Constraints on the Self-Interaction Cross-Section of Dark Matter from
  Numerical Simulations of the Merging Galaxy Cluster 1E 0657-5
Constraints on the Self-Interaction Cross-Section of Dark Matter
from Numerical Simulations of the Merging Galaxy Cluster
1E 0657-56
Scott W. Randall1, Maxim Markevitch1,2, Douglas Clowe3,4, Anthony H. Gonzalez5, and
Marusa Bradač6
ABSTRACT
We compare recent results from X-ray, strong lensing, weak lensing, and
optical observations with numerical simulations of the merging galaxy cluster
1E 0657-56. X-ray observations reveal a bullet-like subcluster with a prominent
bow shock, which gives an estimate for the merger velocity of 4700 km s−1, while
lensing results show that the positions of the total mass peaks are consistent
with the centroids of the collisionless galaxies (and inconsistent with the X-ray
brightness peaks). Previous studies, based on older observational datasets, have
placed upper limits on the self-interaction cross-section of dark matter per unit
mass, σ/m, using simplified analytic techniques. In this work, we take advantage
of new, higher-quality observational datasets by running full N-body simulations
of 1E 0657-56 that include the effects of self-interacting dark matter, and com-
paring the results with observations. Furthermore, the recent data allow for a
new independent method of constraining σ/m, based on the non-observation of
an offset between the bullet subcluster mass peak and galaxy centroid. This
new method places an upper limit (68% confidence) of σ/m < 1.25 cm2 g−1.
If we make the assumption that the subcluster and the main cluster had equal
mass-to-light ratios prior to the merger, we derive our most stringent constraint
of σ/m < 0.7 cm2 g−1, which comes from the consistency of the subcluster’s ob-
served mass-to-light ratio with the main cluster’s, and with the universal cluster
1Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02138, USA
2Space Research Institute, Russian Academy of Science, Profsoyuznaya 84/32, Moscow 117997, Russia
3Steward Observatory, University of Arizona, 933 N. Cherry Ave., Tuscon, AZ 85721, USA
4Department of Physics and Astronomy, Ohio University, Clippinger Lab 251B, Athens, OH 45701, USA
5Department of Astronomy, University of Florida, 211 Bryant Space Science Center, Gainesville, FL
32611, USA
6Kavli Institute for Particle Astrophysics and Cosmology, P.O. Box 20450, MS-29, Stanford, CA 94309,
http://arxiv.org/abs/0704.0261v1
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value, ruling out the possibility of a large fraction of dark matter particles being
scattered away due to collisions. Our limit is a slight improvement over the pre-
vious result from analytic estimates, and rules out most of the 0.5 − 5 cm2 g−1
range invoked to explain inconsistencies between the standard collisionless cold
dark matter model and observations.
Subject headings: dark matter — clusters: individual (1E0657-56) — methods:
numerical — large scale structure of universe
1. Introduction
The nature of dark matter, which accounts for the majority of the mass in the Universe,
is one of the major outstanding problems of modern astrophysics. Although it is often
assumed that dark matter is collisionless, there is no a priori reason to believe that this is the
case, and it has been noted by other authors that a non-zero self-interaction cross-section can
have important astrophysical implications (e.g., Spergel & Steinhardt 2000). In particular,
self-interacting dark matter (SIDM) has been invoked to alleviate some apparent problems
with the standard cold dark matter (CDM) model, such as the non-observation of cuspy
mass profiles in galaxies (e.g., Moore 1994; Flores & Primack 1994; cf. Navarro et al. 1997;
Moore et al. 1999b) and the overprediction of the number of small sub-halos within larger
systems (e.g., Klypin et al. 1999; Moore et al. 1999a). Previous simulations and theoretical
studies suggest that a self-interaction cross-section per unit mass of σ/m ∼ 0.5− 5 cm2 g−1
is needed to explain the observed mass profiles of galaxies (e.g., Davé et al. 2001; Ahn &
Shapiro 2003, though see also Ahn & Shapiro 2005). Earlier studies have found stringent
upper limits on σ/m, inconsistent with the above range (e.g., Yoshida et al. 2000a; Hennawi
& Ostriker 2002; Miralda-Escudé 2002, though see also Sand et al. 2002). However, in
general these studies require non-trivial assumptions or statistical samples of clusters and
full cosmological simulations.
