Answering a Basic Objection to Bang/crunch Holography

The current cosmic acceleration does not imply that our Universe is basically de Sitterlike: in the first part of this work we argue that, by introducing matter into anti-de Sitter spacetime in a natural way, one may be able to account for the acceleration just as well. However, this leads to a Big Crunch, and the Euclidean versions of Bang/Crunch cosmologies have [apparently] disconnected conformal boundaries. As Maldacena and Maoz have recently stressed, this seems to contradict the holographic principle. In the second part we argue that this “double boundary problem” is a matter not of geometry but rather of how one chooses a conformal compactification: the appearance of disconnectedness can be regarded as a coordinate effect. With the kind of matter we have introduced here, namely a Euclidean axion, the underlying compact Euclidean manifold has an unexpectedly non-trivial topology: it is in fact one of the 75 possible underlying manifolds of flat compact four-dimensional Euclidean spaces. 1. de Sitter . . . or Anti-de Sitter? From the fact [1][2] that our Universe is accelerating, it is often deduced that our spacetime basically resembles de Sitter spacetime. But is this justified? There are in fact general reasons [3][4][5] for thinking that, despite appearances, our world is basically unlike de Sitter spacetime. For example, holography does not work very well for de Sitter-like spacetimes. Clearly de Sitter spacetime itself does not have a holographic Euclidean version, since its standard Euclidean version has no boundary [see however [6]], and its Lorentzian holography [7][8] leads to problems [9] which have only been incompletely resolved [10][11]. It is therefore important to note that a Universe which accelerates only temporarily — and we have no reason to believe that it does otherwise — need not be related to de Sitter spacetime at all: in fact, it can be modelled by a suitably modified version of anti-de Sitter spacetime. In view of all this, it is reasonable to ask: can we dispense with de Sitter spacetime altogether, and construct a complete cosmology using only anti-de Sitter physics? AdS4 itself is of course completely unacceptable as a cosmological model, since it does not expand, has no Big Bang, and is not globally hyperbolic [so that it cannot be understood as a FRW cosmology — it has a FRW coordinate system, but this only covers a small part of the spacetime]. All of these blemishes are removed, however, by introducing matter into AdS4. In fact, by introducing scalar matter into AdS4 [12] one can produce a globally hyperbolic spacetime which does have a Bang and also a temporary burst of acceleration. [See also [13].] Such spacetimes are therefore ideally suited to answering questions about the global structure of spacetimes which accelerate and yet are not asymptotically de Sitter, because the acceleration is only temporary. [For another approach to using AdS in cosmology, see [14].] The cosmological models obtained in this way underline a fundamental fact which deserves more attention than it has received: AdS4 is unstable to singularity formation. The introduction of any amount of any form of matter [other than vacuum energy itself] satisfying the Strong Energy Condition causes singularities to develop, producing a spacetime which is not a small perturbation of AdS4. [This follows from the HawkingPenrose cosmological singularity theorem [15]. See also [16] for a good recent discussion of this theorem. Note that singularities will usually form even if the matter does not satisfy the SEC.] Singular versions of AdS4 are therefore the generic spacetimes with negative cosmological constants, and so it is quite natural for an AdS4-derived spacetime to accommodate a Big Bang [and to be globally hyperbolic]. However, putting a Bang into AdS4 has a price: it almost inevitably produces a Crunch. This is because, as the Universe expands, it dilutes all forms of matter and radiation [except phantom matter [17], which we do not consider here] more rapidly than a cosmological constant, so the latter will become increasingly important as the expansion proceeds. The negative cosmological constant of AdS4 will inevitably halt the expansion and cause a contraction. If the Universe is still contracting when matter and radiation again become dominant, there will be a Crunch at the end just as there was a Bang at the beginning. Hence a quasi-realistic cosmological model arising from AdS4 has both a Crunch and a Bang. [For the basic theory of Crunch cosmologies, see [18].]