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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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slide1

Non-extensive statistics and cosmology:

a case study

Ariadne Vergou

with Nikolaos Mavromatos

and Sarben Sarkar

Theoretical Physics Department

King’s College London

slide2

Outline:

  • Introduction
  • Tsallis p-statistics
  • p-statistics effects on SSC
  • Discussion
slide3

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.

slide4

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
slide5

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!
slide6

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

slide7

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)

slide8

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)
slide9

-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
slide10

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

slide11

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

slide12

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:

slide13

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

slide15

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
slide16

“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])

slide17

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)

slide18

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

slide19

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
slide20

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

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