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Phase Spaces for asymptotically de Sitter Cosmologies

William R. Kelly, Donald Marolf

TL;DR

The authors construct two phase spaces of asymptotically de Sitter gravity, Γ(dS_k=0) and Γ(dS_k=-1), in any d≥3, each with a finite, conserved symplectic structure and well-defined charges that generate their asymptotic diffeomorphisms. In general, the ASG matches the isometries of the corresponding dS patch, but in d=3, k=−1 they uncover a Virasoro structure and an extended two-Virasoro algebra with imaginary central charges, connected to dS/CFT expectations. The d=3, k=−1 sector hosts nontrivial wormhole spacetimes that yield a BTZ-like mass gap despite the absence of local degrees of freedom. The charges agree with Strominger–Balasubramanian-type constructions on appropriate frames, clarifying how dS/CFT charges act on a physical phase space and revealing rich global structures in de Sitter cosmologies.

Abstract

We construct two types of phase spaces for asymptotically de Sitter Einstein-Hilbert gravity in each spacetime dimension $d \ge 3$. One type contains solutions asymptotic to the expanding spatially-flat ($k=0$) cosmological patch of de Sitter space while the other is asymptotic to the expanding hyperbolic $(k=-1)$ patch. Each phase space has a non-trivial asymptotic symmetry group (ASG) which includes the isometry group of the corresponding de Sitter patch. For $d=3$ and $k=-1$ our ASG also contains additional generators and leads to a Virasoro algebra with vanishing central charge. Furthermore, we identify an interesting algebra (even larger than the ASG) containing two Virasoro algebras related by a reality condition and having imaginary central charges $\pm i \frac{3\ell}{2G}$. Our charges agree with those obtained previously using dS/CFT methods for the same asymptotic Killing fields showing that (at least some of) the dS/CFT charges act on a well-defined phase space. Along the way we show that, despite the lack of local degrees of freedom, the $d=3, k=-1$ phase space is non-trivial even in pure $Λ> 0$ Einstein-Hilbert gravity due to the existence of a family of `wormhole' solutions labeled by their angular momentum, a mass-like parameter $θ_0$, the topology of future infinity ($I^+$), and perhaps additional internal moduli. These solutions are $Λ> 0$ analogues of BTZ black holes and exhibit a corresponding mass gap relative to empty de Sitter.

Phase Spaces for asymptotically de Sitter Cosmologies

TL;DR

The authors construct two phase spaces of asymptotically de Sitter gravity, Γ(dS_k=0) and Γ(dS_k=-1), in any d≥3, each with a finite, conserved symplectic structure and well-defined charges that generate their asymptotic diffeomorphisms. In general, the ASG matches the isometries of the corresponding dS patch, but in d=3, k=−1 they uncover a Virasoro structure and an extended two-Virasoro algebra with imaginary central charges, connected to dS/CFT expectations. The d=3, k=−1 sector hosts nontrivial wormhole spacetimes that yield a BTZ-like mass gap despite the absence of local degrees of freedom. The charges agree with Strominger–Balasubramanian-type constructions on appropriate frames, clarifying how dS/CFT charges act on a physical phase space and revealing rich global structures in de Sitter cosmologies.

Abstract

We construct two types of phase spaces for asymptotically de Sitter Einstein-Hilbert gravity in each spacetime dimension . One type contains solutions asymptotic to the expanding spatially-flat () cosmological patch of de Sitter space while the other is asymptotic to the expanding hyperbolic patch. Each phase space has a non-trivial asymptotic symmetry group (ASG) which includes the isometry group of the corresponding de Sitter patch. For and our ASG also contains additional generators and leads to a Virasoro algebra with vanishing central charge. Furthermore, we identify an interesting algebra (even larger than the ASG) containing two Virasoro algebras related by a reality condition and having imaginary central charges . Our charges agree with those obtained previously using dS/CFT methods for the same asymptotic Killing fields showing that (at least some of) the dS/CFT charges act on a well-defined phase space. Along the way we show that, despite the lack of local degrees of freedom, the phase space is non-trivial even in pure Einstein-Hilbert gravity due to the existence of a family of `wormhole' solutions labeled by their angular momentum, a mass-like parameter , the topology of future infinity (), and perhaps additional internal moduli. These solutions are analogues of BTZ black holes and exhibit a corresponding mass gap relative to empty de Sitter.

Paper Structure

This paper contains 21 sections, 104 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The phase space $\Gamma(\textrm{dS}_{k=0})$
  3. Asymptotic Symmetries
  4. Conserved Charges
  5. Familiar Examples
  6. Comparison with Brown-York methods at future infinity
  7. The $k=-1$ phase space
  8. Asymptotic Symmetries
  9. Familiar Examples
  10. Asymptotically dS${}_{k=-1}$ wormhole spacetimes
  11. Comparison with Brown-York methods at future infinity
  12. Discussion
  13. Finiteness of the charges for $\Gamma(\textrm{dS}_{k=0})$.
  14. The radial Hamiltonian constraint
  15. $k=0$
  16. ...and 6 more sections

Figures (2)

  • Figure 1: A conformal diagram of dS${}_d$. The region above the diagonal line is the $k=0$ cosmological patch. Each point represents an $S^{d-2}$. Our boundary conditions are applied at the $S^{d-2}$ labeled $i^0$, which we call spatial infinity. The dashed line shows a representative $(t= {\rm constant})$ slice.
  • Figure 2: A conformal diagram of dS${}_d$. The region above the diagonal line is the expanding hyperbolic patch. The $S^{d-2}$ labeled $i^0$ is the spatial infinity at which our boundary conditions are applied. The dashed line shows a representative $(T= {\rm constant})$ slice.