The return of the merging galaxy subclusters of El Gordo?
Karen Y. Ng, William A. Dawson, D. Wittman, M. James Jee, John P. Hughes, Felipe Menanteau, Cristóbal Sifón
MMon. Not. R. Astron. Soc. , 1–20 (2014) Printed 1 October 2018 (MN L A TEX style file v2.2)
The return of the merging galaxy subclusters of El Gordo?
Karen Y. Ng, William A. Dawson, D. Wittman, M. James Jee, John P. Hughes, Felipe Menanteau, , Crist´obal Sif´on Department of Physics, University of California Davis, One Shields Avenue, Davis, CA 95616, USA Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, CA 94551-0808, USA Department of Physics & Astronomy, Rutgers University, 136 Frelinghysen Rd., Piscataway, NJ 08854, USA National Center for Supercomputing Applications, University of Illinois at Urbana-Champaign, 1205 W. Clark St, Urbana IL, 61801, USA Department of Astronomy, University of Illinois at Urbana-Champaign, W. Green Street, Urbana, IL 61801, USA Leiden Observatory, Leiden University, PO Box 9513, NL-2300 RA Leiden, Netherlands arXiv 1412.1826
ABSTRACT
Merging galaxy clusters with radio relics provide rare insights to the merger dynamicsas the relics are created by the violent merger process. We demonstrate one of the firstuses of the properties of the radio relic to reduce the uncertainties of the dynamicalvariables and determine the 3D configuration of a cluster merger, ACT-CL J0102-4915, nicknamed El Gordo. From the double radio relic observation and the X-rayobservation of a comet-like gas morphology induced by motion of the cool core, it iswidely believed that El Gordo is observed shortly after the first core-passage of thesubclusters. We employ a Monte Carlo simulation to investigate the three-dimensional(3D) configuration and dynamics of El Gordo. Using the polarization fraction of theradio relic, we constrain the estimate of the angle between the plane of the sky andthe merger axis to be α = 21 ◦ ± . We find the relative 3D merger speed of El Gordoto be 2400 ± km s − at pericenter. The two possible estimates of the time-since-pericenter are 0 . ± . . Gyr and 0 . ± . . Gyr for the outgoing and returning scenariorespectively. We put our estimates of the time-since-pericenter into context by showingthat if the time-averaged shock velocity is approximately equal to or smaller than thepericenter velocity of the corresponding subcluster in the center of mass frame, thetwo subclusters are more likely to be moving towards, rather than away, from eachother, post apocenter. We compare and contrast the merger scenario of El Gordo withthat of the Bullet Cluster, and show that this late-stage merging scenario explains whythe southeast dark matter lensing peak of El Gordo is closer to the merger center thanthe southeast cool core.
Key words: gravitational lensing – dark matter – cosmology: observations – galaxies:clusters: individual (ACT-CL J0102-4915) – galaxies: high redshift – methods: statistical
Mergers of dark-matter-dominated galaxy clusters probeproperties of the cluster components like no other systems.In terms of mass content, dark matter makes up ∼
80% ofthe mass of clusters of galaxies, a smaller portion of themass consist of intercluster gas( ∼
15% in mass content), andsparsely spaced galaxies ( ∼
2% in mass content). During amerger of clusters of the high mass range of 10 M (cid:12) to thelow mass end of 10 M (cid:12) , the subclusters are accelerated tohigh speeds of ∼ − − (Lage & Farrar 2014,Dawson et al. 2012). The offsets of different componentsof the subclusters reflect the differences in the strengths ofinteractions between various components. Galaxies are ex- pected to lead the gas due to their negligible interaction crosssections with other components. The intracluster medium(ICM) is expected to lose momentum through electromag-netic interactions. On the other hand, offsets between darkmatter and galaxies may suggest dark matter self-interaction(Kahlhoefer et al. 2013, Randall et al. 2008).The galaxy cluster ACT-CL J0102-4915, (nicknamed “ElGordo”, at z=0.87), was discovered via its Sunyaev-Zel’dovich(SZ) effect by the Atacama Cosmology Telescope (ACT;Menanteau et al. 2010; Marriage et al. 2011); it has thestrongest SZ effect of the full ACT survey (Hasselfield et al.2013), and was discovered to be undergoing a major mergerapproximately in the plane of the sky (Menanteau et al. 2012,hereafter M12). El Gordo possesses a range of noteworthy fea- c (cid:13) a r X i v : . [ a s t r o - ph . C O ] J u l Karen Y. Ng et al.
RA (J2000) D e c (J ) X-ray
C.M.NWSEE relic NW relicSE relic
Figure 1.
Configuration of El Gordo showing overlay of dark matter distribution in blue, and X-ray emission in red. (Image credit:NASA, ESA and Jee et al. 2014). The cross markers show the positions of the northwest (NW) and southeast (SE) dark matter densitypeaks, and the center of mass (CM) locations respectively. Note that the mass ratio of the NW subcluster to the SE subcluster is ∼ tures that allow us to constrain the merger dynamics in mul-tiple ways. From the spectroscopy and Dressler-Schectmantest for the member galaxies in Sif´on et al. (2013), it is shownthat El Gordo does not have complicated substructures in itsgalaxy velocity distribution. El Gordo is further confirmedto be a binary merger from the weak lensing analysis by Jeeet al. (2014). The weak lensing analysis shows a mass ratioof ∼ c (cid:13)000
Configuration of El Gordo showing overlay of dark matter distribution in blue, and X-ray emission in red. (Image credit:NASA, ESA and Jee et al. 2014). The cross markers show the positions of the northwest (NW) and southeast (SE) dark matter densitypeaks, and the center of mass (CM) locations respectively. Note that the mass ratio of the NW subcluster to the SE subcluster is ∼ tures that allow us to constrain the merger dynamics in mul-tiple ways. From the spectroscopy and Dressler-Schectmantest for the member galaxies in Sif´on et al. (2013), it is shownthat El Gordo does not have complicated substructures in itsgalaxy velocity distribution. El Gordo is further confirmedto be a binary merger from the weak lensing analysis by Jeeet al. (2014). The weak lensing analysis shows a mass ratioof ∼ c (cid:13)000 , 1–20 he return of the merging galaxy subclusters of El Gordo? outskirts, strong radio emission is detected in the NW andthe SE respectively. These radio emitting regions show steepspectral index gradients and are identified as radio relics as-sociated with shock waves created from the merger (Lindneret al. 2014). El Gordo is one of ∼
50 galaxy clusters thathave been associated with a radio relic and show dissociationbetween the X-ray gas and the DM subclusters.In this paper, we combined most of the available infor-mation of El Gordo with the main goal of giving estimatesof the dynamical parameters after taking into account allconstraints and uncertainties due to the missing variables.Since mergers of clusters proceed on the time-scale of manymillions of years, one of the most important missing variablesto infer is the time-since-pericenter (TSP) † , which is definedto be the time when the mass peaks of the DM subclustersare at minimum separation. Determining the TSP of similarclusters helps us reconstruct different stages and recover thephysics of a cluster merger. In particular there is a degen-eracy between the following two possible scenarios: We callthe scenario for which the subclusters are moving apart afterpericenter to be “outgoing”, and the alternative scenario“returning” for which the subclusters are approaching eachother after turning around from the apocenter for the firsttime (See Fig. 2).Another crucial, missing piece of information is the 3Dconfiguration, i.e. the angle between the plane of the skyand the merger axis called the projection angle α . Sincemost of the dynamical observables are projected quantitieswhile the modelling of physics requires 3D variables, thedeprojection contributes the largest amount of uncertaintyto the dynamical variables (Dawson 2013, hereafter D13).The morphology of the double relic of El Gordo suggeststhat α should be small. For mergers with a large projectionangle, the radio emission would be projected towards thecenter of the merger, which inhibits detection (Vazza et al.2012). However, the only quantitative constraint on α for ElGordo is from the analysis of the radio relic from Lindneret al. (2014) with a lower bound of α ≥ . ◦ . A tighterconstraint on α is needed for us to reduce uncertainty of thedynamical variables.We employed a data-driven approach that thoroughlyprobes parameter space by directly drawing samples fromthe probability density functions (PDFs) of the observables.This work based on Monte Carlo simulation is particularlyimportant since the phase space of possible merger scenariosis large. Previous attempts at modeling El Gordo with hydro-dynamical simulations such as Donnert (2014) and Molnar& Broadhurst (2015) provided only in total a dozen possibleconfigurations of El Gordo, which do not reflect the rangeof input uncertainties. Another approach for estimating dy-namical parameters would be to look for multiple analogsof El Gordo in cosmological simulations. However, underthe hierarchical picture of structure formation in the ΛCDMmodel, there is a rare chance for massive clusters like ElGordo to have formed at a redshift of z = 0 .
