Non-extensive statistics and cosmology

Ariadne VergouAriadne Vergou

Theoretical Physics DepartmentTheoretical Physics DepartmentKing’s College LondonKing’s College London

Photo of thePhoto of the Observatory Museum Observatory Museum in Grahamstown, South Africa in Grahamstown, South Africa

Outline:Outline:

Part 1: Tsallis statistics Part 1: Tsallis statistics frameworkframework+cosmology+cosmology

Part 2: Tsallis effects on Part 2: Tsallis effects on supercritical string cosmology supercritical string cosmology (SSC)(SSC) ( a case study)( a case study)

Part 3: A physical examplePart 3: A physical example

ConclusionsConclusions

Tsallis formalism is based on considering entropies of the general form:

1- pi

1-1

w q

iS kq q

standard1 1lim p lnp

wq i iq iS k S

• denotes the i-microstate probability

• q is Tsallis parameter in general ,labels an infinite family of such entropies

are non-extensive: if A and B independent systems the entropy for the total system A+B is :

ip

1

are a natural generalization of Boltzmann-Gibbs entropy which is acquired for q=1 :

qS

( ) ( ) ( ) (1 ) ( ) ( )q q q q qS A B S A S B q S A S B departure from extensitivity

Part 1: Tsallis statistics

are positive, concave (are positive, concave (crucial for thermodynamical stabilitycrucial for thermodynamical stability), and preserve the Legendre ), and preserve the Legendre transform structure of thermodynamics transform structure of thermodynamics

give power law probabilities instead of the standard exponential lawsgive power law probabilities instead of the standard exponential laws

qS

qS

qS

Tsallis approach may be applied to describe physical systems: Tsallis approach may be applied to describe physical systems:

• with long-range interactionswith long-range interactions

• with long memory effectswith long memory effects

• evolving in fractal space-timeevolving in fractal space-time

ExamplesExamplesself-gravitating systems, electron-positron annihilation, classical and quantum chaos, linear response theory, Levy-type anomalous super diffusion, low dimen-sional dissipative systems , non linear Focker- Planck equations etc

Assumptions usually made for Assumptions usually made for qq::- is taken to beis taken to be sufficiently close to 1 sufficiently close to 1 (calculations to leading order in (calculations to leading order in qq))- is taken to be constantis taken to be constant

By extremizing By extremizing ((M. E. Pessaha, Diego F. Torres and H. Vuceticha) one can obtain : one can obtain :

the the generalizedgeneralized microstates probabilities and partition functions for a state R

where

the generalized Bose- Einstein, Fermi-Dirac and Boltzmann- Gibbs distribution functions (to first order in q)

where applies for bosons, for fermions and corresponds to the Maxwell-Boltzmann distribution

the q-corrected number density, energy density and pressure for relativistic and non-relativistic speciese.g. the energy density of relativistic matter (m<<T) is found to be:

Note: the equation of state of ordinary matter remains the same, i.e

qS

24 4

2

( 1)5! (5)

30 22b

b i ibosonsq

g Tg

T

3q

qp

11] /[1 ( 1) ( 1) q

R R qRP Zq E q N

11][1 ( 1) ( 1) q

q R RR

Z q E q N

( )2

( ) ( ) 2

1 1 ( ( ))

2 ( )

r

r r

rq en

e e

1 1 0

24 4

2

7 ( 1) 155! (5)

8 30 2 162f

f j ifermionsq

g Tg

T

• If we consider only the relativistic species, the total energy density of all species in equilibrium will If we consider only the relativistic species, the total energy density of all species in equilibrium will be the sum be the sum ((M. E. Pessaha, Diego F. Torres and H. Vuceticha) : :

This sum can be expressed in terms of the photon temperature:This sum can be expressed in terms of the photon temperature:

defines the corrected effective number of degrees of freedomdefines the corrected effective number of degrees of freedom

, , ,R q i q j qbosons fermions

2

4, *30

qR q g T

*qg

* *q stg g

44

, ,

7

8ji

b i f jbosons fermions

TTg g

T T

44

, ,, ,

( 1) 9.58 8.98 jib i j f

i bosons j fermions

TTq g g

T T

q-correctionq-correction

Due to the same temperature dependence of one can have the same evolution equations as in Due to the same temperature dependence of one can have the same evolution equations as in the standard case:the standard case:

