Dark Matter Caustics in Galaxy Clusters

Transcription

1 Dark Matter Cautic in Galaxy Cluter V. Onemli a and P. Sikivie b a Department of Phyic, Univerity of Crete, GR Heraklion, Crete, Greece b Department of Phyic, Univerity of Florida, Gaineville, FL 3611, USA We interpret the recent gravitational lening obervation of Jee et al. [1] a firt evidence for a cautic ring of dark matter in a galaxy cluter. A cautic ring unavoidably form when a cold colliionle flow fall with net overall rotation in and out of a gravitational potential well. Evidence for cautic ring of dark matter wa previouly found in the Milky Way and other iolated piral galaxie. We argue that galaxy cluter have at leat one and poibly two or three cautic ring. We calculate the column denity profile of a cautic ring in a cluter and how that it i conitent with the obervation of Jee et al. arxiv: v [atro-ph] 15 Apr 009 PACS number: d INTRODUCTION Uing trong and weak gravitational lening method, Jee et al. [1] contructed a column denity map of the central region of the galaxy cluter Cl The map how a ring of dark matter of radiu 400 k, width 150k and maximum column denity 58 M. Remarkably, the overdenity in dark matter i not accompanied by an analogou tructure in x-ray emitting ga or luminou matter. In thi regard, Jee et al. dicovered a new intance where dark and ordinary matter have dramatically different patial ditribution. The well-known obervation of the Bullet Cluter 1E [] provided an earlier example. Jee et al. interpret the dark matter ring in Cl a the product of a near head-on colliion, along the line of ight, of two ubcluter. They performed a imulation of the repone of the dark matter particle to the time-varying gravitational field and found that, after the colliion ha occurred, the dark matter particle move outward and form hell-like tructure which appear a a ring when projected along the colliion axi [1]. The interpretation of Jee et al. fit with independent line of evidence that Cl , an apparently relaxed cluter, i a colliion of two ubcluter. However, it i hown in ref. [3] that the oberved dark matter ring i reproduced only for highly fine-tuned, and hence unlikely, initial velocity ditribution. The purpoe of our paper i to propoe an alternative interpretation, to wit that Jee et al. have oberved a cautic ring formed by the in and out flow of dark matter particle falling onto the cluter for the firt time. We how that the oberved ring i explained auming only that the dark matter falling ontothe cluter ha net overall rotation, with angular momentum vector cloe to the line of ight, and velocity diperion le than 60 km/. When cold colliionle dark matter fall from all direction into a mooth gravitational potential well, the phae pace ditribution of the dark matter particle i characterized everywhere by a et of dicrete flow [4]. The flow form outer and inner cautic. The outer cautic are formed by outflow where they turn around before falling back in. Each outer cautic i a fold catatrophe (A ) located on a topological phere urrounding the potential well. The inner cautic [5, 6] are formed near where the particle with the mot angular momentum in a given inflow reach their cloet approach to the center before going back out. The catatrophe tructure of the inner cautic [7] depend on the angular momentum ditribution of the infalling particle. If that angular momentum ditribution i characterized by net overall rotation, the inner cautic are ring (cloed tube) whoe cro-ection i a ection of the elliptic umbilic catatrophe (D 4 ) [6]. Thee tatement are valid independently of any aumption of ymmetry, elf-imilarity, or anything ele. It i argued in ref. [8] that dicrete flow and cautic are a generic and robut property of galactic halo if the dark matter i colliionle and cold. The radii of the outer cautic phere are predicted by the elf-imilar infall model [9, 10] of halo formation. The radii of the inner cautic ring are predicted [5], in term of a ingle parameter j max, after the model i generalized [11] to allow angular momentum for the infalling particle. Evidence for inner cautic ring ditributed according to the prediction of the elf-imilar infall model ha been found in the Milky Way [5, 1, 13] and in other iolated piral galaxie [5, 14]. The reolution of mot preent numerical imulation i inadequate to ee dicrete flow and cautic. However uch feature are een in dedicated imulation which increae the number of particle in the relevant region of phae pace [15, 16]. They hould alo become apparent in fully general imulation of tructure formation through the ue of pecial technique [17]. DARK MATTER CAUSTICS IN GALAXY CLUSTERS With regard to our propoal that Jee et al. have oberved a cautic ring of dark matter, the firt quetion

