Extensive Composable Entropy for the Analysis of Cosmological Data

Abstract

Along 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 $S_{BH}\propto A/l_P^2$ ($A \equiv$ area; $l_P \equiv$ 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 $W(N)\sim \mu^{N^\gamma}\;\;(\mu >1;\gamma >0)$, $W$ and $N$ respectively being the total number of microscopic possibilities and the number of components; $\gamma=1$ corresponds to standard Boltzmann-Gibbs (BG) statistical mechanics. For this $W(N)$ asymptotic behavior, we introduce here, on a group-theory basis, the entropic functional $S_{\alpha,\gamma}=k \Bigl[ \frac{\ln \Sigma_{i=1}^W p_i^\alpha}{1-\alpha} \Bigr]^{\frac{1}{\gamma}} \;(\alpha \in \mathbb{R};\,S_{1,1}=S_{BG}\equiv-k\sum_{i=1}^W p_i \ln p_i)$. This functional simultaneously is {\it extensive} (as required by thermodynamics) and {\it composable} (as required for logic consistency), $\forall (\alpha,\gamma)$. We further show that $(\alpha,\gamma)=(1,2/3)$ 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.