BULLET CLUSTER: A CHALLENGE TO ΛCDM COSMOLOGY

The bow shock in the merging cluster 1E0657-57 (also known as the "bullet cluster") observed by Chandra indicates that the subcluster (found by Barrena et al. 2002) moving through this massive (10¹⁵ h⁻¹ M_☉) main cluster creates a shock, and the shock velocity is as high as 4700 km s⁻¹ (Markevitch et al. 2002; Markevitch 2006). A significant offset between the distribution of X-ray emission and the mass distribution has been observed (Clowe et al. 2004, 2006; Markevitch et al. 2004), also indicating a high-velocity merger with gas stripped by ram pressure.

Several groups have carried out detailed investigations of the physical properties of 1E0657-57 using non-cosmological 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 high-velocity merging system is expected in a Λ cold dark matter (ΛCDM) 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 (0.5 h⁻¹ Gpc)³, there are only five cluster-size halos with M₂₀₀ > 10¹⁵ h⁻¹ M_☉, and one cluster with M₂₀₀ > 2 × 10¹⁵ h⁻¹ M_☉ at z = 0.28 (close to the redshift of 1E0657-57, z = 0.296). Therefore, Hayashi & White (2006) had to extrapolate their results for M₂₀₀ > 10¹⁴ h⁻¹ M_☉ assuming that the likelihood of finding the bullet-cluster systems scales with V_sub/V₂₀₀, where V_sub is the subcluster velocity in the rest frame of the main cluster, and V₂₀₀ = (GM₂₀₀/R₂₀₀)^1/2. Here, R₂₀₀ is the radius within which the mean mass density is 200 times the critical density of the universe, and M₂₀₀ is the mass enclosed within R₂₀₀.

While Hayashi & White (2006) concluded that the existence of 1E0657-57 is consistent with the standard ΛCDM cosmology, this conclusion was later challenged by Farrar & Rosen (2007) who showed that, once an updated mass of the main cluster of 1E0657-57 is taken into account, the probability of finding 1E0657-57 is as low as 10⁻⁷. This conclusion still relies on the extrapolation of the likelihood derived for M₂₀₀ > 10¹⁴ h⁻¹ M_☉.

As the probability of finding high-velocity 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 4700 km s⁻¹). Milosavljević et al. (2007) found that the subcluster velocity can be 4050 km s⁻¹, whereas Springel & Farrar (2007) found that it can be as low as 2700 km s⁻¹. Mastropietro & Burkert (2008) showed that the subcluster velocity of 3100 km s⁻¹ best reproduces the X-ray data of 1E0657-57.

These varying results are in part due 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 an initial separation of 4.6 Mpc (which is two times R₂₀₀ of the main cluster, 2.3 Mpc). The velocity is about 1600 km s⁻¹ at a separation of 3.5 Mpc (≃1.5R₂₀₀)³; Springel & Farrar (2007) used an initial velocity of 2057 km s⁻¹ when the separation was 3.37 Mpc (≃1.5R₂₀₀); and Mastropietro & Burkert (2008) explored various initial velocities such as 2057 km s⁻¹ at the initial separation of 3.37 Mpc and 2000, 3000, and 5000 km s⁻¹ at the initial separation of 5 Mpc (≃2.2R₂₀₀). Mastropietro & Burkert (2008) found that the simulation run with an initial velocity of 3000 km s⁻¹ best reproduces the X-ray data.

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 ΛCDM 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 X-ray and weak lensing data of 1E0657-57, and Mastropietro & Burkert (2008) argue that one needs an initial velocity of 3000 km s⁻¹ to explain the data. If this is true, the existence of 1E0657-57 is incompatible with the prediction of a ΛCDM model.

As high-velocity systems are rare, it is crucial to use a large-volume simulation to derive a reliable probability distribution. The previous study is somewhat inconclusive due to the limited volume of the Millennium Run simulation (0.5 h⁻¹ Gpc)³. We calculate the probability of finding bullet-like systems using a simulation with substantially larger volume (3 h⁻¹ Gpc)³.

