with an Einstein radiuslarger than arcsec and arcsec, respectively. In each panel, the blue circles represent the median of the distribution, while the blueand the light-blue lines enclose the quartiles and of the data, respectively. between the c − M relation predicted in numerical simula-tions and the best-fit on the observed dataset.As a last case, we compare our predictions to those ob-tained by analyzing the weak lensing data of clustersfrom the LoCuSS survey, listed in Table 6 of Okabe et al.(2010b). In this case, to be consistent with their analysis, wehave considered only the situation in which the mass and theconcentration are estimated by fitting the reduced tangen-tial shear profile. The MOKA sample has been selected from our database to match the redshift of the LoCuSS clusters(see Table 1 by Okabe et al. (2010b). In the left panel ofFig. 17, we show the median estimated concentration-massrelation obtained for 10,000 realizations of the sample. Thesolid black line represents the least-squares fit to our data.In the central panel we have considered only clusters withan Einstein radius larger than arcsec, while in the rightthose with θ E > arcsec. In all panels the magenta crossesrepresent the location in the mass-concentration diagram c (cid:13) , 1–18 alo properties from weak and strong lensing Figure 16.
Median concentration-mass relation for 10,000 re-alizations of clusters selected using the convergence ellipticityand the Einstein radius to match the sample of Oguri et al. (2012).The red circles show the median of the distribution while the or-ange and pink lines enclose the quartiles and of the datapoints, respectively. The black solid line is the least-squares fit tothe selected MOKA clusters. Dashed curves, as in Fig. 15, showthe theoretical c − M predictions at the corresponding redshifts. Table 3.
Best fit parameters of the c − M for 10,000 realizationsof the LoCuSS galaxy cluster sample: c vir = c N (cid:0) M vir / (cid:1) − α c N α all clusters . ± .
08 0 . ± . θ E > . ± .
05 0 . ± . θ E >
10 13 . ± .
08 0 . ± . of LoCuSS clusters and the filled circle the values obtainedby performing a stacking analysis of them as described byOkabe et al. (2013). As already discussed, the strong lens-ing selection tends to increase both the normalization andthe slope of the concentration-mass relation. In Table 3 wesummarize the slope and the zero point of best fitting therecovered c − M relations for the three cases. In this work we have studied how well galaxy cluster massesand concentrations are recovered using strong and weak lens-ing signals. Using an NFW halo model as reference, we re-cover mass and concentration using only weak lensing dataor combining them with a measurement of the size of theEinstein radius. In addition, for the case in which we com-bine both WL and SL measurements we have made esti-mates of the mass and concentration using a generalizedNFW model, which reduces both mass and concentrationbiases while introducing a new free parameter (the innerslope of the density profile). We summarize our study andmain results as follows.- On average, lensing analysis provides biases on the clus-ter mass that depend on the host halo mass. Small systemstypically present a mass bias of about while for themore massive ones the bias almost vanishes. The scatter hasa log-normal distribution with a σ log M ≈ . . For the most massive systems, adding the constraint on the size of theEinstein radius reduces the bias to a few percents.- The estimated concentration is slightly positively bi-ased and decreasing with the halo mass. This behavior canbe attributed to the presence of the BCG at the center ofthe cluster and to the adiabatic contraction (here includedunlike in the previous work (Giocoli et al. 2012c)).- Adopting a generalized NFW model for fitting weak andstrong lensing data reduces both mass and concentration bi-ases. However, this introduces an additional free parameter,the inner slope of the density profile, β .- The bias and the scatter in the estimated mass modifythe shape of the mass function with respect to the theoreticalprediction, from which the cosmological sample is drawn.However, for mass bins in which cluster count has a S/N (cid:62) the residuals between the input and the recovered massfunction are smaller by a factor of − .- The uncertainties on the cluster mass and concentra-tion also change the mass-concentration relation. On aver-age, when an NFW model is used to fit the clusters, thenormalization of the recovered c − M relation has a normal-ization higher by − . The use of a generalized NFWhelps to recover a c − M relation in better agreement withthe theoretical expectations.- The biases in the concentration-mass relation, as re-ported by the analyses of different galaxy cluster surveys,are clarified when selecting from our MOKA sample a re-laxed sub-sample of systems.To conclude, in this work we have presented a de-tailed and systematic analysis of the estimation of massand concentration of clusters using lensing data (weak andweak+strong). We have studied how the mass and concen-tration biases depend on the halo mass and redshift. We havepresented how the bias and the scatter in the estimated massand concentration influence the halo mass function and the c − M relation. In particular, we have discussed how differentselection criteria affect the concentration-mass relation andwe have found that strong lens clusters may have a concen-tration as high as − above the average, at the fixedmass. ACKNOWLEDGEMENTS
