Bullet Cluster: A Challenge to LCDM Cosmology
Abstract
To quantify how rare the bulletclusterlike highvelocity merging systems are in the standard Cold Dark Matter (CDM) cosmology, we use a largevolume () cosmological body MICE simulation to calculate the distribution of infall velocities of subclusters around massive main clusters. The infallvelocity distribution is given at of the main cluster (where is similar to the virial radius), and thus it gives the distribution of realistic initial velocities of subclusters just before collision. These velocities can be compared with the initial velocities used by the noncosmological hydrodynamical simulations of 1E065756 in the literature. The latest parameter search carried out by Mastropietro and Burkert have shown that the initial velocity of 3000 km s at about is required to explain the observed shock velocity, Xray brightness ratio of the main and subcluster, Xray morphology of the main cluster, and displacement of the Xray peaks from the mass peaks. We show that such a high infall velocity at is incompatible with the prediction of a CDM model: the probability of finding 3000 km s in is between and . A lower velocity, 2000 km s at , is also rare, and moreover, Mastropietro and Burkert have shown that such a lower initial velocity does not reproduce the Xray brightness ratio of the main and subcluster or morphology of the main cluster. Therefore, we conclude that the existence of 1E065756 is incompatible with the prediction of a CDM model, unless a lower infall velocity solution for 1E065756 with at is found.
1 Introduction
The bow shock in the merging cluster 1E065757 (also known as the “bullet cluster”) observed by Chandra indicates that the subcluster (found by Barrena et al., 2002) moving through this massive () main cluster creates a shock, and the shock velocity is as high as (Markevitch et al., 2002; Markevitch, 2006). A significant offset between the distribution of Xray emission and the mass distribution has been observed (Clowe et al., 2004, 2006), also indicating a highvelocity merger with gas stripped by ram pressure.
Several groups have carried out detailed investigations of the physical properties of 1E065757 using noncosmological hydrodynamical simulations (Takizawa, 2005, 2006; Milosavljević et al., 2007; Springel & Farrar, 2007; Mastropietro & Burkert, 2008). One of the key input parameters for all of these simulations is the initial velocity of the subcluster, which is usually given at somewhere near the virial radius of the main cluster.
An interesting question is whether the existence of such a highvelocity merging system is expected in a CDM universe. Hayashi & White (2006) were the first to calculate the likelihood of subcluster velocities using the Millennium Run simulation (Springel et al., 2005). As the volume of the Millennium Run simulation is limited to , there are only 5 clustersize halos with , and 1 cluster with at (close to the redshift of 1E065757, ). Therefore, Hayashi & White (2006) had to extrapolate their results for assuming that the likelihood of finding the bulletcluster systems scales with , where is the subcluster velocity in the rest frame of the main cluster, and . Here, is the radius within which the mean mass density is 200 times the critical density of the universe, and is the mass enclosed within .
While Hayashi & White (2006) concluded that the existence of 1E065757 is consistent with the standard CDM cosmology, this conclusion was later challenged by Farrar & Rosen (2007) who showed that, once an updated mass of the main cluster of 1E065757 is taken into account, the probability of finding 1E065757 is as low as . This conclusion still relies on the extrapolation of the likelihood derived for .
As the probability of finding highvelocity merging systems decreases exponentially with velocities, an accurate determination of the subcluster velocity, rather than the shock velocity, is crucial. Milosavljević et al. (2007) and Springel & Farrar (2007) used hydrodynamical simulations to show that the subcluster velocity can be significantly lower than the shock velocity (which is ). Milosavljević et al. (2007) found that the subcluster velocity can be , whereas Springel & Farrar (2007) found that it can be as low as . Mastropietro & Burkert (2008) showed that the subcluster velocity of best reproduces the Xray data of 1E065757.
These varying results are due in part to the varying assumptions about
the initial velocity given to the subcluster at the beginning of their
hydrodynamical simulations:
Milosavljević et al. (2007) used zero relative velocity between the main cluster
and subcluster at the initial separation of 4.6 Mpc (which is 2 times
of the main cluster, 2.3 Mpc). The velocity is about
1600 km s at a separation of 3.5 Mpc ()
In this paper, we demonstrate that the initial velocities used by Milosavljević et al. (2007) and Springel & Farrar (2007) are consistent with the prediction of a CDM model, but those of Mastropietro & Burkert (2008) at 5 Mpc are not. The simulations of Milosavljević et al. (2007) and Springel & Farrar (2007) do not reproduce details of the Xray and weak lensing data of 1E065757, and Mastropietro & Burkert (2008) argue that one needs the initial velocity of 3000 km s to explain the data. If this is true, the existence of 1E065757 is incompatible with the prediction of a CDM model.
2 Finding Clusters of Clusters in Simulation
