simplifies to $$L = \sqrt{-g}\left( R+\frac{1}{2}\sum_A{\partial}_\m\f_A{\partial}^\m\f_A+8
     \sum_ie^{-{\bf P}_i\cdot\bph}{\partial}_\m z_i{\partial}^\m z_i\right) \label{L}$$ where $${\bf P}_i = -\m_i{\bf p}_i\; .$$ The obvious advantage of this form is that the rescaled blow-up moduli $z_i$ (unlike their counterparts $u_i$) only appear through their derivatives. This implies that their equations of motion can be immediately integrated once, leading to a set of first integrals. We will use this property in the following section when we discuss the cosmological evolution based on the Lagrangian . Since we have neglected higher order terms in $\e_{(\t ,n)}$ in its derivation, it is equivalent to the one directly obtained from the Kähler potential  only provided fields are in the region of moduli space defined by Eq. . Also note that this equivalence holds for an arbitrary choice of the signs $\m_i$.

Cosmological evolution equations
================================

To study the cosmological evolution based on the Lagrangian  we start with a metric $$ds^2=-e^{2\n (\t )}d\t^2+e^{2\a (\t )}d{\bf x}^2$$ of Friedmann-Robertson-Walker type, with the spatial sections taken to be flat for simplicity, and time-dependent moduli fields $$\f_A=\f_A (\t )\; ,\qquad z_i=z_i(\t )\; .$$ Here, $\a (\t )$ is the scale factor of the universe and $\n (\t
)$ is the lapse function, which 