The Bullet Cluster is not a Cosmological Anomaly
PPrepared for submission to JCAP
The Bullet Cluster is not aCosmological Anomaly
Craig Lage and Glennys R. Farrar
Center For Cosmology and Particle PhysicsDepartment of PhysicsNew York University, New York, NY 10003, USAE-mail: [email protected], [email protected]
Abstract.
The Bullet Cluster (1E0657-56) merger is of exceptional interest for testing thestandard cold-dark-matter plus cosmological constant cosmological model, and for investigat-ing the possible existence of a long- or short-range “fifth-force” in the dark sector and possibleneed for modifications of general relativity or even of Newtonian gravity. The most recentprevious simulations of the Bullet Cluster merger required an initial infall velocity far in excessof what would be expected within the standard cosmological model, at least in the absence ofadditional forces or modifications to gravity. We have recently carried out much more detailedsimulations than previously had been done, making pixel-by-pixel fits to 2D data-maps of themass distribution and X-ray emission, allowing for triaxial initial configurations and includ-ing MHD and cooling. Here, we compare the initial conditions of the Bullet Cluster mergeras found in our new simulations to the initial conditions in similar-mass merging clusters inthe Horizon cosmological simulation. We conclude that our initial infall velocity, 2900 km/sat a separation of 2.5 Mpc, is consistent with
ΛCDM , given the inferred main cluster massof × M (cid:12) . The initial concentration and shape found for the smaller (Bullet) clusterare typical for clusters of this mass range, but both quantities seem somewhat low for thelarger (Main) cluster. Due to the paucity of examples of clusters with such a high mass insimulations, these features of the main cluster cannot presently be used to test ΛCDM . a r X i v : . [ a s t r o - ph . GA ] M a r ontents The “Bullet Cluster” is well-known and widely cited as a poster-child for Dark Matter (DM).The observations show a clear separation between the 2D-projected mass distribution ofgas revealed by the X-ray emission and the projected total mass revealed by gravitationallensing. Interpreted in the standard cosmology with Dark Matter, this separation is a naturalconsequence of the shock-slowing of gas while the collisionless DM is unimpeded. At thesimplest level, this detachment of normal matter from the locus of gravitational lensing seemsto rule out Modified Newtonian Dynamics (MOND) [1] as the explanation of Galactic rotationcurves, but this conclusion has been contested [2]. A second means to test conventional physicswith the Bullet Cluster, is to ask how rare its initial conditions are, in
ΛCDM simulations.Several simulation studies of the Bullet Cluster collision have already been performed. Leeand Komatsu [3] reviewed the simulations that had been done as of 2010, and found that,while the studies of Milosavljevic et al ([4], hereafter M+07), and Springel and Farrar ([5],hereafter SF07) were consistent with a
ΛCDM cosmology, the study of Mastropietro andBurkert ([6], hereafter MB08) was not. Lee and Komatsu estimated that the large initialinfall velocity of the two clusters seen in MB08 had a probability of between . × − and . × − of occurring in a ΛCDM universe.Partially motivated by a desire to add clarity to this situation, we performed a detailedsimulation of the Bullet Cluster where we compared the simulation to the observational dataon a pixel-by-pixel basis in order to better constrain the initial conditions of the cluster colli-sion. This simulation study included triaxial initial clusters, magnetohydrodynamics, plasmacooling, and adaptive mesh refinement, and is reported on in detail in Lage and Farrar 2014([7] - hereafter LF14). Our simulations in LF14 led to a best fit set of initial conditions witha somewhat lower initial velocity and a significantly larger mass for the combined clusters, ascompared to the MB08 study. As reported below, because of these differences the initial infallvelocity of the Bullet Cluster collision deviates from the mean of
ΛCDM
N-body simulationsby less than two standard deviations. We also examine other aspects of the initial condi-tions, including concentrations, triaxiality ratios and impact parameter, and find no majorinconsistencies with observations and N-body simulations.The paper is organized as follows. We begin by placing all of the existing simulationson a common footing with respect to initial infall velocity at a fixed separation. Then weidentify analog clusters in a large N-body simulation of the growth of cosmological structure,– 1 –n order to quantitatively answer the question of whether the initial conditions of the Bullet-Cluster-merger precursor clusters are consistent with
ΛCDM . Finally, we compare the clusterconcentrations to observations and the shapes of the initial clusters to N-body simulations,and conclude.
