NON-EXTENSIVE THEORY OF DARK MATTER AND GAS DENSITY DISTRIBUTIONS

							                  COSMO-05, BONN 2005


NON-EXTENSIVE THEORY OF
  DARK MATTER AND GAS
DENSITY DISTRIBUTIONS IN
 GALAXIES AND CLUSTERS


         M. P. LEUBNER
     Institute for Astrophysics
   University of Innsbruck, Austria
  core         –      halo                 NON-GAUSSIAN
                                         DISTRIBUTIONS
leptokurtic long-tailed

      PERSISTENT FEATURE OF DIFFERENT
        ASTROPHYSICAL ENVIRONMENTS

  standard Boltzmann-Gibbs statistics not applicable
    thermo-statistical properties of interplanetary medium
    PDFs of turbulent fluctuations of astrophysical plasmas
    self – organized criticality ( SOC ) - Per Bak, 1985
             stellar gravitational   equilibrium
      Empirical fitting relations - DM

Burkert, 95 / Salucci, 00            DM ~
                                                          1
non-singular                                 (1  r / rs )(1  r 2 / rs2 )

Navarro, Frenk & White, 96, 97                            1
NFW, singular                        DM ~
                                               (r / rs )(1  r / rs ) 2


Fukushige 97, Moore 98, Moore 99…


Zhao, 1996                           DM ~
                                                            1
singular                                     (r / rs ) (1  r / rs ) (3 )


Ricotti, 2003: good fits on all scales: dwarf galaxies  clusters
      Empirical fitting relations - GAS

Cavaliere, 1976:   single β-model        GAS ~ (1  r / rc )3/ 2 

                        Generalization
       convolution of two β-models  double β-model
     Aim: resolving β-discrepancy: Bahcall & Lubin, 1994

   good representation of hot plasma density distribution

                      galaxies / clusters

            Xu & Wu, 2000, Ota & Mitsuda, 2004

      β ~ 2/3 ...kinetic DM energy / thermal gas energy
             Dark Matter - Plasma
   DM halo  self gravitating system of weakly interacting

               particles in dynamical equilibrium

   hot gas  electromagnetic interacting high temperature

               plasma in thermodynamical equilibrium


                any astrophysical system
                               
long-range gravitational / electromagnetic interactions
       FROM EXPONENTIAL DEPENDENCE
      TO POWER - LAW DISTRIBUTIONS
Standard Boltzmann-Gibbs statistics
based on extensive entropy measure             S B  k B  pi ln pi

pi…probability of the ith microstate, S extremized for equiprobability

 Assumtion:                                   no correlations
                particles independent from e.o.
 Hypothesis: isotropy of velocity directions  extensivity
 Consequence: entropy of subsystems additive  Maxwell PDF

   microscopic interactions short ranged, Euclidean space time

         not applicable accounting for long-range interactions
                               THUS
         introduce correlations via non-extensive statistics
            derive corresponding power-law distribution
       NON - EXTENSIVE STATISTICS
Subsystems A, B:                        EXTENSIVE
                                                    
non-extensive statistics                                      1
Renyi, 1955; Tsallis,85      Sq ( A  B)  Sq ( A)  Sq ( B)  Sq ( A)Sq ( B)
                                                              
     1 /(1  q )
                                                                   
PSEUDOADDITIVE NON-EXTENSIVE ENTROPY BIFURKATION
Dual nature   + tendency to less organized state, entropy increase
              - tendency to higher organized state, entropy decrease

generalized entropy (kB = 1, -     )       S   ( pi11/   1)

       1/       long – range interactions / mixing
                 quantifies degree of non-extensivity /couplings
                 accounts for non-locality / correlations
FROM ENTROPY GENERALIZATION TO PDFs
 S … extremizing entropy under conservation of mass and energy
                     power-law distributions, bifurcation   0

                                                         
                                                v 
                                                  2
     HALO  > 0                 f ch  Bch 1                    CORE  < 0
                                                2 
                                                     

       N          ( )                                              N (  3 / 2)
Bh  1 / 2                        normalization               Bc  1 / 2
     vth 1 / 2 (  1 / 2)                                      vth  1 / 2 (  1)

                                 different                                          
h  vth                                                               c  vth
            3/ 2        generalized 2nd moments                                   3/ 2

  3/ 2                                                                   vmax  vth 
                              Leubner, ApJ 2004
restriction               Leubner & Vörös, ApJ 2005                     thermal cutoff
       EQUILIBRIUM OF N-BODY SYSTEM
              NO CORRELATIONS
         spherical symmetric, self-gravitating, collisionless