Furlanetto & Loeb (2002) pointed out that if one observes an offset between the gas and
dark matter in a merging cluster, arising because of the ram pressure acting on the gas but not
the dark matter, it can be used to constrain the collisional nature of dark matter. Markevitch
et al. (2002, hereafter M02) found just such a cluster, 1E 0657-56, which in the Chandra
image shows a bullet-like subcluster exiting the core of the main cluster, with prominent
bow shock and cold front features, and a uniquely simple merger geometry (Markevitch et
al. 2002, hereafter M02). This gas bullet lags behind the subcluster galaxies, which led
M02 to suggest that this cluster could be used to determine whether or not dark matter is
collisional. If dark matter were collisionless, one would expect the subcluster dark matter
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halo to be coincident with the collisionless subcluster galaxies. A map of the dark matter
distribution was subsequently derived by Clowe et al. (2004) using weak lensing observations,
which showed that the subcluster dark matter clump lay ahead of the gas bullet, close to the
centroid of the subcluster galaxies (see also Clowe et al. 2006a, hereafter C06). The X-ray
image of 1E 0657-56 is shown in Fig. 1 with the most recently derived weak lensing mass
contours of C06 overlain. The weak lensing contours are shown instead of the strong lensing
contours since they are better for showing the overall structure, as they are derived from
a wider field of view. The positions of the total mass peaks from strong and weak lensing
analyses are consistent with one another, and the general structures are similar. The more
massive main cluster is on the left and the high-velocity merging bullet subcluster is on the
right. The main and subcluster mass peaks are clearly visible in the mass map, as is the
offset between the gas bullet and the corresponding dark matter (DM) peak. C06 argued
that this offset is direct evidence for the existence of dark matter.
The weak lensing mass map of Clowe et al. (2004) was used by Markevitch et al. (2004,
hereafter M04), in conjunction with the X-ray and optical observations available at the time,
to analytically estimate upper limits on the self-interaction cross-section per unit mass of DM,
σ/m, using three independent methods. These methods were based on the observed offset
between the gas bullet and the DM subclump, the high merger velocity of the subcluster,
and the survival of the DM subclump (more precisely, the subcluster’s M/L ratio being
equal to that observed in other clusters and in the main cluster). M04 assume a King mass
profile, based on the original weak lensing mass map, and that the subcluster has passed
only once through the main cluster, close to the main cluster core, as indicated by the X-ray
image. Their most stringent limit comes from the observed survival of the DM subclump,
from which they infer that σ/m < 1 cm2 g−1.
Although the analytic estimates performed by M04 provide useful upper limits on σ/m,
several conservative simplifying assumptions were necessary. For instance, the effects of
dynamical friction as the subcluster disturbs the main cluster mass distribution were ignored,
as was the possibility of multiple scatterings per particle. Although these effects are relatively
small, their inclusion may lead to tighter constraints. Furthermore, the analytic estimates
cannot address any structure that may be found in a high-resolution mass map (e.g., tails in
the DM distribution, similar to the gas tails seen in the bullet, due to collisional stripping
of DM, as described by M04). This argues for full N-body simulations that would include
the effects of SIDM with varying cross-sections.
Additionally, new data (both X-ray and lensing) have become available for 1E 0657-56.
Analysis of data from 450 ks of total exposure with Chandra gives a more accurate shock
Mach number of M = 3.0 ± 0.4 (all uncertainties 68%), which corresponds to a shock (and
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bullet) velocity of 4700±630 km s−1 (Markevitch 2005). Recent weak and strong lensing
analyses of a much larger optical dataset, which includes HST observations, give a higher
quality mass map and a more accurate determination of the subcluster dark matter and
galaxy centroids (Bradač et al. 2006, hereafter B06; C06). In particular, the accuracy of the
total mass and galaxy centroids is now sufficient for an additional method of constraining
σ/m. In this paper, we will concentrate on the most sensitive method from M04, which is
based on the observed mass-to-light (M/L) ratios, and on this new test. The best way to
interpret the new high-quality data is through comparisons with detailed numerical simula-
tions of the merger which allow for SIDM with varying cross-sections. We present results
from such simulations and give constraints on the self-interaction cross-section of dark matter
particles. We assume Ω0 = 0.3, ΩΛ = 0.7, and H0 = 70 km s
−1 Mpc−1, for which 1′′ = 4.42
kpc at the cluster redshift of z = 0.296.
2. The Simulations
2.1. Simulation Code and Parameters
All simulations were performed using a modified version of the publicly available TreeSPH
code GADGET2 (Springel 2005). To model the self-interaction of the DM particles, we
adopted a Monte Carlo method used previously by other authors (e.g., Burkert 2000; Yoshida
et al. 2000b). At each simulation time step, the scattering probability for the ith particle is
given by
Pi = ρiσvrel∆t, (1)
where ρi is the local density, vrel is the relative velocity between the ith particle and its nearest
neighbor, and ∆t is the time step size. The local density is determined using GADGET2’s
smoothed particle hydrodynamic (SPH) capabilities. Collisions are assumed to be elastic
and scattering isotropic in the center-of-mass frame. In order for this relation to be valid,
∆t must be chosen such that Pi ≪ 1.
We ran a series of merger simulations with σ/m varying between 0 and 1.25 cm2 g−1.
