| De Sitter space is the maximally symmetric cosmology satisfying Einstein's equations with a positive cosmological constant. It has played a crucial role in the theory of inflationary cosmology. Recent astronomical observations indicate our universe is entering a new asymptotically de Sitter phase, with a mysteriously small value for the cosmological constant. We study several aspects of de Sitter and de Sitter-esque geometries in three and four spacetime dimensions. Particularly, we discuss the asymptotic symmetry group (ASG) of four-dimensional de Sitter space at future infinity, I+ , in Einstein gravity with positive cosmological constant. We find, very much unlike its anti-de Sitter cousin, an infinite dimensional group consisting of the three-dimensional diffeomorphisms acting on I+ . We then move on to rotating black holes in de Sitter space and focus on a limit where the black hole and cosmological horizons coincide. We compute the ASG of the near (cosmological) horizon geometry, the rotating Nariai geometry, which has its own future boundary I+RN and find a Virasoro algebra. This is suggestive of a holographically dual interpretation in terms of a two-dimensional CFT. Scalar waves in the rotating Nariai geometry are studied to provide further evidence for the proposal. Finally, we find toy models of the rotating Nariai geometry in three-dimensional theories of gravity with a gravitational Chern-Simons term and further explore the possibility of a holographic duality. Interestingly, we find a de Sitter like vacuum, warped dS3, whose smooth quotients contain both a cosmological as well as an internal event horizon. In contrast, quotients of Lorentzian dS3 always contain conical singularities. |