The studies of SF07 and MB08 only attempted to constrain a small number of extractedparameters, specifically the spacings between the mass centroids and the X-ray peaks, andto qualitatively resemble the general shape of the X-ray flux maps. Our work in LF14 was amore detailed study which minimized a χ figure of merit between the two-dimensional obser-vational data sets and the simulation. (The χ parameter is used only as a figure of merit tooptimize and compare quality of different fits; since the σ in the denominator does not includeastrophysical noise such as expected from small scale inhomogeneities, even an excellent fitwill not have χ = 1 .) This work included triaxial initial clusters, magnetohydrodynamics,plasma cooling, and adaptive mesh refinement. More than 1000 sets of initial conditions weretested in order to find the best fit, with the result that the simulation in LF14 fits clusterobservations significantly better than past simulations. Figures 1 shows the improved fit weachieved in LF14 as compared to SF07 and MB08. k p c Data Simulation k p c
500 0 500kpc 0204060
DataSim
Mass Lensing, T = 0.87 Gy (a) LF14 - Mass k p c Data Simulation k p c
500 0 500kpc 0204060
Mass Lensing, T = 0.93 Gy (b) SF07 - Mass k p c Data Simulation k p c
500 0 500kpc 0204060
Mass Lensing, T = 0.64 Gy (c) MB08 - Mass k p c Data Simulation k p c
500 0 500kpc 0123
DataSim
X-ray Flux - 500-2000eV, T = 0.87 Gy (d) LF14 - X-ray k p c Data Simulation k p c
500 0 500kpc 0123
DataSim
X-ray Flux - 500-2000eV, T = 0.93 Gy (e) SF07 - X-ray k p c Data Simulation k p c
500 0 500kpc 0123
DataSim
X-ray Flux - 500-2000eV, T = 0.64 Gy (f) MB08 - X-ray
Figure 1 : Mass lensing and 0.5-2.0 keV X-ray data as compared to best fits of LF14, SF07,and MB08. – 2 –he figure of merit χ values are calculated by comparing the mass lensing and lowestenergy X-ray data to the simulations as described in LF14. The resulting χ values are 3.92for LF14, 13.67 for SF07, and 19.93 for MB08. The greatly improved fit motivates us torevisit the question of consistency with ΛCDM , within the framework of the LF14 best fitsimulation.Table 1 shows a comparison of the cluster masses, initial cluster separations and initialinfall velocities found in the various studies. (For brevity, in what follows we refer to thelarger cluster as the main cluster, and the smaller cluster as the subcluster.) To facilitatecomparison, we also give a standardized initial infall velocity calculated assuming that theclusters move as point masses on a ballistic trajectory from their starting separation to aseparation of 2500 kpc; since there is very little interaction between the clusters at separationslarger than 2500 kpc, assuming a ballistic trajectory of these widely separated clusters is avery good approximation, as shown in Figure 2. We have included the simulation study ofMilosavljevic [4] in Table 1, although since it is a 2D axisymmetric simulation, it is not in thesame category as the other studies. Also included is another early study, the dark-matter-onlysimulation of Randall et al ([8], hereafter R+08). The initial velocity and separation of theRandall study were obtained from personal communication with the author; the masses arecalculated from halo parameters given in R+08. A more recent dark-matter-only study hasbeen performed by Dawson [9], but comparable initial condition information is not available;from Figures 2 and 4 of [9], the results appear to be largely consistent with SF07 (W. Dawson,private communication).Authors M Main M Sub R Initial V Initial V χ (M (cid:12) ) (M (cid:12) ) (kpc) (km/sec) (km/sec)M+07 . × . × . × . × . × . × . × . × . × . × Table 1 : Comparison of initial infall velocities from different simulation studies. The V column gives a standardized initial infall velocity calculated assuming that the clusters moveas point masses on a ballistic trajectory from their starting separation to a separation of 2500kpc. The calculation of the χ parameter, which measures the fit between the simulationsand the observations, is described in detail in LF14. To estimate whether the initial velocities of these simulations are consistent with a