Each simulation run included 106 DM particles (gas was not included in the simulations,
see discussion in § 4.5). Additionally, we performed a convergence test run with 107 DM
particles and σ/m ≈ 1 cm2 g−1, which agreed well with the lower resolution run for all
tests we performed. We interpret this agreement as indicating that the effects of individual
self-interacting DM particles are well modeled by the large computation particles used in
the simulations, and that the results we present here are not seriously affected by numerical
resolution effects. The ratio of DM particles in the main cluster and subcluster was set equal
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to the initial total mass ratio of the clusters, which is known analytically from the King
models used to build the clusters.
In this work, we apply a new method for constraining σ/m based on the absence of an
offset between the subcluster total mass and galaxy centroids. For this, we added another
family of particles to the simulations to represent the collisionless galaxies. We choose 105
“normal” galaxy particles for the main cluster and 2.5×104 for the subcluster throughout.
The ratio of the number of normal galaxy particles was estimated based on galaxy counts
given by Barrena et al. (2002). The galaxies were initially distributed like the DM in each run.
The mass in galaxies was assumed to be roughly 5% of the total mass for each cluster, which,
combined with the number of galaxy particles, gives a low mass per galaxy (2.15×108 M⊙).
Using a large number of light-weight galaxies was chosen over using a more realistic mass
per galaxy so that accurate galaxy centroids could be determined. Test simulations run with
the more realistic average mass per galaxy of 1011 M⊙, also determined from results given by
Barrena et al. (2002), showed similar results to those given below in §3, though, as expected,
with a larger scatter. Two cD galaxies, one at the center of each cluster, were also included in
the simulations (though we note that three cD galaxies are observed, two associated with the
main cluster and one with the subcluster). Their inclusion leads to conservative estimates of
the effects of DM self-interaction, since a lower central DM density is required to reproduce
the observed total mass profile (and the scattering probability depends on the local DM
density via Eqn. 1). The cD galaxies were each given a mass of 1013M⊙.
The gravitational softening length was chosen to be 2 kpc throughout, which is on the
order of the mean inter-particle separation in the densest region in each simulation (i.e., at
core passage). The softening length for the cD galaxy particles was set to 60 kpc throughout.
This large softening length was chosen since on the scale of the simulations cD galaxies are
significantly extended objects and treating them as concentrated point-like masses would
be unrealistic. Since the lensing observations do not give an accurate mass profile for the
subcluster, King models with density profiles of ρ(r) = ρ0(1 + r
2/r2c )
−3/2, where ρ0 is the
central density and rc is the core radius, were conservatively chosen for the mass profiles of
each cluster. Such a choice gives conservative limits for the effects of self-interacting DM,
since King models do not have strongly concentrated “cuspy” cores, as compared to NFW
(Navarro et al. 1995) and Hernquist (1990) profiles. Thus the central density is lower and the
total number of DM particle collisions is conservatively reduced. Further discussion of the
impact of the bullet mass profile on our results can be found in §4.2. As suggested by the
X-ray morphology (M04), all simulated mergers were head-on collisions with zero impact
parameter and an initial separation of 4 Mpc. The effects of a possible non-zero impact
parameter are discussed in §4.1.
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2.2. Initial Conditions
For each run, the initial conditions were chosen such that the projected mass profiles of
the main cluster and the bullet subcluster, after core passage and at the observed separation
(720 kpc), roughly matched those from the most recent combined strong and weak lensing
results derived by B06, which are given in the last row of Table 2. A relatively small
contribution from the observed distribution of gas mass was subtracted from the B06 total
mass, so that the resulting values could be directly compared with the simulations. The gas
masses were computed from the X-ray observations. The details of the derivation of the gas
mass map will be given in a future paper. A summary of the parameters for each simulation
run is given in Table 1, which gives the initial central density and core radius for the main
cluster (ρc,1, rc,1) and the bullet subcluster (ρc,2, rc,2), and σ/m for each run. The mass
profiles were truncated at 20 rc.
As pointed out by C06, weak lensing is expected to underestimate the mass of the lens
by 10-20% in the dense central regions. Furthermore, weak lensing can underestimate masses
due to mass-sheet degeneracy, where the mass map is affected by the non-detection of mass
at the edges of the field of view. The effect can be seen by comparing the total (i.e., without
the gas mass subtracted) weak lensing mass estimates of C06 to the mass profiles derived by
B06, which combine strong and weak lensing observations. We chose to match the projected
mass profiles in the simulations to those given by B06, since strong lensing is expected to
give better results in the central core regions. We are most interested in these regions since
most of the particle scattering depth accumulates near the center. We explore the effects of
decreasing the total mass of the system on our results in § 4.3.
