Equivalence of Emergent de Sitter Spaces from Conformal Field Theory

Recently, two groups have made distinct proposals for a de Sitter space that is emergent from conformal field theory (CFT). The first proposal is that, for two-dimensional holographic CFTs, the kinematic space of geodesics on a spacelike slice of the asymptotically anti-de Sitter bulk is two-dimensional de Sitter space (dS$_2$), with a metric that can be derived from the entanglement entropy of intervals in the CFT. In the second proposal, de Sitter dynamics emerges naturally from the first law of entanglement entropy for perturbations around the vacuum state of CFTs. We provide support for the equivalence of these two emergent spacetimes in the vacuum case and beyond. In particular, we study the kinematic spaces of nontrivial solutions of $3$d gravity, including the BTZ black string, BTZ black hole, and conical singularities. We argue that the resulting spaces are generically globally hyperbolic spacetimes that support dynamics given boundary conditions at future infinity. For the BTZ black string, corresponding to a thermal state of the CFT, we show that both prescriptions lead to an emergent hyperbolic patch of dS$_2$. We offer a general method for relating kinematic space and the auxiliary de Sitter space that is valid in the vacuum and thermal cases.