Steve Carlip - Einstein manifolds, spacetime foam, and the cosmological constant

In Wheeler's ``spacetime foam'' model, the quantum mechanical state of the universe is a superposition of many different spacetime topologies. In a saddle point approximation for the partition function, each Einstein manifold M contributes a term of the form , where is the cosmological constant and is the volume of M with scalar curvature normalized to . For , one therefore naively expects the sum over topologies to be dominated by a few manifolds with small volumes. Contrary to this expectation, I show that the number of hyperbolic manifolds grows so fast with volume that the sum over topologies is dominated by arbitrarily large manifolds with arbitrarily complicated topologies. This phenomenon indicates an instability in the sum over topologies that may be relevant to the ``cosmological constant problem,'' the observation that the vacuum energy density of the universe is at least 120 orders of magnitude smaller than one would expect from quantum fluctuations at the Planck scale. Processes that would normally increase may instead merely drive the production of more and more complicated spacetime foam.