We use the publicly available simulated dark matter halo catalogs at z = 0 and 0.5, which are constructed from the largest-volume N-body Marenostrum Institut de Ciències de l'Espai (MICE) simulations (Crocce et al. 2010). They used the publicly available GADGET-2 code (Springel 2005), with the cosmological parameters of Ω_m = 0.25,  Ω_Λ = 0.75,  Ω_b = 0.044,  h = 0.7,  n_s = 0.95, and σ₈ = 0.8. These numbers are consistent with those derived from the seven-year data of the Wilkinson Microwave Anisotropy Probe (Komatsu et al. 2010).

The MICE simulation that we shall use in this paper has a particle mass of M_par = 23.42 × 10¹⁰ h⁻¹ M_☉ and a linear box size of L_box = 3072 h⁻¹ Mpc. The standard friends-of-friends (FoF) algorithm (Davis et al. 1985) with a linking length parameter of b = 0.2 was employed to find the cluster halos from the distribution of 2048³ dark matter particles. See Fosalba et al. (2008) and Crocce et al. (2010) for a detailed description of the MICE simulations and the halo-identification procedure.

The halos identified in the MICE simulation contain at least 143 N-body particles. The derived halo catalog contains the center-of-mass positions (X) and velocities (V) of halos, as well as the number of particles in each halo (N_par). Note that the number of particles in each halo has been corrected for the known systematic effect of the FoF algorithm, using N^corr_par = N_par(1 − N^−0.6_par) (Warren et al. 2006; Crocce et al. 2010).

The mass of each halo is calculated as N_par times the mass of each particle, M_par. The mass of halos identified by FoF with a linking length of 0.2 approximately corresponds to M₂₀₀, i.e., the mass within R₂₀₀, within which the overdensity is 200 times the critical density of the universe at a given redshift, $M_{200}=\frac{4\pi }{3}[200\rho _c(z)]R_{200}^3$. It is, however, known that the FoF mass tends to be larger than M₂₀₀, especially for high-mass clusters which are less concentrated (Lukić et al. 2009). As a result, R₂₀₀ we quote in this paper may be an overestimate.

The difference between the FoF mass and M₂₀₀ decreases as the number of particles per halo, N_par, increases (Lukić et al. 2009). For the main halo masses of our interest, M_main ⩾ 0.5 × 10¹⁵ h⁻¹ M_☉, the average value of N_par is 3355 and 3160 at z = 0 and 0.5, respectively. Using this, we estimate that, on average, our R₂₀₀ 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 high-velocity subclusters even smaller, we shall ignore the difference between R₂₀₀ 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 re-apply the FoF algorithm with a linking length of 0.2. This time, the linking length of 0.2 means the length of 0.2 times L_box/(N_cl)^1/3, where N_cl is the total number of clusters found in the simulation (2.8 and 1.7 million clusters at z = 0 and 0.5, 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 1 shows the total number of cluster-size 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 R₂₀₀ from its mass as R₂₀₀ = [3M₂₀₀/(4π × 200ρ_c(z))]^1/3. Most of the satellite clusters are located at r ≳ 2R₂₀₀ from the main cluster, where r is the distance between the main cluster and its satellites.

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 (2–3)R₂₀₀ (Mastropietro & Burkert 2008), at 1.5R₂₀₀ (Milosavljević et al. 2007; Springel & Farrar 2007), and at R₂₀₀.

We define the pairwise velocity of a satellite cluster, V_c, as the velocity of the satellite relative to that of the main cluster, V_c ≡ V_main − V_sat. When satellite clusters are close to the main cluster, V_c must be strongly influenced (if not completely determined) by the gravitational potential of the main cluster. Thus, V_c should depend on the main cluster mass, M_main. If V_c is solely determined by the gravitational potential of the main cluster, then V_c ∝ M^1/2_main. In reality, however, it is not only the gravity of the main cluster but also the influences from the surrounding large-scale structures that should determine V_c (Benson 2005; Wang et al. 2005; Wetzel 2010).

