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Consistent Evaluation of the No-Boundary Proposal

Ahmed I. Abdalla, Stefano Antonini, Raphael Bousso, Luca V. Iliesiu, Adam Levine, Arvin Shahbazi-Moghaddam

TL;DR

The paper analyzes the Hartle-Hawking no-boundary proposal through the gravitational path integral (GPI) and shows that extracting probabilities requires both the amplitude and the norm, which drastically changes predictions. In the conventional interpretation, normalized probabilities for a closed universe are nearly 1 due to dominant disconnected saddles with nonperturbative connected contributions suppressed, while in the statistical ensemble interpretation all amplitudes are exactly 1, reflecting a one-dimensional Hilbert space. The authors apply these ideas to a slow-roll inflationary model, deriving the saddle structure, reproducing historic pathologies under the naive approach, and demonstrating how consistent evaluation with smeared data yields near-unit probabilities compatible with inflationary cosmology. The work highlights a fundamental tension in quantum cosmology: without modifying the GPI or the state construction, no-boundary predictions cannot meaningfully differentiate cosmological data, and it suggests a path forward via projected states or ensemble-based interpretations, with connections to holographic ensemble ideas.

Abstract

We revisit the Hartle-Hawking no-boundary proposal. To extract probabilities, one must use the gravitational path integral (GPI) to compute not only the no-boundary amplitude, but also the norms by which its square is divided. We find that this dramatically alters predictions: the probability for any closed universe is either nearly 1, or exactly 1. That is, in the Hilbert space of closed universes defined by the GPI, the states of interest in cosmology are all nearly parallel to the Hartle-Hawking state up to nonperturbative corrections in $G_N^{-1}$. We also consider a statistical interpretation of the GPI, as an average of arbitrary products of amplitudes. We find that all amplitudes are exactly 1 in this case, consistent with recent arguments that the statistical approach to the GPI with a closed boundary computes an average over one-dimensional Hilbert spaces. As an example, we illustrate the consistent evaluation of the no-boundary proposal in inflationary cosmology.

Consistent Evaluation of the No-Boundary Proposal

TL;DR

The paper analyzes the Hartle-Hawking no-boundary proposal through the gravitational path integral (GPI) and shows that extracting probabilities requires both the amplitude and the norm, which drastically changes predictions. In the conventional interpretation, normalized probabilities for a closed universe are nearly 1 due to dominant disconnected saddles with nonperturbative connected contributions suppressed, while in the statistical ensemble interpretation all amplitudes are exactly 1, reflecting a one-dimensional Hilbert space. The authors apply these ideas to a slow-roll inflationary model, deriving the saddle structure, reproducing historic pathologies under the naive approach, and demonstrating how consistent evaluation with smeared data yields near-unit probabilities compatible with inflationary cosmology. The work highlights a fundamental tension in quantum cosmology: without modifying the GPI or the state construction, no-boundary predictions cannot meaningfully differentiate cosmological data, and it suggests a path forward via projected states or ensemble-based interpretations, with connections to holographic ensemble ideas.

Abstract

We revisit the Hartle-Hawking no-boundary proposal. To extract probabilities, one must use the gravitational path integral (GPI) to compute not only the no-boundary amplitude, but also the norms by which its square is divided. We find that this dramatically alters predictions: the probability for any closed universe is either nearly 1, or exactly 1. That is, in the Hilbert space of closed universes defined by the GPI, the states of interest in cosmology are all nearly parallel to the Hartle-Hawking state up to nonperturbative corrections in . We also consider a statistical interpretation of the GPI, as an average of arbitrary products of amplitudes. We find that all amplitudes are exactly 1 in this case, consistent with recent arguments that the statistical approach to the GPI with a closed boundary computes an average over one-dimensional Hilbert spaces. As an example, we illustrate the consistent evaluation of the no-boundary proposal in inflationary cosmology.
Paper Structure (19 sections, 86 equations, 1 figure)

This paper contains 19 sections, 86 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Outline
  3. Inner Products From the Gravitational Path Integral
  4. Conventional approach
  5. Statistical approach
  6. Probabilities in the Hartle-Hawking State
  7. Hartle-Hawking probabilities in the conventional approach
  8. Hartle-Hawking probabilities in the statistical approach
  9. Inflationary Cosmology as an Explicit Example
  10. Single-field Inflation in the Slow-roll Limit
  11. Historical Application of Hartle-Hawking
  12. Consistent Evaluation
  13. Discussion
  14. Assessment
  15. Constructing distinguishable states.
  16. ...and 4 more sections

Figures (1)

  • Figure 1: During inflation, the universe undergoes exponential expansion as the inflaton slowly rolls down its potential. When $\phi\sim\phi_{\rm end}$, the potential energy of the inflaton becomes comparable to its kinetic energy, and inflation ends. The inflaton then oscillates around the minimum of the potential and decays into standard model particles (reheating phase). The value $V(\phi_{\text{min}})$ of the potential at the minimum sets the value of the cosmological constant.