As highvelocity systems are rare, it is crucial to use a largevolume simulation to derive the reliable probability distribution. The previous study is somewhat inconclusive due to the limited volume of the Millennium Run simulation, . We calculate the probability of finding bulletlike systems using a simulation with substantially larger volume, .
We use the publicly available simulated darkmatter halo catalogs at and , which are constructed from the largestvolume body Marenostrum Institut de Ciències de l’Espai(MICE) simulations (Crocce et al., 2010). They used the publicly available GADGET2 code (Springel, 2005), with the cosmological parameters of , and . These numbers are consistent with those derived from the sevenyear data of the Wilkinson Microwave Anisotropy Probe (Komatsu et al., 2010).
The MICE simulation that we shall use in this paper has the particle mass of and the linear box size of Mpc. The standard friendsoffriends (FoF) algorithm (Davis et al., 1985) with the linking length parameter of was employed to find the cluster halos from the distribution of dark matter particles. See Fosalba et al. (2008) and Crocce et al. (2010) for a detailed description of the MICE simulations and the haloidentification procedure.
The halos identified in the MICE simulation contain at least 143 body particles. The derived halo catalog contains the centerofmass positions () and velocities () of halos, as well as the number of particles in each halo (). Note that the number of particles in each halo has been corrected for the known systematic effect of the FoF algorithm, using (Warren et al., 2006; Crocce et al., 2010).
The mass of each halo is calculated as times the mass of each particle, . The mass of halos identified by FoF with the linking length of 0.2 approximately corresponds to , i.e., the mass within , within which the overdensity is 200 times the critical density of the universe at a given redshift, . It is, however, known that the FoF mass tends to be larger than , especially for highmass clusters which are less concentrated (Lukić et al., 2009). As a result, we quote in this paper may be an overestimate.
The difference between the FoF mass and decreases as the number of particles per halo, , increases (Lukić et al., 2009). For the main halo masses of our interest, , the average value of is 3355 and 3160 at and , respectively. Using this, we estimate that, on average, our may be 10% too large. This error is insignificant for our purpose. Moreover, as correcting this error strengthens our conclusion by making the probability of finding highvelocity subclusters even smaller, we shall ignore the difference between and the radius estimated from the FoF mass.
To find the “clusters of clusters” (i.e., groups of clusters with one massive main cluster surrounded by many less massive satellite clusters), we treat each cluster in the catalog as a particle and reapply the FoF algorithm with the linking length of 0.2. This time, the linking length of 0.2 means the length of 0.2 times , where is the total number of clusters found in the simulation (2.8 and 1.7 million clusters at and , respectively). Each cluster of clusters has the “main cluster,” or the most massive member of each cluster of clusters. All the other clusters are called “satellite clusters” or “subclusters.” Table Bullet Cluster: A Challenge to CDM Cosmology shows the total number of clustersize halos found in the simulation, the number of clusters of clusters having at least two members, and the mean mass of main clusters. For each main cluster, we calculate from its mass as . Most of the satellite clusters are located at from the main cluster, where is the distance between the main cluster and its satellites.
3 Deriving the Infall Velocity Distribution
Our goal in this paper is to derive the distribution of infall velocities around the main clusters. To compare with the initial velocities used by the hydrodynamical simulations in the literature (Milosavljević et al., 2007; Springel & Farrar, 2007; Mastropietro & Burkert, 2008), we calculate the infall velocity distribution within (Mastropietro & Burkert, 2008), at (Milosavljević et al., 2007; Springel & Farrar, 2007), and at .
We define the pairwise velocity of a satellite cluster, , as the velocity of the satellite relative to that of the main cluster, . When satellite clusters are close to the main cluster, must be strongly influenced (if not completely determined) by the gravitational potential of the main cluster. Thus, should depend on the main cluster mass, . If is solely determined by the gravitational potential of the main cluster, then . In reality, however, it is not only the gravity of the main cluster but also the influences from the surrounding largescale structures that should determine (Benson, 2005; Wang et al., 2005; Wetzel, 2010).
Figure 1 shows the distribution of satellite clusters in the plane (dotted line) at . There is a clear correlation between and (the larger the is, the larger the becomes), although it is not simply . The dotted line in Figure 1 shows the distribution of all satellite clusters. Next, we shall select the satellite clusters that belong to bulletlike systems. We define the bulletlike system as follows: the main cluster exerts dominant gravitational force on satellite clusters, and at least one satellite cluster is on its way of headon merging with the main cluster. More specifically, the following three criteria are used to select the candidate bullet cluster systems from the clusters of clusters at a given :