2.3. Matching Simulations to Observations
Columns 4 & 5 of Table 2 give the total projected mass within 150 kpc of the mass peak
for the main cluster (M1(r < 150kpc)) and the subcluster (M2(r < 150kpc)) at the observed
separation for each simulation run and from the lensing observations. Some trial-and-error
was necessary to determine what initial conditions gave the desired projected mass profiles at
the desired separation. In general, larger σ/m values required a more concentrated subcluster
initially. This effect occurs because the self-interaction of the DM particles causes them to
be scattered away, particularly in high density regions. Consequently, the subcluster mass
profile is spread out during core passage. For the purposes of comparing the simulations
with observations, we take the simulation snapshot where the offset between the clusters is
closest to the observed separation. The time resolution of the snapshots was small enough
to match this value to within a few kpc (< 1′′), which is within the observational error. We
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require the simulated mass profiles at this moment to be consistent with the strong lensing
mass map to within 10% in the inner regions (r < 500 kpc).
2.4. Stability of Simulated Halos
It is of interest to evaluate the stability of our simulated clusters, particularly in the pres-
ence of SIDM. In the case of non-self-interacting DM, the phase-space distribution function
can be computed and used to generate a gravitationally stable King model density profile.
However, in the case of SIDM, particle collisions will tend to transfer kinetic energy from one
region of the cluster to another, consequently altering the density profile (see, e.g., Burkert
2000). In section § 3.2, we will draw conclusions based on the fraction of particles scattered
away from the core of the subcluster due to the merger event. It is therefore necessary to
determine what fraction of particles might flow from the central region of the bullet due to
the instability resulting from SIDM collisions. To this end, we ran simulations of the subclus-
ter DM halo, allowing it to evolve in isolation over the timescale of the merger simulations
(about 1 Gyr). We used the same cluster parameters as the subcluster that has the highest
central density of all the clusters in Table 1. The results are shown in Figure 3, which gives
the initial density profile (solid line) and the density profile after 1 Gyr for σ/m = 0 cm2 g−1
(dotted line) and σ/m = 0.7 cm2 g−1 (dashed line). The density in the inner regions is
marginally enhanced in the case of SIDM. This result is similar to the core-collapse phase
seen by Burkert (2000), where weak interactions between the kinematically hot core and the
cooler outer regions result in an outward transport of kinetic energy (though this effect is
expected to be somewhat curtailed here due to the near isothermality of the King profile
at small radii). For the purposes of the test described in § 3.2, we are only concerned with
the total mass within projected radii. This quantity is plotted for each run in Figure 4. The
above effect of SIDM on the projected mass profile is negligible, particularly for projected
radii x ≥ 150 kpc, which is the minimum radius considered for the test described in § 3.2.
Thus, if we find that a large fraction of SIDM particles scattered outside this radius, it can
be assumed to be caused by the merger event as opposed to any halo instability. Further-
more, the collisionless galaxies are expected to adjust to any change in the overall potential
(which is dominated by the DM), thereby acting to further stabilize the mass-to-light ratio.
Indeed, in these isolated subcluster runs, the mass-to-light ratios within a projected radius
of 150 kpc from the cluster centers stay within 2% of their initial values, regardless of the
DM self-interaction.
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3. Results
3.1. Galaxy – Dark Matter Centroid Offset
For non-self-interacting DM, the centroids of the subclump DM and galaxy distributions
are expected to be coincident throughout the simulation, since gravity is the only operating
force. However, when σ/m > 0, the subcluster DM halo experiences a drag force as it passes
through the main cluster, and subsequently lags the collisionless galaxies, just as the fluid-
like subcluster gas core is observed to lag the DM halo (see Fig. 1). We ran simulations with
a range of values for σ/m and calculated the centroids for each particle type by taking the
average projected position of the particles in some large region, centering on this position
with a smaller region, and repeating with smaller and smaller regions (down to a region with
a radius of 200 kpc). Column 6 of Table 2 gives ∆x, the offset between the subcluster galaxy
and DM centroids, for each run, for the moment when the subcluster is close to the observed
separation of 720 kpc from the main cluster. The dependence of ∆x on σ/m is also plotted
in Figure 5 (solid line). Results from the run with σ/m = 0 indicate that the offsets from the
simulations are accurate to about ±2 kpc (0.5′′). It is clear from Table 2 that the centroid
offset is a strong function of σ/m.
An X-ray image close-up of the bullet region with error contours for the subcluster total
mass and galaxy centroids overlain is shown in Figure 2. Details of the derivation of the
total mass centroids are given in C06. The centroid of the galaxy distribution was calculated
from the ACS photometry, using all galaxies for which the F814W-F606W color is within
0.15 mag of the red sequence. We used an Epanechnikov kernel with h = 30′′ (Merritt &
Tremblay, 1994; Gonzalez et al. 2002) to determine the centroid, and a bootstrap technique
to quantify the uncertainty. The centroid of the subcluster galaxies is found to be 5.7′′±6.6′′
(25 ± 29 kpc) west of the corresponding weak lensing mass peak. Given the observational
errors on the centroid positions (roughly 5′′, or 22 kpc, on the subcluster mass peak and
galaxy centroid), the absence of a larger offset means that σ/m < 1.25 cm2 g−1. We note
that, although this upper limit is greater than the best constraint of σ/m < 1 cm2 g−1 found
by M04, it is more robust, since it does not rely on the assumption that the subcluster and
the main cluster had equal M/L ratios prior to the merger, as is the case with the limit from
M04 (see § 3.2). This distinction is relevant since, although there is evidence for a universal
M/L ratio for clusters, the level scatter for individual clusters is not negligible (see Dahle
2000).