87. The number TSP in this paper is completely identical to the variable time-since-collision (TSC) in Dawson et al. (2012). We have renamedthe variable to avoid confusion about how we define collision aspericenter.
Figure 3.
Points showing the locations of the member galaxiesand the division of the member galaxies among the two subclustersof El Gordo by a spatial cut (black line). The color of the pointsshows the corresponding spectroscopic redshift of the membergalaxies (see color bar for matching of spectroscopic values), withthe redder end indicating higher redshift. The galaxy numberdensity contours in the background in green indicate a bimodaldistribution. density of analogs with mass comparable to El Gordo in acosmological simulation is as low as 10 − Mpc − (M12).In the following sections, we adopt the following conven-tions: (1) we assume the standard ΛCDM cosmology withΩ m = 0 .
3, Ω Λ = 0 . H = 70 km / s / Mpc. (2) All confi-dence intervals are quoted at the 68% level unless otherwisestated. (3) All quoted masses ( M c ) are based on masscontained within r where the mass density is 200 timesthe critical density of the universe at the cluster redshift of z = 0 . We gathered and analyzed data from multiple sources. SeeTable 1 for descriptions of the PDFs of the input variables.We examined the spectroscopic data obtained from the VeryLarge Telescope (VLT) as described in M12 and Sif´on et al.(2013) for estimating the relative velocity differences betweenthe subclusters. We adopted the identification of galaxymembership of El Gordo given by Sif´on et al. (2013) witha total count of 89 galaxies. To further distinguish membergalaxies of each subcluster, we adopted the spatial cut fromM12. The adopted spatial cut is approximately perpendicularto the 2D merger axis (M12) and is consistent with thebimodal number density contours (See Fig. 3). There are 51members identified in the NW subcluster and 35 membersin the SE subcluster.For the weak-lensing mass estimation, we used the MonteCarlo Markov Chains (MCMC) mass estimates from J13.The Hubble Space Telescope data used in J13 were obtainedfrom two programs PROP 12755 and PROP 12477. PROP12755 consisted of two pointings in the F625W, F775W,and F850LP. F850LP for a 6’ x 3’ strip, while PROP 12477provided a 2 × c (cid:13) , 1–20 Karen Y. Ng et al.
Velocity
Pericenter
Key: subcluster
Velocity
Outgoing scenario
Apocenter
Returning scenario shockfront
Figure 2.
Illustration of the spatial location of different components of El Gordo at different stages of the merger. Earlier stages (with asmaller TSP) are on the left side of later stages. The rightmost returning scenario is preferred from our simulation. referred to the properties of the radio relics from Lindner et al.(2014). El Gordo shows radio emission on the periphery ofboth subclusters (M12). The two radio relics, the northwest(NW) relic and the southeast (SE) relic, of El Gordo weretentatively identified in the Sydney University MolongloSky Survey (SUMSS) data in low resolution at 843 MHz(Mauch et al. 2003) as shown in M12. Higher resolution radioobservations conducted by Lindner et al. (2014) at 610 MHzand 2.1 GHz later confirmed the identities of the NW andthe SE relic, and found another extended source of radiorelic in the east (E) (See Fig. 1). Among the radio relics,the NW relic possesses the most extended geometry (0.56Mpc in length), and its physics, including the polarizationand Mach number, were studied in the greatest detail. Suchinformation allows us to constrain α and the merger scenario.The E relic was also reported to have a resolved length of0.27 Mpc, while the SE relic was found to overlap with apoint source (Lindner et al. 2014). Both the E and SE relicare closer to the SE DM subcluster, so we considered themto originate from the same merger shock in the followingwork. We used the collisionless dark-matter-only Monte Carlo mod-eling code written by D13, to model the dynamics of the DMsubclusters of their first core-passage. In the D13 code, thetime evolution of the head-on merger was computed basedon an analytical, dissipationless model assuming that theonly dominant force is the gravitational attraction from themasses of two Navarro-Frenk-White (hereafter NFW) DMhalos (Navarro et al. 1996). The gravitational attraction wasevaluated and summed at 10 000 fixed grid points of each ofthe analytic NFW halo profiles out to the respective r c .In the simulation, many realizations of the collision arecomputed by drawing random realizations of the PDFs ofthe inputs. Most input variables are obtained from previousobservations ( (cid:126)D ). One unknown model variable, which is theprojection angle between the plane of the sky and the mergeraxis, α , is drawn from the PDF of α being observed: α ( j ) ∼ f ( α ) = cos α. (1) and the calculation of the output variables of the j -th real-ization can be denoted as:( (cid:126)θ (cid:48) ) ( j ) = g ( α ( j ) , (cid:126)D ( j ) ) , (2)for a suitable function g that describes conservation of en-ergy during the collision of the two NFW halos due tothe mutual gravitational attraction. In particular, the re-quired (cid:126)D includes the masses ( M NW , M SE ) the red-shifts ( z NW , z SE ) and the projected separation of the twosubclusters ( d proj ). See Table 1 for quantitative descriptionsof the sample PDFs, and the outputs with physical impor-tance are described in detail in Section 3.2.Finally, we excluded realizations that produce any un-physical output values, such as realizations with time-since-pericenter larger than the age of universe at the clusterredshift. We refer to this process of excluding unphysical re-alizations as applying weights. To ensure convergence of theoutput PDFs, in total, 2 million realizations were computed.However, the estimates would agree up to 1% just from 20000 runs (D13). Even though we describe the weights for onevariable at a time (See Appendix A), the correlations betweendifferent variables are properly taken into account since wediscarded all the variables of the problematic realizations.The system of El Gordo satisfies several major assump-tions in the Monte Carlo simulation. One of the strongestassumptions is that all realizations correspond to gravitation-ally bound systems. The simulation excludes all realizationsthat would result in relative pericenter velocities of the sub-clusters higher than the free-fall velocity. We justify ourassumption of modeling only gravitationally bound systemsby noting that the relative escape velocity of the subclustersfor El Gordo is 4500 km s − (based on the mass estimatesof Jee et al. (2014)). Studies from cosmological simulationsgiving the PDFs of the pairwise velocities of massive mergingclusters ( > M (cid:12) ) indicate that it is highly unlikely thatthe pairwise velocities would be > − under ΛCDM(Thompson & Nagamine 2012, Lee & Komatsu 2010).Other major assumptions for modeling systems withthis code include negligible impact parameter. Several pa-pers have noted that the X-ray morphology of a bimodalmerger is sensitive to the impact parameter (Springel & Far-rar 2007, Ricker 1998, Mastropietro & Burkert 2008); an c (cid:13)000
