We develop a general framework for effective equations of expectation values in quantum cosmology and pose for them the quantum Cauchy problem with no-boundary and tunneling wavefunctions. Cosmological configuration space is decomposed into two sectors that give qualitatively different contributions to the radiation currents in effective equations. The field-theoretical sector of inhomogeneous modes is treated by the method of Euclidean effective action, while the quantum mechanical sector of the spatially homogeneous inflaton is handled by the technique of manifest quantum reduction to gauge invariant cosmological perturbations. We apply this framework in the model with a big negative non-minimal coupling, which incorporates a recently proposed low energy (GUT scale) mechanism of the quantum origin of the inflationary Universe and study the effects of the quantum inflaton mode.
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