  Extensive studies of pre-big bang scenarios for Bianchi-I type universe have
been made, at various approximation levels. Knowing the solution of the
equations for the post-big bang universe, the symmetries of the equations (time
reversal and scale dual transformations) allow the study of pre-big bang
solutions. However, the proposed solutions are unable to explain the actually
observed acceleration of the expantion of the universe.Calculating the $\beta $
equations for the Non-Linear Sigma model, at the first loop approximation and
imposing conformal invariance at this level, lead to equations of motion that
simply state that the curvature must be nil, which in turn allows the
utilization of groups to solve the $\beta $ equations. This is what is done in
the Weiss-Zumino-Witten (WZW) model. In this article, I will show that using
the WZW model on $SU_2$, some of the difficulties encountered in the
determination of the pre and post big-bang solutions are eliminated. Combining
the general results obtained with the $\Lambda $CDM parameters lead to
realistic solutions for the evolution of the universe, giving an explanation to
the actually observed acceleration of the expansion in terms ot the dilation
field $\phi $(t).
