Slow-roll approximation in quantum cosmology
Oliver Janssen
Abstract
Abstract In minimally coupled scalar field theories with a potential of the slow-roll type, we give a detailed description of the complex O (4)-symmetric solutions to Einstein’s equations on the four-ball which contribute to the no-boundary amplitude Ψ NB ( b , χ ) for a closed universe to contain a round three-sphere spatial slice of size b covered homogeneously with the scalar field at value χ . Our derivation demonstrates a result anticipated by in Hartle et al (2008) Phys. Rev. D 77 123537, sharpens result in Vilenkin’s (1988) Phys. Rev. D 37 888 in the classical regime of the minisuperspace and makes use of a complexified slow-roll approximation. Our technique applies to both the Hartle–Hawking and Vilenkin wave functions, which both predict a family of inflationary universes but weight each member exponentially differently in the semiclassical approximation.