do this because linearized gravitational perturbations (gravitons) in a special gauge can be shown to be equivalent to a pair of minimally coupled scalar fields (Lifshitz 1946). Quantum cosmology of similar models of gravitational perturbations and scalar fields have been studied by several authors (Halliwell and Hawking 1985, Kiefer 1987, Vachaspati and Vilenkin 1988). Our basic strategy will be to try to derive an “effective" Wheeler-DeWitt equation for the minisuperspace sector which contains the “averaged" effect of the higher modes as a backreaction term. We will then explicitly calculate the backreaction term using the effective action and consequently present a criterion for the validity of the minisuperspace approximation.

Effective Wheeler DeWitt Equation
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The gravitational and matter actions in our model are given by S\_g & = & da\^[2]{}(1 - [a’\^[2]{}a\^2]{})\
S\_m & = & - d\^[4]{}x\[+ m\^2\^2 +[2ł4!]{}\^4 + [R6]{} \^2\] where $a$ is the scale factor of a closed Robertson-Walker universe, a factor $l_p^2 = 2/(3\pi m_p^2) $ is included in the metric for simplification of computations, and $\eta$ is the conformal time. $\Box$ is the Laplace Beltrami operator on the metric $g_{\m\n}$, and $m$ is the mass of the conformally coupled scalar field. Defining a conformally related 