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arXiv:0801.1875v3 [astro-ph] 19 Aug 2008

Mon. Not. R. Astron. Soc. 000, 1–9 (2008)Printed 19 August 2008(MN LATEX style file v2.2)

Large Einstein Radii: A Problem for ΛCDM

Tom J. Broadhurst1and Rennan Barkana2,3⋆

1Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel

2Institute for Cosmic Ray Research, University of Tokyo, Kashiwa 277-8582, Japan

3Guggenheim Fellow; on sabbatical leave from the School of Physics and Astronomy, Tel Aviv University, Israel

19 August 2008

ABSTRACT

The Einstein radius of a cluster provides a relatively model-independent measure

of the mass density of a cluster within a projected radius of ∼ 150 kpc, large enough

to be relatively unaffected by gas physics. We show that the observed Einstein radii

of four well-studied massive clusters, for which reliable virial masses are measured,

lie well beyond the predicted distribution of Einstein radii in the standard ΛCDM

model. Based on large samples of numerically simulated cluster-sized objects with

virial masses ∼ 1015M⊙, the predicted Einstein radii are only 15−25′′, a factor of two

below the observed Einstein radii of these four clusters. This is because the predicted

mass profile is too shallow to exceed the critical surface density for lensing at a sizable

projected radius. After carefully accounting for measurement errors as well as the

biases inherent in the selection of clusters and the projection of mass measured by

lensing, we find that the theoretical predictions are excluded at a 4-σ significance.

Since most of the free parameters of the ΛCDM model now rest on firm empirical

ground, this discrepancy may point to an additional mechanism that promotes the

collapse of clusters at an earlier time thereby enhancing their central mass density.

Key words:

cosmological parameters – dark matter

galaxies: clusters: general – cosmology:theory – galaxies:formation –

1 INTRODUCTION

The standard picture of the basic cosmological framework

has recently come to rest firmly on detailed empirical ev-

idence regarding the cosmological parameters, the propor-

tions of baryonic and non-baryonic dark matter, together

with the overall shape and normalization of the power spec-

trum (e.g., Astier et al. 2006; Spergel et al. 2007; Perci-

val et al. 2007). This framework has become the standard

ΛCDM cosmological model, with the added simple assump-

tions that the dark matter reacts only to gravity, is initially

sub-relativistic, and possesses initial density perturbations

which are Gaussian distributed in amplitude. This is a very

well defined and relatively simple model, with clear predic-

tions which are amenable to examination with observations.

The cooling history of baryons complicates the interpreta-

tion of dark matter on galaxy scales, especially for dwarf

galaxies that traditionally have been a major focus of stud-

ies of halo structure. Clusters have the advantage that the

virial temperature of the associated gas is too hot for effi-

cient cooling, so the majority of the baryons must trace the

overall gravitational potential and hence we may safely com-

⋆E-mail: tjb@wise.tau.ac.il; barkana@wise.tau.ac.il

pare lensing-based cluster mass measurements to theoretical

predictions that neglect gas physics and feedback.

Lensing-based determinations of the mass profiles of

galaxy clusters rely on detailed modeling of the strong lens-

ing region to define the inner mass profile, and also a care-

ful analysis of the outer weak lensing regime. The latter

involves substantial corrections for instrumental and atmo-

spheric effects (Kaiser et al. 1995), and a clear definition of

the background, free of contamination by the lensing clus-

ter (Broadhurst et al. 2005b; Medezinski et al. 2007). In

the center we may make use of the Einstein radius of a

cluster which is often readily visible from the presence of

giant arcs and provides a relatively model-independent de-

termination of the central mass density. In the case of axial

symmetry, the projected mass inside the Einstein radius θE

depends only on fundamental and cosmological constants:

