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 $S_{BH}\propto A/l_P^2$ (A≡ 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 $W\left(N\right)\sim {\mu }^{N^{\gamma }}(\mu >1;\gamma >0)$, 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 $W\left(N\right)$ asymptotic behavior, we make use of the group-theoretic entropic functional $S_{\alpha ,\gamma }=k{\left[\frac{\ln {\Sigma }_{i=1}^W{p}_i^{\alpha }}{1-\alpha }\right]}^{\frac{1}{\gamma }}(\alpha \in R;S_{1,1}=S_{BG}\equiv -k{\sum }_{i=1}^W{p}_i\ln p_i)$, first derived by P. Tempesta in Chaos 30,123119, (2020). This functional is extensive (as required by thermodynamics) and composable, $\forall (\alpha ,\gamma )$. 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: $S\left(N\right)\propto N$ for $N\to \infty$. An entropy is composable if it satisfies that the entropy $S_A$ of a system $A=B\times C$ consisting of two statistically independent parts B and C is given in a consistent way as $S_A=\Phi (S_B,S_C)$ where the composition function $\Phi (x,y)$ is obtained from group-theory.

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.