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Simulated Dark-Matter Halos as a Test of Nonextensive Statistical Mechanics ∗ Chloé Féron and Jens Hjorth Dark Cosmology Centre, Niels Bohr Institute, University of Copenhagen Juliane Maries Vej 30, DK-2100 Copenhagen, Denmark (Dated: March 21, 2008) Intheframeworkofnonextensivestatisticalmechanics,theequilibriumstructuresofastrophysical self-gravitating systemsare stellar polytropes, parameterized bythepolytropicindexn. Bycareful comparison to the structures of simulated dark-matter halos we find that the density profiles, as wellasotherfundamentalproperties,ofstellarpolytropesareinconsistentwithsimulationsfor any value of n. This result suggests the need to reconsider the applicability of nonextensive statistical mechanics (in its simplest form) to equilibrium self-gravitating systems. 8 PACSnumbers: 0 0 2 I. INTRODUCTION II. NONEXTENSIVE STATISTICAL n MECHANICS AND STELLAR POLYTROPES a J Nonextensive statistical mechanics is a generalization Nonextensivestatisticalmechanicsisbasedonthe for- 6 of thermodynamics and statistical mechanics proposed mulation of a generalized entropy 1 by C. Tsallis in 1988 [1], aiming to overcome the limita- tionsofthe Boltzmannentropyinits conventionalappli- 1−PW pq ] cations [2, 3, 4, 5]. The theoryhas considerablywidened S =k i=1 i (q ∈R), (1) h q B q−1 thefieldsofapplicationofstatisticalphysics,allowingthe p - description of systems affected by nonlocal effects, such whichreducesinthelimitq →1totheBoltzmann-Gibbs ro as long-range forces and memory effects. entropySBG =−kBPWi=1pilnpi (pi istheprobabilityof st Thestudyofastrophysicalself-gravitatingsystemswas findingthesysteminthemicrostatei,W isthenumberof a one of the first applications of the theory [6]. This ap- microstatesiforagivenmacrostate,andk istheBoltz- B [ proach has recently witnessed a renewed interest in the mann constant). When the index q 6= 1, the entropy of 1 hope that it may provide a theoretical basis for the de- the system is nonextensive, i.e., it is not possible to sum v scription of the universal structure of dark-matter (DM) the entropies S (A) and S (B) of two subsystems A and q q 4 halos [7, 8, 9, 10]. Nonextensive statistical mechanics B,butthegeneralizedentropyS isconvenientlypseudo- q 0 predicts their equilibrium states to be stellar polytropes additive: S (A+B)=S (A)+S (B)+(1−q)S (A)S (B). q q q q q 5 (SP) [6, 11], which have been claimed to fit DM halos as The third term accounts for coupling between the sub- 2 well as the usual Navarro, Frenk & White model [9, 10] systems, due to nonlocal effects. . 1 (hereafter NFW [12]). Moreover, SP have the distinct In this framework, astrophysical self-gravitating sys- 0 advantage of being analytically derived from nonexten- tems at equilibrium are described by SP. This solution 8 sive statistical mechanics, while most models describing arises from extremizing the entropy S under the con- 0 q DM halos are empirical fits to N-body simulations. In straintsoffixedtotalmassandenergyasderivedin[6,11] : v this context, nonextensive statistical mechanics appears (seeRef. [13]aboutthe choiceofconstraints),leadingto i X as an attractive framework for providing a theoretical a spherically symmetric isotropic system. The resulting understanding of the structure of DM halos and self- distribution function depends on the parameter q and, r a gravitating systems in general. via the identification n = 1/2+ 1 [32], has the same 1−q In this paper we compare, in a parameter indepen- form as the SP distribution function: f(ǫ) ∝ ǫn−3/2 for dent way, the equilibrium configuration of astrophysical ǫ > 0 and f(ǫ) = 0 for ǫ < 0, where n is the polytropic self-gravitating systems predicted by nonextensive sta- index and ǫ the relative energy per unit mass [14]. tistical mechanics to simulated DM halos. We clarify In order to compare the predictions of nonexten- the issueofwhich centralboundaryconditions to use for sive statistical mechanics to the results of N-body comparing SP to simulated halos. On this basis, we es- simulations, we compute radial profiles of fundamen- tablish that simulated DM halos do not corroborate the tal quantities