2 that arie i whether dicrete flow and therefore cautic hould be expected in galaxy cluter, in view of the fact that gravitational cattering by the galaxie in the cluter tend to diffue the flow. Aflow ofparticle paing through a region populated by a cla of object of ma M and number denity n get diffued by gravitational cattering over a cone of opening angle θ whoe quare [4] bmax ( θ) 4G M dt b min b v 4 nvπb db ( ) 3 ( ) km/ M v 10 1 M ln( b ( ) ( ) max t n ) b min 10 Gyr M 3, (1) where b i impact parameter, v i the velocity of the flow and t i the time over which it encountered the object in quetion. Becaue the denity n fall off with ditance r to the cluter center a a power law 1 r with γ, the γ integral in Eq. (1) i dominated by contribution received while paing through the central part of the cluter. To obtain an etimate, we conider a cluter of 00 galaxie of ma M 10 1 M each, within a region of radiu 1 M, and with velocity diperion < v > 1300 km/. Thee propertie are decriptive of Cl [18]. The velocity of the flow going through the central part for the firt time i approximately v < v >. The time to fall through the central region i approximately t M v year. Since ln( bmax b min ) 1, we find θ 6 10, indicating that the flow and cautic aociated with the firt throughfall (n = 1) are only partially diffued by gravitational cattering. It appear unlikely that the cautic aociated with flow that have paed everal time (n few) through the central part of the cluter urvive. On the other hand, we expect dicreteflowandaociatedcauticforn = 1, andpoibly for n =,3. Mahdavi et al. have already preented evidence for cautic in galaxy cluter. When plotted a a function of poition on the ky, the velocity diperion and urface denity of the galaxie in the group around NGC 5846 ha a harp drop at 840 k = R 1 from the group center, conitent with the occurrence of the firt outer cautic at that radiu[19]. Furthermore, the obervation of Mahdavi et al. allow a tet of the elf-imilar model prediction for the relationhip between velocity diperion and the radiu R 1 of the firt outer cautic. The model depend on a parameter ǫ [9] which i related to the lope of the power pectrum of denity perturbation [0]. ǫ i a lowly increaing function of the object ize. On the cale of galaxie, ǫ i in the range 0.5 to 0.30, wherea for cluter it i in the range 0.35 to 0.45 [11]. Becaue NGC 5846 i a mall cluter, ǫ i near For ǫ = 0.3, 0.35 and 0.40, the model predict repectively <v > R 1 = 60, 590 and 580 km/ M, where < v > i the velocity diperion meaured at the center of the cluter. The reult < v > = 45 km/ of Mahdavi et al. [19] for NGC 5846 agree with the elf-imilar model at the 15% level. Since the flow of infalling dark matter form the firt inner cautic before it form the firt outer cautic, the evidence of ref. [19] for the firt outer cautic ofthe NGC 5846group i alo evidence for it firt inner cautic. The former cannot exit without the latter. We mentioned that the catatrophe tructure of the inner cautic depend on the angular momentum ditribution of the infalling dark matter. If that ditribution i dominated by net overall rotation, the inner cautic are ring whoe cro-ection i an elliptic umbilic catatrophe. Evidence ha been found for cautic ring of dark matter in galaxie [5, 1, 13, 14], implying that the dark matter accreting onto galactic halo ha net overall rotation. Auming that thi i o, there i no reaon to expect differently on the only omewhat larger length cale of galactic cluter. Thu we are led to expect that cluter have one or more cauic ring of dark matter. GRAVITATIONAL LENSING BY CAUSTICS IN GALAXY CLUSTERS Weak lening repond to the column denity ditribution Σ(x,y) along line of ight in the direction ẑ. Cautic have preciely defined denity profile and hence preciely defined column denity profile. Thu one may be able to anwer unambiguouly whether a feature een in a column denity map i due to a cautic or not. For an outer cautic, the denity i d(r) = A R r Θ(R r) () where r i the radial coordinate, R i the radiu of the cautic, A i a contant called the fold coefficient, and Θ i the tep function. The column denity i [1,, 3] Σ(x) = πa R Θ(R x), (3) for x R. x i the projected radial coordinate. For the cluter Cl , appropriate value for the elfimilar infall model input parameter are: age t = 9.4 Gyr, velocity diperion < v > = 1300 km/ and ǫ = 0.4. The model predict then and {R n : n = 1,,...6} = (1.6, 0.9, 0.7, 0.55, 0.45, 0.40) M (4) {Σ n πa n Rn : n = 1,...6} = (1, 11, 10, 10, 9.5, 9.5) M. (5)

