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 . 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 θ0, 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.