Figure 1 shows the distribution of satellite clusters in the log V_c–log M_main plane (dotted line) at z = 0. There is a clear correlation between V_c and M_main (the larger the M_main is, the larger the V_c becomes), although it is not simply V_c ∝ M^1/2_main. The dotted line in Figure 1 shows the distribution of all satellite clusters. Next, we shall select the satellite clusters that belong to bullet-like systems. We define the bullet-like system as follows: the main cluster exerts dominant gravitational force on satellite clusters, and at least one satellite cluster is on its way to head-on 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 z.

• 1.  
  Satellite clusters lie between 2R₂₀₀ ⩽ r ⩽ 3R₂₀₀, and thus their motion is predominantly determined by the gravitational potential of the main cluster.
• 2.  
  Satellite clusters are about to undergo nearly head-on collisions with the main cluster: |V_c · r|/(|V_c||r|) ⩾ 0.9.
• 3.  
  The mass of satellites is less than or equal to 10% of that of the main cluster, M_sat/M_main ⩽ 1/10, and the main cluster mass is greater than some value, M_main ⩾ M_crit.

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The third criterion is motivated by the observation of 1E0657-57 indicating that the mass of the bullet subcluster is an order of magnitude lower than that of the massive main cluster, and the mass of the main cluster is ∼10¹⁵ h⁻¹ M_☉ (Springel & Farrar 2007). As the latest simulation by Mastropietro & Burkert (2008) showed that the mass ratio of 6:1 best reproduces the observed data of 1E0657-56 (also see Nusser 2008), we have also studied the case with M_sat/M_main ⩽ 1/5, finding similar results; thus, our conclusion is insensitive to the precise value of the mass ratio. In Figures 2 and 3, we show the distribution of the mass ratio, M_sat/M_main, at z = 0 and 0.5, respectively. As expected, larger-M_sat/M_main (i.e., closer-to-major-merger) collisions are exponentially rare. This makes 1E0657-57 even rarer, if the mass ratio is as large as M_sat/M_main = 1/6. For the rest of the paper, we shall study the case of M_sat/M_main < 1/10, keeping in mind that 1E0657-57 can be even rarer than our study indicates.

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In Figure 1, we show the distribution of satellite clusters satisfying the condition 1 (dashed line), the conditions 1 and 2 (dot-dashed line), and the conditions 1, 2, and 3 (solid line). Note that V_c of the satellite clusters that satisfy all of the above conditions approximately follows V_c ∝ M^1/2_main. 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 z = 0.5.

In Tables 2 and 3, we show the number of bullet-like systems satisfying all of the above conditions at z = 0 and 0.5, respectively. At z = 0, about one in three clusters of clusters with M_main ⩾ 0.7 × 10¹⁵ h⁻¹ M_☉ contains a nearly head-on collision subcluster. At z = 0.5, about one in five clusters of clusters with M_main ⩾ 0.7 × 10¹⁵ h⁻¹ M_☉ contains a nearly head-on collision subcluster. Therefore, head-on collision systems are quite common, but what about their infall velocities?

We calculate the probability density distribution of log V_c using the selected bullet-like systems (within (2–3)R₂₀₀) at z = 0 and 0.5. The results for M_main ⩾ 0.7 × 10¹⁵ h⁻¹ M_☉ are shown in Figures 4 (z = 0) and 5 (z = 0.5). A striking result seen from Figure 4 is that, of 1135 bullet-like systems shown here for z = 0, none have an infall velocity as high as 3000 km s⁻¹, which is required to explain the X-ray and weak lensing data of 1E0657-56 (Mastropietro & Burkert 2008). A lower velocity, 2000 km s⁻¹, is also rare: none (out of 1135) within (2–3)R₂₀₀ have V_c ⩾ 2000 km s⁻¹ at z = 0.

