1. Non-extensive statistical theory of dark matter and plasma density distributions in clustered structures DARK 2007, SYDNEY M. P. LEUBNER Institute for Astro- and Particle Physics University of Innsbruck, Austria

2. 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 PRONOUNCED NON-GAUSSIAN DISTRIBUTIONS  GRAVITATIONAL EQUILIBRIA of CLUSTERED SRUCTURES

3. Empirical fitting relations – DM density profiles 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

4. Empirical fitting relations – GAS density profiles 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

5. Dark Matter - Hot Gas 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 develop theory…

6. 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 isotropy of velocity directions  “ extensivity“ Consequence: entropy of subsystems additive  Maxwell PD

7. NON - EXTENSIVE STATISTICS Subsystems A, B: EXTENSIVE     non-extensive statistics Renyi, 1955; Tsallis,85       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, - ∞ ≤ κ ≤ + ∞ ) 1/κ  long – range interactions / mixing 1/κ  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

8. Bifurcation manifest in   Equilibrium power-law velocity distributions, bifurcation   0 restrictionthermal cutoff HALOCORE different normalization and different generalized higher moments  > 0  < 0 FROM ENTROPY GENERALIZATION TO PDFs NO GRAVITY S κ … extremizing entropy under conservation of mass and energy Leubner, ApJ 2004 Leubner & Vörös, ApJ 2005

9. STANDARD EQUILIBRIUM OF N-BODY SYSTEM NO CORRELATIONS but GRAVITY spherical symmetric, self-gravitating, collisionless Equilibrium via Poisson’s equation f(v 2 + ) = f(E) … energy distribution f(v 2 + Φ ) = f(E) … energy distribution Available by extremizing BGS entropy, conservation of mass and energy exponential energy distribution extensive, independent particles (relative potential Ψ = - Φ + Φ 0, vanishes at systems boundary) After integrating over all velocities: isothermal, self-gravitating sphere of gas == phase-space density distribution of collisionless system of particles

10. GRAVITATIONAL EQUILIBRIUM OF N-BODY SYSTEM; NON-EXTENSIVE CORRELATIONS long-range interactions long-range interactions  non-extensive systems extremize non-extensive entropy, conservation of mass and energy in gravitational potential Ψ:  equilibrium distribution integration over v ∞ limit κ = ∞ : expo – form of extensive statistics BIFURCATION  > 0  < 0 Ψ = Ψ(r)

11. NON-EXTENSIVE SPATIAL DENSITY VARIATION combine ρ(r) … radial density distribution of spherically symmetric hot plasma (  > 0 ) and dark matter (  < 0 ) = ∞ … BGS selfduality, conventional isothermal sphere κ = ∞ … BGS selfduality, conventional isothermal sphere Leubner, ApJ, 2005, 2006

12. physics of σ and κ generally variance σ = σ(r) (1) DM: σ(r) … velocity dispersion of members of cluster (2) GAS: σ(r) … thermal speed of plasma v 2 th = 2k B T/m κ, κ, T keep radial dependence σ = σ(r)  relation κ, σ, ρ and κ, T, ρ ρ(r) … radial density distribution of spherically symmetric hot plasma (  > 0 ) and dark matter (  < 0 ) density distribution with spatially varying variance σ = ∞, … BGS selfdual isothermal sphere solution κ = ∞, σ = const … BGS selfdual isothermal sphere solution Κ(r) Du, 2007

13. 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

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 - comparison with favored empirical models 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. Non-extensive DM and GAS density profiles - comparison with DM simulations and observations DM simulations Kronberger Leubner van Kampen A&A, 2006 hydrodynamic simulations Mair and Leubner Integrated mass profile A1413 Pointecouteau et al., A&A 2005

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

18. Hot Plasma Simulation, M. Mair (2005) Dark Matter Simulation, E. van Kampen T. Kronberger (2005) Theory: M. P. Leubner, ApJL 632, L1, 2005

19. 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.