Calculating the Spatial Curvature of the Universe: A New Equation that Relates the Spatial Curvature to Matter Density

Understanding the spatial curvature of our universe is a very important topic in astrophysics. The FLRW metric that determines the evolution of the universe is based on the Cosmological Principle (the universe is homogeneous and isotropic on very large scales)and on Weyl’s Postulate(the universe behaves like a perfect fluid whose components move as temporal geodesics without intersecting each other). This metric is specified in two equations, the Friedmann equations, in which the curvature term Ωk plays an essential role in its resolution. Determining the value of this term with respect to the energy density term Ωρ may mean solving or not solving the equations in many cases .We do not have the solution to this important question, but we have begun to solve it. We have found an equation that relates, in the FLRW metric, the spatial curvature with the energy density and we have found that the spatial curvature is proportional to the matter density with a proportionality factor very similar to that which relates in Einstein’s equations, the Einstein tensor with the energy-momentum tensor, that is, the curvature with the energy-matter. This has important consequences, the first is that, in a universe with matter, the spatial curvature will never be zero, the second is that, for the density of matter in today’s universe, the spatial curvature is very small.

Keywords: Spatial Curvature Constant, Friedmann´s Equation, General Relativity, Cosmic Spacetime