Mock weak lensing analysis of simulated galaxy clusters: bias and scatter in mass and concentration
aa r X i v : . [ a s t r o - ph . C O ] M a y Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 25 October 2018 (MN L A TEX style file v2.2)
Mock weak lensing analysis of simulated galaxy clusters:bias and scatter in mass and concentration
Yannick M. Bah´e ⋆ , Ian G. McCarthy , and Lindsay J. King , Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, United Kingdom Kavli Institute for Cosmology Cambride, University of Cambridge, Madingley Road, Cambridge CB3 0HA, United Kingdom
ABSTRACT
We quantify the bias and scatter in galaxy cluster masses M and concentrations c derived from an idealised mock weak gravitational lensing (WL) survey, and theireffect on the cluster mass-concentration relation. For this, we simulate WL distortionson a population of background galaxies due to a large ( ≈ z ≃ .
2. This study takesinto account the influence of shape noise, cluster substructure and asphericity as wellas correlated large-scale structure, but not uncorrelated large-scale structure alongthe line of sight and observational effects (e.g., the source redshift distribution andmeasurement, and measurement of galaxy ellipticities). We find a small, but non-negligble, negative median bias in both mass and concentration at a level of ∼ M > . M ⊙ , alevel of ∼
30% and ∼
20% for concentration and mass respectively and increasesstrongly with decreasing cluster mass. For a typical survey analysing 30 galaxies perarcmin over a radial range from 30 ′′ to 15 ′ from the cluster centre, the derived M - c relation has a slope and normalisation too low compared to the underlying true (3D)relation by ∼
40% and ∼
15% respectively. The scatter and bias in mass are shownto reflect a departure at large radii of the true WL shear/matter distribution of thesimulated clusters from the NFW profile adopted in modelling the mock observations.Orientation of the triaxial cluster haloes dominates the concentration scatter (exceptat low masses, where galaxy shape noise becomes dominant), while the bias in c ismostly due to substructure within the virial radius. Key words: gravitational lensing:weak — galaxies: clusters: general — galaxies:groups: general — cosmology: theory
There are several aspects of galaxy clusters that make themsensitive tracers of cosmic evolution, and thus potentiallypowerful tools for measuring a number of fundamental cos-mological parameters (for recent reviews see Voit 2005 andAllen, Evrard & Mantz 2011). First, in currently-favouredhierarchical models for structure formation where small ob-jects collapse first and merge together to form progressivelylarger ones, clusters are the most recent objects to haveformed, since they are the largest bound and virialized ob-jects in the Universe at present. Consequently, the abun-dance of clusters as a function of mass and redshift is sen-sitive to cosmological parameters that have an influence on ⋆ [email protected] the rate of growth of structure (such as σ , Ω m , and thenature of dark energy). Second, the matter content (specifi-cally the baryon to total mass ratio) of clusters is expectedto reflect that of the universe as a whole (i.e., ≈ Ω b / Ω m ;White et al. 1993), since the potential wells of clusters areincredibly deep and it is difficult to conceive of sources thatare energetic enough to remove baryons from them. Finally,the shape of the total matter density profile (which is dom-inated by dark matter), specifically how concentrated it isas a function of mass (e.g., Buote et al. 2007), is also sensi-tive to the underlying cosmological parameters through thegrowth rate of structure, since the concentration is thoughtto be determined by the background density at epoch offormation (e.g., Navarro, Frenk & White 1996). All of thesecluster-based cosmological tests rely on there being accu-rate estimates of cluster mass and also concentration (in the c (cid:13) Y.M. Bah´e et al. case of the latter test, which is our main focus in the presentstudy), at least in a statistical sense. This therefore requiresthat the bias and scatter inherent in cluster mass and con-centration estimators be carefully quantified and accountedfor. Weak gravitational lensing (hereafter WL) providesone of the most promising methods for deriving themasses and concentrations of large samples of galaxyclusters and thus for measuring cosmological parame-ters such as σ and the evolution of the dark energyto (potentially) high accuracy. For high-mass clusters,masses and concentrations can be obtained individually(e.g., Dahle 2006; Broadhurst et al. 2008; Mahdavi et al.2008; Okabe & Umetsu 2008; Medezinski et al. 2010;Radovich et al. 2008; Corless, King & Clowe 2009),whereas less massive systems can be studied bystacking the lensing profiles of many clusters (e.g.,Johnston et al. 2007; Mandelbaum & Seljak 2007;Sheldon et al. 2009; Rozo, Wu & Schmidt 2011). Inthe near future, the Dark Energy Survey (DES,The Dark Energy Survey Collaboration 2005) is expectedto make observations of ∼ ,
000 clusters with masses inexcess of 2 × M ⊙ . Note that unlike most other methodsthat are used to measure the mass and concentrationof clusters, like hydrostatic equilibrium analyses of thehot intracluster medium (e.g., Vikhlinin et al. 2009 andreferences therein) or virial equilibrium analyses of orbitingsatellite galaxies (e.g., Becker et al. 2007), WL analysesdo not require any assumptions about the dynamical stateof the cluster. As we will show, however, this does notguarantee that the WL-derived masses and concentra-tions are unbiased (see also Becker & Kravtsov 2011 andOguri & Hamana 2011).In the present study we aim to quantify the bias andscatter in galaxy cluster masses, M , and concentrations, c , derived from a mock WL survey and their effect on the de-rived cluster mass-concentration relation. Mock WL obser-vations based on numerical simulations and analytic modelsof clusters offer the unique possibility of comparing the ob-served and true parameters for the same clusters to find theextent to which observational biases contribute to the dis-crepancies. There have been a number of such studies in thepast, focussing on various aspects such as cluster aspheric-ity and substructure (e.g., King, Schneider & Springel 2001;Clowe, De Lucia & King 2004; Corless & King 2007), cor-related (e.g., Metzler et al. 1999; Metzler, White & Loken2001; King & Corless 2007) and uncorrelated large-scalestructure along the line of sight (e.g., Hoekstra 2003;Dodelson 2004; Hoekstra et al. 2011). In particular,Corless & King (2007) found from an analysis of lensed ana-lytic clusters that for projections along a line of sight close tothe major axis concentrations and mass can be overpredictedby factors of up to ≈ can potentially bias the WL-derivedmasses and concentrations, it is equally important to know how likely they will do so. Investigating this requires large,high resolution cosmological simulations containing a large enough number of realistically modelled clusters to allowstatistically robust and meaningful conclusions to be drawnabout those in the real Universe. In a recent study aiming atthis, Becker & Kravtsov (2011) — see also Oguri & Hamana2011 — used a large sample of ∼ haloes and derived atypical scatter in the reconstructed masses of ∼ − ≈ − − strong lensing properties of mas-sive clusters (e.g., Mead et al. 2010), it has recently beenshown by Semboloni et al. (2011) that its effect on WL mea-surements is limited to haloes with a mass below 10 M ⊙ .Analysing exclusively clusters above this mass threshold, wetherefore do not expect the inclusion of a realistic baryoniccomponent to significantly modify our results or conclusions.This paper is structured as follows: In § §
3. We quantify the mass-concentration relation and the spread in the reconstructedparameters in §
4, followed by a detailed analysis of the phys-ical principles responsible for scatter and bias in §
5. We dis-cuss and summarise our findings in §
6. For consistency withthe Millennium Simulation, a flat cosmology with Hubbleparameter h = H / (100 km s − Mpc − ) = 0 .