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3.2. Subcluster M/L Ratio
In a merger scenario, SIDM is expected to give a lower M/L ratio for the subcluster that
has just passed through a dense core as compared to collisionless DM. This is because during
the merger, DM particles are scattered away due to collisions, while the collisionless galaxies
are relatively unaffected. To estimate the change in the M/L ratio in the simulations due
to the merger, we simply take the ratio of the total mass to galaxy mass within 150 kpc
(projected) of the bullet DM centroid and compare the values at the start of the simulations
and at the observed separation. The results are tabulated in Column 7 of Table 2, which
gives f , the fractional decrease in the bullet M/L ratio within 150 kpc, and also plotted in
Figure 5 (dashed line). We note that for σ/m ≈ 1 cm2 g−1, the subcluster loses about 38%
of its mass within 150 kpc, which is in agreement with a conservative estimate of 20 - 30%
given by M04. As expected, the numerical results yield somewhat tighter constraints on
σ/m as compared to the analytic estimates when using the same method and observational
constraints.
Using the latest lensing mass map from B06, we rederived M/L ratios for each of the
two subcluster within a projected 150 kpc of the total mass peaks (for previous results see
Clowe et al., 2004). For the subcluster, the mass contribution from the outskirts of the
main cluster has been approximately subtracted, whereas for the main cluster, the total
mass is used, since the contribution from the subcluster is negligible. The projected mass
contribution from the main cluster to the subcluster is estimated by taking the average
mass in an annulus at the distance of the subcluster (excluding the region of the subcluster
itself). This gives a conservative estimate for the upper limit on σ/m, since scattering
due to putative DM collisions is expected to result in an anomalously low M/L value for
the subcluster as compared to the main cluster, and by reducing the observed mass of
the subcluster we minimize the effect of the collisions that we want to constrain. We find
M/LB = 471± 28, 422± 25 and M/LI = 179± 11, 214± 13 for the subcluster and the main
cluster, respectively (for a discussion of the errors on the mass measurements, see B06). The
ratios agree with one another to within about the 68% confidence intervals. From the I band
data, we find that the ratio of M/L ratios of the subcluster and main cluster is 0.84± 0.07.
We conservatively choose to use the I band data only, since we want to put a firm lower limit
on this ratio, andM/LB is larger for the subcluster than for the main cluster. Assuming each
cluster started out with similar M/L values, which appears to be a reasonable assumption
for clusters in general (e.g., Mellier 1999; Dahle 2000), we conclude that the subcluster could
not have lost more than ∼ 23% of its initial mass. A comparison with the results from
simulations plotted in Figure 5 (dashed line) shows that this implies σ/m . 0.6 cm2 g−1,
which is a slight improvement over the previous best limit of σ/m . 1 cm2 g−1 from the
conservative estimates of M04.
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3.3. Structure in Subcluster Dark Matter Distribution
M04 suggested that scattered DM particles, which would account for about 1/5 of the
total subcluster mass, might form tail features in the DM distribution, similar to the tails seen
in the X-ray image of the gas bullet (see Figure 1). The simulations allow us to determine
whether the non-observation of such tails in the mass map could be used to constrain σ/m.
We find that, rather than forming a tail, the scattered particles are mostly deposited in
the core of the main cluster, and do not form any features at a level that is interesting for
constraining σ/m.
4. Discussion
4.1. Non-zero Impact Parameter
As M04 argue, the morphology of the X-ray image, in particular, the symmetry of the
North-South X-ray bar (most likely an oblate spheroid viewed edge-on) between the main
cluster and subcluster mass peaks around the axis of symmetry set by the shape of the X-ray
bullet (which gives its present velocity direction), combined with the line-of-sight velocity
and X-ray derived Mach number, indicate a merger axis that is ∼ 10◦ from the plane of the
sky, and that the cluster cores must have passed close to one another, certainly within the
∼ 200 kpc core radius of the main cluster. In all simulations previously discussed, it was
assumed that the bullet subcluster passed directly through the center of the main cluster
core, i.e., that the impact parameter of the merger, b, is zero. For b > 0, we expect that the
effects of self-interacting DM will be reduced, since the density is at a maximum when the
core centers pass directly through one another, and the scattering probability is proportional
to the density (Eqn. 1). To test the strength of this effect, we re-ran the simulation R4 (see
Table 1) with an impact parameter of b = 200 kpc. Aside from the impact parameter,
the initial mass and velocity distributions were identical to those for run R4, so that the
relative effects of b > 0 could be investigated (specifically, no adjustments were made to the
initial conditions to more closely match the current observed mass profiles). The resulting
projected total mass profiles for the subcluster within 150 kpc and 250 kpc at the observed
separation agreed with those from the b = 0 run to within 4%. For the main cluster, the
match was better than 1%.
