Effective equations of motion and initial conditions for inflation in quantum cosmology

We obtain effective equations of inflationary dynamics for the mean inflaton and metric fields - expectation values in the no-boundary and tunneling quantum states of the Universe. The equations are derived in the slow-roll approximation taking the form of the local Schwinger-DeWitt expansion. In this approximation effective equations follow from the Euclidean effective action calculated on the De Sitter gravitational instanton - the basic element of the no-boundary and tunneling cosmological wavefunctions. Effective equations are applied in the model of the inflaton scalar held coupled to the GUT sector of matter fields and also having a strong nonminimal coupling to the curvature, The inverse of its large non-minimal coupling constant, -xi = \xi\ much greater than 1, serves as a small parameter of the slow-roll expansion and the semiclassical expansion of quantum gravitational effects. As a source of initial conditions for effective equations we use a sharp probability peak recently obtained in the one-loop approximation for the no-boundary and tunneling quantum states and belonging (in virtue of large \xi\) to the GUT energy scale much below the Planck scale. Cosmological consequences of effective equations in the tunneling quantum state predict a finite duration of the inflationary stage compatible with the observational status of inflation theory, whereas for the no-boundary state they lead to an infinite inflationary epoch with a constant inflaton field, (C) 1998 Elsevier Science B.V.