Quantum cosmology aspects of D3 branes and tachyon dynamics

We investigate aspects of quantum cosmology in relation to string cosmologysystems that are described in terms of the Dirac-Born-Infeld action. Using theSilverstein-Tong model, we analyze the Wheeler-DeWitt equation for the rollingscalar and gravity as well for $R\times{S^3}$ universe, by obtaining the wavefunctions for all dynamical degrees of freedom of the system. We show, that insome cases one can construct a time dependent version of the Wheeler-DeWitt(WDW) equation for the moduli field $\phi$. We also explore in detail theminisuperspace description of the rolling tachyon when non-minimal gravitytachyon couplings are inserted into the tachyon action.Comment: 29 pages, 3 figures, REVTeX 4; v2 clarifications, comments and references added; v3 more typos corrected, additional comments on the minisuperspace description of unstable universes, version published in JHE