73, dark en-ergy density parameter Ω Λ = 0 .
75 (dark energy equationof state parameter w = − M = 0 .
25 is used throughout this paper.
Gravitational lensing describes the deflection of light bymassive objects. It is sensitive only to the projected massdensity Σ of the lens, with higher Σ generally correspondingto larger distortions, for centrally condensed objects. Unlikestrong lensing, which occurs in cluster centres and leads tomultiple images of background galaxies which may be highlydistorted (eg. Soucail et al. 1987), weak gravitational lens-ing can be employed to study the outer ( r > ∼
100 kpc), less c (cid:13) , 000–000 eak lensing simulated clusters dense cluster regions. Due to its weak nature, backgroundgalaxies are only slightly distorted, an effect that must bestudied statistically. Assuming that the light from galaxiescan be approximated by elliptical isophotes, one can quan-tify their shape by the complex quantity ellipticity ( ǫ ), itsmodulus related to the ratio between the minor and majoraxis r = b/a by | ǫ | = 1 − r r and with a phase of twice the position angle of the ellipse’smajor axis. In the WL regime, the observed and source el-lipticities of a background galaxy, in the following denotedby ǫ and ǫ s respectively, are related by ǫ = ǫ s + g g ∗ ǫ s (1)where the complex quantity g is the “reduced shear”, withits complex conjugate denoted by g ∗ . The reduced shear isthen related to the (complex) shear, γ (the tidal gravita-tional field), and the convergence, κ , by g = γ − κ . (2)Equation (1) implies that in the ideal case of a perfectlycircular background galaxy with r = 1 and ǫ s = 0, the lensedellipticity ǫ = g .In the above, κ is proportional to the projected massdensity of the lens, Σ: κ = ΣΣ cr , (3)where Σ cr = c πG D s D d D ds (4)is the “critical density” and D s , D d and D ds are the angulardiameter distances between observer – source, observer –lens and lens – source respectively. In this notation, the WLregime is where Σ << Σ cr , i.e. κ << ψ , is related to γ and κ by a set of partial differential equations (e.g.Bartelmann & Schneider 2001): κ ( θ ) = 12 ( ψ , + ψ , ) (5) γ ( θ ) = 12 ( ψ , − ψ , ) γ ( θ ) = ψ , = ψ , , where γ ( θ ) = γ + iγ and the indices following the comma denote partial deriva-tives with respect to the components of the position vector θ . A secondary effect is a slight modification in the ob-served number density of background galaxies, n lensed , dueto magnification changing the apparent fluxes of galaxies, aswell as changing the apparent area of sky in which they areobserved. The net effect depends on the slope of the num-ber counts of the galaxies ( β ), with the lensed and unlensednumber counts being related by (Canizares 1982): n lensed = µ β − n unlensed (6) where the magnification µ is given by µ = 1(1 − κ ) − | γ | . (7)Since β ≈ . We extract simulated galaxy clusters from the MillenniumSimulation (hereafter MS; Springel et al. 2005), a large cos-mological N-body simulation that follows 2160 dark mat-ter particles from z = 127 to z = 0 in a periodic box of500 h − Mpc on a side. The cosmology adopted for the MS,which we also adopt for our analysis, is a flat ΛCDM modelwith h = 0 .
73, Ω M = 0 .
25, and a power spectrum normali-sation on a scale of 8 h − Mpc of σ = 0 . h − Mpc extrapo-lated to present-day). These parameters are consistent withthe latest measurements of temperature and polarizationanisotropies in the cosmic microwave background (CMB)with the
Wilkinson Microwave Anisotropy Probe (WMAP;Komatsu et al. 2011), although the value of σ adopted forthe MS is larger than the maximum likelihood CMB valueby ≈ σ . The larger value of σ means that the MS will havemore massive clusters than a universe with the WMAP 7-year cosmology. However, as we are interested in the lensingsignal of individual clusters (specifically how well WL canrecover their mass and concentration) and not their abun-dances, this discrepancy will not affect the validity of ourresults.From the snapshot at z ≃ .
2, a typical redshift of ob-served WL clusters, we select all of the simulated clusterswith masses M ≥ M ⊙ for analysis. Note that M is defined as the mass within a radius ( r ) that enclosesa mean density equal to 200 times the critical density ofthe universe — in our adopted cosmology, this density is ρ crit = 1 . · M ⊙ / Mpc at our chosen redshift. Fromhere on, we will refer to these masses obtained directly fromthe MS particle distribution without any profile fitting as‘true’ mass, denoted by M Mill . A large sample of 2678 sim-ulated cluster haloes satisfies this criterion, which will allowus to robustly quantify the mean trends and scatter in thederived mass-concentration relationship.We extract all of the dark matter particles within a10 h − Mpc (comoving) box, centered on the most boundparticle, for producing WL maps of each cluster. For refer-ence, r is typically ≈ local environment around theclusters. This was deliberate, so that we can isolate the ef-fects of cluster triaxiality, substructure and connecting fil-aments (i.e., “correlated” signals) from uncorrelated line-of-sight structures at much larger distances. While thesecorrelated structures can in general extend beyond a dis-tance of 5 h − from the cluster centre, we verify in ap- c (cid:13) , 000–000 Y.M. Bah´e et al. pendix B1 (Fig. B1) that, in agreement with the findingsby Becker & Kravtsov (2011), our results are robust to in-creasing the line-of-sight integration length by a factor of fiveto 50 h − comoving Mpc. Much longer integration lengths,which would capture uncorrelated large scale structure morefully, would in principle be desirable for the purpose of im-proving the prediction accuracy, but require the use of ray-tracing algorithms which is beyond the scope of this work.In a complementary study, Hoekstra et al. (2011) have usedray-tracing of the MS to examine the effect of such uncorre-lated structures on the derived masses and concentrations ofanalytic clusters and found that it contributes to scatter, butnot bias in mass and concentration. We discuss the resultsof these authors in the context of our findings in Section 6. For each cluster we produce surface mass density maps byprojecting the 10 h − Mpc box along one side and interpolat-ing to a regular grid of 2000 pixels using a triangular shapedclouds (TSC) algorithm (Hockney & Eastwood 1988). It waschecked that other interpolation algorithms, such as cloud-in-cells or smoothed particle hydrodynamics (SPH) inter-polation, did not yield significantly different results. At theadopted resolution, 1 pixel length corresponds to 5 h − kpc,equivalent to the gravitational softening length of the MS,which makes it the highest sensible resolution.To further boost our statistics, and explore the role ofviewing angle for individual objects, we repeated this pro-cedure for each simulated cluster five times, each time se-lecting a random orientation and thus produce five differentmaps for each cluster. In the DES, observations will be madeof ∼ ,
