The Spatially Closed Universe

The general world model for homogeneous and isotropic universe has been proposed. For this purpose, we introduce a global and fiducial system of reference (world reference frame) constructed on a 4 + 1 Minkowski space-time that is embedding the universe, and define the line element as the separation between two neighboring events that are distinct in space and time, as viewed in the world reference frame. The effect of cosmic expansion on the measurement of physical distance has been included in the new metric, which differs from the Friedmann-Robertson-Walker metric where the spatial separation is measured on the hypersurface at a constant time, regardless of the dynamical state of space. The Einstein’s field equations with the new metric imply that closed, flat, and open universes are filled with positive, zero, and negative energy, respectively. We have demonstrated that the flat universe is empty and stationary, equivalent to the Minkowski space-time, and that the universe with positive energy density is always spatially closed and finite. In the closed universe, the proper time of a comoving observer does not elapse uniformly as judged in the world reference frame, in which both cosmic expansion and time-varying light speeds cannot exceed the limiting speed of the special relativity. We have also reconstructed cosmic evolution histories of the closed world models that are consistent with recent astronomical observations, and derived useful formulas such as energy-momentum relation of particles, redshift, total energy in the universe, cosmic distance and time scales, and so forth. It has been also shown that the inflation with positive acceleration at the earliest epoch is improbable.