3 3 Let u briefly conider the propoal that the feature een by Jee et al. i due to an outer cautic in Cl An advantage of thi interpretation i that the circular appearance of the feature doe not require an accident. Since outer cautic are approximately pherically ymmetric, they appear approximately axially ymmetric from any vantage point. However, there are eriou difficultiewith thihypothei. Firt, forthe radiur n tofit the oberved radiu of the feature (400 k), n hould be of order 6; ee Eq. (4). But, a we dicued above, the cautic with uch high n are almot entirely degraded by gravitational cattering off the galaxie in the cluter. Second, the feature identified by Jee et al. ha, at it maximum, a column denity of order 58 M which i a factor five or o larger than the column denity contrat Σ n produced by outer cautic; ee Eq. (5). Finally, the predicted profile doe not fit the oberved profile. A a function of projected ditance x to the cluter center, the oberved Σ(x) how a bump at x 400 k, wherea the Σ(x) due to outer cautic i a tep function. Cautic ring of dark matter were decribed in detail in ref. [6]. The propertie of an axially ymmetric cautic ring are determined entirely in term of ix parameter: the cautic ring radiu a, the ize p and q of it croection in the direction parallel and perpendicular to the plane of the ring, the peed v and the ma infall rate per unit olid angle dm of the particle contituting the cautic, and a parameter which i of order a. Let z be the coordinate perpendicular to the plane of the ring and x the radial coordinate at a particular location on the ring. ŷ i in the direction tangent to the ring. The denity d(x,z) ha a unique expreion in term of the tated parameter [6]. It i traightforward to integrate d(x,z) along any line in the xz plane. For the line of ight perpendicular to the plane of the ring, the reult i Σ(x) = dz d(x,z) = 4 v where I( ) = π 0 a dm I( ) ha a logarithmic ingularity I( ) 1 ( ) 3. ln 1 x I ( ) (x a) (6) dα Θ(α coα + ) α +. (7) (8) when 0. The finite velocity diperion of the particle in the flow forming the cautic moothe the cautic and therefore alo the logarithmic ingularity of Eq. (8). Fig. 1 how the function I( ), a well a I( ) averaged over two different cale. The column denity doe not depend harply on the angle of view. One can how in particular that from any vantage point in the xz plane there i a line of ight for which the column denity i logarithmically divergent in the cold flow limit. I( ) FIG. 1: The function I( ) defined by Eq. (7) (olid line), the function I( ) convoluted with a Gauian of FWHM 0.07 (dot-dahed), and FWHM (dahed). COMPARISON WITH OBSERVATION The elf-imilar model predict v and dm in term of the age and velocity diperion of the object. For Cl , uing the previouly mentioned input parameter, thi implie Σ(x) = 31 M a a x I ( ) (x a). (9) To determine a, information mut be provided about the dependence of pecific angular momentum on declination α near α = 0 [6]. In the abence of uch information, it i only poible to tate that a i of order one, i.e. that it i likely between 0.5 and. A wa mentioned, the logarithmic ingularity in the column denity profile at x = a i moothed by the velocity diperion of the dark matter flow which make the cautic. We now dicu two ource of velocity diperion: the preence of mall cale tructure in the dark matter falling onto the cluter, and gravitational cattering by inhomogeneitie in the cluter. In the hierarchical clutering cenario of tructure formation, the dark matter falling onto a galaxy cluter ha previouly formed maller cale object. The infalling flow of dark matter ha effective velocity diperion equal to the velocity diperion of the mall cale object in the flow. If thi effective velocity diperion i δv, the pread in pecific angular momentum i δl δv R where R i the turnaround radiu. Hence the pread in the cautic ring radiu i δa δl v δv v R [5]. For Cl , R 6.7 M. Since v < v > 600 km/, the requirement that the cautic ring not be pread in radiu over more than the oberved width ( 150 k) of the