of astrophysical self-gravitating systems: predictions of nonextensive statistical mechanics. This the matter density ρ(r), the logarithmic density slope resultcallsintoquestionthedirectapplicabilityofnonex- dlog(ρ)/dlog(r), the integrated mass M(r), and the cir- tensivestatisticalmechanicstoequilibriumastrophysical cular velocity Vc(r) (see Fig. 1). SP are solutions of the self-gravitating systems. Lane-Emden equation [14], depending on three free parameters: the polytropic index n, the central density ρ0 and the central velocity dispersion σ0. The value of n (andq)is anintrinsicpropertyofeachsystem, reflecting ∗Electronicaddress: [email protected] itsdegreeofnonextensivity,buthasnotbeendetermined 2 fromfirstprinciplesandisthereforeafreeparameter. SP with n>3/2 are stable due to Antonov’s stability crite- rion, i.e., df(ǫ)/dǫ < 0 [14]. Other values of n are un- realistic: n =3/2 corresponds to a distribution function independent of ǫ, and n<3/2 to a distribution function divergingattheescapeenergy,ǫ=0. Therefore,wecom- putenumericalsolutionsoftheLane-Emdenequationfor n > 3/2, choosing values of n ranging from 2 to ∞. We verified the consistency of our profiles using the analytical solutions of the Lane-Emden equation, as well as the isothermalspherewhichis the asymptoticsolutionwhen n→∞ [33]. Nonextensivestatisticalmechanicsdoesnotspecifythe boundary conditions for SP, which are needed to solve numerically the Lane-Emden equation. Simulated ha- losnecessarilyhaveafinite gravitationalpotentialatthe center, therefore we look for SP fulfilling this condition. For SP the density ρ is linked to the gravitational potential ψ by ρ = cst×ψn [14], so we require solutions of the Lane-Emden equation with finite density at the center. Chandrasekhar proved that such solutions have dlog(ρ)/dlog(r)=0atr =0[15]. Therefore,theclassof SP we can compare consistently to simulated halos have the natural boundary conditions at the center: ρ = ρ0 and dlog(ρ)/dlog(r)=0. We present the corresponding profiles in Fig. 1. In [9, 16], the authors arbitrarily fix a non-zero logarithmic density slope near the center, cho- sen to match the NFW density profile [34]. This class of solutions must be treated with caution, as it does not ensure physical boundary conditions at the center. III. COMPARISON TO SIMULATED DM HALOS Colddark-matter(CDM) halosresultingfromN-body simulations provide an excellent way of testing the predictions of nonextensive statistical mechanics. They are collisionless systems whose particles interact via long- range gravitational interaction only, allowing a direct comparison to theory. DM structures in N-body simula- tionsevolvefromprimordialfluctuationsviahierarchical clusteringtoformhalosofuniversalshape,welldescribed bytheNFWmodel[12]. Thesehalosformedthroughac- cretion and mergers are sufficiently similar to those ob- tainedbymonolithic collapse,i.e.,isolatedhalos,sothat wedonottakeintoaccountinthisstudytheinfluenceof formation history and cosmological environment on the final state of simulated halos [17]. Collisionless systems have very long collisional relax- FIG. 1: From top to bottom are presented the density, the ation timescales to reach the ’true’ equilibrium, but ac- logarithmicdensityslope,theintegratedmassandthecircular cess a stable quasi-equilibrium state faster (dynamical velocity. Theprofilesarescaledtor−2,theradiusatwhichthe timescalesareshortcomparedtocosmologicaltimescales, slopeequals−2. Stellarpolytropespredictedbynonextensive i.e., the age of the universe) through the processes of statistical mechanicsarerepresentedbythesolid curveswith violent relaxation and phase-mixing [14]. The quasi- theshadesofgreencorrespondingtodifferentpolytropicindex equilibria predicted by numerical cosmological simula- n. Simulated dark-matterhalos arerepresentedbytheNFW tions and by nonextensive statistical mechanics, respec- model(pinkdashedcurve),the3DSérsicmodel(pinkdotted curve; α=0.17) and the Hernquist model (pink dash-dotted tively,mayineithercasebeconsideredthemostprobable curve;a=0.45). 