000 clusters with masses in excess of 2 × M ⊙ .The total number of projections we produce is compara-ble to this survey. For the most massive clusters (with M ≥ . M ⊙ ) in our sample, we have elected to boostour statistics even further by producing maps correspond-ing to 50 different random projections. The logic behindthis is that i) such massive systems are the most commonly-targetted in present WL observational studies (and thereforeare the systems for which we can presently make compar-isons between theory and observations); and ii) very highmass ( ∼ M ⊙ ) clusters are rare even in a 500 h − Mpcvolume.Convergence ( κ ) maps are then computed according toeqn. (3) with Σ cr as above, and the shear maps formed ac-cording to equations (5) and (2). To solve these equations,we adopt the same FFT method as Clowe, De Lucia & King(2004), computing˜ γ = ˆ k − ˆ k ˆ k + ˆ k ˜ κ, k ˆ k ˆ k + ˆ k ˜ κ ! (8)where ˜ γ and ˜ κ are Fourier transforms of γ and κ , and ˆ k arethe appropriate wave vectors. As an example, Fig. 1 showsthree projections of one high-mass cluster with the corre-sponding shear fields as tick marks. It is apparent that dif-ferent projections of the same cluster can look very differentand lead to different shear signals.With shear maps for all the projections generated, anarray of background galaxies, extending by default to a pro-jected distance of r out = 15 ′ from the cluster centre, was then simulated for each individual projection. This limitin projected distance excludes potential numerical artifactsnear the edges of the shear field; it also matches the typicalfield-of-view of observations (e.g. Clowe & Schneider 2001;see also Hoekstra 2003), where for the most massive clus-ters the shear signal is detected out to roughly this radius.Although galaxies used in real weak lensing surveys have arange of redshifts, we consider only the simplified case of auniform source redshift of z = 1 . . This is a value typical forobservational studies such as CFHTLS (Ilbert et al. 2006)and COSMOS (Ilbert et al. 2009) and has also been used inother theoretical work concerning the accuracy of weak lens-ing (e.g., Clowe, De Lucia & King 2004; Oguri & Hamana2011; Gruen et al. 2011; Becker & Kravtsov 2011). In realsurveys the dispersion in source redshifts and its finite sam-pling (Hoekstra et al. 2011) has to be considered as anadditional error source. We note that for this choice oflens and source redshift, the critical surface mass densityΣ cr = 3 . · M ⊙ / Mpc .To exclude the strong lensing regime in the cluster cen-tre, we excised the central region within r = r in , with adefault inner limit radius r in = 30 arcsec, a value typical ofthe extent of strongly lensed arcs in observational studies(e.g., Hennawi et al. 2008). We consider the effect of chang-ing the values of both r in and r out in appendix B3; whilethe exact amount of bias and scatter depends somewhat onthe exact choice of r in and r out (see also Becker & Kravtsov2011) our overall conclusions are unaffected by their exactchoice.The background galaxy ellipticities ǫ s were drawn ran-domly from a Gaussian distribution with standard deviation σ = 0 . σ is also similar to that used by Hoekstra et al.(2011). The effect of increasing the value of σ is shown inappendix B2; it does not affect our results significantly. Thepositions of galaxies were allocated randomly, with averageunlensed density from which shapes can be measured accu-rately n = 30 arcmin − , and accounting for Poisson noise.This is comparable to background densities achievable withcurrent ground-based pointed observations of galaxy clusters(e.g., Dahle 2006) and similar to what will be achieved withupcoming large surveys such as the DES and that from theLarge Synoptic Survey Telescope (LSST) .The distortion of these simulated background galaxiesby the cluster-induced shear field was then computed ac-cording to eqn. (1) to yield the lensed background galaxyellipticities ǫ . To account for number count depletion as de-scribed in Section 2, the lensed source density was adjusted Note that at the lens redshift of z ≃ .
2, 1 ′ corresponds to ∼ . Even though the expected source density for the DES, of ∼ − , is somewhat lower than our adopted fiducal back-ground density, we have experimented with lowering our sourcedensity from 30 to 10 arcmin − and find that it results in only amodest increase in the scatter in the derived masses and concen-trations. Our conclusions in Section 5, that it is halo triaxialityand substructure, rather than low signal-to-noise, that dominatethe error in the reconstructed masses and concentrations are ro-bust to lowering the background source density to this level. (cid:13) , 000–000 eak lensing simulated clusters Figure 1.
Three projections of a high-mass cluster with M = 4 · M ⊙ . The red tick marks represent the reduced shear field inducedby these projections in the WL regime (shown for κ < . ′ around thecluster centre which was analysed in this study. Different projections of the same cluster can give rise to very different shear signals. locally according to equations (6) and (7). At this stage wehave the final synthetic catalogues of lensed galaxies. We derive the best-fit mass, M , and concentra-tion, c , by fitting projected NFW mass distributions(Navarro, Frenk & White 1995; Navarro, Frenk & White1996; Navarro, Frenk & White 1997) to our mock WL data.The NFW profile, which has been shown to reproduce the3D spherically-averaged mass density profiles of dark matterhaloes spanning a wide range of masses, is described by thetwo parameter form: ρ ( r ) = ρ s ( r/r s )(1 + r/r s ) , (9)where ρ s /ρ crit = 2003 c ln(1 + c ) − c/ (1 + c ) (10)and the concentration c ≡ r /r s . Thus, the density dis-tribution is determined once the concentration (or equiva-lently the scale radius, r s ) and M (or equivalently r )are specified.Analytic expressions for the NFW profile integratedalong the line of sight, Σ NFW , and the resulting shear γ aregiven by Bartelmann (1996) and Wright & Brainerd (2000)respectively, from which the WL distortion can be calculatedas shown above. We point out that the analytic expressionsfound in these studies were derived by integrating an NFWprofile extending to infinity along the line of sight, ratherthan to some distance characteristic of the cluster size (e.g., r ). We discuss the effect of this model assumption furtherin Section 5.2.2.We derive the best-fit NFW parameters by first calcu-lating the tangential ellipticity ǫ t of all background galaxieswith cluster-centric distance r in < r < r out , defined by ǫ t = − Re n ǫe − iφ o (11)which are then fit by the tangential reduced shear generated by an NFW profile, parameterised by M and c , using astandard least-squares method (see Press et al. 1992) withmetric χ = N X i =0 [ ǫ t,i − g NFW ( r i , M, c )] . (12)Equation (12) implies that we take into account, equallyweighted, the tangential ellipticity of each individual galaxywhen fitting. Other commonly used methods employed toanalyse weak lensing signals are maximum-likelihood tech-niques (e.g., Corless & King 2008) and fitting a 1D shearprofile derived from binning up the individual galaxy ellip-ticities (e.g., Clowe, De Lucia & King 2004). We have ex-plicitly verified that our particular choice of fitting methodhas no significant influence on our results (see also appendixB3). The best-fit concentration and mass are then taken asthe reconstructed cluster parameters. From here on, we willrefer to the values obtained in this way as c WL and M WL respectively.Our aim is to compare WL reconstructions to the realcluster parameters to see how projection effects and the ap-plication of a simplified mass model for the halo modify thededuced values. For this comparison it is also necessary toobtain a ‘true’ concentration — a model-independent ref-erence value for the mass is already provided by M Mill —which we do by fitting an NFW profile to the 3D clusterdata. In the following, we will denote the parameters M and c derived in this way by c and M .Our 3D fitting procedure is similar to that described byNeto et al. (2007) for haloes in the Millennium simulation:the cluster particles are grouped into 32 radial bins fromlog ( r/r ) = − . ( r/r ).For each bin, the average density ρ i is then computed andthe NFW parameters obtained by performing a least squaresfit of log ρ i to log ρ NFW , taking into account only binsat a radius log ( r i /r ) ≥ − .