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We find a similar result for z = 0.5 (Figure 5): none (out of 78) have the infall velocity as high as 3000 km s⁻¹, and only one has V_c ⩾ 2000 km s⁻¹. However, we would need better statistics (i.e., a bigger simulation) at z = 0.5 to obtain more accurate probability. In any case, Mastropietro & Burkert (2008) argued that an infall velocity of 2000 km s⁻¹ is not enough to explain the X-ray brightness ratio of the main and subcluster or the X-ray morphology of the main cluster. These results indicate that the existence of 1E0657-56 rules out ΛCDM, unless a lower infall velocity solution for 1E0657-56 is found.

The significance increases if we lower the minimum main cluster mass. Mastropietro & Burkert (2008) argue that M_main ∼ 0.5 × 10¹⁵ h⁻¹ M_☉ fits the data of 1E0657-56 better. For a lower minimum main cluster mass, M_main ⩾ 0.5 × 10¹⁵ h⁻¹ M_☉, none of the 2189 bullet-like systems at z = 0 have V_c ⩾ 2000 km s⁻¹, none of the 186 systems at z = 0.5 have V_c ⩾ 3000 km s⁻¹, and only one system at z = 0.5 has V_c ⩾ 2000 km s⁻¹.

To examine whether or not the above results depend on the value of the linking length parameter, b, of the FoF algorithm used for finding clusters of clusters, we have repeated all the analyses by varying the values of b 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 b 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 1.5R₂₀₀. As most of the subclusters are located at r ≳ 2R₂₀₀, we have much fewer subclusters in (1–2)R₂₀₀. (There are only 191 subclusters within (1–2)R₂₀₀ at z = 0.) To solve this problem and keep the good statistics, we shall use the following simple dynamical model to convert the results in (2–3)R₂₀₀ to those at 1.5R₂₀₀ as well as at R₂₀₀.

The motion of the subclusters located in (2–3)R₂₀₀ is predominantly determined by the gravitational potential of the main halo. This is especially true for those in a nearly head-on collision course (i.e., nearly a radial orbit); thus, one may treat a selected sub-main cluster system as an isolated two-body system. Under this assumption, the pairwise velocity at r_in < 2R₂₀₀ is given in terms of the velocity at r_out ⩾ 2R₂₀₀ (which is measured from the simulation) and the mass of the main halo (which is also measured from the simulation):

$$\begin{equation} V^2_{c}(r_{\rm in})= V_{c}^{2}(r_{\rm out}) + \frac{2GM_{\rm main}}{R_{200}} \left(\frac{R_{200}}{r_{\rm in}}-\frac{R_{200}}{r_{\rm out}} \right), \end{equation} \tag{ 1 }$$

where G = 4.3 × 10⁻⁹ km² s⁻²M⁻¹_☉ Mpc is Newton's gravitational constant.

In Figures 4 and 5, we show the probability density distribution of log V_c at z = 0 and 0.5, respectively. The dashed lines show the original distribution for (2–3)R₂₀₀, while the dotted and solid lines show the distribution at 1.5R₂₀₀ and R₂₀₀, respectively, computed from Equation (1). We find that the initial velocities used by Milosavljević et al. (2007; ≈1600 km s⁻¹) and Springel & Farrar (2007; ≈2000 km s⁻¹) are consistent with the predictions of a ΛCDM model: at 1.5R₂₀₀, 9 (out of 1135) subclusters have V_c ⩾ 2000 km s⁻¹ at z = 0, and 16 (out of 117) subclusters have V_c ⩾ 2000 km s⁻¹ at z = 0.5. However, these simulations do not reproduce the details of the X-ray and weak lensing data of 1E0567-56 (Mastropietro & Burkert 2008), and thus this agreement does not imply that the existence of 1E0567-56 is consistent with ΛCDM.