Illustration of the spatial location of different components of El Gordo at different stages of the merger. Earlier stages (with asmaller TSP) are on the left side of later stages. The rightmost returning scenario is preferred from our simulation. referred to the properties of the radio relics from Lindner et al.(2014). El Gordo shows radio emission on the periphery ofboth subclusters (M12). The two radio relics, the northwest(NW) relic and the southeast (SE) relic, of El Gordo weretentatively identified in the Sydney University MolongloSky Survey (SUMSS) data in low resolution at 843 MHz(Mauch et al. 2003) as shown in M12. Higher resolution radioobservations conducted by Lindner et al. (2014) at 610 MHzand 2.1 GHz later confirmed the identities of the NW andthe SE relic, and found another extended source of radiorelic in the east (E) (See Fig. 1). Among the radio relics,the NW relic possesses the most extended geometry (0.56Mpc in length), and its physics, including the polarizationand Mach number, were studied in the greatest detail. Suchinformation allows us to constrain α and the merger scenario.The E relic was also reported to have a resolved length of0.27 Mpc, while the SE relic was found to overlap with apoint source (Lindner et al. 2014). Both the E and SE relicare closer to the SE DM subcluster, so we considered themto originate from the same merger shock in the followingwork. We used the collisionless dark-matter-only Monte Carlo mod-eling code written by D13, to model the dynamics of the DMsubclusters of their first core-passage. In the D13 code, thetime evolution of the head-on merger was computed basedon an analytical, dissipationless model assuming that theonly dominant force is the gravitational attraction from themasses of two Navarro-Frenk-White (hereafter NFW) DMhalos (Navarro et al. 1996). The gravitational attraction wasevaluated and summed at 10 000 fixed grid points of each ofthe analytic NFW halo profiles out to the respective r c .In the simulation, many realizations of the collision arecomputed by drawing random realizations of the PDFs ofthe inputs. Most input variables are obtained from previousobservations ( (cid:126)D ). One unknown model variable, which is theprojection angle between the plane of the sky and the mergeraxis, α , is drawn from the PDF of α being observed: α ( j ) ∼ f ( α ) = cos α. (1) and the calculation of the output variables of the j -th real-ization can be denoted as:( (cid:126)θ (cid:48) ) ( j ) = g ( α ( j ) , (cid:126)D ( j ) ) , (2)for a suitable function g that describes conservation of en-ergy during the collision of the two NFW halos due tothe mutual gravitational attraction. In particular, the re-quired (cid:126)D includes the masses ( M NW , M SE ) the red-shifts ( z NW , z SE ) and the projected separation of the twosubclusters ( d proj ). See Table 1 for quantitative descriptionsof the sample PDFs, and the outputs with physical impor-tance are described in detail in Section 3.2.Finally, we excluded realizations that produce any un-physical output values, such as realizations with time-since-pericenter larger than the age of universe at the clusterredshift. We refer to this process of excluding unphysical re-alizations as applying weights. To ensure convergence of theoutput PDFs, in total, 2 million realizations were computed.However, the estimates would agree up to 1% just from 20000 runs (D13). Even though we describe the weights for onevariable at a time (See Appendix A), the correlations betweendifferent variables are properly taken into account since wediscarded all the variables of the problematic realizations.The system of El Gordo satisfies several major assump-tions in the Monte Carlo simulation. One of the strongestassumptions is that all realizations correspond to gravitation-ally bound systems. The simulation excludes all realizationsthat would result in relative pericenter velocities of the sub-clusters higher than the free-fall velocity. We justify ourassumption of modeling only gravitationally bound systemsby noting that the relative escape velocity of the subclustersfor El Gordo is 4500 km s − (based on the mass estimatesof Jee et al. (2014)). Studies from cosmological simulationsgiving the PDFs of the pairwise velocities of massive mergingclusters ( > M (cid:12) ) indicate that it is highly unlikely thatthe pairwise velocities would be > − under ΛCDM(Thompson & Nagamine 2012, Lee & Komatsu 2010).Other major assumptions for modeling systems withthis code include negligible impact parameter. Several pa-pers have noted that the X-ray morphology of a bimodalmerger is sensitive to the impact parameter (Springel & Far-rar 2007, Ricker 1998, Mastropietro & Burkert 2008); an c (cid:13)000 , 1–20 he return of the merging galaxy subclusters of El Gordo? Table 1.
Properties of the input sampling PDFs ( (cid:126)D ) of the MonteCarlo simulation. We obtained estimates of the inputs via differentmethods. a We made used of the MCMC chains from J13 for mass-related estimates (See Section 3.1.2 ). b The redshift distributionswere obtained from bootstrapping (See Section 3.1.1). c We approx-imated the positions of the centroids with 2D Gaussians beforewe calculated the projected separations of the subclusters (SeeSection 3.1.3). Even though the distributions of the mass estimatesand the redshift were not estimated via parametric methods (e.g.fitting mean and variances of Gaussians), they resemble Gaussiandistributions due to the Central Limit Theorem.Data Units Location Scale Ref M c NW h − M (cid:12) a c NW a M c SE h − M (cid:12) a c SE a z NW b z SE b d proj Mpc 0.74 0.007 J13 c impact parameter as small as 0.1 Mpc can result in sub-stantial asymmetry. The X-ray morphology of El Gordo isapproximately symmetric about the merger axis. On theother hand, the dynamics of the merger is not as sensitiveto the impact parameter as the X-ray morphology. The sim-ulations of Ricker (1998) of bimodal mergers of 10 M (cid:12) halos, showed that the resulting relative velocity would beapproximately 2000 km s − , relatively insensitive to impactparameters between 0 to 5 times the scale radius (5 r s = 3 . . r = 0 .
14 Mpc affected mergerdynamics only at the ∼
10% level. Molnar & Broadhurst 2015indicated that the impact parameter of El Gordo may be aslarge as 40% r s ≈ . r s is the correspondingcharacteristic core radius of the NFW halo with the mass ofthe SE subcluster. We attribute the result from Molnar &Broadhurst 2015 to incomplete exploration of the parameterspace, and note that other impact parameter values may alsomatch the X-ray observables of El Gordo.Other assumptions in this simulation include negligibledynamical friction during the merger, negligible mass accre-tion and negligible self-interaction of dark matter. Discussionof the effects of each of these assumptions is included in D13. After identifying members of each subcluster, we performed10, 000 bootstrap realizations to estimate the biweight loca-tions of the spectroscopic redshifts of the respective membersin order to obtain the samples of the PDFs of the redshiftsof each subcluster. The spectroscopic redshift of the sub-clusters were determined to be z NW = 0 . ± . z SE = 0 . ± . σ bias-corrected confidence levelrespectively (Beers et al. 1990). Both the estimated redshiftsof the subclusters and the uncertainties are consistent withestimates of z = 0 . ± . Figure 4.