M(< θE) = θ2

tion of angular diameter distances (observer-lens, observer-

source, and lens-source) leads to a relatively weak depen-

dence on the lens and source redshifts. More generally, an

effective Einstein radius can be defined by axially averag-

ing the projected surface density, which itself is well deter-

mined when there are a large number of constraints. Vir-

tually all known massive clusters at intermediate redshifts,

E(c2/4G)DOLDOS/DLS, where this combina-

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2Tom J. Broadhurst and Rennan Barkana

0.15 < z < 0.8, show multiple images including obvious arcs

in sufficiently deep high-resolution data. The derived Ein-

stein radius of these massive clusters typically falls in the

range 10′′< θE < 20′′(Gioia et al. 1990; Smith et al. 2005),

with the largest known case of ∼ 50′′for A1689.

Increasingly large simulations have helped to specify the

evolution of the halo mass function and the form of the

mass profile predicted in the context of the ΛCDM model

(Navarro et al. 1997; Bullock et al. 2000). These simulations

are now becoming sufficiently large and detailed to define the

predicted spread of halo structure over a wide range of halo

mass, and to quantitatively assess the inherent bias in ob-

serving clusters in projection and selecting them by lensing

cross-section (Hennawi et al. 2007; Neto et al. 2007). For the

most massive collapsed objects in these simulations (virial

mass Mvir ∼ 1015M⊙), a mean observed concentration of

c200 ∼ 6 is predicted for lenses, where c200 is defined pre-

cisely in the next section. Such profiles are relatively shallow

and seem at odds with recent careful lensing studies of mas-

sive clusters; although the Navarro et al. (1997) (hereafter

NFW) profile provides acceptable fits to the observations,

relatively high concentrations of cvir ∼ 10 − 15 are derived

for several well-studied massive clusters (Kneib et al. 2003;

Gavazzi et al. 2003; Broadhurst et al. 2005b; Kling et al.

2005; Limousin et al. 2007; Bradaˇ c et al. 2007; Halkola et al.

2008; Umetsu & Broadhurst 2008). These values are larger

than expected based on simulations of the standard ΛCDM

model. Given the relatively shallow mass profile predicted

for cluster-mass CDM halos, the question arises whether the

projected critical surface density for lensing can be exceeded

within a substantial radius for this model.

In this paper we compare observations of well-

constrained massive clusters with the predictions of ΛCDM

simulations. The idea is to compare directly the projected

2-D mass distributions in the observations and the simu-

lations. In the observations, the projected surface density

is obtained from the lensing analysis, which we emphasize

does not assume any symmetry of the lensing mass. In the

simulations, the 2-D density is directly measurable from the

simulation outputs. We summarize each 2-D density distri-

bution with two defined quantities, the virial mass and the

effective Einstein radius. In the simulations, the 2-D density

distribution of each halo was already axially averaged by

fitting to a projected NFW profile, so we use this profile to

obtain the effective θE (which, consistently, is also defined

through an axial average). The virial mass is measured di-

rectly in the simulations (with its usual spherically-averaged

definition). In the observations, we obtain Mvir directly in

A1689 (though in projection), and measure it using NFW

fits to the final distribution in the other clusters.

This paper is structured as follows. In section 2 we first

summarize the theoretical predictions, including a brief re-

view of the NFW profile and its lensing properties, and of

the halo concentrations measured by Neto et al. (2007) and

Hennawi et al. (2007) in large numerical simulations. Note

that the conflict between high observed concentrations and

lower ones determined for the numerical halos was noted in

both of these papers (see also Williams et al. (1999)). We

then present the observational data for the four clusters,

followed by a model-independent method for measuring the

mass, which we apply to A1689. In section 3 we confront the

theoretical predictions with the data, finding a clear discrep-

ancy. We discuss the possible implications in section 4.