The resulting offset between the galaxy and DM centroids during the post core passage
phase was systematically smaller than the offset seen in the b = 0 run. At the observed
separation, the difference in the centroid offsets, as compared to the b = 0 run, was about
4 kpc, which is on the order of both the observational error and the accuracy of our numerical
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technique. The fractional change in the M/L ratio of the subcluster was similarly affected;
for the b = 0 case, the M/L ratio within 150 kpc drops by about 27%, whereas for the run
with b = 200 kpc, it drops by 22%. Assuming, as we did in § 3.2, that the subcluster could not
have lost more than ∼ 23% of its initial mass, we find the constraint that σ/m < 0.7 cm2 g−1.
We therefore conclude that, although a non-zero impact parameter reduces the effects of self-
interacting DM as expected, the level of the effect is relatively small. This is likely due to
the assumed King mass profile of the main cluster. The radial density gradient is relatively
small within the core radius of the main cluster (which in this case is 151 kpc), so it is not
surprising that the effects of self-interacting DM are not significantly reduced by increasing
the impact parameter, so long as it is comparable to the core radius of the main cluster.
Naturally, the effects of a non-zero b would be increased if the main cluster had a strongly
peaked mass profile, though the current lensing data suggest otherwise. We conclude that
any value of impact parameter that is consistent with the observations will only slightly alter
our results.
4.2. Alternative Bullet Mass Profiles
As noted in §2, the choice of a King mass profile for the subcluster is expected to give
conservative estimates on the effects of collisional DM, based on the M/L ratio, since the
central density is low as compared to models with cuspy cores such as NFW and Hernquist
models. However, in the case of the galaxy/DM centroid offset test, one might argue that
since the subcluster galaxies are more tightly bound in the center for more highly concen-
trated mass profiles, it will be more difficult to displace them, which could lead to a smaller
offset between the centroids despite the increased action of DM collisions. We therefore ran
a test simulation, using a King model for the main cluster and a Hernquist model for the
subcluster, with σ/m = 0.72 cm2 g−1. The Hernquist profile is given by
ρ(r) =
(r + a)3
, (2)
whereM is the total cluster mass and a is the scale length (Hernquist 1990). As before, initial
parameters were chosen such that the bullet mass profile roughly matches the observed profile
at the current separation (we used M = 3.13× 1014 M⊙, a = 100 kpc). This is expected to
be the most conservative model combination for this test, since the main cluster King model
minimizes the effects of DM self-interaction while the subcluster Hernquist model maximizes
the binding energy of the subcluster galaxies. The results show that, when comparing to run
R4 in Table 2, the galaxy/DM centroid offset was only slightly less than that found with the
King model subcluster, on the order of the accuracy of the simulation offset values (less than
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1 kpc). The change in the subcluster’s M/L was similarly only weakly affected (f = 0.27
for the King model bullet subcluster, whereas for the Hernquist model we find f = 0.31,
consistent with the King profile being the conservative case). The agreement is likely due
to the fact that the centroids become offset from one another after core passage, and it is
during core passage that the central density peak of the bullet is mostly “smoothed away”
due to DM collisions (recall that DM scattering is more frequent in high density regions,
so high density structures are more efficiently destroyed by DM self-interactions). Although
strongly peaked density profiles have been found to be unstable to SIDM (e.g., Burkert 2000;
Yoshida et al. 2000b), in our simulation a significant change in density only occurred at small
radii, such that the total projected mass of the subcluster within 50 kpc remained stable
up until the merger event. Therefore the subcluster mass distribution remained significantly
more peaked than a King profile cluster with the same projected mass within 150 kpc. We
conclude that our results are only weakly dependent on the mass profile chosen for the bullet,
so long as we require that the observed mass profile is reproduced. For the initially more
centrally concentrated profiles, the effects of the increased binding energy in the core are
balanced by the increased scattering frequency in this region.
4.3. Mass Profile Dependence
As mentioned in § 2, weak lensing is expected to underestimate the mass of the lens by
10-20%. There are two separate effects that contribute to this underestimation. First, near
the core of a cluster there is a large region without weak lensing galaxies, and this region
is effectively smoothed over when computing the mass map. Additionally, galaxies near the
regions where strong lensing dominates are measured in the weak lensing approximation,
which also leads to an underestimate of the mass in the core. Second, the total cluster mass
can be underestimated due to mass-sheet degeneracy, where the mass-map is affected by the
non-detection of mass at the edge of the field of view. Although projection of foreground
and/or background structures unassociated with the clusters will artificially increase the
mass, it is highly unlikely that such projected structures significantly contributed to the de-
tected lensing signal (C06). Results from strong lensing, which is not susceptible to the same
systematic underestimation as weak lensing, do indeed give systematically higher projected
masses for this system, by about a factor of 2 within the inner few hundred kpc, which is the
region we are most interested in for this analysis (compare B06 and C06; see C06 for further
discussion of this discrepancy). Though we chose to use the mass estimates from strong
lensing, since it should give a more reliable estimate of the projected mass near the cluster
cores, it is interesting to explore the dependence of our results on the lensing mass estimates.