Bootstrapped location of the redshift estimates and v rad estimates for each subcluster using the selected spectroscopicmembers. The shaded histograms represent the bootstrappedsamples. ity dispersion among all the ACT galaxy clusters, as reportedby M12. We estimated the radial velocity differences of thesubclusters by first calculating the velocity of each subclusterwith respect to us, using v i = (cid:20) (1 + z i ) − z i ) + 1 (cid:21) c, (3)where i = 1 , c is thespeed of light. The relative radial velocity was calculated by:∆ v rad ( t obs ) = | v − v | − v v c . (4)We obtained a low radial velocity difference of the two sub-clusters to be 476 ±
242 km s − (See Fig. 4). The radialvelocity difference of 586 km s − reported by M12 is higherthan our estimates but within the 68% bias-corrected confi-dence interval. We obtained 40, 000 samples of the joint PDFs of the massesof the two dark matter halos as the outputs of the MonteCarlo Markov Chain (MCMC) procedure from Jee et al. 2014.Discussion of the handling of the weak lensing source galaxiesand the details of the MCMC procedure for mass estimationcan be found in Jee et al. 2014. d proj ) To be consistent with our MCMC mass inference, our MonteCarlo simulation takes the projected separation of the NFWhalos to be those of the inferred DM centroid locations in Jeeet al. 2014. We drew random samples of the location of cen-troids from two 2D Gaussians centered at R.A.=01:02:50.601,Decl.= − bias from the location estimator of the bootstrapped distributionnot giving the maximum likelihood value was corrected for.c (cid:13) , 1–20 Karen Y. Ng et al. − d proj )of 0 . ± .
007 Mpc.
We outline the outputs of the simulation here to facilitatethe discussion of the design of the weights used in the simu-lation. The simulation provides PDF estimates for 8 outputvariables. Variables of the highest interest include the time-since-pericenter and the angle α , which is defined to be theprojection angle between the plane of the sky and the mergeraxis. Other output variables are dependent on α and time.Specifically, the simulation denotes the time dependence byproviding several characteristic time-scales, including thetime elapsed between consecutive collisions ( T ) and the time-since-pericenter of the observed state ( T SP ), with the timeof pericenter defined to be when the centers of the two NFWhalos coincide.We provide two versions of the time-since-pericentervariables
T SP out and
T SP ret to denote different possiblemerger scenarios. 1) The TSP for the“outgoing” scenariocorresponds to the smaller
T SP out value, and 2) the “return-ing” scenario corresponds to the larger
T SP ret . We describehow we make use of properties of the radio relic to evaluatewhich scenario is more likely in section 3.4. Evolution of themerger after the second passage is not considered. Outputsfrom our dissipationless simulation for a “second” passagewill not differ from the first passage, and the predicted relicposition would be so far for us to rule this out.The simulation also outputs estimates of variables thatdescribe the dynamics and the characteristic distances of themerger. The relative 3D velocities of the subclusters, bothat the time of the pericenter ( v D ( t per )) and at the timeof observation ( v D ( t obs )) are provided. The characteristicdistances included in the outputs are the maximum 3D sepa-ration ( d max ), which is the distance between the subclustersat the apocenter and the 3D separation of the subclusters atobservation ( d D ). One of the strengths of the Monte Carlo simulation by D13is its ability to detect and rule out extreme input values thatwould result in unphysical realizations via the applicationof weights. Our default weights are described in D13 andwe include them in Appendix B for the convenience of thereaders. In addition, we have devised a new type of weightsof the projection angle α based on the polarization fractionof the radio relic. We can relate the polarization fraction of the radio relic tothe projection angle by examining the generating mechanismof the radio relic. The observed radio relic was generatedby synchrotron emission of free electrons in a magnetic field.
Figure 5.
Predictions of polarization percentage of the radio relicat a given projection angle from different models, reproduced from(Ensslin et al. 1998) or equation 5. Each model assumes electronsproducing the radio emission to be accelerated inside uniformmagnetic field of various strengths ( strong or weak ). The curvesare plotted with spectral index of the radio emission ( α radio ),spectral index of the electrons ( γ ) and compression ratio of themagnetic field ( R ) corresponding to the estimated values fromLindner et al. (2014). We highlight the observed polarizationpercentage of the main NW radio relic of El Gordo by the dottedvertical line with the greyed out region indicating the uncertainty(Lindner et al. 2014). If the magnetic field was uniform, the observed polariza-tion fraction of the synchrotron emission of the electronsdepends on the viewing angle (or equivalently the projec-tion angle) with respect to the alignment of the magneticfield. Synchrotron emission from electrons inside unorganizedmagnetic field is randomly polarized. The high reported in-tegrated polarization fraction from Lindner et al. (2014) canbe explained by a highly aligned magnetic field, compressedalong with the ICM during a merger (Ensslin et al. 1998,van Weeren et al. 2010, Feretti et al. 2012).We designed the weights to reflect how α decreases mono-tonically as the maximum observable integrated polarizationfraction ( (cid:104) P (cid:105) ). This assumption is based on the class of mod-els given by Ensslin et al. (1998)(See Figure 5). In particular,we refer to a model from Ensslin et al. (1998) that wouldgive the most conservative estimate on the upper bound of α : α = 90 ◦ − arcsin (cid:118)(cid:117)(cid:117)(cid:116)
215 13 R − R − γ +7 / γ +1 (cid:104) P strong (cid:105) γ +7 / γ +1 (cid:104) P strong (cid:105) , (5)This model corresponds to the case of a strong field with therelic being supported by magnetic pressure only, with thespectral index of the radio emission being α radio = 0 .
