Cosmology in Five Dimensions: Is Our Universe the Surface of a Four-Dimensional Hypersphere? Application to Dark Matter and Dark Energy

Two of the main problems of cosmology today are dark matter and dark energy. While dark energy was introduced to try to explain the accelerating expansion of the universe, dark matter was postulated as a solution to the high rotation speeds of stars around galaxies, among other observed experimental results. In this paper, we will assume that our universe is contained on the surface of a 4D hypersphere, we will propose its metric so that its radius R_ be dependent on cosmological time ₜ_. In this way, a 4D metric similar to the FLRW will be obtained in which the role of its scale factor will be played by the radius R (t) _ of the hypersphere, that is, its size. With the new metric obtained, when applying the equations of Einstein's General Relativity Theory (GRT) new Friedman equations will be obtained but changing ₐ (t) _ for R (t). _ In this way, and assuming the cases of a universe dominated by matter and energy, it will be obtained that both universes will be in decelerated expansion, a deceleration that is due to the gravitational force of the initial mass and energy of the universe. The novelty of this paper and of the use of the new metric is the proposal that this gravitational force is transmitted to our 3D universe so that it is possible, in the case of a universe dominated by matter, to explain the rotation curves of galaxies and the Tully-Fisher relationship without having to resort to dark matter. Finally, in this paper, we will propose that the expansion rate of the universe will generate a relativistic effect of time dilation that, without resorting to dark energy, will be able to explain the experimental results that have so far been interpreted as an accelerated expansion when in fact they correspond to a decelerated expansion of the universe.