EFFECTIVE EQUATIONS OF MOTION AND INITIAL CONDITIONS FOR INFLATION INQUANTUM COSMOLOGY

We obtain effective equations of inflationary dynamics for the mean in flaton and metric fields - expectation values in the no-boundary and t unneling quantum states of the Universe. The equations are derived in the slow-roll approximation taking the form of the local Schwinger-DeW itt expansion. In this approximation effective equations follow from t he Euclidean effective action calculated on the De Sitter gravitationa l instanton - the basic element of the no-boundary and tunneling cosmo logical wavefunctions. Effective equations are applied in the model of the inflaton scalar held coupled to the GUT sector of matter fields a nd also having a strong nonminimal coupling to the curvature, The inve rse of its large non-minimal coupling constant, -xi = \xi\ much greate r than 1, serves as a small parameter of the slow-roll expansion and t he semiclassical expansion of quantum gravitational effects. As a sour ce of initial conditions for effective equations we use a sharp probab ility peak recently obtained in the one-loop approximation for the no- boundary and tunneling quantum states and belonging (in virtue of larg e \xi\) to the GUT energy scale much below the Planck scale. Cosmologi cal consequences of effective equations in the tunneling quantum state predict a finite duration of the inflationary stage compatible with t he observational status of inflation theory, whereas for the no-bounda ry state they lead to an infinite inflationary epoch with a constant i nflaton field, (C) 1998 Elsevier Science B.V.