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Comments on the no boundary wavefunction and slow roll inflation

Juan Maldacena

TL;DR

This work analyzes the Hartle-Hawking no-boundary wavefunction in the context of slow-roll inflation and shows that, while mathematically natural, it generically assigns a weight $|\Psi|^2 \propto \exp\left( 8 \pi^2 \cdot {3 \over V(\phi_*)} \right)$ to closed reheating geometries, driving a prediction of large positive spatial curvature that conflicts with observations. It develops analytic approximations to the no-boundary geometry in the slow-roll regime via a Hamilton-Jacobi framework and connects the reheating surface to horizon-crossing dynamics, while highlighting the tension between theory and data. The paper then discusses AdS/CFT intuition and the speculative dS/CFT angle, and surveys several proposed resolutions, including stochastic eternal inflation, alternative initial states, tunneling wavefunctions, selection principles, and possible quantum corrections. Together these insights illustrate a compelling but unsettled tension: the no-boundary proposal is elegant and natural, yet its curvature predictions challenge empirical constraints, motivating further explorations of initial conditions and quantum gravitational effects.

Abstract

We review aspects of the Hartle-Hawking no boundary geometry in the context of slow roll inflation. We give an analytic approximation to the geometry and we explain the rationale for the proposal. We also explain why it gives a prediction for the curvature of the universe that is in disagreement with observations and give a quick review of proposed ways to resolve that disagreement.

Comments on the no boundary wavefunction and slow roll inflation

TL;DR

This work analyzes the Hartle-Hawking no-boundary wavefunction in the context of slow-roll inflation and shows that, while mathematically natural, it generically assigns a weight to closed reheating geometries, driving a prediction of large positive spatial curvature that conflicts with observations. It develops analytic approximations to the no-boundary geometry in the slow-roll regime via a Hamilton-Jacobi framework and connects the reheating surface to horizon-crossing dynamics, while highlighting the tension between theory and data. The paper then discusses AdS/CFT intuition and the speculative dS/CFT angle, and surveys several proposed resolutions, including stochastic eternal inflation, alternative initial states, tunneling wavefunctions, selection principles, and possible quantum corrections. Together these insights illustrate a compelling but unsettled tension: the no-boundary proposal is elegant and natural, yet its curvature predictions challenge empirical constraints, motivating further explorations of initial conditions and quantum gravitational effects.

Abstract

We review aspects of the Hartle-Hawking no boundary geometry in the context of slow roll inflation. We give an analytic approximation to the geometry and we explain the rationale for the proposal. We also explain why it gives a prediction for the curvature of the universe that is in disagreement with observations and give a quick review of proposed ways to resolve that disagreement.
Paper Structure (21 sections, 109 equations, 4 figures)

This paper contains 21 sections, 109 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Short summary of the no boundary geometry in the slow roll inflation situation
  3. Preliminaries: computing wavefunctions in flat space and de Sitter
  4. Scalar field in de Sitter in the flat slicing
  5. Scalar field in de Sitter in the global slicing
  6. The no boundary geometry for slow roll inflation
  7. Comments on the connection to AdS/CFT
  8. Proposals for dealing with the problem
  9. Slow roll eternal inflation and the Hartle Hawking proposal
  10. Perhaps the path integral is ill defined because gravity is not renormalizable or the action is not bounded below
  11. Perhaps we are not in the Hartle Hawking state
  12. Perhaps we should consider a tunneling wavefunction
  13. Perhaps we should apply a selection principle
  14. Perhaps the quantum corrections become important
  15. Final Comments
  16. ...and 6 more sections

Figures (4)

  • Figure 1: Cartoon for the evolution of the universe. We have a period of inflation followed by the standard hot big bang picture. We treat the inflationary period using quantum mechanics up to the reheating surface. The universe that follows is treated classically and as a measurement apparatus that measures the shape of the reheating surface.
  • Figure 2: We can use complex classical solutions to compute the wavefunctions for universes of various shapes. (a) The usual case of almost flat slices with small fluctuations. (b) A situation with a large fluctuation. (c) A closed universe.
  • Figure 3: Contours in the complex plane for various wavefunction computations. (a) For a harmonic oscillator or quantum fields in flat space a contour in imaginary time from infinity to $t=0$ gives us the wavefunction at $t=0$. (b) For a quantum field in de-Sitter with flat slicing (\ref{['Fsli']}) we start from imaginary values of $\eta$ and we evolve to real $\eta$ up to $\eta=0$ to find the wavefunction in the asymptotic future. (c) In de-Sitter in global slicking (\ref{['dSGlob']}) we start from $t= i { \pi \over 2}$ and then we go to $t=0$ and continue to real times. In (d) we see the argument of the function in (\ref{['HypSph']}) as we follow the contour in (c).
  • Figure 4: (a) Plot of the numerical result for $\sigma(u)$ solution of (\ref{['SigEq']}). (b) We plot $\sigma e^{3 u}$ to show that indeed $\sigma \sim \tilde{c} e^{ - 3 u }$ for large $u$. The horizontal line is $\tilde{c} = 5.333$.