Oscillating universes

Some properties of cyclic closed universes are investigated under the assumption that the total entropy of the universe increases from cycle to cycle. Exact solutions are given for the increase in maximum size of a cyclic universe filled with matter and radiation. Asymptotically, a cyclic closed universe approaches 'flatness'. If a positive cosmological constant exists then these oscillations will eventually cease and be replaced by an era of expansion which will continue unless the cosmological constant is associated with a form of vacuum energy that ultimately decays away. If that occurs, then universal oscillations will be resumed. Oscillations will also be ended by the indefinite presence of any non-zero stress that violates the strong energy condition. A stable stress of this sort will always dictate the ultimate evolution of a cyclic closed universe. We also consider what occurs when the Hawking entropy associated with the cosmological constant is included in the entropy budget, and describe situations in which the sequence of universal oscillations can decrease in amplitude. Eventually, they will become too small for the cosmological constant to affect the dynamics, and then small oscillations will approach this marginally stable state. We examine some particular closed anisotropic universes, and investigate how the evolution of the anisotropy is affected by the universal oscillations. In the most general known homogeneous closed universes it appears that the asymptotic behaviour is for the cycles to increase in size, as entropy increases, for any dust-dominated era to occupy an increasingly longer period of the total evolution, and hence for the anisotropy to decrease in influence at late times with increasing cycle number when the cosmological constant and other stresses violating the strong energy condition are absent.