Back reaction of quantum fields in an Einstein universe

I study the back-reaction effect of the finite-temperature massless scalar field and the photon field in the background of the static Einstein universe. In each case I find a relation between the temperature of the universe and its radius. This relation exhibits a minimum radius below which no self-consistent solution for the Einstein field equation can be found. A maximum temperature marks the transition from the vacuum dominated era to the radiation dominated era. An interpretation of this behavior in terms of Bose-Einstein condensation in the case of the scalar field is given