A Unified Cosmology in a 5D Hypersphere: A Geometric Framework Without Inflation, Dark Matter, or Dark Energy

The standard cosmological model, ΛCDM, despite its observational success, relies on three components whose physical nature remains unconfirmed: inflation, dark matter, and dark energy. This work proposes an alternative, purely geometric framework that offers a unified solution to these enigmas based on a single and fundamental hypothesis: our universe is a three-dimensional hypersphere in expansion, embedded in a five-dimensional spacetime. We argue that the intrinsic perspective of standard cosmology (the FLRW metric) provides an incomplete description of reality, forcing the introduction of “dark” components to explain effects that arise naturally from the extrinsic geometry and dynamics of the fifth dimension. In our model, the “dark” phenomena are not exotic substances or epochs, but rather the manifestations in our 4D spacetime of this higher-dimensional geometric reality. Moreover, the model requires only three initial parameters—the baryonic mass of the universe, its radiation content, and the current value of H₀ that fixes the proper time τ—highlighting its simplicity compared to the ΛCDM paradigm. First, we postulate a fundamental relation between curvature and inhomogeneity (|Ωk| = 1/2δρ), which resolves the flatness and horizon problems without the need for an inflationary epoch. When applied to the CMB, this hypothesis allows us to determine the fundamental parameters of the universe and to reproduce the angular scale of the first acoustic peak with an error of about 8%. Second, the global deceleration of the hypersphere projects an additional acceleration into our 3D space, which—using the parameters derived from the CMB—quantitatively explains galactic rotation curves, the dynamics of clusters, and provides a new Tully–Fisher relation of the form M ∝ v³. From this formulation, the MOND law and the Virial-like relation M ∝ σv³ for galaxy clusters naturally emerge without the need to invoke dark matter. Third, the non-trivial temporal component (gtt ≠ 1) of the proposed 5D metric induces a gravitational redshift that reproduces the Hubble diagram of supernovae, eliminating the need for dark energy. In summary, we present a self-consistent, purely geometric cosmological model that addresses several of the major puzzles of modern cosmology within a unified framework, offering a potentially simpler alternative to ΛCDM and making testable predictions for a range of astrophysical and cosmological observations. While the model provides a radical alternative to the current paradigm, it should be regarded as an initial and basic proposal, requiring further mathematical development and more detailed confrontation with observations. Its simplicity, together with the breadth of phenomena it accounts for, suggests it may serve as a viable starting point for dialogue and further research aimed at testing and refining this geometric approach.