wave modes with a Hamiltonian $H_\Phi(\Phi, \pi_\Phi, r^m, p_m)$. These constitute the environment, or the ‘irrelevant’ part in our model. Thus, the wave function of the Universe is a function $\Psi=\Psi(r^m, \Phi)$ which satisfies the Wheeler-De Witt equation (Wheeler 1968, DeWitt 1967) : $$H\Psi= \bigl(H_g + H_\Phi\bigr)\Psi=0
$$ where the momenta are now replaced by operators and $H_{\Phi}$. We use the same factor ordering as before. If one is interested in making predictions only about the behavior of the minisuperspace variables, a suitable quantity for such a coarse-grained description is the reduced density matrix defined as: $$\rho_{red}(r_2,r_1) = \int d\Phi \Psi^{*}(r_1,\Phi) \Psi(r_2,\Phi)$$ We will consider an ansatz for the wave function of the following form: $$\Psi(r,\Phi) = \sum\limits_n e^{iMS_{(n)}(r)}C_{(n)}(r)\chi_{(n)}(r,\Phi).
                                                             $$ where $S_n$ is a solution of a Hamilton-Jacobi equation with respect to the $r$ coordinate which is the same as (2.12) without the backreaction term and with the gradient given by $G^{mm'}{{\partial }\over{\partial r^{m'}} }$. The prefactor $C_{(n)}
$is determined through the equation $2G^{mm'} {\partial_mC_{(n)}} \partial_{m'}S_{(n)} + \Box_G S_0 = 0$, and $\chi_{(n)}$ is a solution of a Schrödinger equation with respect to the WKB time ${d\over dt} = G^{mm'}{{\partial S_0}\over{\partial r^{m'}} }
{{\partial}\over{\partial r^m}}$ and with Hamiltonian $H_{\Phi}$. These 