with Nikolaos Mavromatoswith Nikolaos Mavromatosand Sarben Sarkarand Sarben Sarkar

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

Non-extensive statistics and cosmology: a case study

Ariadne Vergou

OutlineOutline::

IntroductionIntroduction

Tsallis p-statisticsTsallis p-statistics

p-statistics effects on SSCp-statistics effects on SSC

DiscussionDiscussion

The original motivation for our work has been the idea of fractal-exotic, cosmological scalings suggested by some authors [1,2] Such models are compatible with current astrophysical data from high-redshift supernovae, distant galaxies and baryon oscillations [2]

what is exotic scaling?theoretical and/or observed “extra” energy density contribution scaling as with and is a fractalUsually referred to as “exotic” matter

A possible source for fractality is : “exotic” particle statistics Tsallis statistics

where it comes from?

a

3,4

e.g.

Tsallis statisticsBasic ideas and results

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

1- pi1

-1

w p

iS kp p

standard1 1lim p lnp

wp i ip iS k S

• denotes the i-microstate probability

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

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

ip

p 1

( ) ( ) ( ))p AB p A p B

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

pS

pS

( ) ( ) ( ) (1 ) ( ) ( )p p p p pS A B S A S B p S A S B

departure from extensitivity

Throughout all this analysis p is considered constant and sufficiently close to 1

By extremizing (subject to constraints) one obtains, as shown in [3]:

• the generalized microstates probabilities and partition functions

• the generalized Bose- Einstein , Fermi-Dirac and Boltzmann- Gibbsdistribution functions

• the p-corrected number density, energy density and pressure

pS

e.g. the energy density for a relativistic species of fermions or bosons with internal degrees of freedom and respectively, is found to be:

42

1 5!(1.04 0.97 )( 1)

2 2p st b fg g p T

p-correction2

47

30 8b fg g T

fg bg

It can be proven that the equation of state for radiation remains despite the non-extensitivity! 3

ppp

Following the methods of conventional cosmology, we can also derive as in [3]:

the corrected effective number of degrees of freedom

, ,, ,

( 1) 9.58 8.98p b i j fi bosons j fermions

g g p g g

44

, ,, ,

7

8ji

b i f ji bosons j fermions

TTg g

T T

p-correction

the corrected entropy degrees of freedom 33

, ,, ,

( 1) 7.18 6.73 jip st b i f j

i bosons j fermions

TTh h p g g

T T

33

, ,, ,

7

8ji

b i f ji bosons j fermions

TTg g

T T

p-correction

Properties of Tsallis entropies (comparison with standard B.G. entropy)

are positive are concave (crucial for thermodynamical stability) preserve the Legendre transform structure of thermodynamics (shown in [4]) Differences are non-additive give power law probabilities

Physical applications of Tsallis p-statistics In general, Tsallis formalism can be used to describe physical systems which: • have any kind of long-range interactions • have long memory effects• evolve in fractal space-times

Similarities

Examples self-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 (see [5] and references within)

Tsallis statistics effects on SSCp-statistics affects ordinary cosmological scaling

We investigate the modification of non-critical ,Q- cosmology as established in [1] .The original set of dynamical equations for a flat FRW universe in the E.F. 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 ( today critic. density)

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

• with , ,

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

0ˆ 3H H H 03 Et H t ,0i i c ,0c

m b r e2

ˆ ( )

2allV

2 2 0ˆˆ ( ) 2allV Q e V 0

ˆ / 3Q Q H 02

03arbitV

VH

Q̂

-Modifications due to non-extensitivity a) all particles will acquire p-statistics, i.e , , ,b) and

-Questions1. for radiation and matter the on-shell ,equilibrium p-corrected densities

are known from extremization of .Off-shell equilibrium densities?2. p -correction to ?3. p -correction to ?

,b p ,r p ,e p ,p

pS

,p,e p

-Assumptions

entropy constant ( negligible )

off-critical terms are of order less than

we refer to radiation – dominated era

off-shell and source terms are not thermalized

and

2 2( , )ii

dSO G G

dt

( 1)p

pg g ph h

0pdg

dt 0pdh

dt

1. Matter and radiationMatter : the off-shell equilibrium energy density is:

0

33 20 0

p-correction off-shell,dilaton correction, 3

( )( )( ) ( ) ( )

( ) 2eqb p b

m Ta t mTt g m e

a t

standard non.rel.energy density

2

0 0

1 151 3

2 4

p m m

T T

0

exp t

t

dt

Γ includes theoff-shell and source terms (given in [6])

overall scales as (SSC effect), ( )eqb p t

0

3, ( ) ( ) exp

teqb p

t

t a t dt

standardmatter scaling

2. Dilaton field1) define a “generalized” effective number of degrees of freedom ,in orderto include the extra off-shell and dilaton energy contributions (denoted as ) :

(the corresponding eqn. to the last one for the standard case (see [6]) is:

)

2) use the fact that the dilatonic and off-shell degrees of freedom are notthermalized, i.e.