CG and RBM’s research is part of the project GLENCO,funded under the European Seventh Framework Pro-gramme, Ideas, Grant Agreement n. 259349. We acknowl-edge financial contributions from contracts ASI/INAFI/023/12/0 and by the PRIN MIUR 2010-2011 “The darkUniverse and the cosmic evolution of baryons: from currentsurveys to Euclid”. CG and LM also acknowledge the finan-cial contribution by the PRIN INAF 2012 “The Universein the box: multiscale simulations of cosmic structure”. SEacknowledges the financial contribution from contracts ASI-INAF I/088/06/0 and PRIN-INAF 2012. CG would like tothank Giuseppe Tormen and Vincenzo Mezzalira to havehost part of the computer jobs run to produce the sim-ulated galaxy cluster sample. Part of the simulations ofthis project have been run during the Class C Project-HP10CMXLBH (MOKALEN3) CG would like to particu-larly thank Matthias Bartelmann for useful and stimulatingdiscussions. We are also grateful to Maruša Bradač Anja Von c (cid:13) , 1–18 Giocoli et al. 2013
Figure A1.
Average of the rescaled – with respect to the 2Dmass – estimated mass as a function of the 2D cluster mass, forthe systems at redshift z = 0 . . We show the case in which themass is estimated using WL+SL (circles) and WL only (triangles)information using an NFW model as a reference when minimizingthe total χ . The upper and lower curves enclose the σ scatter ofthe distribution. Each mass bin contains cluster realizations.The orange crosses show the same quantity when using gNFWmodel as a reference for both weak and strong lensing. der Linden, Stefano Borgani, Stefano Andreon and MauroSereno for the conversations had during the conference inMadonna di Campiglio in March 2013. We are also gratefulto the anonymous referee for his/her comments and sugges-tions that helped to improve the presentation of our results. APPENDIX A: 2D MASSES
The measurement of gravitational lensing gives an estimateof the projected mass which is causing the distortion of theshape of background galaxies and the creation of multipleimages, without any assumption about the dynamical stateof the system. However some assumption is needed when themass reconstruction is deprojected from 2D to 3D.In Fig. A1 we show the estimated mass derived fromweak and weak + strong lensing rescaled with respect tothe 2D one as a function of the true cluster mass for redshift z = 0 . . We do not show the results for different redshiftssince they are extremely similar this case. From the figure,when we assume an NFW halo as a reference model, wenotice that for smaller masses the bias with respect to the2D mass is of about ten percent, while at larger masses it isalmost negligible, well below − . One more interestingresult is that the scatter is smaller than the one measured inthe relation M est /M . The case WL+SL using generalizedNFW model shows on average no particular bias for anycluster mass sample. APPENDIX B: CONVERGENCE ANDPOTENTIAL ELLIPTICITY
The flexibility of the MOKA code allows us to study therelationship between the convergence and the potential el-lipticity. In order to do so we measure the ellipticity fromconvergence and lensing potential maps, both within R .In this context, we use the relation of Golse & Kneib (2002)to link (cid:15) κ and (cid:15) φ for pseudo-elliptical NFW lens models.However, simulated and real clusters are different from asimple one-component model and we are interested in un-derstanding if, and up to what point, their relation is validfor our MOKA cluster sample. At this aim, we consider twosamples of clusters, one triaxial without BCG and satel-lite population and the second one containing them. Theleft panel of Fig. B1 shows the correlation between the po-tential and the convergence ellipticity for the first sample,when considering together all clusters at all redshifts. Thefilled circles represent the median at fixed (cid:15) κ, , while thesolid lines enclose the first and the third quartiles. In thebottom panel of the figure, we show the residuals of (cid:15) φ, with respect to (cid:15) φ, ,GK that represents the ellipticity ofthe potential directly computed from (cid:15) κ, , using the Golse& Kneib (2002) formalism.In order to estimate (cid:15) φ, ,GK we solve the equation: (cid:15) κ, ,GK = a (cid:15) Φ , + a (cid:15) , , (B1)taken from Golse & Kneib (2002), where: a = 3 .
