Orbifold Physics and De Sitter Spacetime

It now seems probable that the version of de Sitter spacetime which may ultimately emerge from string theory will not be the familiar, maximally symmetric version, since it is likely to be truncated in some way by metastability or otherwise reduced in symmetry so that its isometry group has finite-dimensional representations. We argue that the best way to gain some control over this situation is to embed a suitably modified version of de Sitter spacetime in an anti-de Sitter orbifold bulk, as a braneworld. By requiring them to fit together in this way, we attempt to understand the precise structures of both. We find that tachyonic instabilities of non-supersymmetric AdS orbifolds allow us to constrain the global geometries of these fundamental spacetimes. In the course of doing so, we gain some insights into de Sitter holography and into the way in which de Sitter physics breaks conformal symmetry in the dS and AdS duals. Our results indicate that string theory may rule out the more complex spatial topologies discussed recently. 1. Warping the Cosmological Constant The fact that the expansion of the universe is accelerating (see for example [1][2]) presumably implies that theories of fundamental physics must be capable of leading us to de Sitter spacetime, at least in some approximation. It is not generally appreciated, however, that there are in fact many different topological spaces which can accept the de Sitter metric locally ; this is particularly true if we allow orbifold singularities, and, as we shall see, there are also interesting ways to modify the topology of the standard conformal compactification. We have no reason to believe that the version of de Sitter spacetime which may emerge, possibly in truncated or metastable form (see [3]) from string theory, will necessarily be the most familiar version with symmetry group O(1,4). In fact, just the opposite is true, since a truncated or otherwise mutilated spacetime will not be maximally symmetric; and, furthermore, maximal O(1,4) de Sitter symmetry leads to various problems, as we shall discuss in detail. In searching for de Sitter spacetime in string theory, it is clearly of great importance to know which version we should expect to find. If that version is one of the more unusual ones we shall discuss here, then its unusual features may assist us in finding it. Also, while de Sitter spacetime may not be a good model of the current state of the Universe, it may become more so as cosmic time passes; and as we do not expect the topology of the spatial sections to change, what we can learn in idealized cases may well carry over to more realistic cosmologies. String theory works extremely well on anti-de Sitter space [4], and it is therefore puzzling that the cosmological constant of our Universe is positive. The magnitude of the cosmological constant is also hard to understand from this point of view — the curvature of AdS5 in the best-studied example is far too large (but see [5]). It is nevertheless difficult to believe that anti-de Sitter space is not relevant to de Sitter cosmology in some way. For anti-de Sitter space one has a well-established holographic duality with a conformal field theory, but all manner of difficulties (see for example [6]) have arisen in efforts to find a de Sitter analogue [7][8]. Yet the study of de Sitter physics has convinced many — see for example [9][10][11][12][13][14][15][16][17][18] for several very different approaches — that some kind of holography must hold for de Sitter spacetime. Again, one suspects that AdS holography must somehow underlie or even imply de Sitter holography, in whatever form the latter takes. Now in fact the metric of (part of) five-dimensional anti-de Sitter space is nothing but a local warped product of a one-dimensional space with the four-dimensional de Sitter metric, as we shall explain in detail later. The Euclidean version of this is very clear: the metric of the five-dimensional hyperbolic space (“Euclidean anti-de Sitter”) with curvature −1/L2 is g(H) = dr ⊗ dr + sinh(r/L)g(S), (1) where g(S) is the metric of the 4-sphere (“Euclidean deSitter”) of curvature +1/L. (In an attempt to avoid confusion, we shall throughout this work indicate the dimensions of Euclidean spaces by superscripts, and those of Lorentzian spaces by subscripts.) Thus, the part of H which is covered by this coordinate system is just the warped product of (0, ∞) with S. That is, warping can change the sign of the cosmological constant from positive to negative. Notice that the curvature of an S slice of H at r = c is +1/[Lsinh(c/L)], which can be made very small for a suitable choice of c; thus an S