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

1. Non-extensive statistics and cosmology: a case study Ariadne Vergou with Nikolaos Mavromatos and Sarben Sarkar Theoretical Physics Department King’s College London

2. Outline: • Introduction • Tsallis p-statistics • p-statistics effects on SSC • Discussion

3. 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.

4. 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

5. 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!

6. 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

7. 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)

8. 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)

9. -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

10. 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

11. 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

12. 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:

13. 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

14. Plot for radiation energy density (numerical solution)

15. 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

16. “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])

17. 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)

18. 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

19. 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

20. 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