Self-consistent solutions of the semi-classical back-reaction equations with cosmological constant spatial curvature and perfect fluid

Exact solutions of the semi-classical Einstein equations with cosmological constant for conformally invariant free quantum fields in a general Robertson-Walker metric are found when a classical perfect fluid is present. There exist a one-parameter family of time-symmetric bouncing solutions that avoid the singularity and a one-parameter family which does not have particles horizons. The de Sitter solution is found to be stable, while the Einstein universe is unstable.