M. P. LEUBNER Institute for Astrophysics University of Innsbruck, Austria

1. COSMO-05, BONN 2005 NON-EXTENSIVE THEORY OFDARK MATTER AND GAS DENSITY DISTRIBUTIONS IN GALAXIES AND CLUSTERS M. P. LEUBNER Institute for Astrophysics University of Innsbruck, Austria

3. c o r e –h a l o leptokurtic long-tailed NON-GAUSSIAN DISTRIBUTIONS PERSISTENT FEATURE OF DIFFERENT ASTROPHYSICAL ENVIRONMENTS standard Boltzmann-Gibbs statistics not applicable thermo-statisticalproperties of interplanetary medium  PDFs ofturbulentfluctuations of astrophysical plasmas  self – organized criticality ( SOC ) - Per Bak, 1985 stellar gravitational equilibrium

4. Empirical fitting relations - DM Burkert, 95 / Salucci, 00 non-singular Navarro, Frenk & White, 96, 97 NFW, singular Fukushige 97, Moore 98, Moore 99… Zhao, 1996 singular Ricotti, 2003: good fits on all scales: dwarf galaxies  clusters

5. Empirical fitting relations - GAS Cavaliere, 1976: single β-model 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

6. 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

7. FROM EXPONENTIAL DEPENDENCETO POWER - LAW DISTRIBUTIONS Standard Boltzmann-Gibbs statistics based on extensive entropy measure pi…probability of the ith microstate, S extremized for equiprobability not applicable accounting for long-range interactions THUS  introduce correlations via non-extensive statistics  derive corresponding power-law distribution Assumtion: particles independent from e.o.  no correlations Hypothesis: isotropy of velocity directions extensivity Consequence: entropy of subsystems additive Maxwell PDF microscopic interactions short ranged, Euclidean space time

8. Subsystems A, B: EXTENSIVE   non-extensive statistics Renyi, 1955; Tsallis,85    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, -     ) 1/  long – rangeinteractions / mixing  quantifies degree of non-extensivity /couplings  accounts for non-locality / correlations NON - EXTENSIVE STATISTICS

9. FROM ENTROPY GENERALIZATION TO PDFs S … extremizing entropy under conservation of mass and energy power-law distributions, bifurcation   0 HALO  > 0 CORE  < 0 normalization different generalized 2nd moments Leubner, ApJ 2004 Leubner & Vörös, ApJ 2005 restriction thermal cutoff

10. EQUILIBRIUM OF N-BODY SYSTEM NO CORRELATIONS spherical symmetric, self-gravitating, collisionless Equilibrium via Poisson’s equation f(r,v) = f(E) … mass distribution (1) relative potential Ψ = - Φ + Φ0 , vanishes at systems boundary Er = -v2/2 + Ψ and ΔΨ = - 4π G ρ (2) exponential mass distribution extensive, independent 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

11. EQUILIBRIUM OF N-BODY SYSTEM CORRELATIONS long-range interactions  non-extensive systems extremize non-extensive entropy, conservation of mass and energy  corresponding distribution negative κ again energy cutoff v2/2 ≤ κσ2 – Ψ, integration limit bifurcation integration over v limit κ = ∞

12. 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

13. NON-EXTENSIVE SPATIAL DENSITY VARIATION combine Leubner, ApJ, 2005 ρ(r) … radial density distribution of spherically symmetric hot plasma and dark matter κ = ∞ … BGS selfduality, conventional isothermal sphere

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

15. Non-extensive DM and GAS density profiles Non-extensive GAS and DM density profiles, κ = ± 7 as compared to Burkert and NFW DM models and single/double β-models Integrated mass of non-extensive GAS and DM components, κ = ± 7 as compared to Burkert and NFW DM models and single/double β-models

16. 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

17. 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