3) apply the basic formulae of r.d.e (see [6])

ˆ pg

24

, ˆ30m p p pg T

p

24

, 30m p pg T

2 2

, , ,22, ,

0.3 1( 1) 9.197 8.623

ˆ30 3r

p b i j fi bosons j fermionsp E

p g gg tH

p-correction

24ˆ

30m gT

2ˆ ( )

2allV

1)2)3)

3. Exotic matter we assume that any p-dependence will come into its equation of state parameter w , as in [1] w will be a fitting parameter for our numerical analysis

With the above in hand we can obtain :

the modified continuity equations:

,, 2

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

E E

dd GH p p H H

dt dt a

where 2 2

, , ,22, ,

0.3 1( 1) 9.197 8.623

ˆ30 3r

correction b i j fi bosons j fermionsp E

p g gg tH

It is easy to derive the evolution equation for the radiation energy density:

2 2 2 2

ˆ2 1 1ˆ1 4 1ˆ ˆ ˆ ˆ33 3p pr

r rE E E E

C Cd HH

dt H t Ht Ht H

2 2

,

ˆ3 Ep correction

r

H tC

solve the last equation perturbatively in : pC

42 ln 43( )r E in in

tEt a e a

with 2 ln 43 ln 3

Eta

Numerical estimation

•

But recent astrophysical data have restricted in the range [2] which according to our estimation would require ! ?

0.2885( 1) 0.385( 1)pC p p

3.3 4.3 8.57 1 11.169p

Why? • our analysis, so far, is valid only for early eras, while [2] refers to late eras

(fractal scaling)

Plot for radiation energy density (numerical solution)

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

Non-extensive effects on relic abundances

• “modified” Boltzmann eq. for a species of mass m in terms of parameters

and :p

p nY

s

Tx

m

1

2 21 2,

ˆ0.264 p

A p p pl p p eq p

dY Hg h m m Y Y Y

dx x

• Before the freeze-out yielding

p peqY Y

1

(0)( ) inxp p

xHx

Y x Y e

1 3 2 1 1 21 150.145 1 3

2 4x

s p

pg h x e x x

• “corrected” freeze-out point:by using the freeze-out criterion and the non-extensive equilibrium form ,we get:

f fx H x (0)pY

1 2 1 2, , ,

, ,

1 1 15 13 9.58 8.98

ˆ2 2 4 ( )f

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

px x x x x g g

g x

Comments the correction to the freeze-out point depends only on the point itself! the “standard” satisfies relation:

the correction may be positive or negative ,depending on the last term of the r.h.s. Roughly: at early eras (large ) large relativistic contributions positive correctionat 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 mm gg x

(see [7])

fx

, ,f p f f corx x x

fx

• affected today’s relic abundances

(again to the final result we have separated the non-extensive effects from the source effects in leading order to )

0

1 2

2 20 0

1ˆ h h 1

( )

f

pcorrectionno source

x

x

gdx f

g

Hx

standard result dilaton-

off-shelleffect

non-ext.effect

where:

0

1,

,11 1

1 ( ) 1ˆ2 ( ) ( )

( )

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

1p

9 1

20

1.066 10 h

no sourcepl

GeV

m gJ

1

0( )

xH

dxx

xx xe

0

fx

Ax

dx

, ,, ,

1 9.58 8.98b i j fi bosons j fermions

p g g

(depends only on the freeze-out point)

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

After performing our numerical analysis we see that the modified cosmological equations are in agreement with the data for accelerationexpected at redshifts of around and the evidence for a negative-energy dust at the current era

Fractal scaling for radiation (r.d.e assumption) or for matter m.d.e. assumption) is also naturally induced by our analysis

Today relic abundances are affected by non-extensitivity much more significantly (it can be shown) than by non-critical, dilaton terms

0.2z

Outlook

keep higher order to (p-1) in our calculations

consider the case of non-constant entropy

consider the case of non-negligible off-shell terms

References

[1] G.A. Diamandis, B.C. Georgalas ,A.B. Lahanas, N.E.Mavromatos, D.V.Nanopoulos ,arXiv:hep-th/0605181[2] N.E.Mavromatos, V.A.Mitsou, arXiv:0707.4671 [astro-ph][3] M.E.Pessah, D.F.Torres, H.Vucetich, arXiv:gr-qc/0105017[4] E. M. F. Curado and C. Tsallis, J. Phys. A24, L69 (1991)[5] A. R. Plastino and A. Plastino, Phys. Lett. A177, 177 (1993)[6] E. W. Kolb, M. S. Turner, The early universe[7] A.B. Lahanas, N.E.Mavromatos, D.V.Nanopoulos, arXiv:hep-ph/0608153