Theoretical and observational constraints on early dark energy in F(R) gravity

This work examines an early dark energy (EDE) scenario in the context of $F(R)$ gravity. EDE is introduced to alleviate the Hubble tension by temporarily injecting approximately $10\%$ of the energy fraction around the matter-radiation equality epoch ($z \approx 10^{3}$--$10^{4}$). Building on several benchmark models, we focus on the potential-driven EDE scenario and investigate the conditions required within $F(R)$ gravity. We first introduce a dimensionless quantity to analytically visualize the evolution of the density ratio between EDE and other matter components. Considering several examples, we demonstrate that the desired EDE can indeed be realized in $F(R)$ gravity. However, stringent constraints arising from violations of the equivalence principle could exclude the allowed parameter space. Our result provides a generic constraint on the potential-driven EDE in $F(R)$ gravity at the background level. This work also concludes that nonperturbative effects or nontrivial mechanisms are indispensable for studying EDE in $F(R)$ gravity while maintaining compatibility with local tests of gravity.