ΛCDM cosmology, we use an N-body simulation known as the Horizon Run (Kim [10]). This is alarge dark-matter-only simulation using = 6 . × particles, and covering a volumeof (6 . / h) . We analyze the data from this simulation in the following manner:1. We start with the database of halos from the z = 0 . snapshot. This database containsthe masses, locations, and velocities of approximately 1.1 million halos. The z=0 andz=0.5 snapshots were available to us, and we used the z=0.5 snapshot since it is closeto the redshift at the beginning of the simulation, which is approximately z=0.39.– 3 –. For a range of target masses between × M (cid:12) and × M (cid:12) , we search for acluster within 10% of the target mass. A cluster meeting this criterion is referred to asa main cluster analog.3. For each of these “main” clusters, we search for a neighboring cluster separated fromthe main cluster by a distance between 2500 kpc and 5000 kpc, with a mass between 6times and 10 times less than the main cluster analog. A cluster meeting these criteriais referred to as a subcluster analog.4. We extract the relative velocities of each pair of clusters, and convert to the value at aseparation of 2500 kpc, assuming that the clusters move as point masses along ballistictrajectories from their current separation to a separation of 2500 kpc. Figure 2 showsthat this is a valid assumption.5. We also extract the total energy and impact parameter of these two clusters. Figure 2 : Comparison of simulated vs calculated trajectories. The green circles show thefull Enzo simulation of the clusters detailed in LF14, starting at an initial separation of 5000kpc. The blue crosses show the trajectory calculated assuming that the clusters are twopoint masses. It is seen that the point mass assumption is relatively good down to a clusterseparation of about 2500 kpc, when the virial radii begin to overlap. Thus, it is valid to usethe point mass assumption to normalize the different simulations.Figure 3 shows the initial infall velocities extracted in this way compared to the abovesimulation studies. Since the main cluster mass is much larger than the subcluster mass, weexpect the initial infall velocities to be equal to (cid:112)
Main / R , and this is just what is seenin Figure 3. The fit to the expected behavior improves at lower masses because there are manymore clusters and hence less stochastic variability. While there are 2204 cluster pairs whosemain cluster mass is × M (cid:12) ± there are only 4 cluster pairs at × M (cid:12) ± .Because of the much larger number of cluster pairs at lower cluster masses, we use the meanand standard deviation calculated at a main cluster mass of × M (cid:12) and extrapolate tolarger masses, rather than using the mean and standard deviation calculated at the largermasses. – 4 –he parameters obtained in the simulation studies discussed above are also plotted inFigure 3. The best fit initial infall velocity from LF14 is about 1.24 standard deviations abovethe mean of the ΛCDM distribution, for the mass as determined by LF14.Although two of the earlier studies are 5 standard deviations and one is 3 standarddeviations from the mean, greater reliance can be placed on the LF14 initial conditions, sincethe LF14 simulation and fitting method was superior to earlier efforts, as discussed above.Therefore we conclude that the tension between the initial infall velocity of the Bullet Clusterand expectations from
ΛCDM exposed by Lee and Komatsu [3] can be considered to beresolved.
