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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.

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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


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:


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


  • the corrected entropy degrees of freedom



  • are positive

  • are concave (crucial for thermodynamical stability)

  • preserve the Legendre transform structure of thermodynamics (shown in [4])


  • 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


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)

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)

  • - B.G. entropy)Modifications due to non-extensitivity

    • all particles will acquire p-statistics, i.e , , ,

    • and

  • -Questions

    • 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

    standard non.rel.

    energy density

    Γ includes the

    off-shell and source

    terms (given in [6])

    overall scales as (SSC effect)


    matter scaling

    2. B.G. entropy)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])





    3. B.G. entropy)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:


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

    (fractal scaling)


    • Numerical estimation

    • But recent astrophysical data have restricted in the range [2]

    • which according to our estimation would require ! ?


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

      late eras

    Plot for radiation energy density ( B.G. entropy)numerical solution)

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

    • and :

    • Before the freeze-out yielding

    • “corrected” B.G. entropy)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])

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

    source effectsin leading order to )









    (depends only on the freeze-out point)

    • Conclusions B.G. entropy)

    • 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 B.G. entropy)

    • 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 B.G. entropy)

    [1] G.A. Diamandis, B.C. Georgalas ,A.B. Lahanas, N.E.Mavromatos, D.V.Nanopoulos ,


    [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