Emergent Transition Radius and Acceleration Scale from Cosmological Balance

This work explores the possibility that the characteristic transition radius observed in galactic rotation curves emerges from a balance between local gravitational acceleration and a cosmological background scale. By equating the Newtonian acceleration GM/r² with a cosmological acceleration of order H₀c, we derive a transition radius: r₀ = √(GM / a*) where the effective acceleration scale is given by a* ≈ H₀c / (4π). This naturally reproduces key empirical relations, including:- the scaling r₀ ∝ √M,- the baryonic Tully–Fisher relation (v⁴ ∝ M),- realistic transition radii for spiral galaxies. We further interpret the characteristic acceleration scale as a cosmological quantity: a₀ ≈ c / t_universe ≈ H₀c, suggesting that galactic dynamics may be linked to the age and expansion of the universe. In this framework, the flattening of rotation curves is not attributed to additional mass, but to a change in effective propagation geometry, where the spreading of gravitational influence becomes constrained at large scales. This interpretation provides a geometric perspective on MOND-like behavior without modifying the fundamental force law.