Main Cluster Mass (MSun) V ( k m / s e c ) V = q GM/rV = q GM/r ± , , σ Lage and FarrarMastropietro and BurkertMilosavljevicSpringel and FarrarRandall, 2007Horizon at z = 0.5
Figure 3 : Initial infall velocity of the subcluster relative to the main cluster extracted fromthe z=0.5 snapshot of the Horizon simulation at a separation of 2500 kpc, using the analysistechnique described in the text. The small circles are the mean relative velocity, with 1 σ error bars. The thick solid line shows the expected V = (cid:112)
Main / R behavior. Thethree dotted lines are the V = (cid:112)
Main / R curve offset by 1, 2, and 3 σ , respectively.In Figure 4, we plot the total energy and impact parameter of pairs of clusters extractedas described above, with main cluster mass of × M (cid:12) ± , as compared to the LF14best fit simulation. It is seen that most cluster pairs in the Horizon simulation are near zerototal energy, and the LF14 best fit simulation falls comfortably within the distribution. The concern that the Bullet Cluster is inconsistent with
ΛCDM cosmology has focused onthe initial infall velocity of the colliding clusters, and we have shown in the preceding sectionthat this velocity is in fact not exceptional. However, it is also worthwhile to examine theconsistency of the sizes and shapes of the colliding clusters with observations and N-bodysimulations based on
ΛCDM .First, we examine the concentrations of the colliding clusters and compare these to ob-servations. As used here, the concentration is defined as the ratio of the virial radius to theNFW scale radius as follows: c = R / R s . Figure 5 shows the LF14 best fit masses andconcentrations as compared to two observational studies. Figure 5(a) shows the comparison– 5 – .0 0.8 0.6 0.4 0.2 0.0 Total Energy (ergs) I m p a c t P a r a m e t e r ( k p c ) Figure 4 : Total Energy vs Impact Parameter of cluster pairs in the Horizon Run havingmain cluster mass of × M (cid:12) ± . The large square represents the best fit simulationfrom LF14.to the work of Comerford [11]. While the subcluster is quite typical, the main cluster appearsto have an unusually low concentration for its mass. However, a more recent study of Okabe[12], shown in Figure 5(b), has found a steeper slope for the Mass-Concentration relationship(heavy dashed line in Figure 5(b)) which is more consistent with our findings for the maincluster. We emphasize that in the full simulation study detailed in LF14 a wide range of con-centrations were explored, and the ones reported here are the best fit. The low concentrationof the main cluster found there was necessary in order to fit the observations. (a) Figure reproduced from Comerford et.al.[11]. (b) Figure reproduced from Okabe et.al. [12]. Figure 5 : Comparison of masses and concentrations from the LF14 best fit simulation tomeasured mass-concentration relations. The dotted ellipses represent one-sigma errors aroundthe LF14 best fit initial conditions. In both plots, the subcluster is on the left and the maincluster on the right. – 6 –n order to quantify the shape of these clusters, we introduce a set of axis ratios. Weassume that the clusters are triaxial ellipsoidal shapes, characterized by the lengths of eachof the three axes of the ellipsoid. The shape of the ellipsoid is then completely determinedby the two ratios of these three axes, with P being the shortest axis to longest axis ratio,and Q being the intermediate axis to longest axis ratio. With these definitions, we have . ≥ Q ≥ P ≥ . . If Q = 1 , then the ellipsoid is an oblate spheroid, with a shape like apancake. If
P = Q , then the ellipsoid is a prolate spheroid, with a shape like a cigar. If allthree axes are equal, then the shape is a sphere, and if all three axes are unequal, the shapeis triaxial.Figure 6(a) compares the LF14 best fit axis ratios to an N-body simulation study byBailin [13]. The subcluster is well within the population of clusters, while the small axis ratioof 0.35 found for the main cluster appears somewhat unusual. The more detailed study ofSchneider, Frenk, and Cole [14], shown in Figure 6(b), examines the trends of axis ratios as afunction of cluster mass and finds that more massive clusters tend to have smaller axis ratios,although the large mass of the main cluster (nearly × M (cid:12) ) is actually beyond the rangeconsidered. The lower right panel of Figure 6(b) shows the LF14 best fit axis ratio for themain cluster as compared to the largest masses studied. While we are unable to quantify howlikely the LF14 best fit axis ratio of 0.35 is, the trend of more massive clusters having smalleraxis ratios is in the right direction. Q P (a) Axis ratios of halos extracted from N-bodysimulations by Bailin et.al.