31 + 0 . x (B2) a = − . − . x , and x = R /r s . We recall that this relation has beenobtained by applying the formalism to a pseudo-ellipticalNFW profile, for a limited range of the convergence elliptic-ity: typically (cid:15) κ (cid:46) . . We notice, that since our definitionof 2D ellipticity differs from the one adopted by Golse &Kneib (2002), the parametrization of a and a differs bya constant factor of . . From the bottom left panel wenotice that the Golse & Kneib (2002) formalism perfectlycaptures the potential ellipticity from the convergence for (cid:15) κ, < . with a very small rms. The situation is differentfor the case in which we add BCG and substructures to theconvergence map, see right panel of Fig. B1. In this case, wenotice that on average the analytical prediction is still validbut the scatter of the correlation is much larger, Moreoverthe rms between the ellipticity measured in the potentialmap and the one inferred from the convergence ellipticity islarger than in the previous case by more than one order ofmagnitude. REFERENCES
Applegate D. E., von der Linden A., Kelly P. L., AllenM. T., Allen S. W., Burchat P. R., Burke D. L., EbelingH., Mantz A., Morris R. G., 2012, ArXiv e-printsBartelmann M., 1995, A&A, 299, 11Bartelmann M., 1996, A&A, 313, 697Bartelmann M., 2010, ArXiv e-printsBartelmann M., Schneider P., 2001, Physics Reports, 340,291Bartelmann M., Steinmetz M., 1996b, MNRAS, 283, 431 c (cid:13) , 1–18 alo properties from weak and strong lensing Figure B1.
Potential vs convergence ellipticity correlation. The dots in both panel show halo by the halo correlation while the filledcircles with the error bars correspond to the median and the quartiles of the distribution in bins of convergence ellipticity. Both theconvergence and the potential ellipticities have been measured from the respective maps within R . In the left panel we show thecorrelation for smooth haloes without substructures and central galaxy. In the right panel we consider more realistic clusters wheresubstructures and BCG are included. The bottom panels show the residuals of the measured potential ellipticity with respect to the oneestimated from (cid:15) k, using the Golse & Kneib (2002) relation (see equation (B1)). Becker M. R., Kravtsov A. V., 2011, ApJ, 740, 25Broadhurst T., Benítez N., Coe D., Sharon K., Zekser K.,White R., Ford H., et al. B., 2005b, ApJ, 621, 53Broadhurst T., Takada M., Umetsu K., Kong X., ArimotoN., Chiba M., Futamase T., 2005a, ApJ, 619, L143Bryan G. L., Norman M. L., 1998, ApJ, 495, 80Coe D., Zitrin A., Carrasco M., Shu X., Zheng W., PostmanM., Bradley L., Koekemoer et al. 2013, ApJ, 762, 32Comerford J. M., Natarajan P., 2007, MNRAS, 379, 190Despali G., Tormen G., Sheth R. K., 2013, MNRAS, 431,1143Eke V. R., Cole S., Frenk C. S., 1996, MNRAS, 282, 263Ettori S., Donnarumma A., Pointecouteau E., ReiprichT. H., Giodini S., Lovisari L., Schmidt R. W., 2013,Space Sci.Rev., 