2THEORETICAL AND OBSERVATIONAL

INPUTS

2.1Theoretical Predictions

Our calculations are made in a cold dark matter plus cos-

mological constant (i.e., ΛCDM) universe matching observa-

tions (Spergel et al. 2007), with a power spectrum normaliza-

tion σ8 = 0.826, Hubble constant H0 = 100h km s−1Mpc−1

with h = 0.687, spectral index n = 0.957, and present den-

sity parameters Ωm = 0.299, ΩΛ = 0.701, and Ωb = 0.0478

for matter, cosmological constant, and baryons, respectively.

Unless otherwise indicated, we use physical units that al-

ready include the proper factors of h or h−1, always with

h = 0.687.

Consider a halo that virialized at redshift z in a flat

universe with Ωm+ ΩΛ = 1. At z, Ωm has a value

Ωz

m=

Ωm(1 + z)3

Ωm(1 + z)3+ ΩΛ

,(1)

and the critical density is

ρz

c=3H2

0

8πG

Ωm(1 + z)3

Ωz

m

.(2)

The mean enclosed virial density in units of ρz

and used to define the virial mass and radius in observations

and in simulations. Sometimes a fixed value is used, such as

∆c = 200, although the theoretical value is ∆c = 18π2≃ 178

in the Einstein-de Sitter model, modified in a flat ΛCDM

universe to the fitting formula (Bryan & Norman 1998)

cis denoted ∆c

∆c = 18π2+ 82d − 39d2,

where d ≡ Ωz

z thus has a (physical) virial radius

(3)

m− 1. A halo of mass M collapsing at redshift

rvir= 1.69

?

M

1015M⊙

?1/3?

Ωmh2

Ωz

m

∆c

18π2

?−1/3

1

1 + zMpc .(4)

Numerical simulations of hierarchical halo formation in-

dicate a roughly universal spherically-averaged density pro-

file for virialized halos (Navarro et al. 1997), though with

considerable scatter among different halos (e.g., Bullock et

al. 2000). The NFW profile has the form

ρ(r) = ρz

c

δc

x(1 + x)2,(5)

where x = r/rs in terms of the NFW scale radius rs =

rvir/cvir, and the characteristic density δc is related to the

concentration parameter cvir by

δc =∆c

3

c3

vir

ln(1 + cvir) − cvir/(1 + cvir).

For a halo of mass M at a given redshift z, the profile is

fixed once we know ∆c and cvir. In this paper we denote the

concentration parameter, virial radius and mass by cvir, rvir

and Mvir when using the theoretical value in equation (3),

and by c200, r200, and M200, respectively, when using ∆c =

200.

The lensing properties of a halo are determined by κ, the

(6)

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Large Einstein Radii: A Problem for ΛCDM3

projected surface density Σ measured in units of the critical

surface density Σcr = [c2/(4πG)]DOS/(DOLDLS). For an ax-

isymmetric lens, the Einstein radius (i.e., tangential critical

curve) occurs at a projected radius R where the mean en-

closed surface density satisfies ¯ κ(R) = 1. For an NFW halo,

letting X = R/rs we have ¯ κ(X) = (4/Σcr)ρz

where (Bartelmann 1996)

cδcrsg(X)/X2

g(x) = lnx

2+

1 ,x = 1

2

√

x2−1tan−1?

2

1−x2tanh−1?

x−1

x+1, x > 1

√

1−x

1+x, x < 1

. (7)

We derive the theoretical predictions for cluster lens-

ing in ΛCDM by combining the two largest studies of halo

structure in cosmological numerical simulations. Neto et al.

(2007) studied halo structure within the Millennium simula-

tion, in which over 2000 halos formed with M200 > 1014M⊙

at z = 0. Each halo was resolved with > 80,000 particles, al-

lowing a detailed look at its three-dimensional density struc-

ture using NFW profile fitting. This is therefore the best

available statistical analysis of the cluster halo population

in ΛCDM simulations. However, since Neto et al. (2007)