How reliable is this extrapolation of the infall velocity? To check the accuracy of Equation (1), we compare p(V_c) in 2 ⩽ r/R₂₀₀ ⩽ 2.4 measured from the simulation and p(V_c) at 2.2R₂₀₀ computed from Equation (1). Specifically, we use Equation (1) to calculate the velocity at r_in = 2.2R₂₀₀ from velocities in 2.5R₂₀₀ ⩽ r_out ⩽ 3R₂₀₀. In Figure 6, we show the measured p(V_c) in 2 ⩽ r/R₂₀₀ ⩽ 2.4 (dashed line), the predicted p(V_c) at 2.2R₂₀₀ (solid line), and the original p(V_c) in 2.5 ⩽ r/R₂₀₀ ⩽ 3 (dotted line). We find an excellent agreement between the measured and predicted distribution.

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Mastropietro & Burkert (2008) showed that the subcluster initial velocity of 3000 km s⁻¹ at the separation of 5 Mpc is required to explain the X-ray and weak lensing data of 1E0657-56 at z = 0.296. They argued that a lower velocity, 2000 km s⁻¹, also has to be excluded because it cannot reproduce the observed X-ray brightness ratio of the main and subcluster or the X-ray morphology of the main cluster.

In this paper, we have shown that such a high velocity at 5 Mpc, which is about two times R₂₀₀ of the main cluster, is incompatible with the prediction of a ΛCDM model. Using the results at z = 0 and M_main ⩾ 0.7 × 10¹⁵ h⁻¹ M_☉, ΛCDM is excluded by more than 99.91% confidence level (none of the 1135 subclusters have V_c ⩾ 2000 km s⁻¹ in 2 ⩽ r/R₂₀₀ ⩽ 3). For a lower minimum main cluster mass, M_main ⩾ 0.5 × 10¹⁵ h⁻¹ M_☉, ΛCDM is excluded by more than 99.95% confidence level (none of the 2189 subclusters have V_c ⩾ 2000 km s⁻¹ in 2 ⩽ r/R₂₀₀ ⩽ 3).

The results at z = 0.5 are not yet fully conclusive due to the limited statistics: none of the 78 subclusters have V_c ⩾ 3000 km s⁻¹ in 2 ⩽ r/R₂₀₀ ⩽ 3, while there is one subcluster with V_c ⩾ 2000 km s⁻¹ in 2 ⩽ r/R₂₀₀ ⩽ 3. For M_main ⩾ 0.5 × 10¹⁵ h⁻¹ M_☉, none of the 186 subclusters have V_c ⩾ 3000 km s⁻¹, while there is one subcluster with V_c ⩾ 2000 km s⁻¹.

While these confidence levels are directly measured from the simulation, one can estimate the probability better by fitting the probability density, p(log V_c), to a Gaussian distribution as

$$\begin{equation} p(\log V_c) = \frac{1}{\sqrt{2\pi \sigma ^2_{\nu }}} {\exp \left[-\frac{\left(\log V_c-\nu \right)^2}{2\sigma ^2_{\nu }}\right]}, \end{equation} \tag{ 2 }$$

where V_c is in units of km s⁻¹ and ν and σ_ν are the two fitting parameters. The best-fit values of the two parameters for z = 0 and 0.5 are (ν, σ_ν) = (3.02, 0.07) and (3.13, 0.06), respectively. The mean velocity at z = 0 is smaller than that at z = 0.5 by a factor of 10^3.13–3.02 = 1.29. This may be understood as the effect of Λ slowing down the structure formation at z < 0.5.

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 P(>3000 km s⁻¹) = 3.3 × 10⁻¹¹ and 3.6 × 10⁻⁹ at z = 0 and 0.5, respectively. We also find P(>2000 km s⁻¹) = 2.9 × 10⁻⁵ and 2.2 × 10⁻³ at z = 0 and 0.5, respectively. These numbers pose a serious challenge to ΛCDM, unless one finds a lower velocity solution for 1E0657-56. Here, a "lower velocity" may be somewhere between V_c ≲ 1500 and 1800 km s⁻¹ at r ∼ 2R₂₀₀, which give 1% probabilities at z = 0 and z = 0.5, respectively.