Satellite clusters lie between , and thus their motion is predominantly determined by the gravitational potential of the main cluster,

Satellite clusters are about to undergo nearly headon collisions with the main cluster: , and

The mass of satellites is less than or equal to 10% of that of the main cluster, , and the main cluster mass is greater than some value, .
The third criterion is motivated by the observation of 1E065757 indicating that the mass of the bullet subcluster is an orderofmagnitude lower than that of the massive main cluster, and the mass of the main cluster is (Springel & Farrar, 2007). As the latest simulation by Mastropietro & Burkert (2008) showed that the mass ratio of best reproduces the observed data of 1E065756 (also see Nusser, 2008), we have also studied the case with , finding similar results; thus, our conclusion is insensitive to the precise value of the mass ratio. In Figure 2 and 3, we show the distribution of the mass ratio, , at and , respectively. As expected, larger (i.e., closertomajormerger) collisions are exponentially rare. This makes 1E065757 even rarer, if the mass ratio is as large as . For the rest of the paper, we shall study the case of , keeping in mind that 1E065757 can be even rarer than our study indicates.
In Figure 1, we show the distribution of satellite clusters satisfying the condition 1 (dashed line), the conditions 1 and 2 (dotdashed line), and the conditions 1, 2, and 3 (solid line). Note that of the satellite clusters that satisfy all of the above conditions approximately follows . This is an expected result, as the satellite clusters in this case are basically point masses (nearly) freely falling into the main cluster. We also find similar results for .
In Table Bullet Cluster: A Challenge to CDM Cosmology and Bullet Cluster: A Challenge to CDM Cosmology, we show the number of bulletlike systems satisfying all of the above conditions at and , respectively. At , about 1 in 3 clusters of clusters with contains a nearly headon collision subcluster. At , about 1 in 5 clusters of clusters with contains a nearly headon collision subcluster. Therefore, headon collision systems are quite common  but, how about their infall velocities?
We calculate the probability density distribution of using the selected bulletlike systems (within ) at and . The results for are shown in Figure 4 () and 5 (). A striking result seen from Figure 4 is that, of 1135 bulletlike systems shown here for , none has the infall velocity as high as 3000 km s, which is required to explain the Xray and weak lensing data of 1E065756 (Mastropietro & Burkert, 2008). A lower velocity, 2000 km s, is also rare: none (out of 1135) within has at .
We find a similar result for (Figure 5): none (out of 78) has the infall velocity as high as 3000 km s, and only one has . However, we would need better statistics (i.e., a bigger simulation) at to obtain more accurate probability. In any case, Mastropietro & Burkert (2008) argued that an infall velocity of 2000 km s is not enough to explain the Xray brightness ratio of the main and subcluster or the Xray morphology of the main cluster. These results indicate that the existence of 1E065756 rules out CDM, unless a lower infall velocity solution for 1E065756 is found.
The significance increases if we lower the minimum main cluster mass. Mastropietro & Burkert (2008) argue that fits the data of 1E065756 better. For a lower minimum main cluster mass, , none out of 2189 bulletlike systems at has , none out of 186 systems at has , and only one system at has .
To examine whether or not the above results depend on the value of the linking length parameter, , of the FoF algorithm used for finding clusters of clusters, we have repeated all the analyses by varying the values of from 0.15 to 0.5. We have found similar results at both redshifts, demonstrating that our conclusion is insensitive to the exact values of used for the identification of clusters of clusters with the FoF algorithm.
To compare with the initial velocities used by the other simulations (Milosavljević et al., 2007; Springel & Farrar, 2007), we need to calculate the infall velocity distribution at . As most of the subclusters are located at , we have much fewer subclusters in . (There are only 191 subclusters within at .) To solve this problem and keep the good statistics, we shall use the following simple dynamical model to convert the results in to those at as well as at .
The motion of the subclusters located in is predominantly determined by the gravitational potential of the main halo. This is especially true for those in a nearly headon collision course (i.e., nearly a radial orbit); thus, one may treat a selected submain cluster system as an isolated twobody system. Under this assumption, the pairwise velocity at is given in terms of the velocity at (which is measured from the simulation) and the mass of the main halo (which is also measured from the simulation):
(1) 
where is Newton’s gravitational constant.
In Figure 4 and 5, we show the probability density distribution of at and , respectively. The dashed lines show the original distribution for , while the dotted and solid lines show the distribution at and , respectively, computed from equation (1). We find that the initial velocities used by Milosavljević et al. (2007) ( km s) and Springel & Farrar (2007) ( km s) are consistent with the predictions of a CDM model: at , 9 (out of 1135) subclusters have km s at , and 16 (out of 117) subclusters have km s at . However, these simulations do not reproduce the details of the Xray and weak lensing data of 1E056756 (Mastropietro & Burkert, 2008), and thus this agreement does not imply that the existence of 1E056756 is consistent with CDM.
