Non-extensive statistics and cosmology Ariadne Vergou Theoretical Physics Department King’s College London Photo of the Observatory Museum in Grahamstown, South Africa Photo of the Observatory Museum in Grahamstown, South Africa

Outline: Part 1: Tsallis statistics framework+cosmology Part 1: Tsallis statistics framework+cosmology Part 2: Tsallis effects on supercritical string cosmology (SSC) Part 2: Tsallis effects on supercritical string cosmology (SSC) ( a case study) Part 3: A physical example Part 3: A physical exampleConclusions

Tsallis formalism is based on considering entropies of the general form: denotes the i-microstate probability q is Tsallis parameter in general, labels an infinite family of such entropies are non-extensive: if A and B independent systems the entropy for the total system A+B is : are a natural generalization of Boltzmann-Gibbs entropy which is acquired for q=1 : departure from extensitivity Part 1: Tsallis statistics are positive, concave (crucial for thermodynamical stability), and preserve the Legendre transform structure of thermodynamics are positive, concave (crucial for thermodynamical stability), and preserve the Legendre transform structure of thermodynamics give power law probabilities instead of the standard exponential laws give power law probabilities instead of the standard exponential laws

Tsallis approach may be applied to describe physical systems: with long-range interactions with long-range interactions with long memory effects with long memory effects evolving in fractal space-time evolving in fractal space-time 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 Assumptions usually made for q: - is taken to be sufficiently close to 1 (calculations to leading order in q) - is taken to be constant

By extremizing ( one can obtain : By extremizing (M. E. Pessaha, Diego F. Torres and H. Vuceticha) one can obtain : the generalized the generalized microstates probabilities and partition functions for a state R where the generalized Bose- Einstein, Fermi-Dirac and Boltzmann- Gibbs distribution functions (to first order in q) where applies for bosons, for fermions and corresponds to the Maxwell- Boltzmann distribution the q-corrected number density, energy density and pressure for relativistic and non- relativistic species e.g. the energy density of relativistic matter (m<<T) is found to be: Note: the equation of state of ordinary matter remains the same, i.e

If we consider only the relativistic species, the total energy density of all species in equilibrium will If we consider only the relativistic species, the total energy density of all species in equilibrium will be the sum ( : be the sum (M. E. Pessaha, Diego F. Torres and H. Vuceticha) : This sum can be expressed in terms of the photon temperature:  defines the corrected effective number of degrees of freedom q-correction Due to the same temperature dependence of one can have the same evolution equations as in the standard case:  All non-extensive effects are hidden in ! One follows the same process to obtain the corrected entropy degrees of freedom defined by: is conserved is conserved

Part 2: Tsallis statistics effects on SSC (a case study) The set of dynamical equations for a flat FRW universe in the Einstein frame (Diamandis, Georgalas, Lahanas,Mavromatos,Nanopoulos) is:, and,where is today’s critical density, and,where is today’s critical density accounts for the ordinary matter, along with the exotic matter where and is not constant but evolves with time (Curci-Paffuti equation) off-critical terms

Non-extensive SSC cosmology Modifications: all particles will acquire q-statistics, i.e,,, and Off-shell densities for matter and radiation? q-correction to the dilaton energy density? q-correction to the exotic matter? entropy roughly constant ( negligible) off-critical terms are of order less than (q-1) off-critical and dilaton terms are not thermalized question s assumptions

- -the off-shell energy density for non-relativistic matter in thermal equilibrium is: - - the “corrected” dilaton field energy density is: - -for the exotic matter we considered that any q-dependence comes into its equation of state parameter w, which is treated as a fitting parameter Γ includes the off-shell and dilaton terms Results Standarddensity

  the modified SSC continuity equations are:   it is easy to obtain the evolution equation for radiation: try to solve the last equation perturbatively in : where: with with Numerical estimation Recent astrophysical data have restricted in the range

Plot for radiation energy density

Non-extensive effects on relic abundances the “modified” Boltzmann equation for a species of mass m in terms of the parameters and before the freeze-out yielding corrected freeze-out point: by using the freeze-out criterion we get the correction:

Comments the “standard” is defined through the relation: the correction to the freeze-out point seems to depend only on the point itself! the correction may be positive or negative, depending on the last term of the r.h.s. Roughly we can say: at early eras (large ) large relativistic contributions positive correction at late eras (small ) small relativistic contributions negative correction the corrected today relic abundances are found to be: the corrected today relic abundances are found to be:where standard result dilaton, off-shell effect non-ext. effect See (Lahanas, Mavromatos, Nanopoulos)hep-ph/

Summary- comments   Tsallis statistics is an alternative way to describe particle interactions (natural extension of standard statistics)   Fractal scaling for radiation (under the assumption of a radiation dominated era) or for matter assumption of matter dominated era) is naturally induced in our analysis   Today relic abundances are affected by non-extensitivity as well besides the effects of non-critical, dilaton terms

Part 3: A physical example (work in progress) Proposal: D-particle foam model D-particles (point-like stringy defects) interacting with closed strings D-particles recoil (momentum transfer) “foamy” structure of space-time gravitational fluctuations (fluctuations in the metric) The metric in flat (Minkowskian) space-time can be written as:   represents the fraction of the momentum transferred in the i-th direction due to the recoil “randomly” distributed: take gaussian distribution with, where the standard deviation is considered to be following a chi- distribution (Beck)

obtain distribution functions for fermions and bosons similar to Tsallis results: obtain distribution functions for fermions and bosons similar to Tsallis results: obtain the “corrected” number densities and energy densities for relativistic and non-relativistic obtain the “corrected” number densities and energy densities for relativistic and non-relativisticMatter obtain the “corrected” effective number of degrees of freedom obtain the “corrected” effective number of degrees of freedom obtain the corrections to the Boltzman equation and the relic abundances obtain the corrections to the Boltzman equation and the relic abundances where we have assumed chi-distribution of n degrees of freedom and with variance The equivalent in the above case of (q-1) is the sum of the variances which in general The equivalent in the above case of (q-1) is the sum of the variances which in general are taken to be sufficiently small

Conclusions   Tsallis statistics when applied on cosmology can have many interesting consequences, e.g fractal scaling for radiation, affected today relic abundances etc.   D-particles recoil models could give rise to distribution functions of the same form as those provided by the Tsallis formalism   It would be interesting to seek the modifications to superheavy dark matter relic abundances by such a model