86, thecompression ratio of the magnetic field being R = 2 . γ = 2 . ∼
60% when α →
0. This polariza-tion fraction of ∼
60% predicted by (Ensslin et al. 1998) isconsistent with the upper bound of relic polarization frac-tion in cosmological simulations (Skillman et al. 2013). Fromthis model, the observed integrated polarization fraction of33% ±
1% corresponds to an estimated value of α = 35 ◦ . Noother model of the magnetic field should predict a higher c (cid:13)000
1% corresponds to an estimated value of α = 35 ◦ . Noother model of the magnetic field should predict a higher c (cid:13)000 , 1–20 he return of the merging galaxy subclusters of El Gordo? polarization fraction, thus it is highly unlikely that we see33% integrated polarization at α > ◦ .We cannot rule out α ≤ ◦ because magnetic fieldnonuniformities can lower the polarization below the En-sslin model value. Ensslin et al. (1998) assumes an isotropicdistribution of electrons in an isotropic magnetic field. Cos-mological simulations of radio relics from Skillman et al.(2013) show varying polarization fraction across and alongthe relic assuming α = 0, resulting in a lower integratedpolarization fraction. For example, it is possible to see anedge-on radio relic ( α = 0) with integrated polarization frac-tion of 33%. Furthermore, Skillman et al. (2013) shows thatafter convolving the simulated polarization signal with aGaussian kernel of 4 (cid:48) to illustrate effects of non-zero beamsize, the polarization fraction drops to between 30% to 65%even when α = 0. We examined the effects of perturbing thecutoff value of this weight to ensure the uncertainties do notintroduce significant bias in the estimated output variablesin section 4.2. To summarize, we used a conservative uniformweight to encapsulate the information from the polarizationfraction of the radio relic as: w ( α ) = (cid:40) const. for α < ◦ . (6)We refer to equation 6 as the polarization weights. Unlessotherwise stated, the main results of the paper are obtainedafter applying this polarization weight in addition to thedefault weights. One of the biggest questions involving the merger is whetherEl Gordo is observed during a returning or outgoing phase.We compared the two merger scenarios by making use of theobserved projected separation of the relic from the center ofmass. Simulations of cluster mergers such as the work of Paulet al. (2011), Van Weeren et al. (2011), and Springel & Farrar(2007) showed that merger shock fronts that may correspondto the radio relics 1) are generated near the center of massof the subclusters close to the time of the first core-passage,2) propagate outward with the shock speed decreasing onlyslightly. The propagation speed of the shock wave with respectto the center-of-mass is reported to drop between ∼ T SP out and
T SP ret values. We worked inthe center of mass frame where the shock speed is expectedto drop slightly with TSP. The projected separation of theshock is approximated as: s jproj ≈ (cid:104) v relic (cid:105) j ( t jobs − t jper ) cos( α j ) , (7)where the superscript j of any variable denotes the value of Figure 6.
The marginalized output PDF of the returning time-since-pericenter (
T SP ret ) vs. the 3D velocity at the time of peri-center for El Gordo. The grey set of contours show the confidenceregions before applying the polarization weight and the coloredcontours correspond to the confidence regions after applying theweights. The contours represent the 95% and 68% confidenceregions respectively. the variable from the j-th realization of the simulation, and s proj is the estimated projected separation. We estimatedthe upper and lower bounds of the time-averaged velocity (cid:104) v relic (cid:105) of the shock between the pericenter of the subclustersand the observed time as: (cid:104) v NWrelic (cid:105) j = β v j D,NW ( t per ) (8)= β v j D ( t per ) m jSE m jSE + m jNW , (9)where β is a factor that we introduce to represent the uncer-tainty of the velocity of the relic shock wave, v D,NW ( t per )refers to the pericenter velocity of the NW subcluster inthe center-of-mass frame as a comparison, and m representsthe mass within r c of each subcluster as denoted by thelabels in the subscripts. Likewise, we have also computed theexpected projected separation of the SE relic using: (cid:104) v SErelic (cid:105) j = β v j D ( t per ) m jNW m jSE + m jNW . (10)For the most likely range of β , we refer to the simula-tions of cluster mergers by both Springel & Farrar (2007)and Paul et al. (2011) because those are some of the fewsimulations available that quote shock propagation speeds inthe center-of-mass frame of the cluster, rather than the ICMframe. The simulation of the Bullet Cluster by Springel &Farrar (2007), indicates that the propagation velocity of theshock evolves such that β ≈ .
95 within ∼ . β ≈ . T SP of El Gordois longer. To include the possible range of valid β values, weexamined 0 . ≤ β ≤ .
5. This range of β ≈ β >
1. An exam- c (cid:13) , 1–20 Karen Y. Ng et al.
Table 2.
List of variables that provide quantitative constraints for the merger scenario. For details of the distribution of eachvariable see the corresponding † Section. Calculations were done with all available realizations instead of the best estimate valuelisted here. Variable Best estimate value Unit Section † Time averaged speed of SE relic in the CM frame ( (cid:104) v relic (cid:105) ) 530 km s − (cid:104) v relic (cid:105) ) 310 km s − s proj ) 1.1 Mpc 3.4Projected separation of NW relic from the CM ( s proj ) 0.63 Mpc 3.4Outgoing time-since-pericenter ( T SP out ) 0.61 Gyr 3.2, 4.1Returning time-since-pericenter (
T SP ret ) 1.0 Gyr 3.2, 4.1Age of the universe at z = 0.87 6.30 GyrFree fall velocity of subclusters 4500 km s − (cid:104) P strong (cid:105) ) 33% 3.3.1 Table 3.
Table of the output PDF properties of the model variables and output variables from Monte Carlo simulationDefault weights Default + polarization weightsVariables Units Location 68% CI †
95% CI Location 68% CI 95% CI α (degree) 43 19-69 6-80 21 10-30 3-34 d proj Mpc 0.74 0.74-0.75 0.73-0.76 0.74 0.74-0.75 0.73-0.76 d max Mpc 1.2 0.90-2.2 0.77-4.6 0.93 0.81-1.2 0.75-1.9 d Mpc 1.0 0.79-2.1 0.75-4.3 0.80 0.76-0.88 0.74-0.91
T SP out
Gyr 0.61 0.4-0.95 0.26-2.4 0.46 0.30-0.55 0.21-0.64
T SP ret
Gyr 1.0 0.77-1.7 0.63-4.4 0.91 0.69-1.3 0.59-2.3 T Gyr 1.6 1.3-2.4 1.2-6.4 1.4 1.2-1.6 1.2-2.2 v ( t obs ) km s −
580 260-1200 59-2400 940 360-1800 62-2900 v rad ( t obs ) km s −
360 140-630 27-880 310 110-590 8-840 v ( t per ) km s − † CI stands for confidence interval ple of merging clusters (Merger F) with β ≈ . ∼ . × M (cid:12) in total, and that the coarsetime resolution in Paul et al. 2011 likely underestimates thepericenter velocity and overestimates β . We therefore suggesta most likely range, closer to the value of β inferred fromSpringel & Farrar (2007), as 0 . ≤ β ≤ .
5. In section 4.1,we demonstrate that β has to be larger than 1 . β = 2 .
0. In fig. 7 and 8, we comparedour estimates of the projected locations of the relics to the ob-served location given by Table 3 of L13. The given NW relicand E relic locations are R.A.=01:02:46, Decl.= − − We present an overview of all the estimated variables inTable 3, with results only applying the default weights onthe left hand side of the table and those also applied withthe polarization weight on the right hand side. Furthermore,we include the plots of all the marginalized PDFs with thepolarization weight in Appendix B.We found that the two subclusters collided with a rela-tive velocity of 2400 ± km s − , at an estimated projectionangle of α = 21 ◦ ± . From our analysis of the two scenarios,we found that El Gordo is more likely to be observed at a returning phase with an estimate of T SP ret = 0 . ± . . Gyr(See section 4.1 and Appendix C for a full discussion of theassumed relic propagation speed). This returning scenarioputs the estimate of the time of pericenter to be when theage of the universe was ∼ . v D ( t obs )is 940 ± km s − . (See Fig. 6) This fits comfortably withinthe upper limit of 2500 ± km s − reported by Lindneret al. (2014), which was obtained by making use of the Machnumber of the NW radio relic. The simulation gives two estimates for the time-since-pericenter, with
T SP out = 0 . ± . . Gyr and
T SP ret =0 . ± . . Gyr for the returning model. Both the estimatesof
T SP out and
T SP ret fit within the observable time scale ofthe radio relics, which is on the scale of ∼ d proj us-ing the most likely value of β = 0 . M ret ) holds true for the rel-evant range of β < .