How reliable is this extrapolation of the infall velocity? To check the accuracy of equation (1), we compare in measured from the simulation and at computed from equation (1). Specifically, we use equation (1) to calculate the velocity at from velocities in . In Figure 6, we show the measured in (dashed line), the predicted at (solid line), and the original in (dotted line). We find an excellent agreement between the measured and predicted distribution.
4 Discussion and Conclusion
Mastropietro & Burkert (2008) showed that the subcluster initial velocity of 3000 km s at the separation of 5 Mpc is required to explain the Xray and weak lensing data of 1E065756 at . They argued that a lower velocity, 2000 km s, seems excluded because it cannot reproduce the observed Xray brightness ratio of the main and subcluster or the Xray morphology of the main cluster.
In this paper, we have shown that such a high velocity at 5 Mpc, which is about 2 times of the main cluster, is incompatible with the prediction of a CDM model. Using the results at and , CDM is excluded by more than 99.91% confidence level (none out of 1135 subclusters has km s in ). For a lower minimum main cluster mass, , CDM is excluded by more than 99.95% confidence level (none out of 2189 subclusters has km s in ).
The results at are not yet fully conclusive due to the limited statistics: none out of 78 subclusters has km s in , while there is one subcluster with km s in . For , none out of 186 subclusters has km s, while there is one subcluster with km s.
While these confidence levels are directly measured from the simulation, one can estimate the probability better by fitting the probability density, , to a Gaussian distribution as
(2) 
where is in units of km s and and are the two fitting parameters. The bestfit values of the two parameters for and are and , respectively. The mean velocity at is smaller than that at by a factor of . This may be understood as the effect of slowing down the structure formation at .
Generally, one has to be careful about this approach, as we are probing the tail of the distribution, where the above fits may not be accurate. Using the above Gaussian fits, we find and at and , respectively. We also find and at and , respectively. These numbers pose a serious challenge to CDM, unless one finds a lower velocity solution for 1E065756. Here, a “lower velocity” may be somewhere between and km s at , which give 1% probabilities at and , respectively.
The bullet cluster 1E065756 is not the only site of violent cluster mergers. For example, there are A520 (Markevitch et al., 2005) and MACS J0025.41222 (Bradac et al., 2008). Also, highresolution mapping observations of the SunyaevZel’dovich (SZ) effect have revealed a violent merger event in RX J13471145 at (Komatsu et al., 2001; Kitayama et al., 2004; Mason et al., 2009), which are confirmed by Xray observations (Allen et al., 2002; Ota et al., 2008). The shock velocity inferred from the SZ effect and the Xray data of RX J13471145 is 4600 km s (Kitayama et al., 2004), which is similar to the shock velocity observed in 1E065756 (Markevitch, 2006). The lack of structure in the redshift distribution of member galaxies of RX J13471145 suggests that the geometry of the merger of this cluster is also closer to edgeon (Lu et al., 2010). However, the lack of a bow shock in the Chandra image may suggest that it is not quite as edgeon as 1E065756. In any case, it seems plausible that there may be more clusters like 1E065756 in our universe. This too may present a challenge to CDM.
Since the volume of the MICE simulation is close to the Hubble
volume,
An interesting question that we have not addressed in this paper is how many highvelocity bullet systems are expected for fluxlimited galaxy cluster surveys, such as the South Pole Telescope and eROSITA (extended ROentgen Survey with an Imaging Telescope Array). To calculate, e.g., , one needs the lightcone output of the MICE simulation. While we have not investigated this, we expect two major lightcone effects on the infall velocity distribution. First, the infall velocities at high ’s should be larger since the effect of has yet to kick in at high ’s, which we have already demonstrated here by comparing the mean infall velocity at and . Second, the massive bullet systems with mass greater than are very rare at higher ’s. The first effect will make the highvelocity system more common, while the second effect will make the highvelocity system less common. In order to quantify the net effect, one needs the lightcone output. However, the lightcone effect alone would not be able to reconcile the existence of 1E065756 with the prediction of CDM.
the number of  the number of  the mean mass of  

clusters  clusters of clusters 
main clusters  
2.8 million  0.29 million  
1.7 million  0.20 million 
the number of  the number of  the number of  

clusters of clusters  bulletlike
systems 
bulletlike systems 

at  at  at  
the number of  the number of  the number of  

clusters of clusters  bulletlike
systems 
bulletlike systems 

at  at  at  
Footnotes
 Milosavljević (2010), private communication. All velocities quoted throughout this paper are calculated in the rest frame of the main cluster.
 For example, the comoving volume available from to over the full sky is , which is only twice as large as the volume of the MICE simulation. The comoving volume out to is still , which is nowhere near enough to overcome the probability of .
 A “cluster of clusters” is a group of clustersize halos identified by the FoF algorithm. A useful picture is a massive cluster surrounded by many less massive clusters.
 For . A “bulletlike system” is defined as a nearly headon collision system satisfying all of the conditions (1, 2, and 3) given in Section 3.
 For .
 For .
 For .
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