*qg

21 2

*1.66( )q

pl

TH g

m

All non-extensive effects are hidden in ! All non-extensive effects are hidden in ! *qg

1 2* 2

0.30( ) plq mt g

T

• One follows the same process to obtain the corrected entropy degrees of freedom defined by: qh2

32

45q qs h T

qs is conserved is conserved

Part 2: Tsallis statistics effects on SSC(a case study)(a case study)

The set of dynamical equations for a flat FRW universe in the Einstein frame (Diamandis, Georgalas, Lahanas,Mavromatos,Nanopoulos) is:

22ˆ3

2m

eGH

2ˆ2 i

m miH p p

G

a

2

2

ˆ1 1ˆ322

33

4 2a ill

mi

m

G eG

VH

ap

• , and ,where is today’s critical density, and ,where is today’s critical density

• accounts for the ordinary matter , along with the exotic matter

• where and

• is not constant but evolves with time (Curci-Paffuti equation)

m b r e2

ˆ ( )

2allV

2 2 0ˆˆ ˆ( ) 2allV Q e V

0ˆ 3H H H 03 Et H t ,0i i c ,0c

0ˆ / 3Q Q H

Q̂

off-critical terms

22

6 ( ) ii

E

GdGe H

dt a

Non-extensive SSC cosmologyNon-extensive SSC cosmology

Modifications:Modifications:• all particles will acquire q-statistics, i.e , , ,• and

,b q ,e q,r q ,qqg g

qh h

• Off-shell densities for matter and radiation?

• q-correction to the dilaton energy density?

• q-correction to the exotic matter?

• entropy roughly constant ( negligible)

• off-critical terms are of order less than (q-1)

• off-critical and dilaton terms are not thermalized

2 2( , )ii

dSO G G

dt

questions

assumptions

-the off-shell energy density for non-relativistic matter in thermal equilibrium is:

- the “corrected” dilaton field energy density is:

-for the exotic matter we considered that any q-dependence comes into its equation of state parameter w , which is treated as a fitting parameter

0

33 20 0

-correction off-shell,dilaton correction, 3

( )( )( ) ( ) ( )

( ) 2eqb q b

m Ta t mTt g m e q

a t

2

0 0

1 151 3

2 4

q m m

T T

0

exp t

t

dt

Γ includes the

off-shell and dilatonterms

2 2

, , ,22, ,

0.3 1( 1) 9.197 8.623

ˆ30 3r

q b i j fi bosons j fermionsp E

q g gg tH

2ˆ ( )

2allV

ResultsResults

StandardStandarddensitydensity

1( )

2GRAVt e g

the modified SSC continuity equations are:

it is easy to obtain the evolution equation for radiation:

try to solve the last equation perturbatively in :

2 2

, , ,22, ,

0.3 1( 1) 9.197 8.623

ˆ30 3r

correction b i j fi bosons j fermionsp E

q g gg tH

2 2 2 2

ˆ2 1 1ˆ1 4 1ˆ ˆ ˆ ˆ33 3q qr

r rE E E E

C Cd HH

dt H t Ht Ht H

2 2

,

ˆ3 Eq correction

r

H tC

where:

,, 2

ˆ ˆ ˆ3 ( ) ( 3 ) 6 6( )correctionm iim m m m correction

E E

dd GH p p H H

dt dt a

qC

42 ln 43( )r E in in

tEt a e a

2 ln 43 ln 3

Eta

with with

• Numerical estimation

0.2885( 1) 0.385( 1)qC q q

Recent astrophysical data have restricted in the range 3.3 4.3

Plot for radiation energy density

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.87.5

8

8.5

9x 10

-5

tE

rad.

dens

.