did not study projected halo profiles or the bias in selecting

lenses, we must adjust their results in order to apply them to

lensing. Hennawi et al. (2007) studied 900 simulated cluster

halos at z = 0.41, each resolved into at least 30,000 parti-

cles. They studied projections through on average 15 ran-

dom directions per cluster (more – up to 125 – for the most

massive ones), and fit both 3-D and projected 2-D NFW

profiles. Moreover, they separately studied the distribution

of NFW profile parameters both for the general halo popu-

lation and for the lensing population (i.e., where halos are

weighted by their strong lensing cross section). They showed

that the inherent triaxiality of CDM halos along with the

presence of substructure enhance the projected mass in some

orientations, leading to a bias in the 2-D structure of lenses

compared with the 3-D structure of the general population

of cluster halos (see also Oguri et al. (2005)).

Figure 1 shows various biases in halo concentration pa-

rameters cvir as measured by Hennawi et al. (2007). The

Figure shows (left panel) that the distribution of 3-D con-

centrations of the lens population is the same as that of

the general halo population except for a shift upwards by

a factor of 1.17. For a given real 3-D profile, the 2-D pro-

file measured in projection depends on the orientation, and

is thus given by a probability distribution. The Figure also

shows (right panel) that the ratio c2−D/c3−D follows a log-

normal distribution, i.e., that log10of this ratio is well fitted

by a Gaussian with a mean value of 0.057 and σ = 0.124,

which correspond to factors of 1.14 and 1.33, respectively.

At z = 0.41, r200 and c200 are typically ∼ 15% smaller

than rvir and cvir, respectively, and M200 ≈ 0.9Mvir. We as-

sume that the relative distributions shown in Figure 1 are

approximately the same for c200 as for cvir, and that they are

independent of halo mass within the narrow range consid-

ered, as found by Hennawi et al. (2007). Thus, we can apply

these findings to convert the 3-D c200distributions measured

by Neto et al. (2007) in order to obtain the resulting 2-D

projected distributions of c200 values for the population of

cluster lenses. Specifically, we multiply the c200 values by a

factor of 1.17 (the lensing bias) and then convolve with the

Figure 1. Probability distributions of various concentration pa-

rameters, based on Hennawi et al. (2007). Left panel: When the

concentration parameters are measured relative to the median at

each halo mass (Figure 8 of Hennawi et al. (2007)), the distri-

bution for the general population (solid histogram) matches that

for the lens population when the latter is divided by a factor

of 1.17 (dashed histogram). Right panel: For lensing halos, the

distribution of log10of the ratio of the 2-D to the 3-D concen-

tration (solid histogram) is well fitted by a Gaussian with the

same mean and variance as the histogram (dashed curve). Note

that the non-uniform binning is a result of our conversion of the

linear-axis histogram from Figure 12 of Hennawi et al. (2007) to

one with a logarithmic x-axis.

distribution of c2−D/c3−D for lenses (the projection bias).

Note that we use the values measured by Neto et al. (2007)

at z = 0 when comparing to the observed clusters at red-

shifts z = 0.18 − 0.40. This is a conservative assumption,

since the typical concentration parameter at a given halo

mass declines with redshift. Studies based on large numer-

ical simulations (Zhao et al. 2003; Gao et al. 2007) found

for massive halos a relatively weak decline of ∼ 20% out to

z = 1, which suggests a 5−10% decline out to the redshifts

we consider below, a decline which we do not include here.

In Figure 2 we show the predicted Einstein radii of

ΛCDM cluster halos of mass M200 > 1014M⊙ based on Neto

et al. (2007) and Hennawi et al. (2007). Neto et al. (2007)