63. This criterion ensuresthat the central cluster region, where the density is affectedby smoothing, is excluded from our fit. In this, our methoddiffers slightly from Neto et al. (2007), who adopt an innercut-off radius of log ( r i /r vir ) = − .
3. The reason behindthis is that unlike their study, we only look at massive haloeswith M ≥ M ⊙ with a larger r . It has been shown c (cid:13) , 000–000 Y.M. Bah´e et al.
Figure 2.
Mass-concentration plot obtained from our mock weaklensing analysis showing five projections of each cluster (greypoints). The black triangles show the median concentrations foreach bin with the black dash-dot line giving the correspondingbest-fit power-law as described in the text. The blue circles andsolid line give the corresponding 3D NFW medians and best-fitpower-law respectively. Filled symbols represent median valuesused for constructing the power-law, open ones were discarded.The errorbars (dotted for WL, solid for 3D) indicate binsize inx-direction, whereas in y-direction the 25 th and 75 th percentilesare shown. The weak-lensing derived concentrations are system-atically too low compared to those from our reference 3D fit, withthe discrepancy increasing for lower mass systems. by Gao et al. (2008) that the best-fit concentration in theNFW profile depends somewhat on the exact choice of theinner cut-off radius, with lower values generally leading tohigher concentrations, which we have confirmed for our clus-ter fits. It should therefore be borne in mind that the biaswe derive is strictly applicable only with respect to this par-ticular fitting range. In the future, it may be advisable toadopt more accurate fitting functions such as the Einastoprofile (Einasto 1965) as discussed by Navarro et al. (2004)and Merritt et al. (2006); however, the price to pay for thisimproved accuracy is the introduction of an additional de-gree of freedom and its potential correlations with M and c . We point out in passing that we have not imposed anyrelaxation criteria for selecting our simulated clusters andselect based on mass alone, as it is not trivial to deduce howrelaxed a real cluster is. It is known that the NFW profiledoes not describe obviously unrelaxed cluster haloes well(e.g., Neto et al. 2007) and thus we expect that the accuracyof the reconstructed mass and concentration of real systemswill depend on the dynamical state of the cluster. In Fig. 2 we show the mass-concentration relationship de-rived from our mock WL analysis of MS clusters. Each of thegrey dots represents a single projection of one cluster. Thesolid triangles represent the median WL-derived concentra-tion in 16 equally-spaced bins of log ( M WL /M ⊙ ) from 13.6to 15.2 with bin width ∆ log ( M WL /M ⊙ ) = 0 .
1. The solidcircles represent the median true (3D) concentration in thecorresponding true mass bins .There is considerable scatter in the mass-concentrationrelation derived from the mock WL observations, but thetrend towards higher concentrations for lower masses is ev-ident. A power-law of form α ( M WL / h − M ⊙ ) β (13)was fit to the WL-derived median mass-concentration distri-bution, assuming an error in each bin proportional to 1 / √ n i where n i is the number of cluster projections per mass bin.Only mass bins with M WL > . M Mill = 10 M ⊙ . The best-fit parame-ters obtained from this procedure are α WL = (4 . ± . β WL = ( − . ± . M > M ⊙ dueto the very small number of systems in this range. The pa-rameters of this best-fit 3D powerlaw are α D = (5 . ± . β D = ( − . ± . The results in the previous section underline the need toquantify and account for bias and scatter in observationallyderived cluster masses and concentrations in large surveys.For detailed studies of individual clusters the bias may beless relevant, but knowing the expected scatter is still impor-tant. In this section, we present a quantification of the biasand scatter in our mock WL derived masses and concentra-tions. As variation with halo mass and resulting strengthof the lensing signal can be expected, our sample was firstdivided into five (true) mass bins, as indicated in Table 1.For each of these five bins, a histogram of the relative massesand concentrations, normalised to true mass M Mill and best-fit 3D concentration c , was then created. The results areshown in Fig. 3.The spread in both M WL and c WL is clearly mass-dependent and decreases with increasing cluster mass. Over-concentrations of more than a factor of 2.5 are virtually non-existent except for the lowest mass bin where there is an,albeit small, group of cluster projections whose concentra-tions are overestimated by up to a factor of 3. The masses aresomewhat better constrained, over- and underpredictions bymore than a factor of 2 being rare in all mass bins (apartfrom the very lowest).The error distributions in mass and concentration were c (cid:13) , 000–000 eak lensing simulated clusters Figure 3.
Spread in WL concentrations c WL /c (red, solid line) and masses M WL /M Mill (blue, dash-dot line) for different massranges. Shown is the fraction of projections in each bin. The solid black lines show the corresponding best-fit log-normal distributionsas described in the text, with parameters given in table 1. The numbers in the figure give the average concentration, mass range andnumber of projections per bin. The scatter in the recovered masses and concentrations decreases with increasing halo mass and closelyfollow log-normal distributions. Concentration Masslog ( M /M ⊙ ) µ σ < c/c D > median Q . µ σ < M/M Mill > median Q . > Table 1.
Parameters of the log-normal fit to the distributions of mass and concentration deviations as described in the text. Also givenare the mean, median and Q . quantile for both distributions. Note that µ and σ , the median and standard deviation in the underlyingnormal distribution, are the location and spread in the log-normal fit as defined in equation (14). fit with a log-normal distribution f ( x ) = 0 . x √ πσ e − (ln x − µ )22 σ (14)where the prefactor of 0.1, equal to the bin size in Fig. 3,converts the probability distribution function into the rel-ative histogram density shown in this figure. As shown inFig. 3, the best fit provides an excellent representation ofthe error distributions. The corresponding parameters arequoted in Table 1 together with the median and mean ob-tained directly from the error distributions. Also given are the Q . -quantiles, which give the relative error exceededby only 2.5% of projections as an indication for “reasonably”likely overconcentrations expected in observations.We point out here that, dealing with non-Gaussian dis-tributions, the conventional (arithmetic) standard deviationis only of limited applicability. For the remainder of this pa-per, we will therefore use σ for the spread parameter (thestandard deviation in the underlying normal distribution)as defined in equation (14) and ‘scatter’ for the offset fromunity in the geometric standard deviation, e σ − c (cid:13) , 000–000 Y.M. Bah´e et al. the mass also display a slight bias in the sense that, on aver-age, the reconstructed values are slightly lower than the trueones. This accounts for the lower-than-expected normalisa-tion of the WL-derived mass-concentration relation evidentin Fig. 2. We aim to provide an explanation for these biasesin Section 5 below.Note that we have chosen to use medians, rather thanmeans to quantify bias. The rationale behind this is that, ina non-Gaussian distribution as is the case here (log-normal),the mean, unlike the median, depends on the scatter asoverpredictions can be arbitrarily high, whereas values canclearly not be underpredicted by more than 100%. From theamount of scatter evident in our results (see Figs. 3 and 5)one should not be surprised to find that the mean mass andconcentration show, in general, a positive bias with respectto their median counterparts. For completeness, we showthe difference between mean and median bias in Fig. C1 inappendix C.
