The Dark Matter Distribution in the Central Regions of Galaxy Clusters

Abstract

Cosmological N-body simulations predict that dark matter halos should have a
universal shape characterized by a steep, cuspy inner profile. Here we report
on a spectroscopic study of six clusters each containing a dominant brightest
cluster galaxy (BCG) with nearby gravitational arcs. Three clusters have both
radial and tangential gravitational arcs, whereas the other three display only
tangential arcs. We analyze stellar velocity dispersion data for the BCGs in
conjunction with the arc redshifts and lens models to constrain the dark and
baryonic mass profiles jointly. For those clusters with radial gravitational
arcs we were able to measure precisely the inner slope of the dark matter halo
and compare it with that predicted from CDM simulations.

Full text

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ccsd-00000779 (version 1) : 23 Oct 2003
THE DARK MATTER DISTRIBUTION IN THE CENTRAL REGIONS OF
GALAXY CLUSTERS
D.J. SAND, T. TREU, G.P. SMITH, R.S. ELLIS
Department of Astronomy, Caltech,
Pasadena, CA 91125, USA
Cosmological N-body simulations predict that dark matter halos should have a universal shape
characterized by a steep, cuspy inner profile. Here we report on a spectroscopic study of six
clusters each containing a dominant brightest cluster galaxy (BCG) with nearby gravitational
arcs. Three clusters have both radial and tangential gravitational arcs, whereas the other
three display only tangential arcs. We analyze stellar velocity dispersion data for the BCGs
in conjunction with the arc redshifts and lens models to constrain the dark and baryonic
mass profiles jointly. For those clusters with radial gravitational arcs we were able to measure
precisely the inner slope of the dark matter halo and compare it with that predicted from
CDM simulations.
1 Introduction
The Cold Dark Matter (CDM) paradigm for structure formation is extremely successful in
explaining observations of the universe on large scales (e.g. Percival et al. 2001; Spergel et al.
2003; Croft et al. 2002; Bahcall et al. 2003). A vital tool within the CDM model is that of
N-body simulations which are able to infer the properties of DM halos down to ∼kpc scales and
which have predicted a “universal” shape for DM density profiles (at mass scales ranging from
dwarf galaxies to clusters of galaxies) that goes like ρDM ∝ r−βat small radii (e.g. Navarro,
Frenk & White 1997; Moore et al. 1998; Power et al. 2003; Fukushige et al. 2003). Nearly all
numerical work points to a value of β between 1 and 1.5.
Observational verification of the DM density profile at various mass scales is very important
in confirming the CDM model. At the galaxy cluster scale, we have developed a technique
to measure the DM density profile by combining constraints from gravitational lensing and
the stellar velocity dispersion profile of a centrally located BCG. This allows us to disentangle
luminous and dark components of the mass distribution in the inner regions of galaxy clusters.

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Figure 1: Images of the six clusters in this study. The overlaid “slits” correspond to the actual slit positions and
sizes that were used. The postage stamp insets show zoomed in, BCG-subtracted views of the radial arcs.
2 Sample Selection & Data Analysis
We have undertaken a search of the Hubble Space Telescope (HST) Wide Field and Planetary
Camera 2 (WFPC2) archive to look specifically for radial gravitational arcs in galaxy clusters.
From this catalog of clusters, we picked several candidates to follow up spectroscopically, all of
which met the following criteria: 1) the cluster had a dominant, relatively isolated central galaxy;
2) there was at least one nearby large tangential gravitational arc, although clusters with radial
arcs were preferred; 3) and there were no indications of substantial substructure or a significantly
elongated cluster potential. We collected Keck spectroscopic data for 6 galaxy clusters, three
with radial and tangential arcs and three with only tangential arcs. For each system an extended
stellar velocity dispersion profile of the BCG was measured and, if necessary, gravitational arc
redshifts were found as well. Figure 1 shows an image of each cluster.
We have presented the analysis involved in this work in Sand et al. (2002, 2003); and
only briefly describe the methodology here. We adopted a simple, spherically symmetric two-
component mass model comprising the BCG and cluster DM halo. For the BCG component
we used a Jaffe (1983) mass density profile (which, in projection, describes well the surface
brightness profile of the BCGs) and for the cluster DM halo we used a generalized-NFW profile
which allows the inner slope, β, to be a free parameter. Given this mass model we have effectively
three free parameters: the stellar M/L of the BCG, the amplitude of the DM density profile,
δc, and the crucial inner DM density logarithmic slope, β. For a given set of free parameters,
{M/L,δc,β}, the expected position of the gravitational arcs and stellar velocity dispersion of
the BCG can be calculated. By comparing the expectations from a given mass model with the
observed arc positions and BCG velocity dispersion (along with their uncertainties) a likelihood
can be calculated. In this way, it is easy to marginalize with respect to various parameters to
find confidence limits on individual variables, in particular β. Figure 2 gives the probability

