2024A&A...683A..83B

We study the two-body problem in the context of both dark energy and post-Newtonian modifications. In this unified framework, we demonstrate that dark energy plays the role of a critical period with $ T_{\Lambda} = 2\pi/c \sqrt{\Lambda} \approx 60 $ Gyr. We also show that the ratio between the orbital and critical periods naturally emerges from the Kretschmann scalar, which is a quadratic curvature invariant characterizing all binary systems effectively represented by de Sitter-Schwarzschild space-time. The suitability of a binary system in constraining dark energy is determined by the ratio between its Keplerian orbital period, T_K, and the critical period, T_Λ. Systems with T_K ≈ T_Λ are optimal for constraining the cosmological constant, Λ, such as the Local Group and the Virgo Cluster. Systems with T_K ≪ T_Λ are dominated by attractive gravity (which are best suited for studying modified gravity corrections). Systems with T_K ≫ T_Λ are dominated by repulsive dark energy and can thus be used to constrain Λ from below. We used our unified framework of post-Newtonian and dark-energy modifications to calculate the precession of bounded and unbounded astrophysical systems and infer constraints on Λ from them. We analyzed pulsars, the solar system, S stars around Sgr A*, the Local Group, and the Virgo Cluster, having orbital periods of days to gigayears. Our results reveal that the upper bound on the cosmological constant decreases when the orbital period of the system increases, emphasizing that Λ is a critical period in binary motion.