The No-Boundary Measure of the Universe (0711.4630v4)

Abstract: We consider the no-boundary proposal for homogeneous isotropic closed universes with a cosmological constant and a scalar field with a quadratic potential. In the semi-classical limit, it predicts classical behavior at late times if the initial scalar field is more than a certain minimum. If the classical late time histories are extended back, they may be singular or bounce at a finite radius. The no-boundary proposal provides a probability measure on the classical solutions which selects inflationary histories but is heavily biased towards small amounts of inflation. This would not be compatible with observations. However we argue that the probability for a homogeneous universe should be multiplied by exp(3N) where N is the number of e-foldings of slow roll inflation to obtain the probability for what we observe in our past light cone. This volume weighting is similar to that in eternal inflation. In a landscape potential, it would predict that the universe would have a large amount of inflation and that it would start in an approximately de Sitter state near a saddle-point of the potential. The universe would then have always been in the semi-classical regime.

• The paper explores the no-boundary wave function (NBWF), predicting classical spacetimes and universe histories that favor quantum-to-classical transitions.
• The no-boundary measure predicts cosmological inflation and suggests initial conditions for classical behavior that favor universes with more inflation.
• The no-boundary wave function selects among string landscape vacua, favoring those producing classical spacetime via slow roll inflation.

The No-Boundary Measure of the Universe

The paper "The No-Boundary Measure of the Universe" by Hartle, Hawking, and Hertog presents a detailed exploration of the no-boundary proposal in the context of cosmology. The authors investigate homogeneous isotropic closed universes characterized by a cosmological constant and scalar fields with quadratic potentials. Their analysis explores the implications of the no-boundary wave function (NBWF) concerning the probability measures of classical spacetimes and their correlation with observed cosmological phenomena.

The paper builds on the notion that the string theory landscape potentially includes numerous stable and metastable vacua, but it does not account for why our universe specifically exists in one particular vacuum. Thus, they examine a quantum cosmological framework wherein different classical spacetimes emanate from the NBWF and explore how these can be interpreted as probabilities for various evolutionary histories of the universe.

Classical Prediction and Quantum States

The authors emphasize that classical spacetime is a feature emergent only from particular quantum states within this framework. The NBWF is derived through a path integral formulation defined over geometries with a singular boundary. This translates into the semi-classical approximation, where the probability measures derived are heavily influenced by conformance to the classicality constraint: $|(\nabla S)^2| \gg |(\nabla I_R)^2|$. Consequently, the NBWF forecasts probabilistic measures for entire universe histories, highly favoring those histories characterized by a quantum-to-classical transition.

Predictions and Measures

A principal conclusion of this work is the NBWF’s prediction bias towards cosmological inflation. Specifically, the NBWF, coupled with late-time classicality, naturally implies an inflationary period during the nascent stages of the universe. For a scalar field with a quadratic potential, the authors demonstrate through numerical solutions of the field equations an inherent lower bound on initial conditions for classical behavior at macroscopic scales.

Furthermore, the probability densities obtained anticipate a universe model where initial scalar field values, $\phi_0$, determine distinct evolution paths, with histories initially featuring $\phi_0$ below a critical threshold leading to singular outcomes in the classical limit or non-singular bouncing solutions. The NBWF consequently postulates a universe that inherently favors larger volumes and, hence, more e-folds of inflation when top-down conditions considering our observational data are applied.

Implications for the String Landscape

Considerations within the broader landscape of string theory indicate that the NBWF effectively selects among vacua, discriminating against those that fail to produce classical spacetime due to steep potential curvatures. The classical histories are theorized to initiate in a de Sitter-like state and undergo significant slow roll inflation; predictions aligned with a universe of substantial expansion phase, culminating in a favorable condition for observational cosmology.

Inhomogeneities and Future Directions

Finally, although this paper principally addresses homogeneous models, it discusses the implications for inhomogeneities. It suggests that the NBWF, along with volume weighting, may predict significant large-scale cosmic structure in alignment with notions of eternal inflation, albeit without an explicit requirement for additional measures. Nevertheless, connections to these directions are under continuing examination, as the landscape continues to evolve with new theoretical advancements and empirical discoveries.

Hartle, Hawking, and Hertog’s exploration of the no-boundary measure provides a compelling lens through which to view the origins and structure of our universe, linking quantum cosmological insights to classical spacetime structures observable on cosmological scales.