Exact cosmological solutions of 4D string gravity in the string frame

We have discussed a particular class of exact cosmological solutions of the 4-dimensional low energy string gravity in the string frame. In the vacuum without matter and the 2-form fields, the exact cosmological solutions always give monotonically shrinking universes if the dilaton field is not a constant. However, in the presence of the 2-form fields and/or the radiation-like fluid in the string frame, the exact cosmological solutions show a minimum size of the universe in the evolution, but with an initial cosmological curvature singularity in the string frame