We consider a spatially homogeneous and isotropic cosmological model where Dirac
spinors are coupled to classical gravity. For the Dirac spinors we choose a Hartree–
Fock ansatz where all one-particle wave functions are coherent and have the same
momentum. If the scale function is large, the universe behaves like the classical
Friedmann dust solution. If however the scale function is small, quantum effects lead
to oscillations of the energy-momentum tensor. It is shown numerically and proven
analytically that these quantum oscillations can prevent the formation of a big bang or
big crunch singularity. The energy conditions are analyzed. We prove the existence
of time-periodic solutions which go through an infinite number of expansion and
contraction cycles.