Constraints on Rényi Entropy through Primordial Big-Bang Nucleosynthesis and Baryogenesis

The Rényi entropy, a one-parameter generalization of Boltzmann-Gibbs entropy, offers a promising framework for probing quantum gravitational effects in cosmology. By modifying the entropy-area relation of the apparent horizon, Rényi entropy can alter the expansion dynamics of the early universe, with potential implications for Big-Bang Nucleosynthesis. In this work, we derive the modified Friedmann equations within the Rényi entropy paradigm and investigate their impact on the primordial abundances of light elements Deuterium ($D$), Helium-4 ($_{}^{4}\textit{He}$), Lithium-7 ($_{}^{7}\textit{Li}$) and baryogenesis. Using observational constraints from Planck, primordial abundance data and observational data on baryogenesis, we put stringent bounds on the Rényi parameter. Furthermore, we explore whether Rényi entropy corrections could mitigate the long-standing Lithium discrepancy. This study provides the first systematic constraints on Rényi cosmology from BBN and highlights the role of nonextensive thermodynamics in early-universe physics. Our analysis shows that the obtained ranges for the Rényi parameter $λ$ exhibit a overlap for the Helium-4 and Deuterium, but this overlapping region-despite a small discrepancy-is inconsistent with the range obtained from Lithium-7. This small mismatch between the ranges raises the possibility of alleviating the \textit{Lithium Problem} in the modified cosmology scenarios. Furthermore, we present the relationship between the cosmic time and temperature within the framework of Rényi cosmology. We observe that an increase in the Rényi parameter increase the temperature of the early universe.