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

3. c o r e – h a l o   leptokurtic long-tailed c o r e – h a l o   leptokurtic long-tailed PERSISTENT FEATURE OF DIFFERENT ASTROPHYSICAL ENVIRONMENTS standard Boltzmann-Gibbs statistics not applicable  thermo-statistical properties of interplanetary medium  thermo-statistical properties of interplanetary medium  PDFs of turbulent fluctuations of astrophysical plasmas  s elf – organized criticality ( SOC ) - Per Bak, 1985  s elf – organized criticality ( SOC ) - Per Bak, 1985 NON-GAUSSIAN DISTRIBUTIONS  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 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 DEPENDENCE TO POWER - LAW DISTRIBUTIONS not applicable accounting for long-range interactions THUS  introduce correlations via non-extensive statistics  derive corresponding power-law distribution Standard Boltzmann-Gibbs statistics based on extensive entropy measure p i …probability of the i th microstate, S extremized for equiprobability 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. NON - EXTENSIVE STATISTICS Subsystems A, B: EXTENSIVE     non-extensive statistics Renyi, 1955; Tsallis,85       PSEUDOADDITIVE NON-EXTENSIVE ENTROPY BIFURKATION PSEUDOADDITIVE NON-EXTENSIVE ENTROPY BIFURKATION Dual nature + tendency to less organized state, entropy increase - tendency to higher organized state, entropy decrease - tendency to higher organized state, entropy decrease generalized entropy (k B = 1, -      ) 1/   long – range interactions / mixing 1/   long – range interactions / mixing  quantifies degree of non-extensivity /couplings  quantifies degree of non-extensivity /couplings  accounts for non-locality / correlations  accounts for non-locality / correlations

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

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 + Φ 0, vanishes at systems boundary E r = -v 2 /2 + Ψ and ΔΨ = - 4π G ρ (2) exponential mass distribution extensive, independent f(E r )… 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 long-range interactions  non-extensive systems extremize non-extensive entropy, conservation of mass and energy  corresponding distribution negative κ again energy cutoff v 2 /2 ≤ κ σ 2 – Ψ, integration limit integration over v ∞ limit κ = ∞ bifurcation

12. DUALITY OF EQUILIBRIA AND HEAT CAPACITY IN NON-EXTENSIVE STATISTICS (A) two families ( of STATIONARY STATES (Karlin et al., 2002) (A) two families ( κ’,κ) of STATIONARY STATES (Karlin et al., 2002) non-extensive thermodynamic equilibria, non-extensive thermodynamic equilibria, Κ > 0 non-extensive kinetic equilibria, non-extensive kinetic equilibria, Κ’ < 0 related by - related by κ’ = - κ limiting BGS state for = ∞ limiting BGS state for κ = ∞  self-duality  extensivity (B) two families of HEAT CAPACITY ( (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 ρ(r) … radial density distribution of spherically symmetric hot plasma and dark matter = ∞ … BGS selfduality, conventional isothermal sphere κ = ∞ … BGS selfduality, conventional isothermal sphere Leubner, ApJ, 2005

14. Non-extensive family of density profiles = 3 … 10 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, = ± 7 as compared to profiles, κ = ± 7 as compared to Burkert and NFW DM models Burkert and NFW DM models and single/double β-models on-extensive Integrated mass of non-extensive GAS and DM components, = ± 7 GAS and DM components, κ = ± 7 as compared to as compared to Burkert and NFW DM models Burkert and NFW DM models and single/double β-models

16. Comparison with simulations DM popular phenomenological: Burkert, NFW DM popular phenomenological: Burkert, NFW GAS popular phenomenological: single / double β-models GAS popular phenomenological: single / double β-models Solid: simulation (  1,  2... relaxation times), dashed: non-extensive dark matter (N – body) gas (hydro) Kronberger, T. & van Kampen, E.Mair, M. & Domainko, W.

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