field $\chi = (al_p)\Phi$ and expanding $\chi$ in scalar spherical harmonics $Q^{k}_{lm}(x)$ on the $S^3$ spatial sections, = [\_[0]{}(t)(2\^2)\^]{} + \_[klm]{}[ ]{}f\_[klm]{} Q\^[k]{}\_[lm]{}(x) where $k=2,3,\ldots \infty, l = 0,1,\ldots k-1, m = -l, -l+1 \ldots
l-1,l$ (henceforth we will use $k$ to denote the set $\{klm\}$). We will make the further assumption that the interactions of orders higher than quadratic of the lowest(minisuperspace) mode $\chi_0$ with the higher modes $f_n$’s ($\sim \chi_0 f_{k}f_{l}f_{m}$) as well as the quartic self interaction of the higher modes ($\sim f_{m}f_{n}f_{k}f_{l}$) are small and can be neglected. With this assumption and with the following redefinitions $ m^{2} \rightarrow {m^{2}/l_p^2} , \l \rightarrow {\l/
2{\pi}^2}$,the matter action can be written as S\_m & = & d{[12]{} - [ł4!]{}\^4 -[12]{}\_[k]{}[f\_k ]{}-[12]{}\_[k]{} [m\^2 a\^2 [f\_k]{}\^2]{}\
\[matter\] & & - [ł4]{}\_[k]{}[[\_0]{}\^2 [f\_k]{}\^2]{} } The Hamiltonian constructed from the action $S = S_g + S_m$ is given by H = -[12]{}[\_a]{}\^[2]{} + [12]{}[\_[\_0]{}]{}\^[2]{} + [12]{}\_[n]{}[\_[f\_n]{}]{}\^2 + V\_[0]{}(a, \_[0]{}) + V(a,\_[0]{}, f\_n) where V\_[0]{}(a,\_[0]{}) = -[12]{}a\^2 + [12]{}m\^[2]{}a\^[2]{}[\_[0]{}]{}\^2 + [ł4!]{}[\_0]{}\^4 and V(a,\_0 ,f\_n) = [ł4]{}\_[k]{}[\_0]{}\^2 [f\_k]{}\^2 + [12]{}\_[k]{}(k\^2 + m\^2 a\^2)[f\_k]{}\^2 where $\pi_a, \pi_{\chi_0},\pi_{f_n}$ are the momenta canonically conjugate to $a, \chi_0, f_n$ respectively. The Wheeler-DeWitt equation for the wave function 