3 state the system reaches. case: the difference between SP and simulated halos in Many empirical models have been proposed as fitting the density, density slope and mass profiles is as high as the universal profile of CDM halos. We use in our study one order of magnitude at 0.1r−2. Therefore, for any three of them to represent simulated halos. The NFW polytropic index n, SP do not properly describe the inner modelisthemorecommonlyused[12]. TheSérsicmodel parts of simulated halos. hasbeenproposedrecentlyasfittingmoreaccuratelythe innerpartofhighresolutionhalos[18,19]. TheHernquist model provides a simple description of self-gravitating IV. DISCUSSION systems, for which all the profiles we use have analytical expressions [20]. As seen in Fig. 1, these profiles form Wehaveestablishedherethatthepredictionsofnonex- a group with similar shapes, and we do not need to use tensivestatisticalmechanicsarenotcorroboratedbysim- actual simulations in our study. ulations of DM halos. Our results are based on readily To compare simulated DM halos to SP, we select the observable quantities, i.e., the matter density itself, and radialrange for which recent N-body simulations are ro- profilesderivedfromit. These findings arein directcon- bustly resolved. Based on the high resolution simula- trast to previous works which found reasonable agree- tionspublishedinNavarroetal.[18],thiscorrespondsto mentbecausetheyeitherconsideredonlytheouterparts 0.1r−2 < r < 10r−2, where r−2 is the radius at which of simulated halos [8], or used a non-zero density slope the logarithmicdensity slope equals −2. To compare SP as initial condition near the center [9], or used values of of all values of n with simulated halos,we need a scaling n<5 that lead to too steep density profiles in the outer suitableforfiniteandinfinitemasshalos,circularvelocity parts [10]. Support for our conclusions, however, comes profileswithandwithoutmaximum,anddensityprofiles from Barnes et al. [16] who, even though fixing a non- withverydifferentsteepness: r−2 providessuchauniver- zero density slope near the center to match the NFW sal scaling [35]. Moreover, it allows to take equally into density profile, found inconsistency betweenSP and DM accounttheinnerandouterpartsofthehalo,respectively halo velocity dispersion profiles. While we are cautious defined as having a slope shallower and steeper than −2 aboutthephysicalrelevanceofsuchprofiles,itisofinter- [12]. Finally, our study is independent of scaling param- esttoemphasizethatnorcoredneithercuspypolytropes eters: SP profiles are scaled depending on the choice of can describe properly simulated DM halos. ρ0 and σ0, but the shape depends only on n. There- These results imply a fundamental difference in the fore, scaling to r−2 makes identical any profiles of same matter distribution of astrophysical self-gravitating sys- nbutdifferentρ0,σ0,keepingonlyinformationaboutthe temspredictedbynonextensivestatisticalmechanicsand shape. All the profiles we present, as scaled to r−2, de- by N-body simulations. Such a discrepancy raises three pend only on structural parameters: range of n for SP; main questions. Is our comparison with simulated DM α = 0.17 for the 3D Sérsic profile [18]; a = 0.45 for the halos valid? Is nonextensive statistical mechanics the Hernquist profile [20]; rs for the NFW profile [36]. In proper theory to describe collisionless long-range inter- Fig. 1 we compare SP with simulated CDM halos, scal- actionsystems at equilibrium? And how does this study ing to r−2. relate to observed DM halos? We briefly address these Intheouterparts,wherer >r−2,SPhaveverydiffer- issues below. entshapesdepending onthe polytropicindexn. Profiles 1a. We studyidealizedsystemswhichdonottakeinto with n < 5 correspond to finite-mass halos, while n ≥ 5 account complex effects in DM halos such as velocity polytropes have infinite mass, tending to the isothermal anisotropy [22] or triaxiality [23]. However, the error we sphere when n → ∞. While finite-mass polytropes may makeby using isotropicandspherically averagedmodels appear more attractive from a physical point of view, to represent DM halos is small compared to the discrep- they also provideworsefits to simulatedDM halos,with ancywefindbetweenSPandsimulatedhalos. Moreover, an outer slope as steep as dlog(ρ)/dlog(r) < −5 