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Density matrix of de Sitter JT gravity

Wilfried Buchmuller, Alexander Westphal

Abstract

Jackiw-Teitelboim (JT) gravity in two-dimensional de Sitter space is an intriguing toy model for a quantum mechanical description of an inflationary phase of the universe, including initial conditions. Starting from exact solutions of the Wheeler-DeWitt equation, we study a conditional density matrix of the system. We find that the ground state is a mixed state, rather than a pure Hartle-Hawking state. Our results are consistent with the semiclassical double-trumpet amplitude, and with recent work on complex geometries containing bra-ket wormholes. We also analyze semiclassical wave functions for metric, dilaton, and an additional inflaton field. The probability distribution for the size of the universe is flat.

Density matrix of de Sitter JT gravity

Abstract

Jackiw-Teitelboim (JT) gravity in two-dimensional de Sitter space is an intriguing toy model for a quantum mechanical description of an inflationary phase of the universe, including initial conditions. Starting from exact solutions of the Wheeler-DeWitt equation, we study a conditional density matrix of the system. We find that the ground state is a mixed state, rather than a pure Hartle-Hawking state. Our results are consistent with the semiclassical double-trumpet amplitude, and with recent work on complex geometries containing bra-ket wormholes. We also analyze semiclassical wave functions for metric, dilaton, and an additional inflaton field. The probability distribution for the size of the universe is flat.

Paper Structure

This paper contains 16 sections, 102 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Wave function vs. density matrix
  3. Hartle-Hawking wave function
  4. JT density matrix
  5. Density matrix from complex geometries
  6. Semiclassical de Sitter JT wave functions
  7. De Sitter JT gravity
  8. Semiclassical wave functions with inflaton
  9. Fluctuations of the inflaton field
  10. WDW equation with an inhomogeneous inflaton
  11. Probability distribution with inflaton
  12. Exponential suppression of large universes
  13. Summary and conclusions
  14. Mixed state density matrix
  15. Prefactor for JT gravity with inflaton
  16. ...and 1 more sections

Figures (2)

  • Figure 1: Left: complex Lorentzian/Euclidean geometry underlying no-boundary wave function. Right: double-trumpet configuration for two different values of $h_0$.
  • Figure 2: Left: transition amplitudes for different values of $h_0$. Right: double-trumpet amplitudes for different values of $h_0$.