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Accelerated Expansion from Negative Λ

James B. Hartle, S. W. Hawking, Thomas Hertog

TL;DR

This work shows that for nonzero cosmological constant, the Wheeler-DeWitt equation imposes a universal semiclassical structure at large volumes, leading to an ensemble of asymptotically classical histories with de Sitter-like expansion even when the fundamental theory has negative Λ. Using minisuperspace and its extension beyond it, the authors demonstrate that asymptotic solutions share leading terms determined by boundary data, with classical histories emerging from imaginary components of the action along appropriate complex contours. The holographic no-boundary wave function provides a concrete realization, where AdS/CFT duality expresses the NBWF in terms of boundary partition functions, yielding dominant classical histories with an effective positive cosmological constant. The results have notable implications for string cosmology, suggesting that accelerating universes need not require a positive Λ at the fundamental level and offering a framework to examine vacuum stability and inflation within AdS compactifications. Overall, the paper unifies quantum cosmology, holography, and semiclassical gravity to show that negative Λ theories can be compatible with the observed cosmic acceleration.

Abstract

Wave functions specifying a quantum state of the universe must satisfy the constraints of general relativity, in particular the Wheeler-DeWitt equation (WDWE). We show for a wide class of models with non-zero cosmological constant that solutions of the WDWE exhibit a universal semiclassical asymptotic structure for large spatial volumes. A consequence of this asymptotic structure is that a wave function in a gravitational theory with a negative cosmological constant can predict an ensemble of asymptotically classical histories which expand with a positive effective cosmological constant. This raises the possibility that even fundamental theories with a negative cosmological constant can be consistent with our low-energy observations of a classical, accelerating universe. We illustrate this general framework with the specific example of the no-boundary wave function in its holographic form. The implications of these results for model building in string cosmology are discussed.

Accelerated Expansion from Negative Λ

TL;DR

This work shows that for nonzero cosmological constant, the Wheeler-DeWitt equation imposes a universal semiclassical structure at large volumes, leading to an ensemble of asymptotically classical histories with de Sitter-like expansion even when the fundamental theory has negative Λ. Using minisuperspace and its extension beyond it, the authors demonstrate that asymptotic solutions share leading terms determined by boundary data, with classical histories emerging from imaginary components of the action along appropriate complex contours. The holographic no-boundary wave function provides a concrete realization, where AdS/CFT duality expresses the NBWF in terms of boundary partition functions, yielding dominant classical histories with an effective positive cosmological constant. The results have notable implications for string cosmology, suggesting that accelerating universes need not require a positive Λ at the fundamental level and offering a framework to examine vacuum stability and inflation within AdS compactifications. Overall, the paper unifies quantum cosmology, holography, and semiclassical gravity to show that negative Λ theories can be compatible with the observed cosmic acceleration.

Abstract

Wave functions specifying a quantum state of the universe must satisfy the constraints of general relativity, in particular the Wheeler-DeWitt equation (WDWE). We show for a wide class of models with non-zero cosmological constant that solutions of the WDWE exhibit a universal semiclassical asymptotic structure for large spatial volumes. A consequence of this asymptotic structure is that a wave function in a gravitational theory with a negative cosmological constant can predict an ensemble of asymptotically classical histories which expand with a positive effective cosmological constant. This raises the possibility that even fundamental theories with a negative cosmological constant can be consistent with our low-energy observations of a classical, accelerating universe. We illustrate this general framework with the specific example of the no-boundary wave function in its holographic form. The implications of these results for model building in string cosmology are discussed.

Paper Structure

This paper contains 18 sections, 44 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Asymptotic Semiclassical Structure of the Wheeler-DeWitt Equation in Minisuperspace
  3. Geometry and Action
  4. Wheeler-DeWitt Equation
  5. Asymptotic Semiclassicality
  6. Asymptotic Structure of Solutions to the Einstein Equations
  7. Predictions for Classical Histories
  8. Signature Neutrality
  9. Beyond Minisuperspace
  10. Geometry and Action
  11. Semiclassicality
  12. Classical Histories
  13. Example: The Holographic No-Boundary Wave Function
  14. The No-Boundary State for negative $\Lambda$
  15. Two sets of saddle points
  16. ...and 3 more sections

Figures (1)

  • Figure 1: Left (a): A representation in the complex $\tau$-plane of a real, Euclidean AdS domain wall saddle point, with $H \equiv 1/l$. Right (b): A domain wall with a complex scalar field profile along the horizontal branch of the contour for which the classicality conditions hold at the endpoint $\upsilon$.