divided their halos within each mass bin into “relaxed” and

“unrelaxed” groups of halos, the latter identified as being

disturbed dynamically as indicated by a large amount of

substructure, a large offset between the center of mass and

the potential center, or a high kinetic energy relative to po-

tential. For each group, they found that the statistical dis-

tribution of the concentration parameters was well-fitted by

a lognormal distribution. Thus, after correcting for lensing

and projection bias as explained above, the resulting distri-

butions remain lognormal. As shown in Figure 2, the me-

dian c200 of each group is only weakly dependent on mass,

showing a slight trend of decreasing concentration with in-

creasing halo mass (a trend which is more apparent over

Page 4

4 Tom J. Broadhurst and Rennan Barkana

Figure 2. Dependence of θE (bottom panel), the median c200

(middle panel), and the scatter σlog10c200(top panel) on halo

mass M200 in numerical simulations. We use the 3-D analysis

by Neto et al. (2007) corrected for lensing and projection bias

based on Figure 1. We consider the relaxed (triangles) or unre-

laxed (squares) halo populations. We assume the median c200and

the A1689 redshifts when calculating rE. We show several linear

least-squares fits to help discern trends (solid curves). Also shown

for comparison (narrow box, bottom panel) is the location cor-

responding to the observations of A1689 (where the boxed area

contains the two-sided 1-σ ranges of M200 and θE).

the broader range considered by Neto et al. (2007), down

to M200 = 1012M⊙). As a result, we find that θE ∝ M200

(relaxed) and θE ∝ (M200)1.6(unrelaxed). Note that for a

lognormal distribution, the median and mean are theoret-

ically the same if log10c200 is considered rather than c200.

As the Figure shows, the typical scatter in log10c200 among

halos of the same mass is also fairly independent of mass,

except for the highest mass bin, M200 > 1015M⊙. While this

bin is based on a somewhat small sample (8 relaxed and 11

unrelaxed halos), Neto et al. (2007) suggest that the lower

dispersion is expected since the highest-mass halos are very

rare, thus all formed very recently and should have similar

merger histories and thus internal structures. Note that the

final, effective scatter in our calculations is only ∼ 10% lower

for the last bin compared with the lower mass bins, since we

assume that the projection scatter is independent of halo

mass.

2.2Observational Data

For our data set of lensing clusters we choose four well-

studied clusters with strong constraints available both from

multiply-imaged arcs in the strong-lensing regime and from

distorted arcs and magnification measurements in the weak-

lensing regime. We determine an effective Einstein radius

in each cluster using the 2-D projected mass distribution

obtained from fitting (without assuming symmetry) to the

Table 1. Observational Data

Cluster

Mvir[M⊙]

M200 [M⊙]

θE[′′]

zL

zS

A1689

Cl0024-17

A1703

RXJ1347

1.6 × 1015

8.7 × 1014

1.0 × 1015

1.3 × 1015

1.5 × 1015

8.0 × 1014

9.0 × 1014

1.2 × 1015

52

31

32

35

0.183

0.395

0.258

0.45

3

1.7

2.8

1.8

large number of sets of multiple images; from the obtained κ

we then define θE in the standard way, as the radius enclos-

ing a mean surface density equal to the critical surface den-

sity for lensing. Note that this definition effectively axially

averages the mass distribution. We can define θE relative to

any source redshift, and in each cluster we choose a fiducial

redshift that matches a prominent arc system. Our proce-

dure for determining θEis also a good match to that followed

by Hennawi et al. (2007) for their simulated clusters, where

they fit an axially symmetric model to the projected mass

distribution; we obtain the predicted effective Einstein radii

of the simulated clusters based on their model fits, as de-

tailed in the previous subsection. For the observed clusters,

we also obtain effective virial masses from combined axially-

symmetric fits to the strong and weak lensing data.

