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. what is exotic scaling? where it comes from?
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 is non-extensive: if A and B independent systems (
Basic ideas and results
Tsallis formalism is based on considering entropies 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 constant and 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 relativistic species 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. the corrected effective number of degrees of freedom
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. Physical applications of Tsallis p-statistics
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 not constant 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 ,equilibrium p-corrected densities are known from extremization of Off-shell equilibrium densities?
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 : the off-shell equilibrium energy density is:
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. 3) apply the basic formulae of r.d.e (see [6])
2. Dilaton field
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 valid only for 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 separated the non-extensive effects from the
source effects in 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/
[2] N.E.Mavromatos, V.A.Mitsou, arXiv: [astro-ph]
[3] M.E.Pessah, D.F.Torres, H.Vucetich, arXiv:gr-qc/
[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/