$S$ satisfies \^2 + V\_0 = -\_[n]{}&lt; H\_n&gt; \[HJ\] where $\nabla$ is the gradient operator on minisuperspace. The above equation can be regarded as a Hamilton-Jacobi equation with backreaction. It can be also shown that the $\Psi_n$’s satisfy a Schrödinger - like equation with respect to the WKB time. This approximation can therefore be roughly thought of as the semiclassical limit where the minisuperspace modes behave classically, but the higher modes behave quantum mechanically ( for further subtleties regarding the semiclassical limit see Sec. 3). Identifying ${\partial S\over \partial a} = \p_a$ and ${\partial S\over \partial\chi_0} = \pi_{\chi_0}$, and substituting for the canonical momenta in terms of “velocities" $a'$ and $\chi_0'$ Eq. (12) reduces to a’\^[2]{} -[12]{}[’\_0]{}\^2 + V\_0(a,\_0) = -\_n&lt;H\_n&gt; This is the effective Wheeler- De Witt equation in the WKB limit, which we will compare with the backreaction equation derived in the next section, in order to calculate the term on the right hand side.

Backreaction of the Inhomogeneous Modes
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We would now like to calculate the backreaction term (2.11) explicitly. Since we are making a split of the system from the environment based on the mode decomposition, we should use the coarse-grained effective action where the 