We consider a spatially homogeneous and isotropic system of Dirac particlescoupled to classical gravity. The dust and radiation dominated closedFriedmann-Robertson-Walker space-times are recovered as limiting cases. We finda mechanism where quantum oscillations of the Dirac wave functions can preventthe formation of the big bang or big crunch singularity. Thus before the bigcrunch, the collapse of the universe is stopped by quantum effects and reversedto an expansion, so that the universe opens up entering a new era of classicalbehavior. Numerical examples of such space-times are given, and the dependence onvarious parameters is discussed. Generically, one has a collapse after a finitenumber of cycles. By fine-tuning the parameters we construct an example of aspace-time which is time-periodic, thus running through an infinite number ofcontraction and expansion cycles.
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