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DM density profiles in non-extensive theory Eelco van Kampen Institute for Astro- and Particle Physics Innsbruck University In collaboration with Manfred Leubner and Thomas Kronberger Classical gravity is an extremely rich theory The wonderful world of r –1 : 2 i mi m j p H G i 2mi pairs ri j i j A theory with many equations and approximations from Saslaw (1987) Classical Gravity (including GR) • • • • simple but highly non-linear no equilibrium state (although timescales can be long) long-range, so hard to isolate systems (galaxies & galaxy clusters !) gravitational systems tend to form substructure Gravitational systems are therefore intrinsically hard to model, so approximations are always made If classical gravity is already hard to ‘use’, adding hydrodynamics (gas, stars !) makes things only harder Given this, do we really need alternative theories ? Have we properly solved the highly non-linear classical equations yet ? Bottom line: Astrophysical systems are messy and simply hard to model even 2 mm p with ‘just’ H i G i j i 2mi pairs i j ri j Empirical fitting relations for DM density profiles Burkert (1995), Salucci (2000) DM ~ 1 (1 r / rs )(1 r 2 / rs2 ) Moore et al. (1999) 1 ~ (r / rs )(1 r / rs ) 2 Navarro, Frenk & White (1996, 1997) DM Zhao (1996) DM ~ Kravtsov et al. (1998) and others … 1 (r / rs ) (1 r / rs )(3 ) From exponential dependence to power-law distributions Standard Boltzmann-Gibbs statistics based on extensive entropy measure SB kB pi ln pi pi…probability of the ith microstate, S extremized for equiprobability Assumtions: particles independent from e.o. no correlations isotropy of velocity directions extensivity Consequence: entropy of subsystems additive Maxwell PDF microscopic interactions short ranged, Euclidean space time This does not account properly for long-range interactions introduce correlations via non-extensive statistics Non-extensive statistical physics Subsystems A, B: EXTENSIVE ENTROPY non-extensive statistics Renyi (1955), Tsallis (1985) Sq ( A B) S q ( A) S q ( B) 1 /(1 q) 1 S q ( A) S q ( B) (PSEUDOADDITIVE) NON-EXTENSIVE ENTROPY Dual nature: + tendency to less organized state, entropy increase - tendency to higher organized state, entropy decrease generalized entropy : with 1/κ S ( pi11/ 1) long–range interactions / mixing quantifies degree of non-extensivity /couplings accounts for non-locality / correlations Equilibrium of a many-body system with no correlations spherical symmetric, self-gravitating, collisionless 1 2 4 G f ( v )d 3v 2 f(r,v) = f(E) from Poisson’s equation: Introduce relative potential Ψ = - Φ + Φ0 Er = -v2/2 + Ψ and (vanishes at boundary) ΔΨ = - 4π G ρ f(Er) from extremizing BGS entropy, conservation of mass and energy exponential energy distribution extensive, independent 0 v2 / 2 f ( Er ) exp( ) (2 2 )3/ 2 2 Equilibrium of a many-body system with correlations long-range gravitational interactions non-extensive systems extremize non-extensive entropy, conservation of mass and energy corresponding distribution bifurcation integration over v 1 v( ) / 2 0 ( Er )B B 1 2 3/ 2 3/ 2 (2 ) ( 3/ 2) 2 2 f κ>0: B 0 ( ) (2 2 )3/ 2 3/ 2( 3/ 2) κ<0: B 0 ( 5 / 2) (2 2 )3/ 2 3/ 2 ( 1) 1 0 1 2 3/ 2 ( κ < 0 energy cutoff v2/2 ≤ κ σ2 – Ψ ) limit κ = ∞: 0 exp( / 2 ) Non-extensive density profiles Combine or 4 G with 1/(3/ 2 ) 1 d 2 d 4 G r 1 r 2 dr dr 0 2 d 2 d 1 1 d 1 2 dr r dr 3 / 2 dr 2 1 0 1 2 2 3/ 2 (Leubner 2005) 4 G 3 / 2 2 2 0 1/ 3/ 2 0 ρ(r) is the radial density distribution of spherically symmetric hot plasma (κ > 0) or dark matter halo ( κ < 0) For κ = ∞ we retrieve the conventional isothermal sphere Non-extensive family of density profiles Non-extensive family of density profiles ρ± (r) , κ = 3 … 10 Convergence to the BGS solution for κ = ∞ Simulation vs. theory vs. empirical fit Kronberger, Leubner & van Kampen (2006) Theory vs. simulation vs. observation Integrated mass profile X-ray data for A1413 (Pointecouteau et al. 2005) Kronberger, Leubner & van Kampen (2006) Final thoughts Classical gravity is an already rich theory full of possibilities to explain astrophysical observations, which have not been all explored yet Hydrodynamics should be added before comparing to observations using gas and stars, adding a whole range of possibilities for explanations A theory like non-extensive statistics should be favoured over empirical fitting relations for density (and other) profiles Non-extensive entropy generalization generates a bifurcation of the isothermal sphere solution into two power-law profiles, controlled by a single parameter accountin for non-local correlations with Κ > 0 for thermodynamic systems Κ < 0 for self-gravitating systems