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

Abstract

We obtain effective equations of inflationary dynamics for the mean inflation 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 field coupled to the GUT sector of matter fields and also having a strong non-minimal coupling to the curvature. The inverse of its large non-minimal coupling constant, − ξ = ∥ ξ ∥ ⪢ 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 ∥ ξ ∥) 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.