at wedonotaddressheregeneralproblemsofstatisticalme- 10r−2. For infinite-mass SP with larger values of the chanicsofself-gravitatingsystemslike,e.g.,infinitemass polytropic index, n = 16.5 (as found in [9]) agrees with [24], so as to focus on the issues specific to nonextensive NFW and Sérsic models, n = 10 with the Hernquist statistical mechanics. model. Therefore, some infinite-mass SP can provide a 1b. N-bodysimulationsdependonvariousparameters, good description of the outer parts of simulated halos. suchasthechoiceofthesofteninglengthandthenumber In the inner parts, where r < r−2, SP share similar of particles,which can introduce numericaleffects in the properties: they have a large core, the shape and the resulting halos. The softening of the gravitational force extent of which depend little on the value of n. This onsmallscale,introducedtoavoidtwo-bodyinteraction, feature is in striking disagreement with the predictions creates a core at the center of the halo, approximately of N-body simulations, which show steeper inner slopes. the size of the softening length. The number of particles While this core structure itself has been advanced as an fixes the maximum phase-space density resolved at red- advantage of SP over the NFW profile [10], solving the shift zero, and if not large enough, a core appears due well-known cusp problem between simulations and ob- to the lack of particles in the center. Both effects lead servations [21], it appears in Fig. 1 that this is not the to a shallowerdensity profile in the center of halos if the 4 simulation is of low resolution [25]. Therefore, the dis- efficientlydrivesthesystemtowardsequilibrium)ismost crepancy we observe between simulated profiles and SP effective. Therefore,the assumptionsb) andc) holdtrue is genuine andincreases with the resolutionofnumerical in the center. However, it is in the inner parts that simulations. the disagreementbetween SP and simulated halos is the 1c. We compare theoretical statistical predictions to strongest. Hence, we suggest that the assumption a), cosmological simulations, but the latter depend on ini- i.e., the choice of the generalized entropy S motivated q tial conditions and cosmologicalparameters. Initial con- by nonextensive statisticalmechanics, cannot be used to ditions are fixed by the shape of the power spectrum of predicttheequilibriumstructuresofsimulatedDMhalos. initial fluctuations P(k) ∝ kns, i.e., by the choice of the 3. Simulated DM halos are in good agreement with indexns. FromCMBobservations,wehaveanindication observations [29], except for the inner core found from that ns = 0.958±0.016 [26], but tests on cosmological spiral galaxy rotation curves [21]. Stellar polytropes, in- simulations proved that they depend very little on the terestingly, have an inner core too, but the discrepancy choice of ns and on the cosmological model used [27]. we observe between SP and simulated DM halos is too Therefore, the universal profiles of simulated halos can large to explain the core of observed DM halos. Adding be compared to statistical theories as a general predic- adiabatic contraction in simulations results in DM ha- tion. loswithsteeperinner parts,whichwouldnotchangeour 1d. The choice of a scaling is necessary to compare conclusions. DMhalosmodels,butcanbringavisualizationbias. We In summary, we have established that nonextensive checked if the use of r−2 leads us to overestimate the statistical mechanics [1, 6, 11], a theory generalizing disagreementbetweennonextensivestatisticalmechanics classicalstatisticalmechanicsandthermodynamics,does and N-body simulations. However, adjusting the fit in not describe the equilibrium state of astrophysical self- the outer parts of the halo, e.g., scaling to the virial ra- gravitating systems, as represented by DM halos formed dius, increases even more the discrepancy in the inner in N-body simulations. parts, while scaling profiles to fit well at smaller radii makes the discrepancy appear in the outer parts of the halo. 2. Nonextensive statistical mechanics relies on three Acknowledgments assumptions: a) the generalized entropy S is the right q form to describe long-range interaction