1, which corresponds to the time-averaged velocity of the relics at (cid:104) v NWrelic (cid:105) < − and (cid:104) v SErelic (cid:105) < − in the center of mass frame. Forcomparison, we found that an extreme, and unlikely rangeof β > . M out )to be preferred. (See Appendix C for plots of the range of β that we examined). We marginalized β to compute the prob-ability of the simulated relic location being compatible with c (cid:13)000
1, which corresponds to the time-averaged velocity of the relics at (cid:104) v NWrelic (cid:105) < − and (cid:104) v SErelic (cid:105) < − in the center of mass frame. Forcomparison, we found that an extreme, and unlikely rangeof β > . M out )to be preferred. (See Appendix C for plots of the range of β that we examined). We marginalized β to compute the prob-ability of the simulated relic location being compatible with c (cid:13)000 , 1–20 he return of the merging galaxy subclusters of El Gordo? projected separation from the center of mass (Mpc) P D F outgoing case observed locationreturning case NW relic location assuming β = Figure 7.
Comparison of the PDFs of the observed positionof the NW relic (red bar includes the 95% confidence interval oflocation of the NW radio relic in the center of mass frame) with thepredicted position from the two simulated merger scenarios (bluefor outgoing and green for the returning scenario). We made useof the polarization weight for producing this figure. The rationaleof picking β = 0 . the observed location P ( S proj ∩ S obs | M ). We then computed P ( S proj ∩ S obs | M ret ) /P ( S proj ∩ S obs | M out ). The ratio of thetwo probabilities is found to be ≈ . ≈
460 for the SE relic, favoring the re-turning scenario despite the uncertainties. (See appendix C).This scenario is further supported by the position of the coolcore in the southeast as discussed in Section 5.Finally, we note that the estimate of NW shock velocityat 2500 ± km s − by Lindner et al. (2014) was inferredfrom the Mach number, thus, this velocity is measured inthe reference frame of the turbulent ICM, not the velocitywith respect to center of mass. Due to the difference thatcould arise from the different frame of references, we havenot made use of the Mach number estimate of Lindner et al.(2014) in this calculation. If there are radio data in morefrequency bands than the radio data available now (Lindneret al. 2014), an alternative constraint of the TSP can beconstructed from the spectral aging of the electrons thatwere involved in the generation of the radio relics, such asshown in Stroe et al. (2014). We performed tests of how the output variables vary accord-ing to the choice of the cutoff of the polarization weightbetween α cutoff = 29 ◦ to 49 ◦ instead of 35 ◦ , that is, shownas the horizontal cut off in Fig. 5. We found that in the mostextreme case, choosing the cutoff values as 29 ◦ , the locationof the v D ( t obs ), is increased by 16%. While the 95% CI of d max is the most sensitive to the weight and it changes by ∼
20% when α cutoff = 49 ◦ . This shows that the exact choiceof the cut off value for α for the polarization weight does notchange our estimates drastically. projected separation from the center of mass (Mpc) P D F outgoing case observed locationreturning case SE relic location assuming β = Figure 8.
Comparison of the PDFs of the observed position of theSE relic (red bar includes the 95% confidence interval of locationof the radio relic in the center of mass frame). We made use ofthe polarization weight for producing this figure. The rationale ofpicking β = 0 . We outline the qualitative agreement and disagreement be-tween our simulations and hydrodynamical simulations ofEl Gordo such as Donnert (2014) and Molnar & Broadhurst(2015). Our simulation focuses on giving PDF estimates ofparticular dynamical and kinematic variables, whereas thehydrodynamical simulations focused on understanding thedetailed gas dynamics required to reproduce the X-ray ob-servables and SZ observables of El Gordo. The goals, assump-tions, and initial conditions of Donnert (2014) and Molnar& Broadhurst (2015) differ substantially with ours. However,our approach has the advantage of considering a much widerrange of geometries and dynamical parameters, and is basedon recently measured lensing masses.Both hydrodynamical simulations were based on a fewsets of initial conditions, instead of thorough sampling ofthe inputs. For example, both simulations made use ofthe mass estimates from the dynamics analysis of M12 at m NW = 1 . × M (cid:12) , which is larger than the upper 95%CI of the mass that we used based on the weak lensing esti-mate. Furthermore, Molnar & Broadhurst (2015) initializedthe relative infall velocity (velocity when the separation ofsubclusters equals the sum of the two virial radii) to be2250 km s − . This corresponds to v D ( t per ) (cid:38) − ,which is close to the escape velocity of the subclusters. Oursimulation shows a negligible number of realizations with v D ( t per ) > − . The range of projection anglessuggested by Molnar & Broadhurst (2015) of α (cid:38) ◦ is alsoexcluded by our polarization weight, whereas we are unableto find information concerning the projection angle of thesimulation from Donnert (2014).With a time resolution of 0.25 Gyr, Donnert (2014) gavean estimate of T ≈ c (cid:13) , 1–20 Karen Y. Ng et al.
TSP / T D i s t an c e f r o m c en t e r o f m a ss DM subcluster peak cool core shock front
TSP ret / T (El Gordo)
TSP out / T (Bullet Cluster& El Gordo)
Figure 9.
Schematic evolution of cool core gas and DM dis-placements relative to the merger center of mass as a function ofthe phase (
T SP/T ), based on simulations of a bimodal clustermerger by Mathis et al. (2005). During and shortly after corepassage, ram pressure (= ρv ) exerts substantial force on the coolcore, which then lags the DM. (This corresponds to the outgoingscenario of T SP out /T indicated by the grey dotted line). Rampressure then declines dramatically as the cool core enters regionsof lower density. The cool core can then fall into (and past thecenter of) the gravitational potential of the corresponding DMsubcluster as what is described as the slingshot effect (Marke-vitch & Vikhlinin 2007). The Bullet Cluster is seen at a phase of T SP out /T ≈ . / . T SP ret /T rather than T SP out /T ), explaining whythe DM of El Gordo is closer to the center of mass than the coolcore. core-passage in Fig. 6 of their work, while our estimate gives T = 1 . ± . .
69 Mpc to thecorresponding observables, Donnert (2014) also reported theirsimulated work to best match observations at ∼ .
15 Gyrafter pericenter. The
T SP out from Donnert (2014) is belowthe estimated 95% CI of
T SP out from our work. On the otherhand, Donnert (2014) obtained a relative pericenter velocitybetween the subclusters at ∼ − , which is compat-ible with our estimate of 2400 ± km s − . This agreementmight be due to the similar assumptions of a low energy orbitand a small impact parameter as the initial conditions inthe work of Donnert (2014) and our work. Ideally, the hydro-dynamic and Monte Carlo dynamical approaches should becombined, with new hydrodynamic simulations seeded withinitial conditions motivated by the results presented here. The hypothesis of El Gordo being in the returning phase ismore plausible when we compare the details of the observ-ables of El Gordo to the Bullet Cluster (Bradaˇc et al. 2006,Springel & Farrar 2007, Mastropietro & Burkert 2008). Manyinferred properties are similar between the two clusters and both clusters were observed in similar wavelengths. Bothclusters are bimodal major mergers of subclusters of substan-tial masses. The inferred merger velocities are comparableat around 2600 km s − and α of both clusters are around20 ◦ . In particular, the inferred outbound T SP out /T ∼ . T SP out for El Gordo is invalid) while the Bullet Cluster isin the outgoing phase, the differences in the observables ofEl Gordo and the Bullet Cluster can be explained.First, the merger shock front of the Bullet Cluster isobserved only in the X-ray, but not via the radio relic, mean-ing that the shock may not have the time to propagate tothe outskirts of the cluster (Br¨uggen et al. 2011, Markevitch& Vikhlinin 2007), and this bow shock is indeed observedto closely lead the corresponding less massive subcluster by ∼ .