Very deep HST/ACS imaging of several massive lensing

clusters has been obtained by the ACS/GTO team (Ford

et al. 1998). This includes the well-studied cluster A1689

(z = 0.183) for which several comprehensive strong lens-

ing analyses have been published. The analysis of Broad-

hurst et al. (2005a) identified over 100 multiply lensed im-

ages of 30 background galaxies by an iterative procedure

in which multiple images are securely identified by delens-

ing and relensing background galaxies. These multiple im-

ages have been used as a basis for other types of modeling,

including fully parametric (Halkola et al. 2006; Zekser et

al. 2006; Limousin et al. 2007) and non-parametric mod-

eling (Diego et al. 2005). Broadhurst et al. (2005b) com-

bined these data on the inner mass profile with wide field

archival Subaru images and obtained a detailed radial mass

profile for the entire cluster, finding an NFW-like projected

mass profile with a surprisingly high value for the concen-

tration parameter (cvir = 13.7 ± 1.2), and a virial mass of

Mvir = (1.9 ± 0.2) × 1015M⊙ (the value in Table 1 is differ-

ent; see the next subsection). This yields an effective Ein-

stein radius of 52′′at a fiducial redshift z = 3, in good

agreement with the radius of ∼ 50′′found for prominent

multiple arcs at this redshift. We adopt a conservative error

estimate of 10% on θE for all the clusters (Table 1). Note

that an Einstein radius of only 31′′is implied by the NFW fit

of Limousin et al. (2007) to independent weak lensing data

from CFHT, but this number is incompatible with their own

strong lensing analysis and is possibly caused by contami-

nation of the lensing signal by unlensed cluster members;

if not thoroughly excluded, this contamination will reduce

the lensing signal preferentially towards the cluster center,

resulting in a flatter mass profile, as pointed out by Broad-

hurst et al. (2005b).

Another well studied cluster that we use here is Cl0024-

17 (z = 0.395), with an effective Einstein radius of 31′′at

z = 1.7 defined by several sets of multiple images iden-

tified in the ACS/GTO images. This agrees in particular

with the mean radius of the famous 5-image system of “θ”

Page 5

Large Einstein Radii: A Problem for ΛCDM5

arcs at a spectroscopically measured redshift of z = 1.685

(Broadhurst et al. 2000). This set of multiple images and

the distortion measurements of background galaxies with

photometric redshifts has been used by Jee et al. (2007) to

constrain the inner mass profile. Their result is in general

consistent in form with the earlier analysis of the inner pro-

file by Broadhurst et al. (2000), but with the addition of a

narrow low-contrast ring that, it is claimed, can be repro-

duced in simulations where merging of two massive clusters

occurs along the line of sight. A line-of-sight merger is also

blamed for the relatively small central velocity dispersion

(Czoske et al. 2002). However, in the extensive weak lensing

analysis of Kneib et al. (2003) only one small subgroup is

visible, offset by 3′in projection from the center of mass and

accounting for only ∼ 15% of the total mass of the cluster.

Kneib et al. (2003) find that the main cluster is well fitted

by an NFW profile with a virial mass of ∼ 6 × 1014M⊙/h

and with a high concentration, cvir ∼ 20. In a more recent

analysis of deep multicolour B,R,Z Subaru images, Medezin-

ski et al. (2007, in preparation) find good agreement with

Kneib et al. (2003) (Table 1).

In addition, we use the very deep ACS/GTO images

of the massive cluster A1703 for which many sets of mul-

tiple images are visible, so that the tangential critical line

is easily identified with a mean Einstein radius of 32′′at

z = 2.8, in good agreement with the radius of the main

giant arc at a similar redshift (Table 1). In the weak lens-

ing analysis of Medezinski et al. (2007, in preparation) a

very good fit to an NFW profile is found with a virial mass

Mvir = 7 × 1014M⊙/h. This cluster appears relaxed and

centrally concentrated, with little obvious substructure. To

date, deep X-ray imaging is unfortunately missing.

Finally, a weak lensing analysis of RXJ1347 (Medezin-

ski et al. 2007, in preparation) shows this cluster to have a

very circular shear pattern, with an estimated virial mass

Mvir = 9 × 1014M⊙/h based on an NFW fit to the radial

distortion profile. This cluster has the highest observed X-

ray temperature of 13 keV and a symmetric X-ray emission

map that indicates that it is relaxed (Vikhlinin et al. 2002).

