Constraining Early Dark Energy cosmological models with Big Bang Nucleosynthesis

Abstract: The recent cosmological picture contains a significant tension indicating that our standard $Λ$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 $Λ$CDM expansion history, while rapidly diluting to obtain standard general relativistic results in the weak freeze-out era.

• The paper constrains multiple Early Dark Energy models using a modified BBN reaction network integrated with Bayesian nested sampling.
• It provides stringent upper bounds on parameters for cosmological constant, linear, polytropic, and temperature-dependent EDE scenarios.
• The findings imply that only highly suppressed or transient dark energy is compatible with observed primordial abundances during nucleosynthesis.

Authoritative Analysis of "Constraining Early Dark Energy cosmological models with Big Bang Nucleosynthesis" (2605.26749)

Context and Motivation

The persistent Hubble tension, revealed by discrepancies between local and early-Universe measurements of $H_0$, necessitates alternatives to the standard $\Lambda$CDM paradigm. Early Dark Energy (EDE) models, characterized by a non-negligible dark energy fraction during the radiation-dominated era, have been extensively posited to address this challenge. Concurrently, Big Bang Nucleosynthesis (BBN) provides rigorous constraints on any modification to expansion history during epoch $T\sim 1~\text{MeV}$, as light element abundances are highly sensitive to deviations in $H(t)$.

This paper presents a systematic investigation of multiple EDE parameterizations—cosmological constant, linear equation of state, polytropic equation of state, and temperature-dependent equation of state—using a customized BBN reaction network, integrated with a robust Bayesian nested sampling framework. The primary objective is quantification of parameter upper limits for EDE models consistent with observed primordial abundances.

Theoretical Models and Modifications

The analysis encompasses four dark energy frameworks:

• Cosmological Constant Model: Incorporates an early-time $\Lambda$ not fixed by the late-time constraint; modifies the Friedmann equation with an additional constant term.
• Linear Equation of State: Dark energy described by $p_\text{DE} = w \rho_\text{DE}$ with $w \in (-1,0)$, yielding a density scaling as $a^{-3(1+w)}$; motivated by scalar field quintessence models.
• Polytropic Equation of State: Dark energy pressure follows $p_\text{DE} = K \rho_\text{DE}^\gamma$; $\gamma$ fixed at $\Lambda$0 (radiation-like) and $\Lambda$1 (stiff fluid), inducing nontrivial scaling.
• Temperature-Dependent Equation of State: $\Lambda$2, leading to dark energy density scaling dynamically with photon temperature.

All models are imposed as minimally coupled, introducing EDE contributions to the background expansion solely via the gravitational sector, without direct plasma coupling.

Computational and Statistical Methodology

A modified PRyMordial BBN code is employed to solve coupled ODEs encompassing both background dynamics (including EDE) and nucleosynthesis reaction network. The implementation allows for arbitrary background modification, including time-dependent/diluting dark energy. Bayesian inference is conducted via nested sampling (dynesty), which efficiently explores multidimensional, non-Gaussian parameter spaces and yields direct evidence estimates for model comparison.

Parameter estimation is constrained primarily through helium ($\Lambda$3), deuterium (D/H), and $\Lambda$4He abundances, excluding $\Lambda$5Li due to unresolved theoretical and observational inconsistencies.

Numerical Results and Physical Bounds

The key results include stringent upper limits on EDE parameters supported by posterior credible intervals:

• Cosmological Constant:
  • $\Lambda$6 MeV$\Lambda$7 ($\Lambda$8 cm$\Lambda$9) at 95% CI
• Linear EDE:
  • $T\sim 1~\text{MeV}$0 MeV$T\sim 1~\text{MeV}$1 ($T\sim 1~\text{MeV}$2 cm$T\sim 1~\text{MeV}$3)
  • $T\sim 1~\text{MeV}$4 at 95% CI
• Polytropic EDE ($T\sim 1~\text{MeV}$5):
  • Converted present-day densities $T\sim 1~\text{MeV}$6 MeV$T\sim 1~\text{MeV}$7 ($T\sim 1~\text{MeV}$8 cm$T\sim 1~\text{MeV}$9) and $H(t)$0 MeV$H(t)$1 ($H(t)$2 cm$H(t)$3)
• Temperature-Dependent EDE:
  • $H(t)$4 MeV$H(t)$5 ($H(t)$6 cm$H(t)$7)
  • $H(t)$8

Model evidences are statistically indistinguishable, with $H(t)$9 across all alternatives compared to the CC model. The temperature-dependent EDE exhibits both minimal deviation from SBBN expansion and optimal statistical preference.

All EDE models preserve compatibility with observed $\Lambda$0, with at most minor shifts ($\Lambda$1), tightly within current CMB and BBN bounds.

Implications and Prospects

The maximal energy density permitted for EDE during BBN is several orders of magnitude above the present-day cosmological constant but remains subdominant relative to the background radiation during epoch $\Lambda$2. Polytropic models allow for transient plateaus but require fast dilution to avoid late-time overclosure or deviation from observed abundances.

The temperature-dependent model demonstrates a physically appealing modulation: substantial EDE at high temperatures, but rapid dilution in the freeze-out era, allowing for significant modification to early expansion without perturbing nucleosynthesis. This flexibility offers a theoretically viable mechanism for resolving the Hubble tension and may motivate further study incorporating both BBN and recombination/conformal horizon constraints.

Numerically robust upper limits provided here impose necessary constraints on any EDE scenario invoked to address cosmological tensions yet ensure concordance with primordial elemental yields.

Conclusion

This work establishes rigorous upper bounds for multiple EDE model parameterizations using nested sampling-informed BBN reaction networks, demonstrating that only highly suppressed or transient dark energy is permissible during nucleosynthesis. The temperature-dependent equation of state emerges as a particularly compelling framework, reconciling substantial early deviations in expansion with rapid dilution to standard background behavior. Continued exploration of such parameterizations in joint cosmological analyses—including CMB and LSS observables—is warranted for resolving ongoing tensions and refining our understanding of dark sector properties.