Extensive composable entropy for the analysis of cosmological data - Constantino Tsallis; Henrik Jeldtoft Jensen (Physics Letters B)
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 SBH∝A/lP2 (A≡ area; lP≡ 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)∼μNγ(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 Sα,γ=k[ln⁡Σi=1Wpiα1−α]1γ(α∈R;S1,1=SBG≡−k∑i=1Wpiln⁡pi), 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 SA of a system A=B×C consisting of two statistically independent parts B and C is given in a consistent way as SA=Φ(SB,SC) 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.