Cosmology in Five Dimensions: A 3-Sphere Universe Without Dark Matter or Dark Energy and Its Astrophysical Applications

Dark matter and dark energy represent two pivotal unresolved problems in modern cosmology, postulated to explain phenomena such as galactic rotation curves and the universe's accelerated expansion, respectively. In this work, we propose a purely geometric model where our universe is a three-dimensional hypersphere embedded in a higher-dimensional space. This framework leads to a 4D spacetime with a generalized metric, where the hyperspherical radius R(t) replaces the standard scale factor a(t) of the FLRW metric. From this geometry, we derive modified Friedmann equations directly from Einstein’s General Relativity, which naturally yield correct pressure-density relations and predict a decelerated cosmic expansion driven by the total mass-energy. A key novelty is the hypothesis that this global deceleration, stemming from the higher-dimensional geometry, projects an additional acceleration onto the observable 3D universe. This effect, rigorously derived from the full 5D metric, manifests as a natural cross term (gtr) in a Schwarzschild-type metric embedded within the expanding hyperspherical background. This gtr term leads to an effective additional radial acceleration. Building on this, we formulate a generalized Schwarzschild–FLRW metric that coherently embeds local gravitational fields within the cosmologically expanding background. This unified spacetime geometry allows for a fully covariant derivation of this additional acceleration, gravitational lensing, and redshift effects. Crucially, this mechanism explains the flattening of galactic rotation curves without invoking non-baryonic dark matter and reproduces a Tully–Fisher-like relation (scaling as v3). The model also successfully accounts for the velocity dispersion in galaxy clusters and the dynamics of wide binary systems, predicting correct mass–velocity relations. Finally, we demonstrate that the 5D metric in a matter-dominated universe naturally implies a gravitational redshift between comoving observers. When combined with the cosmological redshift due to the hypersphere's expansion, this provides a natural explanation for observational effects typically attributed to dark energy, eliminating its need. This model offers specific, testable predictions for velocity curves and redshift-distance relations, providing a compelling, purely geometric alternative to dark components in cosmology.