The bias and scatter in cluster masses derived from weaklensing were recently studied in detail by Becker & Kravtsov(2011, BK11) using clusters formed in a cosmological simu-lation independent of the MS run at somewhat lower reso-lution. In this study, the mass derived from WL was foundto be biased low at a level of ∼ − ′ to 20 ′ from the cluster centre and a shearprofile formed from them, our results are in good quanti-tative agreement (see also Fig. B3 in the appendix, wherewe analyse our simulation using the same radial range asBK11). The scatter determined by BK11 ( ∼ ∼ Biases in both mass and concentration have also been stud-ied by Oguri & Hamana (2011, OH11), who found a massbias similar to that in Becker & Kravtsov (2011) and pre-sented here. However, in contrast to our results, they finda very large, positive concentration bias of ∼ Having established the extent of the scatter and bias in WLreconstructions of cluster haloes, we now aim to find physicalexplanations for them. This is an interesting question in itsown right, but might also allow an identification of possibleways to reduce these systematic errors.Any potential error sources can be broadly grouped intotwo categories: Those due to the background galaxies used inthe reconstruction (i.e., their unknown intrinsic ellipticitiesand finite number, in general also their intrinsic alignmentdue to cosmic shear), and those due to the cluster itself, suchas halo triaxiality and substructure. In this section we showthat, in the case of large statistical samples of clusters suchas we have studied here and those to be derived from theDES and LSST, the latter is dominated by the former onlyfor clusters with masses below a few 10 M ⊙ .Our strategy for assessing the importance of these var-ious error contributions involves making two additional re-constructions of our cluster sample, designed to bridge thegap between the WL analysis based on particles within a 10 h − Mpc box on the one side and the 3D fitting procedurewithin a radius r on the other. In the first of these, whichwe will refer to as “perfect WL”, we use a very high density( n = 300 arcmin − ) of perfectly circular background galax-ies (i.e., σ = 0 . — essentially, we are now analysing the(reduced) shear field g directly.In the second method, we approach the 3D fit even fur-ther by constructing the convergence field κ — and thus theshear — only from those particles that lie within a (3D) dis-tance of r from the cluster centre, the same set of particlesupon which the 3D fit is based. We will refer to this methodas “spherical WL”. One problem with this approach is thatthe expression of Wright & Brainerd (2000) for g assumesa matter distribution extending to infinity, so fitting it toa catalogue of galaxy distortions based only on the matterdistribution inside r alone, which necessarily contains lesstotal mass, will result in severe biases in both mass and con-centration . We overcome this by instead fitting the reducedshear from an NFW profile that is, like our data, truncatedat r (Takada & Jain 2003a; Takada & Jain 2003b). Wenote that it is straightforward to also fit the projected massprofile of Takada & Jain (2003a) directly to the projectedmatter within r . This method approaches the 3D fit evencloser, with the only remaining difference being a fit in 2Dvs. one in 3D. We have done this, and found that the resultsare in very close agreement with those of the ‘spherical WL’method.The bias and scatter as defined in equation (14) inmasses and concentrations resulting from both these meth-ods are shown in Fig. 4 and 5 respectively, the perfect WLfit represented by red lines, the spherical WL fit by blueones. For ease of comparison, we also include the values de- We have explicitly verified that there is no significant differencebetween n = 100 and n = 300 for σ = 0 . The latter because the influence of the ‘missing mass’ is greatestat large projected radii R where only a short section of the lineof sight passes through the inner cluster region as illustrated inFig. A1 in appendix A. c (cid:13) , 000–000 eak lensing simulated clusters Figure 4.
Median values of the distributions derived from ourdefault WL simulation ( n = 30 , σ = 0 .
2, black solid line), “per-fect WL” ( n = 300 , σ = 0 .
0, red dashed line) and “spherical WL”(parameters as in perfect WL, but only based on the matter distri-bution within r , blue dash-dot line). See text for details. Top: concentration, bottom: mass, also showing the median values in M obtained from 3D mass profile fitting (green dash-dot-dot-dot line). The vertical lines at the bottom show the underlyingmass bins. termined from our default WL simulation as found in Table1 as black lines. We can judge the influence of errors associated with thebackground galaxies by comparing the “default” and “per-fect” WL reconstructions (black and red curves in Figs. 4and 5), the latter one, as explained above, not being affectedby them.
Focusing first on scatter in default vs. perfect WL (Fig. 5),we find a very similar picture for both mass and concen-tration: it is comparable for the highest-mass clusters, butwhile the latter is almost mass-independent, the former in-creases considerably with decreasing cluster mass. This is
Figure 5.
As Fig. 4, but instead showing the scatter e σ − σ as defined in equation(14). what would be expected from the influence of shape noise:Less massive haloes, producing a weaker shear signal, yielda lower signal-to-noise ratio than their high-mass counter-parts; in the total absence of shape noise, however, the de-creasing shear signal is irrelevant. As with scatter, the variation in bias between default andperfect WL is similar for both mass and concentration. Inboth cases, it is generally negative and slightly stronger inthe case of perfect WL than in the default simulation (withthe exception of the mass bias for the highest mass clusters): ≈ −
9% vs. ≈ −
5% for concentration and ≈ −
5% vs. ≈ − c (cid:13) , 000–000 Y.M. Bah´e et al. fect weak lensing observations of very large cluster samples slightly less biased than perfect ones. r We now look at the structure of the clusters themselves, fo-cusing first on their outer regions beyond r by comparingthe “perfect” and “spherical” WL reconstructions (red andblue curves in Figs. 4 and 5) — the only difference betweenthem being that the former includes mass beyond r whilethe latter does not. For the concentrations, we find a similar scatter from bothmethods (compare the red and blue curves in the top panelof Fig. 5). In fact, the scatter is even slightly larger for thespherical analysis, a strong indication that it is driven bythe mass distribution inside r as discussed in detail be-low. The bias is even stronger for spherical WL than eitherof the other two methods (compare the blue and black/redcurves in the top panel of Fig. 4), at a level of ≈ − inside r bias the concentration low, whereas those outside tend to increase it again, by ≈ Looking at the masses, a somewhat different pictureemerges: The values reconstructed from the spherical fit(blue lines) are considerably more tightly constrained witha typical scatter of only ∼
5% and a bias at a level of only ≈ − M /M Mill , which we show by green lines in Figs. 4 and5. The very close agreement between these indicates that thespherical WL mass reconstruction is as accurate as could behoped for. The negative bias in the ‘perfect’ and ‘default’simulations is thus due to the mass distribution outside r .As in the case of concentration, a likely explanation for thisbias is the deviation of the mass distribution outside ∼ r from an NFW profile (see, e.g., Fig. 1 of Hayashi & White2008). This leads to less mass along the line of sight thanexpected from an NFW profile extending to infinity, whichexplains the negative mass bias (see also Oguri & Hamana2011). r : Triaxialityand substructure We demonstrated above that most of the scatter and biasin reconstructed concentrations is due to deviations from aspherically symmetric NFW profile within r . The obviousculprits responsible for these deviations are asphericity (e.g.,triaxial haloes) and substructure. We now investigate therole played by each of these sources of error. To begin with, an overall sense of the validity of the(spherically averaged) NFW approximation is given by the M /M Mill distributions shown in Fig. 4 and 5 (green lines).With M Mill being a model-independent quantity, it providesa reference value for M which can be used as an indica-tion of how well the NFW profile describes a cluster. The(logarithmic) scatter and bias are both small at a level of ≈
5% and ≈ −
2% respectively. This confirms that, despitethe obvious presence of substructure and the overall triaxial-ity of the simulated clusters (see, e.g., Fig. 1), the sphericallyaveraged density is still well-described by the NFW profile.But while the 3D mass structure of the haloes may berelatively well described by an NFW profile for many clus-ters, the lensing deflection depends on (the gradient of) the projected density. For realistic, non-spherically-symmetriccluster haloes, deviations from symmetry can be expected toaffect 3D and 2D reconstructions differently, so it is entirelyplausible that the shear signal for a cluster, even one that iswell represented by an NFW profile in 3D, may not be welldescribed by a 2D profile derived from it.
