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with Nikolaos Mavromatos and Sarben Sarkar Theoretical Physics Department King’s College London

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Non-extensive statistics and cosmology:

a case study

Ariadne Vergou

with Nikolaos Mavromatos

and Sarben Sarkar

Theoretical Physics Department

King’s College London

- Outline:
- Introduction
- Tsallis p-statistics
- p-statistics effects on SSC
- Discussion

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 fractal

Usually referred to as “exotic” matter

- where it comes from?

A possible source for fractality is :

- “exotic” particle statistics Tsallis statistics

e.g.

Tsallis statistics

Basic ideas and results

Tsallis formalism is based on consideringentropies of the general form:

- 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 :

departure from extensitivity

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

- Throughout all this analysis p is considered constantand 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- Gibbs
- distribution functions
- the p-corrected number density, energy density and pressure

e.g. the energy density for a relativisticspecies of fermions or bosons with

internal degrees of freedom and respectively, is found to be:

p-correction

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

- Following the methods of conventional cosmology, we can also derive as in [3]:
- the corrected effective number of degrees of freedom

p-correction

- the corrected entropy degrees of freedom

p-correction

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

Similarities

- 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

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 SSC
- p-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:

- , and( today critic. density)
- accounts for the ordinary matter , along with the exotic matter
- with , ,
- is notconstant but evolves with time ( Curci-Paffuti equation)

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

- for radiation and matter the on-shell ,equilibriump-corrected densities are known from extremization of .Off-shell equilibriumdensities?
- p -correction to ?
- p -correction to ?

- 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

- Matter and radiation
- Matter : the off-shell equilibrium energy density is:

standard non.rel.

energy density

Γ includes the

off-shell and source

terms (given in [6])

overall scales as (SSC effect)

standard

matter scaling

2. Dilatonfield

1) define a “generalized” effective number of degrees of freedom ,in order

to 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 not

thermalized, i.e.

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

1)

2)

3)

p-correction

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:

where

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

- solve the last equation perturbatively in :

(fractal scaling)

with

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

Why?

- our analysis, so far, is validonlyfor early eras, while [2] refers to
late eras

Plot for radiation energy density (numerical solution)

- Non-extensive effects on relic abundances

- “modified” Boltzmann eq. for a species of mass m in terms of parameters
- and :

- Before the freeze-out yielding

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

- 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 correction

at late eras (small ) small relativistic contributions negative correction

(see [7])

- affected today’s relic abundances

(again to the final result we have separatedthe non-extensive effects from the

source effectsin leading order to )

standard

result

non-ext.

effect

dilaton-

off-shell

effect

where:

(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 acceleration

expected 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

- 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