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Figure 2: Left: PDF of the DM inner density slope, β, for the three radial arc clusters. Note the wide scatter in
β values from cluster to cluster, ∆β ∼0.3. Right: PDF of the DM inner density slope, β, for the tangential arc
sample. These allow us to place an upper limit on β for each cluster.
distribution function of the DM inner density slope, β, for each of the clusters in our sample.
The left hand panel shows the radial arc cluster constraints, along with the joint distribution
across all three systems (?β? = 0.52+0.05
−0.05; 68% CL). Note not only that the mean DM density
profile is shallower than NFW, but that there is significant scatter in β values across the sample,
∆β ∼0.3. The right hand panel shows the results for the tangential arc sample (β <0.57; 99%
CL), which allows us to place not only an upper limit on the value of β, but serves as a useful
control sample for the radial arcs systems. Note that the radial arc sample is not biased towards
lower values of β as would be expected if they were a biased subset (see Sand et al. 2003).
3 Discussion & Conclusions
Figure 3 illustrates why we are able to place such strong constraints on the shape of the DM
profile. In the left panel is plotted the velocity dispersion profile measured for MS2137-23, along
with the best-fitting velocity dispersion (solid line; β=0.57) that agrees with both the lensing
and dynamics in the system. Plotted as a dashed line is a velocity dispersion profile whose
underlying mass distribution (β=1.30) agrees very well with the lensing data, but does not fit
the measured velocity dispersion. This case illustrates why mass models with too steep an inner
profile cannot match both the velocity dispersion profile and the positions of the gravitational
arcs. In the right panel of Figure 3 is plotted the best-fitting density profile of MS2137-23. The
velocity dispersion measurement of the BCG allows us to probe the mass distribution where
luminous matter is important, while the arcs probe the portion of the mass distribution where
dark matter dominates. The two measurements complement each other.
The observed value of β is expected to lie between 1 and 1.5 on the basis of CDM only simu-
lations, including those with the latest refinements and consideration for numerical convergence
(i.e. Power et al. 2003; Fukushige et al. 2003). We have found a range of acceptable values
of β (see Figure 2), and although individual systems can be consistent with NFW (e.g. RXJ
1133), the average slope is inconsistent with the cuspy profiles expected from CDM simulations.
Is it possible to account for the discrepancy between these observations and numerical predic-
tions? Conventional CDM simulations only include collisionless DM particles. It is not clear
how the inclusion of baryonic matter would affect the DM distribution, especially in regions
where it dominates the total matter density. One possible situation, adiabatic contraction (e.g.
Blumenthal et al. 1986), would steepen the DM distribution through gravitational processes
and exacerbate the current problem. Baryons could also play an important dynamical role by

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Figure 3: Left: An illustration of why including the velocity dispersion profile of BCG with the lensing analysis
improves our constraints on β. The hatched boxes show the observed velocity dispersion profile of MS2137-23.
The dashed curve shows a velocity dispersion profile from a mass model that is compatible with the lensing
analysis, but not the dynamical measurement. The solid curve shows the best fitting profile obtained from the
combined lensing + dynamics analysis. Right: Best-fitting total density profile for the cluster MS2137-23.
driving energy and angular momentum out of the cluster core, thus softening an originally cuspy
profile (e.g. El-Zant et al. 2001,2003). It is also possible that the DM particle is self-interacting.
This would naturally cause the DM density profile to be shallower than that predicted from
standard CDM (Spergel & Steinhardt 2000).
We are in the process of collecting a larger sample of galaxy clusters with radial arcs in
order to further constrain our determination of the mean value of β and its intrinsic scatter. A
clear measurement of both of these parameters will aid in future comparisons to simulations,
especially those that include both baryons and DM.
Acknowledgments
We are grateful to the MACS collaboration for access to MACS 1206. We acknowledge financial support
for proposal number HST-AR-09527 provided by NASA through a grant from STScI, which is operated
by AURA, under NASA contract NAS5-26555.
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