Phase spaces for asymptotically de Sitter cosmologies

Abstract

We construct two types of phase spaces for asymptotically de Sitter Einstein-Hilbert gravity in each spacetime dimension d ≥ 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 ±i 3 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 welldefined 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.

Conserved charges in asymptotically de Sitter spacetimes

“…1 This clearly suggests that these boundary conditions are too restrictive. Alternative boundary conditions have been proposed [6,7,11], though they remain less explored. This needs to be contrasted with the recent developments in linearised gravity in de Sitter, where now there is a good control over many calculations [13,14,15,16,17,18].…”

“…One can give certain indirect arguments for relating these various notions of charges [7]. However, these indirect arguments are hardly illuminating; an explicit comparison between various approaches remain fairly cumbersome as these various approaches are based on very different techniques: ABK use conformal infinity framework whereas counterterm method uses Fefferman-Graham expansion, and Kelly-Marolf use radial expansion in ADM form near spatial infinity.…”

“…In an interesting paper, Kelly and Marolf [7] pointed out that although there has been many discussions of de Sitter charges over the years, most of these discussions do not explicitly construct a phase space on which charges can be viewed as generators of the associated asymptotic symmetries. For the construction such as the one given above, induced metric at future infinity is held fixed and therefore the symplectic structure necessarily vanishes.…”

Asymptotics with a positive cosmological constant. IV. The no-incoming radiation condition

“…We have other definitions of mass at both these points. The one at i o uses space-like surfaces (such as the cosmological slices in the de Sitter space-time) that extend to i o (see, e.g., [41,42]). The one at i − is obtained by working with (the space-like) I − , and imposing the 'no incoming radiation condition' by requiring that the magnetic part of the (appropriately conformally rescaled) Weyl tensor vanishes there (see, e.g.…”

Issues about cosmological Ward identities

“…In the context of gauge-gravity duality, the asymptotic charges are usually defined using the counter-term subtraction method and in [35] these have been shown to generate the desired asymptotic symmetries of the AdS space. On the other hand, in [36] the asymptotic symmetry group and the corresponding charges are specified on appropriately constructed phase space for the asymptotically de Sitter Einstein gravity. Curiously, the spatial fall-off conditions utilized in [36] for the construction of the phase space are very similar to the ones imposed in this paper.…”