We first look at the effect of halo triaxiality. Jing & Suto(2002) suggest a triaxial generalisation of the spherical NFWprofile by replacing the radius r in (9) by r eff where r = X a + Y + Z b (15)where X , Y and Z are the distances along the major, inter-mediate and minor axes respectively; the numbers a and b are the ratio of the major and minor axis to the intermediateaxis, respectively.In a similar way to our standard 3D fitting proceduredescribed above, we now also fit this triaxial NFW profileto our lensing haloes. Each halo is first subdivided into fiveconcentric shells covering a radial range from -1.5 to 0 inlog ( r/r ). Each shell is further divided into 24 sectorsof equal volume. Overall, this procedure divides the clusterinto 2880 cells, for each of which we compute the averagedensity ρ i . The triaxial NFW profile is then fit by least-squares regression .Applying this to the question of the influence of halotriaxiality on lensing reconstructions, we show, in the firstpanel of Fig. 6, how the angle δ between the line of sightand the major halo axis correlates with over- and under-prediction of the lensing cluster’s mass and concentration.Each point corresponds to a mock WL reconstruction of asimulated cluster; we include all simulated clusters with 5projections each. It is immediately obvious that projectionswith small δ lead to overpredictions in both mass and con-centration, the opposite being true for cases with δ ∼ o .The role of orientation becomes even more obvious whenwe move to the spherical WL fit described above and onlyanalyse the region inside r , as shown in the second panel We have also tried computing the axis ratios by diagonalisingthe moment-of-inertia tensor (see e.g., Shaw et al. 2006). In gen-eral the two methods produce results in the same sense (i.e., bothfavour “prolate-like” distributions), with the moment-of-inertiamethod generally yielding slightly smaller axis ratios than foundfrom our default method. c (cid:13) , 000–000 eak lensing simulated clusters Figure 6.
Reconstructed masses and concentrations, both normalised to “true” values ( M Mill and c respectively). The values in theleft-most panel are derived from our default WL analysis, in the other three from the spherical WL reconstruction based only on thematter distribution inside r . Note that the horizontal axes are scaled differently to account for the much smaller mass scatter in thelatter reconstruction. The colour shows, from left to right, the angle between the major halo axis and line of sight ( first two panels ), theratio between major and minor halo axis ( third panel ) and the fraction of the halo mass in bound substructures within r ( last panel ).The influence of halo orientation is clearly dominating the scatter. in Fig. 6. This reduces the scatter along the (1,-1) direc-tion considerably, identifying the influence of backgroundgalaxies on low-mass clusters as its main cause. The findingthat decreasing δ increases c WL and M WL of individual clus-ters simultaneously confirms previous work on the effect ofcluster triaxiality by Clowe, De Lucia & King (2004), whoanalysed a sample of four massive haloes and found a strongcorrelation between concentration and halo orientation, andCorless & King (2007) who derived a similar result usinganalytic cluster haloes. The latter authors concluded thattriaxiality could cause overpredictions in concentration bya factor of 2, in good agreement with the second panel ofFig. 6. The scatter in masses — deviations up to a factorof 1.5 — reported by these authors, on the other hand, isconsiderably larger than ours ( ∼ . r whereas our spherical WL analysis does not.Note that in the first two panels of Fig. 6 the magnitude of the ratio between major and minor axis was not taken intoaccount at all. We explore its influence in the third panel ofFig. 6 and find a much weaker correlation than with haloorientation. Haloes with extremely high axis ratios have atendency to lead to more over- or underpredicted masses andconcentrations, as should be expected, but the influence isclearly much smaller than that of orientation.We noted above that the concentration scatter decreases slightly upon inclusion of matter outside r . One possibleexplanation for this effect is that matter beyond r , whilestill correlated to some extent with the cluster major axis,is more randomly distributed (see also Becker & Kravtsov2011) and therefore reduces the impact of the triaxiality-induced concentration scatter a bit. While we have just demonstrated a strong influence of halotriaxiality on reconstructed NFW parameters of individualclusters , there remains some scatter which is not correlatedwith this. In the right-most panel of Fig. 6 we thereforeinvestigate the influence of substructure, quantified as thefraction f Sub of the halo mass which is found in boundsubhaloes within r identified by the Subfind algorithm(Springel et al. 2001). This fraction is generally small withonly 15% of haloes having f Sub > .
1. The plot shows someinfluence on the reconstructions, most notably on concentra-tion, which tends to be underpredicted in high-substructureclusters. However, as with the axis ratio, the influence ismuch less strong than that of cluster orientation.
We have shown above that the scatter in concentration canbe ascribed largely to the influence of halo orientation. How-ever, for a large sample of clusters, oriented randomly to-wards the observer, this leads to overestimation of concen-tration in some haloes, and underestimation in others (seeFig. 6). While this does not necessarily eliminate all influ-ence of triaxiality on the concentration bias even for arbi-trarily large cluster samples, it can be expected to reducedit to a level where the effect of substructure becomes non-negligible. We explore the role of both effects in this contextin Fig. 7. For this, the sample is split into five equally largequintiles, once in order of increasing axis ratio, and once bysubstructure fraction. For each of the 25 resulting combina-tions, the median concentration, normalised to c , is thenformed. In this way, the figure shows both the variation withaxis ratio at (nearly) fixed substructure level and vice versaand can therefore eliminate potential correlations between c (cid:13) , 000–000 Y.M. Bah´e et al.
Figure 7.
Influence of substructure fraction (different colourlines) and triaxiality (x-axis) on concentration bias (y-axis). It isevident that higher substructure and higher triaxiality both leadto lower concentrations, but the effect of substructure is stronger. these two effects. For added clarity, the black line shows theconcentration bias when the sample is split only accordingto axis ratio.There are two clear trends in this figure: A decrease inconcentration from left to right, corresponding to an increasein axis ratio (of ≈
5% in the case of the lowest substructurequintile), and from blue to red, as the substructure fractionincreases (of ∼
10% for the least elongated clusters). Bothtriaxiality and substructure therefore tend to lower the con-centration, but the influence of substructure appears to bedominant.While the addition of substructure to the outer partsof a smooth NFW halo can be expected to reduce the best-fit concentration of both 2D and 3D analyses, its influenceappears to be larger in 2D. This might be due to the lowsurface mass density in the projected outer cluster regions(see Fig. A1), which is affected more severely by the presenceof substructure than the mass profile obtained in 3D.