magenta: p=0.9green: p=1blue: p=1.1

4r a

Et

Non-extensive effects on relic abundances

• the “modified” Boltzmann equation for a species of mass m in terms of the parameters

and

• before the freeze-out yielding

• corrected freeze-out point:

qq nY s

Tx m

1

2 21 2, ,

ˆ0.264 q

A q q pl q q eq q

dY Hg h m m Y Y Y

dx x

q qeqY Y

1

1 3 2 1 1 21 150.145 1 3

2 4( ) inxx

s pq

xHxq

g h x e xY x ex

by using the freeze-out criterion we get the correction: f fx H x

1 2 1 2, , ,

, ,

1 1 15 13 9.58 8.98

ˆ2 2 4 ( )f

f q f f f f f b i j fi bosons j fermionsf x

qx x x x x x g g

g x

4, 2

30ˆ ˆq qg g T

24

*,ˆ30q qg T

Comments

the “standard” is defined through the relation:

the correction to the freeze-out point seems to depend only on the point itself!

the correction may be positive or negative , depending on the last term of the r.h.s. Roughly we can say: at early eras (large ) large relativistic contributions positive correction

at late eras (small ) small relativistic contributions negative correction

fx

1

1 21 21 1ln 0.038 ln

2ff

ff

in

f A f pl sxx

xx

xg H

x g x m m gg x

fx

fx

• the corrected today relic abundances are found to be:the corrected today relic abundances are found to be:

where where

0

1 2

*2 20 0

*

1ˆ ( ) h h 1

( )

( )

f

fpcorrectionno source

f

x

x

g xdx f

g x

Hx

standard result dilaton, off-shell

effect

non-ext. effect

0

1,

,

* *

11 11 ( ) 1

ˆ2 ( ) ( )

( )

f

fA f corx f cor

correction cor ff f f

x

x

x xf g x dx

g x g x J x

Hx

9 1

20

1.066 10 h

no sourcepl

GeV

m gJ

0

fx

Ax

J dx

1

0( )

xH

dxx

xx xe

See (Lahanas, Mavromatos,See (Lahanas, Mavromatos,Nanopoulos)Nanopoulos)hep-ph/0608153hep-ph/0608153

Summary- comments

Tsallis statistics is an alternative way to describe particle interactions(natural extension of standard statistics)

Fractal scaling for radiation (under the assumption of a radiation dominated era) or for matter assumption of matter dominated era) is naturally induced in our analysis

Today relic abundances are affected by non-extensitivity as well besides the effects of non-critical, dilaton terms

Part 3: A physical example (work in progress)(work in progress)

Proposal: Proposal: D-particle foam modelD-particle foam model

D-particles (point-like stringy defects) interacting with closed strings D-particles recoil (momentum transfer) “foamy” structure of space-time gravitational fluctuations (fluctuations in the metric)

The metric in flat (Minkowskian) space-time can be written as:

1 2 31 2 3

11

22

33

1

1 0 0

0 1 0

0 0 1

k r k r k r

k rg

k r

k r

statistical parameterir

1 2 3( , , ) : momentumk k k k

ir represents the fraction of the momentum transferred in the i-th direction due to the recoil “randomly” distributed: take gaussian distribution with , where the standard deviation is considered to be following a chi- distribution (Beck)

0ir 2 2 0i ir

• obtain distribution functions for fermions and bosons similar to Tsallis results:obtain distribution functions for fermions and bosons similar to Tsallis results:

• obtain the “corrected” number densities and energy densities for relativistic and non-relativisticobtain the “corrected” number densities and energy densities for relativistic and non-relativisticMatterMatter

• obtain the obtain the “corrected”“corrected” effective number of degrees of freedom effective number of degrees of freedom

• obtain the corrections to the Boltzman equation and the relic abundancesobtain the corrections to the Boltzman equation and the relic abundances

( ) 24 4 42 1 2 2 2 3

01 02 03( ) ( ) 2 2

2 21 4

( ) 2 2

r

r r

nen k k k

e e n n

20iwhere we have assumed chi-distribution of n degrees of freedom and with variance

The equivalent in the above case of (q-1) is the sum of the variances which in general The equivalent in the above case of (q-1) is the sum of the variances which in general are taken to be sufficiently small are taken to be sufficiently small

20i

Conclusions

Tsallis statistics when applied on cosmology can have many interesting consequences, e.g fractal scaling for radiation, affected today relic abundances etc.

D-particles recoil models could give rise to distribution functions of the same form as those provided by the Tsallis formalism

It would be interesting to seek the modifications to superheavy dark matter relic abundances by such a model