 Equilibrium via Poisson’s equation                    1
                                          4 G  f ( v 2  )d 3v
 f(r,v) = f(E) … mass distribution                     2

 (1) relative potential Ψ = - Φ + Φ0 , vanishes at systems boundary
              Er = -v2/2 + Ψ    and     ΔΨ = - 4π G ρ

(2) exponential mass distribution                      0            v2 / 2  
                                         f ( Er )             exp(            )
          extensive, independent                    (2 )
                                                        2 3/ 2
                                                                          2



f(Er)… extremizing BGS entropy, conservation of mass and energy

          isothermal, self-gravitating sphere of gas ==
phase-space density distribution of collisionless system of particles
           EQUILIBRIUM OF N-BODY SYSTEM
                   CORRELATIONS

             long-range interactions  non-extensive systems

extremize non-extensive entropy,                                       0 
                                                                                                          
                                                                        2 3/ 2 3/ 1 v / 2   
                                                                                    (2
                                                                                        )
                                                               B 
conservation of mass and energy                       f     ( Er ) (2 ) 1  (  3/ 2)
                                                                     B         
                                                                                
                                                                                   2
                                                                                                      
                                                                                          2        
     corresponding distribution

negative κ again energy cutoff v2/2 ≤ κ σ2 – Ψ,                                     integration limit
          0               ( )                                                   0        (  5 / 2)
B                                        bifurcation                   B 
       (2 2 )3/ 2  3/ 2(  3/ 2)                                           (2 2 )3/ 2  3/ 2(  1)

                                                           3/ 2 
 integration over v                               1 
                                           0 1  2
                                                   

                                             limit κ =    ∞                 0 exp( /  2 )
    DUALITY OF EQUILIBRIA AND HEAT CAPACITY
          IN NON-EXTENSIVE STATISTICS

(A) two families (κ’,κ) of STATIONARY STATES (Karlin et al., 2002)
            non-extensive thermodynamic equilibria, Κ > 0
                non-extensive kinetic equilibria, Κ’ < 0

                          related by κ’ = - κ
limiting BGS state for κ = ∞  self-duality          extensivity

         (B) two families of HEAT CAPACITY (Almeida, 2001)

          Κ > 0 … finite positive … thermodynamic systems
          Κ < 0 … finite negative … self-gravitating systems

   non-extensive bifurcation of the BGS κ = ∞, self-dual state
   requires to identify      Κ > 0 … thermodynamic state of gas
                             Κ < 0 … self-gravitating state of DM
               NON-EXTENSIVE
         SPATIAL DENSITY VARIATION
                                                                                                3/ 2 
                                                                  1 
       4 G                      combine               0 1 
                                                           

                                                                      2 
                                                                           

     1 d  2 d       
                          1/(3/ 2  )
                                       
                                             4 G 
            r    1  
     r 2 dr  dr  0 
                                         
                                                2
                                                                 Leubner, ApJ, 2005
                                     
                                       

                                                                                   1/  3/ 2  
d  2 d                1  d                 4 G  3 / 2           
 2                                            2
                  1
           1                                                   2                          0
dr 2
       r dr  3 / 2      dr                          2
                                                                           0 


ρ(r) … radial density distribution of spherically symmetric
       hot plasma and dark matter

κ = ∞ … BGS selfduality, conventional isothermal sphere
  Non-extensive family of density profiles




Non-extensive family of density profiles ρ± (r) , κ = 3 … 10
     Convergence to the selfdual BGS solution κ = ∞
  Non-extensive DM and GAS density profiles




Non-extensive GAS and DM density    Integrated mass of non-extensive
 profiles, κ = ± 7 as compared to    GAS and DM components, κ = ± 7
   Burkert and NFW DM models                 as compared to
                                      Burkert and NFW DM models
    and single/double β-models         and single/double β-models
               Comparison with simulations
    dark matter (N – body)                           gas (hydro)

Kronberger, T. & van Kampen, E.             Mair, M. & Domainko, W.




    DM         popular phenomenological: Burkert, NFW
   GAS      popular phenomenological: single / double β-models
  Solid: simulation (1, 2 ... relaxation times), dashed: non-extensive
                          SUMMARY

   Non-extensive entropy generalization generates a bifurcation
   of the isothermal sphere solution into two power-law profiles

 The self-gravitating DM component as lower entropy state resides
     beside the thermodynamic gas component of higher entropy

The bifurcation into the kinetic DM and thermodynamic gas branch is
controlled by a single parameter accounting for nonlocal correlations

  It is proposed to favor the family of non-extensive distributions,
  derived from the fundamental context of entropy generalization,
  over empirical approaches when fitting observed density profiles
                     of astrophysical structures