A Unified Theory of Scale- and Environment-Dependent Gravity

Objective: To develop a single, unified theoretical framework that explains both galaxy dynamics without dark matter and large-scale cosmological observations without dark energy, resolving the Hubble tension. Methods: We unify two theoretical approaches into a single cohesive theory. We begin with a scalar field theory grounded in the principle of asymptotic safety, which dictates that the scalar-matter coupling, ξ, must run with the energy scale. We then show that the low-energy effective field theory of this coupling naturally includes higher-order invariants of the stress-energy tensor. This generates an environment-dependent effective potential for the scalar field, which undergoes a phase transition controlled by local baryonic density and angular momentum. Results: The unified theory predicts two distinct gravitational phases. In high angular momentum systems (e.g., spiral galaxies), the scalar field is stabilized in a symmetric phase, recovering standard gravity. In low angular momentum, high-density systems (e.g., elliptical galaxies), the field undergoes a phase transition, leading to an enhanced gravitational force (Geff/G ≈ 1.5) that successfully explains their dynamics without dark matter. On cosmological scales, the evolution of the scalar field in the low-density cosmic background naturally resolves the Hubble tension. Conclusions: This work presents a complete framework from first principles (asymptotic safety) to phenomenology (galaxy dynamics and cosmology). It provides a compelling alternative to the dark sector paradigm, suggesting that dark matter and dark energy are manifestations of a single, environment-dependent modification of gravity.