To this end, we conducted a simulation run similar to run R4, but with the initial cluster
– 13 –
central densities chosen such that the projected mass profiles at the observed separation were
about 2 times lower than the masses derived from strong lensing observations, roughly in
agreement with the weak lensing results given by C06 (the initial core radii were the same
as for run R4). Since the scattering probability depends on the density, we expect these less
massive halos to be more weakly affected by SIDM.
Results obtained from the simulations with the lower mass normalization show that
the effects of SIDM are diminished, as expected for a linear dependence of the scattering
probabilities on the projected mass. In run R4, the M/L ratio dropped by 27%, whereas for
the run with 1/2 the total mass it dropped by 14% (roughly a factor of 2 less). Similarly, the
galaxy/DM centroid offset was 11.1 kpc, again, about a factor of 2 down from the 24.1 kpc
offset seen in run R4. If we assume that this factor of two effect can be applied to all of
values given in columns 6 & 7 of Table 2 and consider the more sensitive M/L test, we find
that requiring f . 0.20 would correspond to σ . 1.25 cm2 g−1. This is done as a test of
the method only, since these low halo mass values are not realistic, as they are insufficient
to produce the system of strong arcs observed in the HST images (C06).
4.4. Low Merger Velocity
All of the simulations discussed so far have assumed a merger velocity that is consistent
with that derived from X-ray observations (Markevitch, 2005), which give a Mach number for
the shock front of M = 3.0±0.4, and it is assumed that the subcluster has the same velocity
as the shock (though see Springel & Farrar, 2007). Since the subcluster could have slowed
down, or the shock front accelerated, it is interesting to ask what effect a lower velocity
would have on the inferred upper limit on σ/m, particularly since the observed velocity is
larger than would be expected from free fall of the subcluster onto the main cluster (Farrar
& Rosen 2007). In order to test the dependence of our upper limit on merger velocity, we
ran a simulation with σ/m = 0.72 cm2 g−1 such that the relative velocity of the cluster DM
halos at observed separation was 1.5 times lower, about 3100 km s−1 (M ≈ 2). This is close
to the expected free-fall velocity of the subcluster, and to the relative velocity of 2860 km s−1
found by Springel & Farrar (2007) from hydrodynamical simulations of this system. The
results showed little difference from the higher velocity run (compare to run R4 in Table 2):
∆x was 30.2 kpc (vs. 24.1 kpc) and f was 0.25 (vs. 0.27). We therefore conclude that our
results are relatively insensitive to merger velocities that are not in large disagreement with
the observations. A weak dependence of the M/L ratio on the subcluster velocity v is easy
to understand: the particle scatters out of the subcluster as long as v/2 is much greater than
the escape velocity from the subcluster, which it is by a large margin (M04).
– 14 –
4.5. Effects of Diffuse Gas
As mentioned in § 2, the intracluster gas observed in the X-ray band was not included
in the simulations (doing so would greatly increase the computing time and the complexity
involved with matching the observations in detail). The only way for the gas to affect the
results is via gravitational interaction (we ignore the possibility of non-gravitational baryon-
DM interactions, the cross-section of which has been shown to be extremely small, e.g.,
Chen et al. 2002). In general, the gas is expected to contribute about 10% of the total
mass of the system, a figure which appears to be consistent with the lensing and X-ray
observations (B06). One might worry that, when matching the observed mass profiles, some
“extra” DM is needed to account for the missing gas. As mentioned in § 2, gas masses
have been subtracted from the lensing masses using a detailed model of the gas distribution
derived from fitting the X-ray observations. In terms of the test involving the decrease in the
subcluster M/L ratio (see § 3.2), we needn’t worry about the subcluster gas for the simple
reason that the gas bullet is far from the subcluster mass peak (roughly 23′′, or 102 kpc).
Therefore, the gas in the region of the bullet mass peak is not centrally concentrated and
will not significantly add to the binding energy of potentially scattered DM particles. For
the galaxy and total mass centroid offset test (see § 3.1), the exclusion of the gas is expected
to give a conservative result: the gas bullet and bar feature seen in Figure 1 will act to
decelerate the subcluster DM halo and galaxies. However, if, as is the case with SIDM, the
DM halo starts to lag behind the galaxies and gets closer to the gas cores than the main
concentration of the galaxies, it will experience a larger deceleration, thereby increasing the
offset between the two. Due to the relatively low mass of the gas components, and the large
distance between the gas peaks and the subcluster DM halo and galaxies (as compared to
the offset of the latter), the strength of this effect will be quite small. We therefore conclude
that including gas in the simulations would not significantly affect our results.