177, 119Ettori S., Gastaldello F., Leccardi A., Molendi S., RossettiM., Buote D., Meneghetti M., 2010, A&A, 524, A68+Ettori S., Morandi A., Tozzi P., Balestra I., Borgani S.,Rosati P., Lovisari L., Terenziani F., 2009, A&A, 501, 61Fasano G., Bettoni D., Ascaso B., Tormen G., PoggiantiB. M., Valentinuzzi T., D’Onofrio M., Fritz J., MorettiA., Omizzolo A., Cava A., Moles M., Dressler A., CouchW. J., Kjærgaard P., Varela J., 2010, MNRAS, 404, 1490Fedeli C., 2012, MNRAS, 424, 1244Gao L., Navarro J. F., Frenk C. S., Jenkins A., Springel V.,White S. D. M., 2012, ArXiv e-printsGiocoli C., Meneghetti M., Bartelmann M., Moscardini L.,Boldrin M., 2012a, MNRAS, 421, 3343Giocoli C., Meneghetti M., Ettori S., Moscardini L., 2012c,MNRAS, 426, 1558Giocoli C., Tormen G., Sheth R. K., 2012b, MNRAS, 422, 185Giocoli C., Tormen G., Sheth R. K., van den Bosch F. C.,2010a, MNRAS, 404, 502Golse G., Kneib J.-P., 2002, A&A, 390, 821Gottlöber S., Yepes G., 2007, ApJ, 664, 117Gottloeber S., Yepes G., Wagner C., Sevilla R., 2006,ArXiv Astrophysics e-printsHajian A., Battaglia N., Spergel D. N., Bond J. R., Pfrom-mer C., Sievers J. L., 2013, JCAP, 11, 64Hernquist L., 1990, ApJ, 356, 359Hirata C. M., Seljak U., 2004, Phys.Rev.D, 70, 063526Hoekstra H., 2003, MNRAS, 339, 1155Hoekstra H., Bartelmann M., Dahle H., Israel H., LimousinM., Meneghetti M., 2013, Space Sci.Rev., 177, 75Host O., 2012, MNRAS, 420, L18Jaffe W., 1983, MNRAS, 202, 995Jing Y. P., 2000, ApJ, 535, 30Jing Y. P., Suto Y., 2002, ApJ, 574, 538Jullo E., Natarajan P., Kneib J.-P., D’Aloisio A., LimousinM., Richard J., Schimd C., 2010, Science, 329, 924Kazantzidis S., Bullock J. S., Zentner A. R., KravtsovA. V., Moustakas L. A., 2008, ApJ, 688, 254Kazantzidis S., Kravtsov A. V., Zentner A. R., Allgood B.,Nagai D., Moore B., 2004, ApJ, 611, L73Kazantzidis S., Zentner A. R., Kravtsov A. V., BullockJ. S., Debattista V. P., 2009, ApJ, 700, 1896Keeton C. R., 2001, ApJ, 561, 46Killedar M., Borgani S., Meneghetti M., Dolag K., FabjanD., Tornatore L., 2012, MNRAS, 427, 533Kneib J.-P., Natarajan P., 2011, A&A Rev., 19, 47Laureijs R., Amiaux J., Arduini S., Auguères J. ., Brinch- c (cid:13) , 1–18 Giocoli et al. 2013 mann J., Cole R., Cropper M., Dabin C., Duvet L., et al.2011, ArXiv e-printsLimousin M., Morandi A., Sereno M., Meneghetti M., Et-tori S., Bartelmann M., Verdugo T., 2013, Space Sci.Rev.Mahdavi A., Hoekstra H., Babul A., Bildfell C., JeltemaT., Henry J. P., 2013, ApJ, 767, 116Meneghetti M., Bartelmann M., Dahle H., Limousin M.,2013, Space Sci.Rev.Meneghetti M., Fedeli C., Pace F., Gottlöber S., Yepes G.,2010a, A&A, 519, A90+Meneghetti M., Fedeli C., Zitrin A., Bartelmann M., Broad-hurst T., Gottlöber S., Moscardini L., Yepes G., 2011,A&A, 530, A17+Meneghetti M., Rasia E., Merten J., Bellagamba F., EttoriS., Mazzotta P., Dolag K., Marri S., 2010b, A&A, 514,A93+Metcalf R. B., Madau P., 2001, MNRAS, 563, 9Morandi A., Pedersen K., Limousin M., 2010, ApJ, 