To summarise the results and interpretations of this section:Within r , substructure and triaxiality cause a negativebias ( ∼ − r , the “correlated large-scale structure”, biasesconcentration positively and mass negatively, due to devia-tions of the mass profile from the NFW model at these largeradii. The scatter in mass is strongly increased by matter inthis region, but the orientation-induced scatter in concen-tration is, to a small degree, cancelled out by it. Imperfectweak lensing observations smooth out part of the influenceof substructure and therefore cause a slight improvementin the mass and concentration bias. Shape noise, while notsolely responsible for the mass and concentration scatter, contributes considerably to it, in particular for lower massclusters. We have analysed a large set of mock WL observations ofgalaxy clusters at redshift z L ≃ . z S = 1 .
0. The aimof our work was to quantify the expected scatter and biasin the reconstructions of masses and concentrations of largecluster samples derived from WL surveys such as DES andLSST. We focus on the effect of the matter within andclose to the lensing clusters, explicitly ignoring the effectof uncorrelated large-scale structure (as studied, e.g., byHoekstra et al. 2011, see below). Furthermore, as we are in-terested in the bias and scatter induced by the clusters andthe weak lensing method itself, we have explicitly ignoredany additional observational sources of error, such as diffi-culties in measuring the background galaxy ellipticities andredshifts. Nevertheless, from the analysis of these mock ob-servations, and subsequent comparison with the true clusterparameters, we can draw the following conclusions: • We confirm that the dark matter haloes in our sample,although not selected using any relaxation criteria, are well-described by the NFW profile. The NFW masses M agreeto within ≈ σ ) with the true halo masses. • The mass-concentration relation derived from our mockobservations has a normalisation of α = (4 . ± .
04) andslope β = ( − . ± . α D =(5 . ± .
08) and β D = ( − . ± . • The spread in the WL parameters M WL /M Mill and c WL /c closely follows a log-normal distribution. Its scat-ter decreases with increasing halo mass, due to an increasedstrength of the lensing signal (for concentration from 0.57 to0.29, and for mass from 0.36 to 0.21 between clusters with M < . M ⊙ and M > . M ⊙ respectively). Bothmass and concentration show in general a negative medianbias at a level of ≈ −
3% and ≈ −
5% respectively. Due tothe presence of scatter, the mean mass and concentrationare generally higher than their median counterparts. • While shape noise due to the unknown intrinsic orien-tation of background galaxies is an important contributionto the parameter scatter (particularly for lower mass haloeswith M ∼ M ⊙ ), we find that physical properties ofthe clusters themselves (e.g., triaxiality, substructure, corre-lated line-of-sight matter) also contribute significantly andare the dominant source of scatter for haloes with massesof M ∼ M ⊙ . Triaxiality of the cluster halo within r , is the main contributor to the scatter in concentra-tions, alignment between major halo axis and the line-ofsight being the dominant factor. The concentration bias isdominated by substructure within the cluster. The bias andscatter in the recovered mass is affected mostly by matterbeyond r , i.e., correlated large-scale structure where de-viations from the NFW profile may be important. c (cid:13) , 000–000 eak lensing simulated clusters We note that this comparison between WL-derived and‘true’ 3D-derived concentrations is somewhat sensitive tothe radial fitting range used in determining the latter; asfound by Gao et al. (2008), a larger fitting range probingregions closer to the cluster centre tends to increase the best-fit concentration. In this work, we have used the smallestpossible truncation radius allowed by the resolution of thesimulation and the mass range of our clusters (0 . r ).For an accurate calibration of the bias and scatter expectedfrom a particular survey, it is therefore not only importantto accurately model the survey, but also the way in whichthe theoretical comparison data is derived. At the price ofintroducing a third fitting parameter, use of the Einastoprofile (Einasto 1965) may be a possible way to remove thisdependence on the radial fitting range.Our analysis of the physical origin of scatter and biassuggests that the most promising way to reduce the uncer-tainty in concentration measurements from WL is to employtriaxial halo models as suggested by Corless & King (2008).It is however important to keep in mind that not only theanalysis of observational data, but also the theoretical pre-dictions used for comparison need to use this triaxial modelbecause the halo ‘mass’ and ‘concentration’ in a triaxialmodel will in general be different from their counterpartsderived using spherical averaging.As stated above, in the present study we have not in-cluded the influence of uncorrelated large-scale structure atlarge distances from the cluster. Using spherical analyticNFW haloes, Hoekstra et al. (2011) have recently studied itseffect on the recovered masses and concentrations and theircorrelation. They found a small bias in the slope of the recov-ered mass-concentration relation, in the sense that the re-covered relation is slightly steeper than the true underlyingone. Additionally, their study addresses the influence of un-correlated large-scale structure on the scatter in both massand concentration: for a cluster of mass M = 10 M ⊙ thisis comparable to the effect of shape noise and can thereforebe expected to contribute significantly to the overall errorbudget, in particular for surveys extending to lower-massclusters. A combined study, taking into account both uncor-related and correlated error sources using realistic clusterswould be a valuable way to make even more realistic pre-dictions of the expected bias and scatter in WL parameterreconstructions.Another potentially important factor that should beexplored in future realistic mock weak lensing surveys isthat of realistic cluster selection. In the present study, wehave imposed a simple (true) halo mass cut to establish firstwhether or not the recovered masses and concentrations arebiased, which they are, for an underlying sample that isunbiased. Cluster selection itself (e.g., based on optical rich-ness, Sunyaev-Zel’dovich effect flux, X-ray luminosity, weaklensing shear signal), however, can potentially introduce bi-ases which may be larger than those we have studied hereand therefore it is important that these potential biases bequantified in the future.While our quantification of scatter and bias in weaklensing measurements was motivated by the use of the mass-concentration relationship to constrain cosmological param-eters, our results can also be applied to outliers in observedconcentration. For instance, we have shown that those clus-ters which are elongated along the line of sight are most liable to having their concentration overestimated in weaklensing studies. We can also make predictions for how many clusters we would expect to exceed a particular concen-tration. For high-mass system with M > . M ⊙ , forwhich the Millennium Simulation includes no systems above c = 5 .
7, we find that 7% of cluster projections yield weaklensing-derived concentrations of c WL >
6, and only 1% ex-ceed c WL = 8. We re-iterate at this point that our study hasnot taken into account uncorrelated line-of-sight structure,which might increase these fractions somewhat.As long as bias and scatter are properly accounted for,future WL surveys are expected to accurately determine themass-concentration relationship of galaxy clusters, even withmodest numbers of background galaxies per cluster. Theform of the mass-concentration relationship, and its evolu-tion with redshift, provide a sensitive probe of the growthof structure and therefore offer an important new and inde-pendent method of testing our cosmological paradigm. ACKNOWLEDGEMENTS
The authors thank the anonymous referee for their manyconstructive suggestions which significantly improved thepaper. The authors thank the Virgo Consortium for pro-viding the Millennium Simulation particle data and SimonWhite for helpful discussions. YMB acknowledges a post-graduate award from STFC. IGM is supported by a KavliInstitute Fellowship at the University of Cambridge. LJKis supported by a University Research Fellowship from theRoyal Society. This research has made use of the
Darwin
High Performance Computing Facility at the University ofCambridge.