08 Mpc, assuming they are propagating outward. Onthe other hand, indirect observables of the merger shocks ofEl Gordo can only be detected through the radio relic, andthe shock is further offset from the corresponding subcluster( ∼ . ∼ . . × M (cid:12) ) and mass ratio (1:1)as El Gordo supports our proposed scenario: it shows theturn-around of the cool core can occur after the apocenter ofthe DM component, resulting in the cool core being furtheraway from the center of mass than the dark matter by asmuch as ∼ . El Gordo possesses a range of special properties that makeit a promising probe of self-interaction of DM. Its high massensures high DM particle density for interactions during thehigh-speed core-passage. Its bimodal configuration makesit relatively simple to interpret the offset and dynamics ofthe different components. The observation of the radio relichas enabled us to constrain the projection angle and reduce c (cid:13)000
08 Mpc, assuming they are propagating outward. Onthe other hand, indirect observables of the merger shocks ofEl Gordo can only be detected through the radio relic, andthe shock is further offset from the corresponding subcluster( ∼ . ∼ . . × M (cid:12) ) and mass ratio (1:1)as El Gordo supports our proposed scenario: it shows theturn-around of the cool core can occur after the apocenter ofthe DM component, resulting in the cool core being furtheraway from the center of mass than the dark matter by asmuch as ∼ . El Gordo possesses a range of special properties that makeit a promising probe of self-interaction of DM. Its high massensures high DM particle density for interactions during thehigh-speed core-passage. Its bimodal configuration makesit relatively simple to interpret the offset and dynamics ofthe different components. The observation of the radio relichas enabled us to constrain the projection angle and reduce c (cid:13)000 , 1–20 he return of the merging galaxy subclusters of El Gordo? uncertainties of other dynamical parameters. Furthermore,El Gordo is likely to be a late-stage merger unlike other wellstudied clusters such as the Bullet Cluster. This gives us abetter picture of how a bimodal merger would behave at alater stage of a merger.This special merger scenario of El Gordo also raisesa question: what phase of a merger or what type of merg-ers would allow the most stringent constraints on the self-interaction cross section of DM ( σ SIDM )? The use of mergingclusters as probes of σ SIDM has been proposed and used invarious papers. (Markevitch et al. 2004, Randall et al. 2008,Merten et al. 2011, Dawson et al. 2012). One common themeamong such work is to make use of the observed offsets ofthe different components of the merging clusters for the es-timation. One of the most popular methods proposed byMarkevitch et al. 2004 (method 1 in the paper) assumesthe gas component would lag behind the corresponding DMsubcluster along the direction of motion due to ram pres-sure stripping. For El Gordo, since the cool core is furtheraway from the center of mass than the SE DM centroid, it isapparent that this particular method does not apply.Alternative methods for determining the self-interactioncross sections, such as from the galaxy-DM offset, are yet tobe perfected. Future work is required to investigate how tobest characterize the spatial distribution of the galaxies. Onepending question is to investigate if the luminosity densitypeak or number density peaks would better represent thegalaxy distributions. The galaxy number density map of ElGordo (J13) shows a noteworthy ∼ . This work has allowed us to examine what information wouldbe needed to better understand the merger dynamics andscenario. Before this work, simulations of merger shocks havefocused on providing estimates of the local conditions of thephysics responsible for the generation of the radio relic or thegas physics. In this work, we demonstrated that the globalproperties of the shocks, are also important for understandingthe merger scenario. Important questions concerning merginggalaxy clusters pending for answers include: • What are the typical propagation velocities of the shockwave that corresponds to the radio relic in the center of mass(CM) frame of the cluster? • What physical properties of the DM subclusters wouldcorrelate the best with the time-evolution of the propagationvelocity of the shock wave (in the CM frame)? • What is the typical duration after the merger for which radio relics are observable in terms of the merger core-passagetime-scales? • How generalizable is the merger scenario in Figure 9? • How would the galaxy-DM offset evolve if we were toadd that information to Fig. 9? • For how long do transcient X-ray features in mergingclusters (such as the wake in El Gordo) persist?We urge simulators to narrow the gap between simulationsand data by investigating these issues.
We provide estimates of the dynamical parameters of ElGordo using Dawson’s Monte Carlo simulation, in particular,we(i) demonstrated the first use of polarization fractioninformation from the radio relics to reduce our estimatesof the projection angle from 43 ◦ ± to 21 ◦ ± (SeeFig. B3). By performing sensitivity analysis, we showedthat this weighting function helps reduce uncertainty forthe dynamical variables without changing the dynamicalvariable estimates drastically ( < relative pericenter velocity between thesubclusters of El Gordo as 2400 ± km s − (iii) showed that a returning scenario is favored if (cid:104) v NWrelic (cid:105) ≤ − and (cid:104) v SErelic (cid:105) ≤ − where the velocities are in the CM frame and angle bracketsdenote averaging over the time since pericenter It takes anunlikely high speed of (cid:104) v relic (cid:105) (cid:29) . v D,sub ( t per ) for theoutgoing scenario to be favored.(iv) showed how our inferred returning scenario may ex-plain the unexpected location of the cool core, namely, thecool core being close to the center of mass of the cluster, andstill be consistent with the wake / gas-tail morphology of thecool core.As large scale sky surveys come online, more cluster mergersat late stages of their merger will be discovered. El Gordowill serve as one of the best studied examples of a bimodalcluster merger for comparison. We thank Franco Vazza, Marcus Br¨uggen and Surajit Paulfor sharing their knowledge on the simulated properties ofradio relic and merger shocks. We extend our gratitude toReinout Van Weeren for first proposing the use of radio relicto weight the Monte Carlo realizations. We appreciate thecomments from Maruˇsa Bradaˇc about using the position ofthe relic to break degeneracy of the merger scenario. KN isgrateful to Paul Baines and Tom Loredo for discussion of theuse of prior information and sensitivity tests. Part of thiswork was performed under the auspices of the U.S. DOEby LLNL under Contract DE-AC52-07NA27344. JPH grate-fully acknowledges support from Chandra (grant numberGO2-13156X) and Hubble (grant number HST-GO-12755.01-A). We would also like to thank GitHub for providing free c (cid:13) , 1–20 Karen Y. Ng et al. repository for version control for our data and analyses. Thisresearch made use of APLpy, an open-source plotting packagefor Python hosted at http://aplpy.github.com; Astropy, acommunity-developed core Python package for Astronomy(Robitaille et al. 2013); AstroML, a machine learning libraryfor astrophysics (VanderPlas et al. 2012), and IPython, asystem for interactive scientific computing, computing inscience and engineering (Perez & Granger 2007).Note: The authors have made the Pythoncode for most of the analyses openly available athttps://github.com/karenyyng/ElGordo paper1.