A very symmetric distribution of arcs is visible around the

cluster center, implying a well-determined Einstein radius

at z = 1.8 of 35′′from the full model, a value which is also

in agreement with a system of 5 multiply-lensed images at

this redshift (Halkola et al. 2008).

These four clusters are particularly useful for our pur-

pose, by virtue of their well-defined Einstein radii and pre-

cise measurements of the virial masses, which allows a com-

parison with the theoretical predictions as a function of halo

mass. We convert Mvir to M200 for each cluster using the

measured value of cvir, and adopt error bars of ±15% on

M200 and ±10% on θE for all four clusters (but see the next

subsection for an alternative measurement of the virial mass

of A1689). It is also interesting to note the many examples

of strong lensing by other galaxy clusters for which the to-

tal mass is not so well constrained. Samples of clusters de-

fined by some reasonable criteria (Smith et al. 2005; Sand

et al. 2005; Comerford et al. 2006) show that invariably the

observed Einstein radius (when detected) for intermediate

redshift clusters does not fall short of 10′′, with a mean

of ∼ 15′′. This may be compared with the predicted typ-

ical Einstein radius of only ∼ 5′′from the simulations of

Neto et al. (2007) and Hennawi et al. (2007) for a cluster of

M200 = several ×1014M⊙ (Figure 2).

2.3Model-Independent Mass

Of the two observational quantities we use to characterize

each cluster, θE is more directly estimated, from the posi-

tions of multiple images. The mass M200 requires a mea-

sured mass profile out to large angles, which can be used

to estimate the angular position corresponding to r200, i.e.,

to an enclosed relative density of 200 times the critical den-

sity. Deep images provide a large density of weakly-lensed

background sources, but weak lensing distortions measure

only the reduced shear and suffer from the well-known mass-

sheet degeneracy. This means that the mass profile can be

measured without degeneracies only by fitting a particular

parametrized density profile model to the data. However,

combining lensing distortions with observations of the vari-

ation in the number density of background sources due to

weak magnification breaks the degeneracy and yields a di-

rect measurement of the projected surface density in each

radial bin (Broadhurst et al. 2005b). Given such indepen-

dent measurements out to large radius, we can derive the

corresponding value of M200 directly from the data, with-

out the intermediary of an assumed model profile, the use of

which inevitably introduces a non-trivial systematic error.

Such accurate measurements are available for A1689, which

we use to illustrate the method, and such data should be ob-

tainable for other clusters as well. We note that this effective

virial mass is defined from deprojecting the projected mass

assuming spherical symmetry, which is the closest lensing

observations can come to the standard theoretical definition

of the virial mass based on a 3-D spherical average.

Lensing by a halo can be analyzed by calculating κ =

Σ/Σcr. The projected surface density is related to the three-

dimensional density ρ by an Abel integral transform. This

implies a relation between the integrated three-dimensional

mass M(r) out to radius r and κ(R) as a function of the

projected radius R:

M(r) = Σcr

?

2π

?r

0

Rκ(R)dR − 4

?∞

r

Rκ(R)f

?R

r

?

dR

?

,(8)

where

f(x) =

1

√x2− 1− tan−1

The first term in equation (8) is the total projected mass

within a ring of projected radius r, and the second term

removes the contribution from mass elements lying at a 3-D

radius greater than r.

To obtain the 3-D mass profile M(r), we apply equa-

tion (8) to the 26 values of κ(R) measured by Broadhurst et

al. (2005b) over the range R = 0.015 − 2.3 Mpc in A1689.

Specifically, we linearly interpolate κ(R) between each pair

of measured points, and extrapolate outside the range. We

extrapolate inward assuming κ(R) = const from the inner-

most point and outward assuming κ(R) ∝ R−2from the

outermost point, where these power laws are motivated by

the NFW profile. However, even varying these powers by ±1

would change the virial mass by only 1%, which is negligible

compared with the effect of the measurement errors. Once

1

√x2− 1

. (9)