for the bulk moduli $\f_A$. Replacing the blow-up moduli with Eq.  and working in the gauge $\n =3\a$ they take the form $$\frac{1}{2}\frac{d^2\f_A}{d\t^2}+\frac{{\partial}V}{{\partial}\f_A} = 0\; . \label{feom}$$ The “effective" potential $$V = 4\sum_i\z_i^2\, e^{{\bf P}_i\cdot\bph} \label{V}$$ in these equations of course originates from integrating out the blow-up moduli. The above equations of motion can be obtained from the Lagrangian $$\cL = \frac{1}{4}\dot{\bph}\cdot\dot{\bph}-V\; . \label{Lf}$$ They have to be supplemented by the Hamiltonian constraint $$\cH = \frac{1}{4}\dot{\bph}\cdot\dot{\bph}+V=E\; , \label{H}$$ which is simply the Friedmann equation rewritten in our language.

For the subsequent discussion it is useful to express the quantities measuring the geometry of the $G_2$ manifold in terms of the redefined fields. The volume $\cV_0$ of the underlying orbifold, defined in Eq. , can then be written as $$\cV_0 = \frac{1}{8}\exp\left(\frac{1}{3}\sum_A\f_A\right)\; ,
 \label{V1}$$ where we have neglected higher order terms. Likewise, the fraction $\e_{(\t ,n)}$ of the total volume taken up by the blow-up $(\t ,n)$ now takes the form $$\e_{(\t ,n)} = \frac{8}{3}\sum_ae^{-{\bf P}_i\cdot\bph}\,
                z_{(\t ,n,a)}^2\; .
 \label{e1}$$

Let us summarise what we have achieved so far. We have integrated out the blow-up moduli and decoupled the scale factor by choosing a particular gauge. This has reduced the problem 