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APPENDIX A: TRUNCATED NFW PROFILE
The NFW profile truncated at some limiting (3D) radius r lim (Takada & Jain 2003a, Takada & Jain 2003b) is ρ = ρ s ( r/r s )(1 + r/r s ) Θ( r lim − r ) (A1)where Θ is the Heaviside Unit Step function and ρ s and r s the scale density and length as defined in section 3.3.The projected mass density of this profile integratedalong the line of sight is then given byΣ( R ) = 2 ρ s r s Z r lim R √ r − R (1 + r/r s ) dr (A2)where R is the distance from the cluster centre perpendicularto the line of sight.Evaluating this integral yieldsΣ = 2 δ c ρ c r s S (A3)where S depends on the value of x = r/r s .For x < S = ln (cid:18) x ( r s + r lim ) r lim + r s x − √ (1 − x )( r − r s x ) (cid:19) − √ (1 − x )( r − r s x ) r s + r lim (1 − x ) / (A4)For x = 1: S = ( r lim − r s )(2 r s + r lim )3( r s + r lim ) p r − r s (A5)For x > S = − arctan (cid:18) √ ( r − r s x )( x − r lim + r s x (cid:19) + √ ( x − r − r s x ) r s + r lim ( x − / (A6) APPENDIX B: EFFECT OF VARIATIONS INLENSING SIMULATIONB1 Line-of-sight integration length
The effect of increasing the line-of-sight integration lengthfrom our default value of 10 h − Mpc by a factor of 5 to50 h − Mpc is explored in Fig. B1. We find very little varia-tion between the two lengths, consistent with the findings byBecker & Kravtsov (2011). Our default integration length of10 h − Mpc is therefore sufficiently large to capture the in-fluence of correlated large-scale structure close to the lensingcluster. c (cid:13) , 000–000 eak lensing simulated clusters Figure A1.
Surface density of an NFW profile integrated alongthe line of sight. The solid black line corresponds to an untrun-cated infinite profile, whereas the blue dashed and red dash-dotlines show profiles truncated at 2 r and r respectively. Trun-cation mostly affects the surface density in the outer cluster re-gions beyond ∼ . r lim . B2 Background galaxy ellipticity
In Fig. B2 we investigate the effect of increasing the value ofthe background galaxy ellipticity dispersion from our defaultof 0.2 to 0.3 and 0.4 per component as well as decreasing itto zero. As can be expected, the scatter in both mass andconcentration increases with increased shape noise. There isalso a very small influence on the mass and concentrationbias, in the sense that a higher ellipticity dispersion leadsto higher bias. As discussed in section 5.1.2, this is mostlikely due to the fact that shape noise, in combination withfinite sampling, smoothes out the influence of triaxiality andsubstructure, which both tend to cause a negative bias.
B3 Weak lensing survey range
Finally, we show in Fig. B3 the effect of varying the ra-dial survey range from our default of 30 ′′ to 15 ′ . The biasesand mass scatter are only affected at a level of a few per-cent. The concentration scatter is somewhat more sensitive,particularly on the choice of inner cut-off radius: increas-ing this to 1 ′ causes a scatter increase by ≈
30% for thelowest-mass clusters. In general, the concentration appearsto be more sensitive to the value of the inner than the outercut-off radius, the opposite being true for mass. For ease ofcomparison with Becker & Kravtsov (2011), we also includean analysis covering the radial range used in their study, inwhich we form a shear profile from 15 bins spaced logarith-mically between 1 ′ and 20 ′ which is then fit to the NFWmodel. There is hardly any difference between the resultsfrom this fit and our default method in which we use eachtangential galaxy ellipticity individually. Figure C1.
Difference between bias definition as median (ourdefault, thick lines) or mean (thin lines) on concentration ( top )and mass ( bottom ). The mean shows, in general, a clear positivebias.
APPENDIX C: VARIATION IN BIASDEFINITION
In Fig. C1, the difference between defining bias as the me-dian or mean is shown. While both are identical in the caseof a Gaussian (normal) distribution, this is not generally thecase for a log-normal parameterisation as employed here (seeFig. 3). In terms of the parameters µ and σ as defined inequation (14), the median is given by˜ x = e µ (C1)and the mean by < x > = e µ + σ / . (C2)Thus, while the median depends only on µ , the mean is ad-ditionally sensitive to the scatter σ . This is confirmed byFig. C1: Distributions with large scatter like concentrationin our default analysis (black lines, top panel), have a meanup to ≈
20% higher than the median. For the very well con-strained spherical WL mass distribution (blue lines, lowerpanel), on the other hand, mean and median are virtuallyidentical. c (cid:13) , 000–000 Y.M. Bah´e et al.
Figure B1.
Effect of increasing the line-of-sight integration length to 50 h − Mpc, shown by solid lines, compared to our default choiceof 10 h − Mpc (broken lines) on concentrations ( left ) and masses ( right ). There is very little variation, neither for high-mass systemswith M > . M ⊙ ( top ), nor for low-mass systems with M < . M ⊙ ( bottom ). c (cid:13)000
Effect of increasing the line-of-sight integration length to 50 h − Mpc, shown by solid lines, compared to our default choiceof 10 h − Mpc (broken lines) on concentrations ( left ) and masses ( right ). There is very little variation, neither for high-mass systemswith M > . M ⊙ ( top ), nor for low-mass systems with M < . M ⊙ ( bottom ). c (cid:13)000 , 000–000 eak lensing simulated clusters Figure B2.
Effect of increasing the dispersion in the intrinsic ellipticities of the background galaxies from our default value of 0.2 percomponent (black, solid line) to 0.3 (red, dashed line) and 0.4 (blue, dot-dashed line) per component respectively. The top two panelsshow the effect on the concentrations, the bottom two on the masses. In both cases medians are shown on the left and scatter on theright; note the different ordinate scalings for the mass and concentration scatter plots. Increased shape noise causes a small positive bias,due to smoothing out of substructure and triaxiality, and a strong increase in scatter.c (cid:13) , 000–000 Y.M. Bah´e et al.
Figure B3.
Effect of varying the inner and outer cutoff radius ( r in and r out respectively); note the different ordinate scalings for themass and concentration scatter plots. Varying r in influences mostly the concentrations, whereas the masses are more sensitive to thevalue of r out . For the first three analyses (black solid, red dashed and blue dash-dot lines), we use our default method of using each galaxyellipticity individually, whereas the last one (purple dash-dot-dot-dot lines, 1 ′ - 20 ′ ) uses 15 logarithmically spaced bins to reproduce theanalysis setup of BK11. There is no indication for significant differences between these two methods.c (cid:13)000