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APPENDIX A: DEFAULT WEIGHTS USEDFOR DAWSON’S MONTE CARLOSIMULATION
The default weight that we employed can be summarized asfollows for most of the output variables: w ( v D ( t per )) = 0 if v D ( t per ) > v free fall . (A1) c (cid:13)000
The default weight that we employed can be summarized asfollows for most of the output variables: w ( v D ( t per )) = 0 if v D ( t per ) > v free fall . (A1) c (cid:13)000 , 1–20 he return of the merging galaxy subclusters of El Gordo? Figure B1.
Matrix of variables used in the simulations. Wepresent them in 4 categories, including, inputs, geometry, timeand velocity. Regions of the same color represent one plot andthe number indicates the corresponding figure number in thisappendix. w ( T SP out ) = (cid:40) const if
T SP out < age of universe at z = 0 .
870 otherwise . (A2)In addition, we apply the following weight on T SP ret : w ( T SP ret ) = (cid:40) const if
T SP ret < age of universe at z = 0 .
870 otherwise . (A3)To correct for observational limitations, we further convolvedthe posterior probabilities of the different realizations with w ( T SP out | T ) = 2 T SP out
T , (A4)to account for how the subclusters move faster at lower
T SP and thus it is more probable to observe the subclusters at astage with a larger
T SP . APPENDIX B: PLOTS OF OUTPUTS OF THEMONTE CARLO SIMULATION
We present the PDFs of the inputs of the dynamical simula-tion and the marginalized PDFs of the outputs after applyingthe polarization weight in addition to the default weights.See Fig. B1 for explanations of the order that we present thefigures containing the PDFs . c (cid:13) , 1–20 Karen Y. Ng et al.
APPENDIX C: COMPARISON OF THEOUTGOING AND RETURNING SCENARIO
Here, we compare the different merger scenarios using thetwo relics independently and show that they consistently givethe conclusion that the returning scenario is favored for theplausible range of β . For each merger scenario, we computethe (marginalized) probability of producing simulated values( s proj ) compatible with the observed location of the radiorelic ( s obs ).Quantitatively, we want to compute and compare theprobability: P ( s proj compatible with s obs | M ) (C1)= (cid:90) (cid:90) f ( S proj ∩ S obs | M, β ) f ( β | M ) ds proj dβ (C2)= (cid:90) (cid:90) f ( s proj | M, β ) f ( s obs ) f ( β | M ) ds proj dβ, (C3)where f indicates the corresponding PDF, M represents oneof the merger scenarios, and β is defined in equation 8, s proj ∈ S proj and s obs ∈ S obs . We set our priors set to beuniform for the marginalization: f ( β | M ret ) = f ( β | M out ) = (cid:40) const if 0 . ≤ β ≤ .
50 otherwise . (C4)which is more conservative than the most likely range of β ,which is 0 . < β < .
1. We found: P ( S proj ∩ S obs | M ret ) /P ( S proj ∩ S obs | M out ) (C5)= (cid:40) . ,
460 for the SE relic , (C6)which shows that the returning scenario is favored over theoutgoing scenario.This test quantity differs from the traditional hypothesistesting or model comparison in several ways:(i) we did not compute a likelihood function. We haveadopted non-parametric PDFs in our Monte Carlo simula-tion,i.e. there is no well-known functional form of the like-lihood in our context. We make use of f ( S proj ∩ S obs ) topenalize the simulated values being different from our ob-served data(ii) with this quantity, we are not asking whether the ex-pected value of the radio relic such as the mean or medianfrom each model match the observation best. Those esti-mators take into account the values that do not match theobserved location of the radio relic.(iii) we marginalized the uncertainty in β to be as con-servative as possible, instead of assuming a fixed value of β . Figure C1.
Probability ratio between the returning model (nu-merator) and the outgoing model at given β . We remind readers β is a factor relating the time-averaged shock velocity and thepericenter velocity of the corresponding subcluster. P D F β = SE subcluster centroid s − outgoings − returning true s proj P D F β = P D F β = P D F β = Projected separation from the center of mass (Mpc) P D F β = Figure C2.
Comparison of the PDFs of the observed position ofthe SE relic (red bar includes 95% confidence interval of location ofrelic in the center of mass frame) with the predicted position fromthe two simulated merger scenarios (blue for outgoing and greenfor the returning scenario). For the plausible values of β < . (cid:13)000
Comparison of the PDFs of the observed position ofthe SE relic (red bar includes 95% confidence interval of location ofrelic in the center of mass frame) with the predicted position fromthe two simulated merger scenarios (blue for outgoing and greenfor the returning scenario). For the plausible values of β < . (cid:13)000 , 1–20 he return of the merging galaxy subclusters of El Gordo? Figure B2.
Marginalized 2-dimensional PDFs of original inputs (vertical axis) and the inputs after applying polarization weight anddefault weights (horizontal axis). The inner and outer contour denote the central 68% and 95% confidence regions respectively. The circularcontours show that the application of weights did not introduce uneven sampling of inputs.
Figure B3.
One-dimensional marginalized PDFs (panels on the diagonal) and two-dimensional marginalized PDFs of variables denotingcharacteristic distances and projection angle of the mergers.c (cid:13) , 1–20 Karen Y. Ng et al.
Figure B4.
Marginalized PDFs of characteristic distances and projection angle of the merger and the inputs of the simulation.
Figure B5.
One-dimensional PDFs of characteristic timescales of the simulation (panels on the diagonal) and the marginalized PDFs ofdifferent timescales. Note that there is a default weight for constraining
T SP out but not for
T SP ret and T , so T SP out spans a smaller range.c (cid:13)000
T SP ret and T , so T SP out spans a smaller range.c (cid:13)000 , 1–20 he return of the merging galaxy subclusters of El Gordo? Figure B6.
Marginalized PDFs of characteristic timescales of the simulation and the characteristic distances and the projection angle ofthe merger.
Figure B7.
Marginalized PDFs of characteristic timescales of the simulation and the inputs.c (cid:13) , 1–20 Karen Y. Ng et al.
Figure B8.
One-dimensional marginalized PDFs of velocities at characteristic times (panels on the diagonal) and marginalized PDFs ofdifferent velocities.
Figure B9.
Marginalized PDFs velocities and the characteristic timescales of the simulation against the inputs.c (cid:13)000
Marginalized PDFs velocities and the characteristic timescales of the simulation against the inputs.c (cid:13)000 , 1–20 he return of the merging galaxy subclusters of El Gordo? Figure B10.
Marginalized PDFs of the velocities at characteristic timescales and the characteristic distances and the projection angle ofthe merger.
Figure B11.
Marginalized PDFs of relative velocities characteristic timescales of the simulation and the inputs.c (cid:13) , 1–20 Karen Y. Ng et al.
This paper has been typeset from a TEX/ L A TEX file preparedby the author. P D F β = SE subcluster centroid s − outgoings − returning true s proj P D F β = P D F β = P D F β = Projected separation from the center of mass (Mpc) P D F β = Figure C3.
Comparison of the PDFs of the observed position ofthe NW relic (red bar includes 95% confidence interval of locationof relic in the center of mass frame) with the predicted positionfrom the two simulated merger scenarios (blue for outgoing andgreen for the returning scenario). For the plausible values of β < .
1, the returning model is preferred. For comparison purpose,we also show that β > . (cid:13)000