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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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2 Tom 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.
2 THEORETICAL AND OBSERVATIONAL
INPUTS
2.1 Theoretical 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
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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.
2.2 Observational 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)
Page 6
6 Tom J. Broadhurst and Rennan Barkana
we have obtained M(r) at all r, we interpolate to find the ap-
propriate r that yields a desired mean enclosed density, and
thus determine the virial radius and mass. The error anal-
ysis is complicated by the fact that the virial radius is not
fixed but rather is itself determined by the data. Thus, to en-
sure self-consistent errors we use a Monte Carlo approach,
generating 1000 random profiles of κ(R) according to the
measurement errors (assumed to be Gaussian distributed
except that κ is constrained to be non-negative). For each
κ(R) profile we find the resulting virial quantities, and then
find the 16%, 50% (median), and 84% percentiles. The result
of this direct, model-independent analysis of A1689 is:
rvir= 2.76 ± 0.2 Mpc, Mvir = (1.6 ± 0.4) × 1015M⊙,
r200 = 2.25 ± 0.2 Mpc, M200 = (1.5 ± 0.4) × 1015M⊙.
The median value obtained for M200 is lower by ∼ 15%
than the best-fit value from the NFW profile. This is con-
sistent with our model-independent analysis of A1689 in
Lemze et al. (2007), where we found that the NFW fit
clearly overshoots the observed density profile at large radii.
Also, the model-independent error of ∼ 25% is larger since
assuming an NFW profile puts a constraint on the den-
sity fluctuations and yields a reduced error. The 25% er-
ror results from allowing completely independent varia-
tions in the κ measurements at different radii, so the er-
ror would be reduced with even a weak assumption of
smoothness in the density profile. Here we adopt the model-
independent mass along with its conservatively large 25%
error (Table 1). The model-independent mass is the most
reliable observationally-determined mass and is more con-
sistent with a comparison to numerical simulations, where
the real virial masses are known without the need to resort
to profile fitting.
The most important assumption in our analysis is
spherical symmetry. Since the measured κ profile goes out
to ∼ r200, what we directly measure includes the full contri-
bution of M200 plus additional projected mass coming from
larger radii. If the halo has non-spherical structure such as
triaxiality, then lensing bias means that the contribution
of mass elements outside the virial radius to the projected
mass will tend to be unusually high in our direction. How-
ever, since we get that contribution by extrapolating from
the measured κ points (which are also enhanced by lensing
bias), the error in our spherical assumption may be small.
The real conclusion is that since the 3-D virial mass is not
directly observable, lensing analyses of halos in numerical
simulations should measure the effective, projected ‘virial’
mass defined by applying equation (8) to the projected pro-
file. This would allow a truly direct comparison with the
observations.
3 CONFRONTING ΛCDM WITH
OBSERVATIONS
For each cluster with known redshifts zL and zS and a reli-
ably measured value of M200, we can use the results of sec-
tion 2.1 to calculate the predicted value of θE in the ΛCDM
model. Figure 3 compares the median expected value with
the observed value for each cluster, showing that in each case
the theoretical expectation falls short of the observed value
Figure 3. Dependence of the Einstein radius θEon the redshifts
zS (left panel) and zL (right panel). We consider A1689 (solid
curves, circles), A1703 (short-dashed curves, squares), Cl0024
(long-dashed curves, triangles), and RXJ1347 (dotted curves,
×’s). In each case, the points correspond to the observed clus-
ter (with a vertical bar indicating the measurement error), while
the curves show the predicted θE based on the median c200 of
relaxed simulated halos as measured by Neto et al. (2007) in the
nearest mass bin, after correction for lensing and projection bias
based on Figure 1.
by about a factor of two. Specifically, the predicted Einstein
radii are 24′′, 15′′, 16′′, and 23′′, for A1689, Cl0024, A1703,
and RXJ1347, respectively. In this Figure we also separately
illustrate the dependence of the predicted θE on the source
and lens redshifts. In general, the predicted θEincreases with
zS, with the dependence weakening as zSmoves further away
from zL. For a fixed zS, θE is maximized at a particular
value of zL, which for the zS values of our four clusters falls
at zL between 0.3 and 0.6. The overall dependence of θE is
weak, illustrating that the discrepancy between the theory
and the observations would not be significantly affected by
small changes in the redshifts.
The results of section 2.1 allow us to make a much
more detailed study of the large Einstein radius problem.
We show in Figure 4 the full predicted probability distri-
bution of θE for each cluster. The main predictions (solid
curves) yield probabilities of 1.5%, 0.56%, 5.0%, and 3.7%
for finding a cluster with as large a value of θE(given the red-
shifts) as observed for A1689, Cl0024, A1703, and RXJ1347,
respectively. In this Figure and the previous one, we have
used the relaxed halo population from Neto et al. (2007).