5. Summary
We have combined results from new X-ray, optical and lensing observations and our
N-body simulations of the merging galaxy cluster 1E 0657-56 in order to derive an upper
limit on the self-interaction cross-section of dark matter particles, σ/m. We give constraints
on σ/m based on two independent methods: from the lack of offset between the total mass
peak and galaxy centroid of the subcluster that would arise during the merger due to drag
on the subcluster halo from DM particle collisions, and from the lack of a decreased mass-
to-light ratio of the subcluster due to scattering of DM particles. From the former, we
find σ/m < 1.25 cm2 g−1, and from the latter, σ/m < 0.7 cm2 g−1, which includes the
– 15 –
uncertainty in the impact parameter of the merger (upper limits are from 68% confidence
intervals). Our best constraint is a modest improvement of the previous best constraint
from conservative analytic estimates of σ/m < 1 cm2 g−1 (M04). Furthermore, our limit of
σ/m < 1.25 cm2 g−1 is more robust than the best analytic limit, since this method does not
depend on the assumption that the subcluster and main cluster M/L ratios were equal prior
to the merger. Previous studies have found that σ/m ∼ 0.5−5 cm2 g−1 is needed produce the
observational effects that self-interacting dark matter has been invoked to explain (e.g., non-
peaked galaxy mass profiles and the underabundance of small halos within larger systems).
Our results rule out almost this full range of values, at least under the assumption that σ is
velocity-independent.
We would like to thank Volker Springel, Naoki Yoshida, Yago Ascasibar, and Alexey
Vikhlinin for useful discussions and for providing access to various private codes. Simulations
were performed on a Beowulf cluster at the ITC in the Harvard-Smithsonian Center for
Astrophysics. Support for this work was partially provided for by the NASA Chandra grants
G04-5152X and TM6-7010X, and NASA contract NAS8-39073.
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Table 1. Initial Simulation Parameters
Run Name NDM σ/m ρc,1 rc,1 ρc,2 rc,2
(cm2 g−1) (106 M⊙ kpc
−3) (kpc) (106 M⊙ kpc
−3) (kpc)
R1 106 0 3.27 213 4.59 149
R2 106 0.24 3.27 213 4.59 149
R3 106 0.48 4.42 183 6.57 129
R4 106 0.72 7.03 151 11.75 108
R5 106 0.96 6.26 167 9.76 124
R6 106 1.25 6.26 167 9.76 124
– 18 –
Table 2. Conditions at Observed Separation
Run Name NDM σ/m M1(r < 150kpc) M2(r < 150kpc) ∆ x
(cm2 g−1) (1013 M⊙) (10
13 M⊙) (kpc)
R1 106 0 12.0 11.1 1.8 0.0
R2 106 0.24 11.5 10.4 5.4 0.08
R3 106 0.48 11.8 10.4 15.0 0.16
R4 106 0.72 12.6 11.0 24.1 0.27
R5 106 0.96 12.4 10.9 37.9 0.32
R6 106 1.25 11.4 9.8 53.9 0.38
Obs. 11.9±1.6 10.6±0.4 25± 29 0.16± 0.07
a∆ x is the offset between the subcluster total mass and galaxy centroids.
bf is the fractional decrease in the mass-to-light ratio of the subcluster within 150 kpc.
– 19 –
Fig. 1.— X-ray image with weak lensing mass contours overlain. The gas bullet lags the
subcluster DM halo. The current separation of the subcluster and main cluster mass peaks
is 720 kpc.
– 20 –
Fig. 2.— Close up of the subcluster bullet region, with the DM (blue) and galaxy (red)
centroid error contours overlain. The contours show the 68.3% and 99.7% error regions. The
left panel shows the X-ray Chandra image, while the right shows the optical HST image.
– 21 –
Fig. 3.— Density profile of an isolated King model cluster at t = 0 (solid line), and after
evolving for 1 Gyr with σ/m = 0 (dotted line) and σ/m = 0.7 cm2 g−1 (dashed line).
– 22 –
Fig. 4.— Total mass within projected radius x for the cluster plotted in Figure 3. Line-type
indications are the same as in Figure 3.
– 23 –
Fig. 5.— The dependence of the subcluster galaxy and total mass centroid offset (∆ x, solid
line) and the fractional change in the subcluster M/L ratio (f , dashed line) on σ/m. Based
on the values given in Table 2.
	Introduction
	The Simulations
	Simulation Code and Parameters
	Initial Conditions
	Matching Simulations to Observations
	Stability of Simulated Halos
	Results
	Galaxy – Dark Matter Centroid Offset
	Subcluster M/L Ratio
	Structure in Subcluster Dark Matter Distribution
	Discussion
	Non-zero Impact Parameter
	Alternative Bullet Mass Profiles
	Mass Profile Dependence
	Low Merger Velocity
	Effects of Diffuse Gas
	Summary