systems; b) the system is at equilibrium, and its most probable state is ItisapleasuretothankS.H.Hansen,E.vanKampen given by the maximum entropy principle; c) S is maxi- and E. I. Barnes for valuable discussions and comments, q mizedatfixedenergy,leadingtoasystemwhosedistribu- andtoacknowledgetherefereesforhelpingustoimprove tion function depends only on the energy per unit mass, this paper. The Dark Cosmology Centre is funded by and has isotropic velocity dispersion. Though simulated the Danish National Research Foundation. This work halos show evidence of velocity anisotropy, the relation was supported in part by the European Community’s betweendensity slope andvelocityanisotropy[22] shows SixthFrameworkMarieCurieresearchTrainingNetwork thatitis isotropicinthe center,whereviolentrelaxation Programme,ContractNo.MRTN-CT-2004-505183“AN- [28] (the process by which strong potential fluctuations GLES”. [1] C. Tsallis, J. Stat. Phys. 52, 479 (1988). [12] J. F. Navarro, C. S. Frenk, and S. D. M. White, Astro- [2] I. V. Karlin, M. Grmela, and A. N. Gorban, Phys. Rev. phys.J. 462, 563 (1996). E 65, 036128 (2002). [13] C. Tsallis, R. S. Mendes, and A. R. Plastino, Physica A [3] A.TaruyaandM.Sakagami,Phys.Rev.Lett.90,181101 261, 534 (1998). (2003). [14] J. Binney and S. Tremaine, Galactic Dynamics (Prince- [4] P.H. Chavanis and C. Sire, Physica A 356, 419 (2005). ton University Press, 1987). [5] D.H. E. Gross, Physica E 29, 251 (2005). [15] S.Chandrasekhar,Anintroduction tothestudy ofstellar [6] A. R. Plastino and A. Plastino, Phys. Lett. A 174, 384 structure (Chicago,Ill.,TheUniversityofChicagopress, (1993). 1939). [7] S. H. Hansen, D. Egli, L. Hollenstein, and C. Salzmann, [16] E.I.Barnes,L.L.R.Williams, A.Babul,andJ.J.Dal- NewAstronomy 10, 379 (2005). canton, Astrophys.J. 655, 847 (2007). [8] M. P. Leubner,Astrophys.J. Lett. 632, 1 (2005). [17] A. Huss, B. Jain, and M. Steinmetz, Astrophys. J. 517, [9] T.Kronberger, M.P.Leubner,andE. vanKampen,As- 64 (1999). tron. Astrophys.453, 21 (2006). [18] J.F.Navarro,E.Hayashi,C.Power,A.R.Jenkins,C.S. [10] J. Zavala, D. Nuń˜ez, R. A. Sussman, L. G. Cabral- Frenk,S.D.M.White,V.Springel,J. Stadel,andT.R. Rosetti, and T. Matos, Journal of Cosmology and As- Quinn,Mon. Not. R.Astron. Soc. 349, 1039 (2004). troparticle Physics 6, 8 (2006). [19] A. W. Graham, D. Merrit, B. Moore, J. Diemand, and [11] A.TaruyaandM.Sakagami,PhysicaA322,285(2003). B. Terz´ıc, Astron.J. 132, 2701 (2006). 5 [20] L. Hernquist,Astrophys.J. 356, 359 (1990). (2005). [21] G. Gentile, P. Salucci, U. Klein, D. Vergani, and [31] M.V.MedvedevandG.Rybicki,Astrophys.J.555,863 P.Kalberla, Mon.Not.R.Astron.Soc.351, 903 (2004). (2001). [22] S. H. Hansen and B. Moore, New Astronomy 11, 333 [32] Anearlierversionofthetheoryusedn=3/2+ 1 ,where q−1 (2006). therelationbetweennandqdependsonthechoiceofthe [23] E. Hayashi, J. F. Navarro, and V. Springel, Mon. Not. constraintswhenextremizingthegeneralized entropySq R.Astron. Soc. 377, 50 (2007). [11, 13]. A third relation n = 5−3q has been proposed 2(1−q) [24] J.HjorthandJ.Madsen,Mon.Not.R.Astron.Soc.253, in [30]. 703 (1991). [33] Analytical solutions exist only for n = 0,1,5. We ap- [25] B. Moore, F. Governato, T. Quinn, J. Stadel, and proximate n→∞ by n=500, as large values of n tend G. Lake, Astrophys.J. Lett. 499, 5 (1998). asymptotically towards theisothermal sphere. [26] D.N.Spergel,R.Bean,O.Doré,M.R.Nolta,C.L.Ben- [34] E. I. Barnes, private communication; E. van Kampen, nett, J. Dunkley, G. Hinshaw, N. Jarosik, E. Komatsu, private communication. L. Page, et al., Astrophys.J. Suppl. 170, 377 (2007). [35] The scaling radius r−2 is determined uniquely in the [27] J. F. Navarro, E, C. S. Frenk, and S. D. M. White, As- case of a monotonically decreasing slope. However, SP trophys.J. 490, 493 (1997). of n ≥ 5 have a logarithmic density slope oscillating [28] D. Lynden-Bell, Mon. Not. R. Astron. Soc. 136, 101 around −2[14,31];wethereforedefiner−2 asthesmall- (1967). est radius at which theslope is −2. [29] E.Pointecouteau,M.Arnaud,andG.W.Pratt,Astron. [36] In the NFW model, thescale radius rs is equal to r−2. Astrophys.435, 1 (2005). [30] J. A. S. Lima and R. E. de Souza, Physica A 350, 303