713,491Narayan R., Bartelmann M., 1996, ArXiv Astrophysics e-printsNavarro J. F., Frenk C. S., White S. D. M., 1996, ApJ, 462,563Neto A. F., Gao L., Bett P., Cole S., Navarro J. F., FrenkC. S., White S. D. M., Springel V., Jenkins A., 2007, MN-RAS, 381, 1450Newman A. B., Treu T., Ellis R. S., Sand D. J., 2011, ApJ,728, L39+Newman A. B., Treu T., Ellis R. S., Sand D. J., RichardJ., Marshall P. J., Capak P., Miyazaki S., 2009, ApJ, 706,1078Oguri M., Bayliss M. B., Dahle H., Sharon K., GladdersM. D., Natarajan P., Hennawi J. F., Koester B. P., 2012,MNRAS, 420, 3213Oguri M., Takada M., 2011, Phys.Rev.D, 83, 023008Oguri M., Takada M., Okabe N., Smith G. P., 2010, MN-RAS, 405, 2215Oguri M., Takada M., Umetsu K., Broadhurst T., 2005,ApJ, 632, 841Okabe N., Smith G. P., Umetsu K., Takada M., FutamaseT., 2013, ApJ, 769, L35Okabe N., Takada M., Umetsu K., Futamase T., SmithG. P., 2010b, PASJ, 62, 811Planck Collaboration Ade P. A. R., Aghanim N., Armitage-Caplan C., Arnaud M., Ashdown M., Atrio-Barandela F.,Aumont J., Baccigalupi C., Banday A. J., et al. 2013,ArXiv e-printsPostman M., Coe D., Benítez N., Bradley L., BroadhurstT., Donahue M., Ford H., Graur O., Graves G., Jouvel S.,et al. 2012, ApJS, 199, 25Rasia E., Borgani S., Ettori S., Mazzotta P., MeneghettiM., 2013, ApJ, 776, 39Rasia E., Ettori S., Moscardini L., Mazzotta P., BorganiS., Dolag K., Tormen G., Cheng L. M., Diaferio A., 2006,MNRAS, 369, 2013Rasia E., Meneghetti M., Martino R., Borgani S., BonafedeA., Dolag K., Ettori S., Fabjan D., Giocoli C., MazzottaP., Merten J., Radovich M., Tornatore L., 2012, New Jour-nal of Physics, 14, 055018Redlich M., Bartelmann M., Waizmann J.-C., Fedeli C.,2012, A&A, 547, A66Rossi G., Sheth R. K., Tormen G., 2011, MNRAS, 416, 248 Sereno M., Zitrin A., 2012, MNRAS, 419, 3280Shaw L. D., Weller J., Ostriker J. P., Bode P., 2006, ApJ,646, 815Shaw L. D., Weller J., Ostriker J. P., Bode P., 2007, ApJ,659, 1082Sheth R. K., Mo H. J., Tormen G., 2001, MNRAS, 323, 1Sheth R. K., Tormen G., 1999, MNRAS, 308, 119Umetsu K., Broadhurst T., Zitrin A., Medezinski E., CoeD., Postman M., 2011, ArXiv e-printsvan den Bosch F. C., Tormen G., Giocoli C., 2005, MNRAS,359, 1029Viola M., Melchior P., Bartelmann M., 2011, MNRAS, 410,2156Zhao D. H., Jing Y. P., Mo H. J., Bnörner G., 2009, ApJ,707, 354Zhao H., 1996, MNRAS, 278, 488Zieser B., Bartelmann M., 2012, ArXiv e-printsZitrin A., Broadhurst T., Barkana R., Rephaeli Y., BenítezN., 2011, MNRAS, 410, 1939Zitrin A., Broadhurst T., Bartelmann M., Rephaeli Y.,Oguri M., Benítez N., Hao J., Umetsu K., 2011c, ArXive-printsZitrin A., Broadhurst T., Coe D., Umetsu K., Postman M.,Benítez N., Meneghetti M., et al. M., 2011, ApJ, 742, 117Zitrin A., Menanteau F., Hughes J. P., Coe D., BarrientosL. F., Infante L., Mandelbaum R., 2013b, ArXiv e-printsZitrin A., Meneghetti M., Umetsu K., Broadhurst T.,Bartelmann M., Bouwens R., Bradley L., Carrasco M.e. a., 2013a, ApJ, 762, L30 c (cid:13)000