Instead of the hypothesis of the dark matter and dark energy, curvature of a four dimensional Lorentzian manifold can be used to explain the accelerating expansion of the universe today and the shrink of that in the future, as well as two critical points for each period life of the universe.Therefore, the Big Bang can be treated as the beginning point of the process that the total energy changes to total mass along the positive direction of entropy on the cosmic scales,where the curvature is positive and, conversely, the Big Crunch can be treated as the beginning point of the process that total mass changes to total energy along the negative direction of entropy, where the curvature is negative.By derivative, sinusoidal functions obtained to describe the change rate of the scale and that of the curvature along the periodic life of the universe.
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