be viewed as generic. Hence, once moduli are large it appears to be difficult to evolve towards a singular state of the internal space. This suggests that a successful cosmological evolution leading to phenomenologically interesting singular $G_2$ spaces should stabilise moduli at small values from the outset.

Kähler potential for $G_2$ moduli
=================================

In Ref. [@Lukas:2003dn] the moduli Kähler potential has been calculated for a specific manifold with $G_2$ holonomy constructed from a $G_2$ orbifold by blowing up the orbifold singularities. The chiral moduli multiplets for this manifold split into two categories, namely the bulk moduli $T^A$, where $A,B,C,\dots =
1,\dots ,7$, which encode the radii of the seven-torus $T^7$ underlying the orbifold, and the blow-up moduli $U^i$, which measure the volume of the blow-ups. The general structure of this Kähler potential is $$K = -\sum_{A=1}^7\ln\left(T^A+\bar{T}^A\right) + \sum_iF_i(T^A,\bar{T}^A)
     \left( U^i+\bar{U}^i\right)^2+c\; , \label{K}$$ where the functions $F_i$ are given by $$F_i = \frac{8}{(T^{A_i}+\bar{T}^{A_i})(T^{B_i}+\bar{T}^{B_i})}$$ and $c$ is a constant. Here, $A_i$ and $B_i$ specify the two particular bulk moduli by which the blow-up modulus $i$ is divided in the above Kähler potential. The values for these indices reflect the underlying structure of the orbifold and they can be conveniently encoded in constant 