Constraining Early Dark Energy cosmological models with Big Bang Nucleosynthesis

The recent cosmological picture contains a significant tension indicating that our standard $\Lambda$CDM picture may be incomplete. Early Dark Energy models can alleviate the Hubble tension, by assuming an early acceleration that could explain the divergence between the early and late-time cosmological data. We investigate the implications of Early Dark Energy models on the Big Bang Nucleosynthesis processes by considering several cosmological models, including a model assuming a simple cosmological constant, alongside with varying equations of state dark energy models. We construct a simulator through a nested sampling algorithm, with the help of which we estimate the upper bounds for model parameters, and determine the maximum allowable dark energy density contribution during the radiation-dominated era. Our results are obtained through the \href{https://github.com/croi900/eden}{eden} program. We show that for a linear or polytropic equation of state, the dark energy density is constrained to less than $10^{-13}$ MeV$^4$ and $10^{-5}$ MeV$^4$, respectively, at the 95\% confidence level. Furthermore, we identify a temperature-dependent equation of state of dark energy as the most physically compelling framework, which remains consistent with primordial abundances for coupling parameters $\lesssim 10^{-2}$. This model successfully allows for high-temperature deviations from the standard $\Lambda$CDM expansion history, while rapidly diluting to obtain standard general relativistic results in the weak freeze-out era.