Friedmann equations with the generalized logarithmic modification of Barrow entropy and Tsallis entropy

Abstract

Barrow, inspired by the Covid-19 characteristics and its effect on human cells, explained that quantum gravitational effects might create complex and fractal properties on the black hole horizon. In this perspective, the entropy of a black hole no longer follows the law of area, but can instead be represented by exponential raising of the area Δ, where 0≤Δ≤1, and it indicates the quantum gravitational deformation effects. In this paper, starting from the first law of thermodynamics dE=ThdSh+WdV, at apparent horizon of a FRW universe, and using the entropy of Tsallis, Sq=A4G+αlnq⁡A4G+β where lnq⁡x=x1−q−11−q, we derive modified Friedmann equations describing the dynamics of the universe. The Tsallis parameter q measures the nonextensivity, or extensivity, level of any specific system. In fact, we modify the cosmological field equations in order to include new additional expressions that act as an effective dark energy part. We also put constrain on the parameters of the model using observational data (Union2 and updated Pantheon data) For both cases; absence and presence of cosmological constant Λ. For the case where we do not consider an explicit cosmological constant the resulting cosmology is very interesting and in agreement with the observed behavior. The interesting feature of the model in this case is that modified geometry term of the Friedman equations caused by Barrow Tsallis entropy correction plays the role of dark energy and could explain the acceleration of the universe. For the case Λ≠0, the dark energy equation-of-state parameter deviates from ΛCDM cosmology at small redshifts, while lying always in the phantom regime, and at asymptotically large times the Universe moves to the dark-energy dominated, de Sitter phase. For the best fitted of the parameters in both cases, the model implying a universe which undergoes first a radiation dominated phase, followed by a matter dominated phase and allowing the intermediate matter epoch to happen slowly and finally reaching a dark energy dominated phase. This is exactly the expected behavior of the observed universe which is well modeled by a universe filled with radiation, dark matter and dark energy.