The bullet cluster 1E0657-56 is not the only site of violent cluster mergers. For example, there are A520 (Markevitch et al. 2005) and MACS J0025.4-1222 (Bradac et al. 2008). Also, high-resolution mapping observations of the Sunyaev–Zel'dovich (SZ) effect have revealed a violent merger event in RX J1347-1145 at z = 0.45 (Komatsu et al. 2001; Kitayama et al. 2004; Mason et al. 2010), which are confirmed by X-ray observations (Allen et al. 2002; Ota et al. 2008). The shock velocity inferred from the SZ effect and the X-ray data of RX J1347-1145 is 4600 km s⁻¹ (Kitayama et al. 2004), which is similar to the shock velocity observed in 1E0657-56 (Markevitch 2006). The lack of structure in the redshift distribution of member galaxies of RX J1347-1145 suggests that the geometry of the merger of this cluster is also closer to edge-on (Lu et al. 2010). However, the lack of a bow shock in the Chandra image may suggest that it is not quite as edge-on as 1E0657-56. In any case, it seems plausible that there may be more clusters like 1E0657-56 in our universe. This too may present a challenge to ΛCDM.

Since the volume of the MICE simulation is close to the Hubble volume,⁴ our results can be compared directly with observations, provided that detailed follow-up observations are available for us to calculate the shock velocity, gas distribution, and dark matter distribution. These three observations would then enable us to estimate the mass ratio and initial velocity of the collision which, in turn, can be compared to the probability distribution we have derived in this paper. Note also that the probabilities obtained in our work are the conditional ones. That is, the probability for which a fitting formula is provided is the probability of the velocity of bullet systems that are nearly head-on, with 1:10 or more mass ratio, and with M_main ⩾ 0.7 × 10¹⁵ h⁻¹ M_☉. If we computed the probability of finding high-velocity bullet systems among all clusters from the simulation, then the probability would be even smaller than those estimated above. Such a conditional probability is relevant to the observation, if we have sufficient amount of data for estimating the mass ratio and the initial velocity, as mentioned above. Note that we have precisely such data for 1E0657-56.

An interesting question that we have not addressed in this paper is how many high-velocity bullet systems are expected for flux-limited galaxy cluster surveys, such as the South Pole Telescope and eROSITA (extended ROentgen Survey with an Imaging Telescope Array). To calculate, e.g., dN_bullet/dz, one needs the light-cone output of the MICE simulation. While we have not investigated this, we expect two major light-cone effects on the infall velocity distribution. First, the infall velocities at high z's should be larger since the effect of Λ has yet to kick in at high z's, which we have already demonstrated here by comparing the mean infall velocity at z = 0 and 0.5. Second, the massive bullet systems with mass greater than 10¹⁵ h⁻¹ M_☉ are very rare at higher z's. The first effect will make the high-velocity system more common, while the second effect will make the high-velocity system less common. In order to quantify the net effect, one needs the light-cone output. However, the light-cone effect alone would not be able to reconcile the existence of 1E0657-56 with the prediction of ΛCDM.

We acknowledge the use of data from the MICE simulations that are publicly available at http://www.ice.cat/mice. We thank C. Mastropietro, M. Milosavljević, and P. R. Shapiro for discussions. We also thank an anonymous referee for helpful comments. J.L. is very grateful to the members of the Texas Cosmology Center of the University of Texas at Austin for the warm hospitality during the period of her visit when this work was initiated and performed. J.L. acknowledges the financial support from the Korea Science and Engineering Foundation (KOSEF) grant funded by the Korean Government (MOST, No. R01-2007-000-10246-0). This work is supported in part by the NASA grant NNX08AL43G and the NSF grant AST-0807649.