A generalized metric for the average distribution of matter on the cosmological scale: toward a clear and flat universe?

We propose a generalized metric, denoted G, applicable to the universe as a whole, considered in the first order to be homogeneous, without any particular viewpoint on it, and filled with matter at its average density. The low density of this matter allows for the weak-field approximation and the summation of terms of single masses M/r (where r is the distance from the observer), described by the Schwarzschild metric, extended to a continuous cosmological medium. This leads to a global gravitational potential proportional to ρuRu2, where ρu is the average density and Ru is a cosmological scale of the universe (its Hubble radius). The resulting metric can be interpreted in terms of the effective refractive index of vacuum on the cosmological scale: the coefficients of the metric and the index are functions of quantities associated with the universe as a whole (summarized in the gravitational potential Φ and the deviation Ω from flatness). The calculation of the index value is performed in two steps. In the first step, we consider a critical universe such as our own (homogeneous approximation). The index is exactly equal to 2. In the second step, an increase in the index, up to a value possibly equal to 2.4, results from the effect of deviations from the homogeneity of the universe, and from the nonlinear relationship between the index and density. The propagation of light on a cosmological scale at a speed 2.4 times slower than its value c0 in the “local” vacuum (our laboratories, our solar system) provides a single solution to the dozen problems on the dark side of astrophysics, which authors currently address separately. The incorporation into a metric of the gravitational potential created by the mass of the universe offers a fresh perspective on Mach’s principle and Newton’s first law: the law of inertia is the response to the influence of the total mass of the universe. The equality of inertial mass and gravitational mass is also understood for a critical universe (the gravitational potential is of the order of c02). Conversely, the criticality of the universe is not the result of a chance encounter between two quantities unrelated to one another (the speed of light on one hand, the mass of the universe on the other) but the expression of their natural connection within a physical regularity that can be expressed only in relational terms (they cannot be described separately). Flatness is an expression of Mach’s principle revisited and extended, and does not require inflation. The text proposes a preliminary explanatory framework and orders of magnitude for the physical parameters, as a basis for future, more comprehensive revisions of existing models (expansion models without dark matter or dark energy).