Re-evaluation of $\Omega_k$ of the normalised Friedmann-Lema\^{\i}tre-Robertson-Walker model: Implications for Hubble constant determinations

Abstract

The description of spacetime is an fundamental problem of cosmology. We explain why the current assignments of spacetime geometries for $\Omega_k$ of the Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) model are probably incorrect and suggest more useful descriptions. We show that $\Omega_k$ represents not only curvature but the influence of matter density on the extent of spacetime between massive objects. Recent analyses of supernovae type Ia (SNe Ia) and HII/GEHR data with the FLRW model present the best fits with a small value for $\Omega_m$ and a large $\Omega_k$. These results are consistent with our Universe exhibiting sparse matter density and quasi-Euclidean geometry and the small $\Omega_m$ value agrees with Big Bang nucleosynthesis calculations. We suggest the geometry of our current Universe is better described by a value for $\Omega_k\approx$1 rather than 0. As an example we extend the FLRW model towards the Big Bang and discover a simple explanation of how matter creation developed into the currently geometrically flat Universe with sparse, homogeneous, isotropic matter and energy distributions. Assigning $\Omega_k\approx1$ to describe quasi-Euclidean spacetime geometry is also useful for estimating $H_0$ and should help resolve the tension surrounding current estimates by different investigators.