[13]. The dottedellipses represent one-sigma errors around theLF14 best fit axis ratios, with the main clusteron the left and the subcluster on the right. PQ (b) Axis ratios of halos of different masses ex-tracted from N-body simulations by Schnei-der, Frenk, Cole [14], showing that more mas-sive clusters have smaller axis ratios. Thenumbers in parentheses are the mass rangesin log(M (cid:12) ) , with masses increasing from up-per left to lower right. The dotted lines in thelower right panel are the LF14 best fit axisratios for the main cluster. Figure 6 : Comparison of the LF14 best fit axis ratios to those extracted from N-bodysimulations. – 7 –
Conclusions
Ever since the initial report of a high relative velocity between the components of the mergingBullet Cluster, 4740 km/s at the time of observation [15], the possibility that the BulletCluster may require a new force between dark matter particles [16], or be incompatible with
ΛCDM cosmology[17], has been a topic of interest. Ref. [5] pointed out that the relativevelocity estimated from the bow shock[15] is not the relative velocity of the Dark Matterclusters, mitigating the issues raised in [16, 17]. However the more recent analysis by [3]comparing the initial conditions reported in detailed merger simulations to configurationsfound in
ΛCDM cosmological simulations, found a seemingly serious problem: the best-fitinitial conditions of [6] (MB08) – then the most recent and in principle comprehensive attemptto simulate the merger – had a probability of × − − × − of occurring in ΛCDM .We recently performed a detailed simulation of the Bullet Cluster using a new approach,fitting the simulation to the observational data on a pixel-by-pixel basis. The simulationincluded triaxial initial clusters, magnetohydrodynamics, plasma cooling, and adaptive meshrefinement [7] (LF14). Besides being more complete in terms of physical modeling than earliersimulations, the fit achieved to the mass-lensing and X-ray data by LF14 is significantly betterthan in previous simulations. In the present work, we compare the initial infall velocities,impact parameters, and shapes of the intial clusters found in the LF14 fit to the Bullet Clusterobservations, to the distributions found in the Horizon N-body simulation of structure growthin
ΛCDM cosmology.Our most important result is that the initial infall velocity as determined by LF14 isentirely compatible with expectations from
ΛCDM cosmology. Since the LF14 simulationgives the (by far) best description of the observations [7], the tension between the BulletCluster infall velocity and
ΛCDM cosmology is now removed. However this is not the onlylesson to be learned. Figure 3 collects the initial masses and infall velocities determined by thevarious simulations of the Bullet Cluster merger, at a common initial separation of 2.5 Mpc,and compares them to the distribution for analog-clusters found in the Horizon simulation.Most of the earlier simulations dramatically disagree with the
ΛCDM expectations, with twobeing 5-sigma above the mean relationship and one being 3-sigma below. This highly disparatebehavior arises because the inferred initial conditions of the various merger simulations differwidely (c.f., Table 1). The mass of the main cluster varies by a factor of 3: × to × M (cid:12) , and the mass of the smaller cluster even more: − × M (cid:12) . The relativevelocity at 2.5 Mpc in the different simulations ranges from . − . × km/s. This largedisparity in inferred initial conditions demonstrates that the observations are in fact highlyconstraining vis-a-vis the initial conditions; conversely, rough agreement between simulationand observations is insufficient to deduce the initial conditions and accurate modeling isneeded.Comparing other properties of the Bullet Cluster initial conditions inferred by SF14, wefind that the impact parameter of the Bullet Cluster merger is compatible with the rangefound in analogous systems in simulated ΛCDM cosmology. The concentration and shapeof the less-massive initial cluster are also unremarkable, given its mass. Available
ΛCDM simulations do not have enough clusters in the extremely high mass range of the main cluster, ≈ × M (cid:12) , to permit a critical test of ΛCDM predictions for its halo concentration(1.3) and the ratio of the shortest to longest axes ( ≈ . ), although they are not obviouslyincompatible with extrapolations from lower mass systems.We conclude that the initial conditions of the Bullet Cluster are compatible within– 8 –ncertainties, with the range expected to occur in a ΛCDM cosmology.