As the Figure shows, including the unrelaxed halos would
only strengthen the inconsistency with the observations. The
results from Neto et al. (2007) are uncertain due to the rel-
atively small halo samples used. Specifically, since log10c200
is Gaussian distributed, the fractional sampling error with
N ≫ 1 halos is 1/√N in the measurement of the mean of
the distribution and 1/√2N in the standard deviation. Fig-
ure 4 shows the result of increasing the standard deviation
Page 7
Large Einstein Radii: A Problem for ΛCDM7
Figure 4. Cumulative probability distribution P(> θE), assum-
ing the lognormal c200 distribution measured by Neto et al.
(2007) for halos from a numerical simulation. We consider A1689,
Cl0024, A1703, and RXJ1347 as indicated, assuming in each case
the best-fit mass from observations, and the c200 distribution as
measured from the simulation for the nearest mass bin of relaxed
halos (solid curves). We also consider several possible sources of
statistical or systematic error, and we illustrate the result of as-
suming a cluster mass higher by the 1-σ measurement error (dot-
ted curves), or a scatter σlog10c200higher by the 1-σ sampling
noise (see text; long-dashed curves). In all cases shown, P(> θE)
for unrelaxed halos would lie below the corresponding curve for
relaxed halos, throughout the plotted region. Thus, we only il-
lustrate the main case with the c200distribution as measured for
unrelaxed halos (short-dashed curves). Also shown for comparison
for each cluster (dotted vertical line) is the observed θE.
by its 1-σ sampling error; increasing the mean by its 1-σ er-
ror would have a smaller effect. While larger simulations will
reduce the sampling noise, the Figure shows that this uncer-
tainty is already smaller than the effect of the measurement
errors in the cluster masses.
Including the rather conservative observational uncer-
tainties that we have assumed in M200 and θE, and aver-
aging over Gaussian error distributions in these two observ-
ables, we obtain probabilities of 8.5%, 3.9%, 7.9%, and 13%,
for agreement between the ΛCDM simulations and A1689,
Cl0024, A1703, and RXJ1347, respectively. If we considered
just one of the clusters, the large Einstein radius problem
would only constitute around a 2-σ discrepancy. However,
we have four independent objects selected from the popu-
lation of cluster lenses, and all four are discrepant (in the
same direction). The total probability of the theoretical pre-
diction yielding four clusters with such large values of θE is
3 × 10−5, which corresponds to a 4-σ discrepancy. We em-
phasize that we have included in this calculation the lensing
and projection biases, as well as the measurement errors in
the cluster masses and Einstein radii.
4 DISCUSSION
We have presented perhaps the clearest, most robust cur-
rent conflict between observations and the standard ΛCDM
model. This model is highly successful in fitting large scale
structure measurements, which in turn strongly constrain
the free parameters of the model and thus produce pre-
cise predictions for comparison with data on smaller scales.
Structure on these scales is non-linear and potentially af-
fected by gas physics, but clusters provide perhaps the best
opportunity for a robust comparison between the models
and the theory. Clusters are so large and massive that their
evolution is dominated by gravity, especially since their high
virial temperature prevents most of the intracluster gas from
cooling. The evolution of clusters including gravitational
collapse and virialization can now be accurately numeri-
cally simulated, with sufficient resolution for studying clus-
ter structure and for simulating lensing in projection, and
in sufficient volumes to produce large samples in a cosmo-
logical context. At the same time, observations of clusters
combining weak and strong lensing now produce accurate
virial mass determinations. Given the virial mass, the clean-
est measure of the halo structure is the effective Einstein ra-
dius, which is easily obtained observationally from a model
constrained by large numbers of arcs and directly measures
the central mass density.
We derived the theoretical predictions for cluster lens-
ing in ΛCDM using the distribution of 3-D halo profiles mea-
sured by Neto et al. (2007) in the Millennium simulation,
after correcting it for lensing and projection biases based on
Hennawi et al. (2007). These analyses of numerical samples
expressed halo structure in terms of the NFW concentra-
tion parameter. We found two key results (Figure 1) based
on the halo analysis by Hennawi et al. (2007): first, that the
distribution of 3-D concentrations 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; and second, that the con-
centrations measured in projection are related to the 3-D
concentrations, such that the ratio follows a lognormal dis-
tribution which corresponds to a factor of 1.14 shift plus a
factor of 1.33 spread. The concentration parameter is higher
for relaxed halos than for unrelaxed, and it declines slowly
with halo mass, resulting in a predicted Einstein radius that
increases roughly linearly with mass for relaxed halos (Fig-
ure 2).
We compared the theoretical predictions with the ob-
served θE for four clusters, A1689, Cl0024, A1703, and
RXJ1347. For the latter three we used the virial mass as
given by NFW fits to the lensing observations, but for
A1689 we obtained a model-independent mass directly from
the lensing data, only assuming spherical symmetry (sec-
tion 2.3). For each object, the predicted θE values came up
short by a factor of two compared with the observations
(Figure 3). After including the measurement errors, the full
probability distribution functions of the predicted Einstein
radii excluded the theoretical model at 2-σ for each object.
