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 :

ji

pairs ji

ji

i i

i

r

mmG

m

pH

2

2

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

with ‘just’

ji

pairs ji

ji

i i

i

r

mmG

m

pH

2

2

Empirical fitting relations for DM density profiles

(3 )

1~( / ) (1 / )DM

s sr r r r

2 2

1~(1 / )(1 / )DM

s sr r r r

Burkert (1995), Salucci (2000)

Navarro, Frenk & White (1996,1997)

Moore et al. (1999)

Zhao (1996)

Kravtsov et al. (1998)

2

1~( / )(1 / )DM

s sr r r r

and others …

From exponential dependenceto power-law distributions

This does not account properly for long-range interactions

introduce correlations via non-extensive statistics

iiBB ppkS lnStandard Boltzmann-Gibbs statisticsbased on extensive entropy measure

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

Non-extensive statistical physics

Subsystems A, B: EXTENSIVE ENTROPY

non-extensive statistics Renyi (1955), Tsallis (1985)

(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 long–range interactions / mixing quantifies degree of non-extensivity /couplings

accounts for non-locality / correlations

)1( /11 ipS

)1/(1 q

1( ) ( ) ( ) ( ) ( )q q q q qS A B S A S B S A S B

Equilibrium of a many-body systemwith no correlations

spherical symmetric, self-gravitating, collisionless

f(r,v) = f(E) from Poisson’s equation: 2 314 ( )

2G f v d v

Introduce relative potential Ψ = - Φ + Φ0 (vanishes at boundary)

Er = -v2/2 + Ψ and ΔΨ = - 4π G ρ

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

exponential energy distribution extensive, independent

202 3/ 2 2

/ 2( ) exp( )

(2 )r

vf E

Equilibrium of a many-body systemwith correlations

long-range gravitational interactions non-extensive systems

extremize non-extensive entropy,conservation of mass and energy corresponding distribution

02 3/ 2 3/ 2

( )

(2 ) ( 3 / 2)B

( κ < 0 energy cutoff v2/2 ≤ κ σ2 – Ψ )

3/ 2

0 2

11

2

2

1 / 2( ) 1r

vf E B

02 3/ 2 3/ 2

( 5 / 2)

(2 ) ( 1)B

02 3/ 2 3/ 2

( )

(2 ) ( 3 / 2)B

integration over v

limit κ = ∞ :∞ : 20 exp( / )

bifurcation κ > 0 :

κ < 0 :

Non-extensive density profiles

1/(3/ 2 )

22 2

0

1 41

d d Gr

r dr dr

1/ 3/ 2222

2 20

4 3/ 22 1 11 03/ 2

Gd d d

dr r dr dr

3/ 2

0 2

11

Combine

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 we retrieve the conventional isothermal sphere

4 G

(Leubner 2005)

with

or

Non-extensive family of density profiles

Non-extensive family of density profiles ρ± (r) , κ = 3 … 10 = 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

X-ray data for A1413

(Pointecouteau et al. 2005)

Integrated mass profile

Kronberger, Leubner & van Kampen (2006)

Final thoughts

Classical gravity is an already rich theory full of possibilities to explainastrophysical observations, which have not been all explored yet

Hydrodynamics should be added before comparing to observations using gasand stars, adding a whole range of possibilities for explanations

A theory like non-extensive statistics should be favoured over empiricalfitting relations for density (and other) profiles

Non-extensive entropy generalization generates a bifurcationof the isothermal sphere solution into two power-law profiles,controlled by a single parameter accountin for non-local correlationswith

Κ > 0 for thermodynamic systemsΚ < 0 for self-gravitating systems