Acknowledgments
Thanks to Jeremy Tinker for valuable discussions and advice in several areas, and to ScottRandall and Will Dawson for helpful consultations. Thanks also to the anonymous referee forpointing out a number of items requiring clarification and making suggestions for improve-ments. This work has been supported in part by grants NNX08AG70G, NSF PHY-1212538,NSF PHY-0900631 and NSF PHY-0970075.
References [1] M. Milgrom,
The MOND paradigm , ArXiv e-prints (Jan., 2008) [ arXiv:0801.3133 ].[2] J. R. Brownstein and J. W. Moffat,
The Bullet Cluster 1E0657-558 evidence shows modifiedgravity in the absence of dark matter , "Monthly Notices RAS" (Nov., 2007) 29–47,[ astro-ph/ ].[3] J. Lee and E. Komatsu, Bullet Cluster: A Challenge to Λ CDM Cosmology , "Astrophys. J." (July, 2010) 60–65, [ arXiv:1003.0939 ].[4] M. Milosavljević, J. Koda, D. Nagai, E. Nakar, and P. R. Shapiro, The Cluster-Merger Shock in1E 0657-56: Faster than a Speeding Bullet? , "Astrophys. J. Lett." (June, 2007)L131–L134, [ astro-ph/ ].[5] V. Springel and G. R. Farrar, The speed of the ‘bullet’ in the merging galaxy cluster 1E0657-56 , "Monthly Notices RAS" (Sept., 2007) 911–925, [ astro-ph/ ].[6] C. Mastropietro and A. Burkert, Simulating the Bullet Cluster , "Monthly Notices RAS" (Sept., 2008) 967–988, [ arXiv:0711.0967 ].[7] C. Lage and G. Farrar, Constrained Simulation of the Bullet Cluster , "Astrophys. J." (June, 2014) 144, [ arXiv:1312.0959 ].[8] S. W. Randall, M. Markevitch, D. Clowe, A. H. Gonzalez, and M. Bradač, Constraints on theSelf-Interaction Cross Section of Dark Matter from Numerical Simulations of the MergingGalaxy Cluster 1E 0657-56 , "Astrophys. J." (June, 2008) 1173–1180, [ arXiv:0704.0261 ].[9] W. A. Dawson, The Dynamics of Merging Clusters: A Monte Carlo Solution Applied to theBullet and Musket Ball Clusters , "Astrophys. J." (Aug., 2013) 131, [ arXiv:1210.0014 ].[10] J. Kim, C. Park, J. R. Gott, III, and J. Dubinski, The Horizon Run N-Body Simulation:Baryon Acoustic Oscillations and Topology of Large-scale Structure of the Universe , "Astrophys. J." (Aug., 2009) 1547–1559, [ arXiv:0812.1392 ].[11] J. M. Comerford and P. Natarajan, The observed concentration-mass relation for galaxyclusters , "Monthly Notices RAS" (July, 2007) 190–200, [ astro-ph/ ].[12] N. Okabe, M. Takada, K. Umetsu, T. Futamase, and G. P. Smith, LoCuSS: Subaru WeakLensing Study of 30 Galaxy Clusters , "Pub. ASJ" (June, 2010) 811–, [ arXiv:0903.1103 ].[13] J. Bailin and M. Steinmetz, Internal and External Alignment of the Shapes and AngularMomenta of Λ CDM Halos , "Astrophys. J." (July, 2005) 647–665, [ astro-ph/ ].[14] M. D. Schneider, C. S. Frenk, and S. Cole, The shapes and alignments of dark matter halos , "J.Cosmology Astropart. Phys." (May, 2012) 30, [ arXiv:1111.5616 ].[15] M. Markevitch, Chandra observation of the most interesting cluster in the universe , astro-ph/0511345 . – 9 –
16] G. R. Farrar and R. A. Rosen,
A New Force in the Dark Sector? , Physical Review Letters (Apr., 2007) 171302–+, [ astro-ph/ ].[17] E. Hayashi and S. White, How rare is the bullet cluster? , astro-ph/0604443 (2006).(2006).