The total probability of the standard ΛCDM model yielding
four clusters with such large θE is 3 × 10−5, a 4-σ discrep-
ancy.
Lensing work is now being extended to larger samples
of clusters, so that in the near future we may examine more
fully the relation between the Einstein radius and virial mass
Page 8
8 Tom J. Broadhurst and Rennan Barkana
Figure 5. Total number of observable clusters in the universe
above redshift z, obtained by integrating the halo mass function of
Sheth, Mo, & Tormen (2001) over our past light cone. We consider
all cluster halos above virial mass M = 1, 2, 3, 4, 5, or 6×1015M⊙
(top to bottom). Also listed for each M (top-right corner) is the
redshift above which there is a 50% chance of observing at least
one halo of mass greater than M. For M = 6×1015M⊙we instead
list the probability of observing at least one halo at any z > 0.
and its scatter, over a wider range of cluster masses. The
theoretically predicted triaxiality of CDM halos implies a
particular scatter in the projected concentration parameter
(and thus in the Einstein radius) for a given halo mass. This
scatter, which we included in our calculations, is apparently
insufficient to explain the observations. If the scatter is ob-
servationally determined to be relatively small, then this
would further highlight the problem we have discussed and
leave the high concentrations unexplained. Determining the
scatter observationally will also statistically probe the de-
gree of triaxiality of CDM halos. To ensure the most direct,
unbiased comparison, the simulated distributions should be
calculated not at a fixed 3-D virial mass, but at a fixed pro-
jected, effective virial mass, defined as in section 2.3. We
expect this projected virial mass to also be observationally
measured in more clusters.
Numerical simulations show a clear correlation between
the concentration of a halo and its formation time, i.e., the
time at which a significant portion of the halo mass first
assembled (e.g., Neto et al. 2007). This agrees with the in-
tuitive notion that a dense halo core must have assembled at
high redshift, when the cosmic density was high. Thus, the
fact that observed cluster halos are apparently more cen-
trally concentrated than is predicted in ΛCDM suggests an
additional mechanism that promotes the collapse of cluster
cores at an earlier time than expected. Baryons are unlikely
to help. Central cD galaxies contribute only a small frac-
tion of the mass within the Einstein radius, which for our
four clusters is ∼ 150 kpc enclosing a projected mass of
∼ 2 × 1014M⊙, or a 3-D mass of ∼ 1 × 1014M⊙. We can es-
timate the effect of baryons on the total mass profile using
the simple model of adiabatic compression (Blumenthal et
al. 1986). Within this model, conservation of angular mo-
mentum implies that the quantity rM(r) (assuming spher-
ical symmetry) is fixed. Assuming that we start out with a
halo with the mean expected theoretical concentration (Fig-
ure 2), the observed 3-D mass within the Einstein radius can
be obtained through adiabatic compression if the enclosed
baryonic mass within this radius is ∼ 3 × 1013M⊙ in the
four clusters we considered. Thus, explaining the discrep-
ancy through adiabatic compression requires the baryonic
fraction within the Einstein radius to be ∼ 1/3, twice the
cosmic baryon fraction. This seems highly unlikely, as the
observed X-ray emission yields at these radii gas fractions
well below the cosmic value (e.g., see Figure 12 of Lemze
et al. (2007) for A1689), and a cD galaxy contains only
∼ 1×1012M⊙ in baryons (e.g., see Matsushita et al. (2002)
for M87).
Modifications in the properties of dark matter or the
slope of the primordial power spectrum are generally ex-
pected to have a smaller effect on clusters than on smaller-
scale objects which are predicted in ΛCDM to have earlier
formation times and higher concentrations. On the other
hand, since clusters are rare objects in the standard model,
primordial non-Gaussianity would significantly affect them
and allow clusters to form earlier, which may also help ex-
plain other observations (Mathis et al. 2004; Sadeh et al.
2007). Regardless of the mechanism, early collapse of cluster
cores may have observable consequences if it is accompanied
by star formation.
Finally, we note that the fact that clusters are now be-
ing detected with masses ∼ 1015M⊙ is completely consis-
tent with the ΛCDM model. Indeed, Figure 5 shows that
large numbers of clusters are expected out to significant red-
shifts, including M = 2 × 1015M⊙ halos out to z ∼ 1, as
well as more massive halos up to ∼ 5 × 1015M⊙ at lower
redshift. Clearly, while large samples of halos with precise,
profile-independent lensing determinations of both θE and
Mvirwill make our results completely conclusive, the highly-
significant discrepancy we have identified already represents
a substantial challenge for ΛCDM.
ACKNOWLEDGMENTS
We thank Masataka Fukugita and Masahiro Takada for use-
ful discussions. RB is grateful for support from the ICRR in
Tokyo, Japan and from the John Simon Guggenheim Memo-
rial Foundation. We acknowledge Israel Science Foundation
grants 629/05 (RB) and 1218/06 (TJB).
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