Extensive composable entropy for the analysis of cosmological data

Abstract

In recent decades, an intensive worldwide research activity is focusing both black holes and cosmos (e.g. the dark-energy phenomenon) on the basis of entropic approaches. The Boltzmann-Gibbs-based Bekenstein-Hawking entropy S B H ∝ A / l P 2 ( A ≡ area; l P ≡ Planck length) systematically plays a crucial theoretical role although it has a serious drawback, namely that it violates the thermodynamic extensivity of spatially-three-dimensional systems. Still, its intriguing area dependence points out the relevance of considering the form W ( N ) ∼ μ N γ ( μ > 1 ; γ > 0 ) , W and N respectively being the total number of microscopic possibilities and the number of components; γ = 1 corresponds to standard Boltzmann-Gibbs (BG) statistical mechanics. For this W ( N ) asymptotic behavior, we make use of the group-theoretic entropic functional S α , γ = k [ ln ⁡ Σ i = 1 W p i α 1 − α ] 1 γ ( α ∈ R ; S 1 , 1 = S B G ≡ − k ∑ i = 1 W p i ln ⁡ p i ) , first derived by P. Tempesta in Chaos 30 ,123119, (2020). This functional is extensive (as required by thermodynamics) and composable , ∀ ( α , γ ) . Being extensive means that in the micro-canonical, or uniform, ensemble where all micro-state occur with the same probability, the entropy becomes proportional to N asymptotically: S ( N ) ∝ N for N → ∞ . An entropy is composable if it satisfies that the entropy S A of a system A = B × C consisting of two statistically independent parts B and C is given in a consistent way as S A = Φ ( S B , S C ) where the composition function Φ ( x , y ) is obtained from group-theory. We further show that ( α , γ ) = ( 1 , 2 / 3 ) satisfactorily agrees with cosmological data measuring neutrinos, Big Bang nucleosynthesis and the relic abundance of cold dark matter particles, as well as dynamical and geometrical cosmological data sets.

Topics & Concepts