Ancient orbital structures: simulating Local Group evolution under full Hyperspherical Expansion Acceleration

In previous work, an alternative cosmological framework was developed based on a four‑dimensional hyperspherical geometry undergoing strict linear expansion. Within this setting, the observable universe is treated as a three‑dimensional hypersurface on which standard gravitational physics applies locally, without invoking Friedmann–Lemaître–Robertson–Walker dynamics. Building on the kinematic properties of this manifold, that analysis derived the full Hyperspherical Expansion Acceleration (HEA) equation, which naturally introduces an effective cosmic drag together with a velocity‑dependent coupling term. To test this framework, we perform a forward-in-time numerical simulation of the Local Group orbit (the Milky Way–Andromeda system). We model the post-recombination epoch as a non-linear phase in which a cosmic surface-tension mechanism drives matter into the expanding flow. Although this transient phase is explosive, the requirement of matter homogeneity over cosmic time constrains the resulting recession velocities to remain generally much smaller than the speed of light. Our numerical results show that the velocity-dependent HEA coupling naturally moderates the strong primordial expansion, allowing ancient trajectories to evolve consistently within the cosmic flow. As gravity progressively decelerates this initial separation, the system naturally passes through a turnaround phase and stabilizes into quasi-Newtonian orbits. Once the initial conditions are constrained by present-day observations and by the cosmological principle, the simulation reproduces the current Milky Way–Andromeda separation of approximately 2.5 million light-years and the observed approaching velocity of about 110 km/s. These results, which accurately reproduce the orbital parameters of the Local Group, suggest that local structural scales may emerge from mass evolving within a linearly expanding hyperspherical geometry.