Review of the No-Boundary Wave Function

TL;DR

This review discusses the no-boundary wave function in quantum cosmology, explaining its theoretical foundations, implications for the universe's initial conditions, and connections to inflation, perturbations, and string theory.

Contribution

It provides a comprehensive overview of the no-boundary proposal, including analytical and numerical methods, with insights into its observational and theoretical implications.

Findings

Illustrates how the no-boundary wave function predicts initial conditions for the universe

Explains the inclusion of perturbations and classicality assessment in quantum cosmology

Connects the no-boundary proposal with string theory and observational data

Abstract

When the universe is treated as a quantum system, it is described by a wave function. This wave function is a function not only of the matter fields, but also of spacetime. The no-boundary proposal is the idea that the wave function should be calculated by summing over geometries that have no boundary to the past, and over regular matter configurations on these geometries. Accordingly, the universe is finite, self-contained and the big bang singularity is avoided. Moreover, given a dynamical theory, the no-boundary proposal provides probabilities for various solutions of the theory. In this sense it provides a quantum theory of initial conditions. This review starts with a general overview of the framework of quantum cosmology, describing both the canonical and path integral approaches, and their interpretations. After recalling several heuristic motivations for the no-boundary proposal, its consequences are illustrated with simple examples, mainly in the context of cosmic inflation. We review how to include perturbations, assess the classicality of spacetime and how probabilities may be derived. A special emphasis is given to explicit implementations in minisuperspace, to observational consequences, and to the relationship of the no-boundary wave function with string theory. At each stage, the required analytic and numerical techniques are explained in detail, including the Picard-Lefschetz approach to oscillating integrals.