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The Wave Function of Quantum de Sitter

Alejandra Castro, Alexander Maloney

TL;DR

The paper computes the Hartle-Hawking wave function for quantum gravity in 2+1 dimensions with a positive cosmological constant by using Maldacena’s contour, relating the Lorentzian wave function at future infinity to the Euclidean AdS partition function with torus boundary. The analysis identifies an infinite family of saddles, $dS_3/\mathbb{Z}$, labeled by $\mathbb{Z}\backslash SL(2,\mathbb{Z})$, whose conformal boundary data is encoded by the torus modulus $\tau$, and expresses the wave function as a modular-invariant sum over BTZ-like saddles via $Z(\tau,\bar{\tau})$. Upon analytic continuation $c_{AdS}\to i c_{dS}$, the resulting wave function develops a non-perturbative divergence at cusps, yielding a non-normalizable peak at highly inhomogeneous future conformal structures ${\cal I}^+$ and implying a quantum instability of the de Sitter vacuum in this setting. The results illuminate how non-perturbative saddles shape the quantum cosmology of de Sitter in three dimensions and suggest possible endpoints or extensions when additional degrees of freedom are included.

Abstract

We consider quantum general relativity in three dimensions with a positive cosmological constant. The Hartle-Hawking wave function is computed as a function of metric data at asymptotic future infinity. The analytic continuation from Euclidean Anti-de Sitter space provides a natural integration contour in the space of metrics, allowing us -- with certain assumptions -- to compute the wave function exactly, including both perturbative and non-perturbative effects. The resulting wave function is a non-normalizable function of the conformal structure of future infinity which is infinitely peaked at geometries where I^+ becomes infinitely inhomogeneous. We interpret this as a non-perturbative instability of de Sitter space in three dimensional Einstein gravity.

The Wave Function of Quantum de Sitter

TL;DR

The paper computes the Hartle-Hawking wave function for quantum gravity in 2+1 dimensions with a positive cosmological constant by using Maldacena’s contour, relating the Lorentzian wave function at future infinity to the Euclidean AdS partition function with torus boundary. The analysis identifies an infinite family of saddles, , labeled by , whose conformal boundary data is encoded by the torus modulus , and expresses the wave function as a modular-invariant sum over BTZ-like saddles via . Upon analytic continuation , the resulting wave function develops a non-perturbative divergence at cusps, yielding a non-normalizable peak at highly inhomogeneous future conformal structures and implying a quantum instability of the de Sitter vacuum in this setting. The results illuminate how non-perturbative saddles shape the quantum cosmology of de Sitter in three dimensions and suggest possible endpoints or extensions when additional degrees of freedom are included.

Abstract

We consider quantum general relativity in three dimensions with a positive cosmological constant. The Hartle-Hawking wave function is computed as a function of metric data at asymptotic future infinity. The analytic continuation from Euclidean Anti-de Sitter space provides a natural integration contour in the space of metrics, allowing us -- with certain assumptions -- to compute the wave function exactly, including both perturbative and non-perturbative effects. The resulting wave function is a non-normalizable function of the conformal structure of future infinity which is infinitely peaked at geometries where I^+ becomes infinitely inhomogeneous. We interpret this as a non-perturbative instability of de Sitter space in three dimensional Einstein gravity.

Paper Structure

This paper contains 5 sections, 47 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Classical Geometries
  3. The Wave Function
  4. Euclidean BTZ
  5. Euclidean AdS$_3$ path integral and its analytic continuation

Figures (2)

  • Figure 1:
  • Figure 2: The norm $|\Psi(\tau)|$ of the wave function \ref{['EPSI']} for purely imaginary values of $\tau$ ($\tau_1=0$). In order to obtain stable numerical results it was necessary to include the ranges $n=-3\ldots 3$ and $m=0\ldots30$ in equations \ref{['E00terms']}, \ref{['Enterms']}.