4 4 ring implie δv < 60 km/. Thi place an upper limit on the ize of the mall cale tructure compoing the ring. They cannot be larger than mall galaxie. Since no mall cale tructure are obervedin the ring, δv may be much le than 60 km/. The flow of dark matter falling onto a cluter for the firt time i diffued through gravitational cattering at leat omewhat by the time it form it firt inner cautic. Uing Eq. (1), we etimate that the reulting velocity diperion in Cl i of order δv 0.035v. The aociated pread in pecific angular momentum i of order δl a δv. Hence the firt inner cautic i moothed over a minimum ditance cale of order (400 k) = 14 k. Thi correpond to a moothing cale of 0.07 in = (x a) if = a. Fig.1 how I( ) defined by Eq. (7), and I( ) averaged over moothing cale of 0.07 and in, by convoluting with Gauian with thoe full width at half maximum (FWHM). The oberved dark matter ring i alo moothed by the reolution of the Cl ma recontruction, which i limited by the number denity of background galaxieand the finite grid ize. If the actual ring column denity were a delta function in the radial coordinate, it would be oberved a a Gauian bump with FWHM of order 7 k [4]. Let u call I av ( ) the function I( ) moothed over the correponding cale (0.133) in. I av ( ) i hown by the dahed curve in Fig.1 for the cae = a. The dark matter ring in Cl appear a a bump in the graph of the azimuthally averaged column denity plotted a a function of ditance x to the center of the ring [1]. Fig. how the data in the region of the bump. Each data point i the meaured average column denity at the correponding radiu with the 1 σ error bar [1]. The tep in x between ucceive data point i the bin ize (1 k). The ignificance of the bump over an aumed contant background i approximately 8 σ [1]. The data point in Fig. were fitted with the function F(x) = B 1 x +C +A a a x I av ( ) where A = 31 M, I (x a) av i I ( ) (x a), (10) ( ) (x a) averaged over 7 k in x, and B, C, a and are fitting parameter. The firt two term decribe the background column denity due to the cluter a a whole and due to other matter that i moothly ditributed on the cale of the ring thickne. The lat term repreent the contribution of the cautic ring of dark matter moothed over the reolution of the obervation. The leat quare fit returned a = 394 k, = 171 k, B = M, and C = M. The χ value wa. for 8 degree of freedom, indicating an acceptable fit. The χ value i low becaue the data point do not fluctuate relative to F(x) (10 3 M Sun / ) x (k) FIG. : The azimuthally averaged column denity of Cl in the range 70 < x < 50 k, fitted to the function defined in Eq. (10). one another a much a independent data point with the tated error bar would. The fitted function i plotted in Fig.. The value 1.5 of a, although lightly high, i acceptable in the context of the model ince a i expected to be of order one. We alo fitted the data with F(x) defined in Eq. (10) impoing the contraint = a, and uing A, B, C and a a fitting parameter. In thi cae, the leat quare fit returned a = 395 k, A = 46 M, B = M, and C = M, with χ =.0 for 8 degree of freedom. The fitted function i almot the ame a the one plotted in Fig.. Thu, if one impoe = a a a contraint, the meaured column denity i 50% higher than the column denity predicted by the elf-imilar infall model. The cautic ring radiu a i determined in the elfimilar infall model in term of a free parameter called j max which i the pecific angular momentum of the particle in the plane ofthe ring. For a to equal the oberved radiu (395 k) of the dark matter ring, one mut require j max CONCLUSIONS We conclude that the darkmatter ring obervedby Jee etal. hapropertieconitentwith acauticringofdark matter. The column denity agree both in hape and overall amplitude. At preent the data are too imprecie to infer with confidence the nature of the oberved ring. However, our propoal make a ditinct prediction for the column denity profile acro the ring, a illutrated in Fig.1. Thi may be teted by future obervation. In thi regard, let u emphaize that the 1 x I a( (x a) ) profile applie to each azimuth. If the cautic ring interpretation

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