Spacetime Foam and the Cosmological Constant

In the saddle point approximation, the Euclidean path integral for quantum gravity closely resembles a thermodynamic partition function, with the cosmological constant Λ playing the role of temperature and the “density of topologies” acting as an effective density of states. For Λ < 0, the density of topologies grows superexponentially, and the sum over topologies diverges. In thermodynamics, such a divergence can signal the existence of a maximum temperature. The same may be true in quantum gravity: the effective cosmological constant may be driven to zero by a rapid rise in the density of topologies. PACS numbers: 04.60.Gw, 98.80.Hw email: [email protected] The cosmological constant Λ—in modern language, the energy density of the vacuum—is observed to be less than 10−47GeV, or 10−120 in Planck units. The cosmological constant problem [1,2], the problem of explaining the smallness of this number, is one of the central puzzles of modern physics. A natural guess is that some symmetry forces Λ to vanish, but the two obvious candidates, supersymmetry and conformal symmetry, are both badly broken. One can, of course, set Λ to zero by fiat, but this requires fine-tuning over a vast range of energies, and is in any case time-dependent, since phase transitions in the early universe can change the value of Λ. One can search for dynamical mechanisms to relax the cosmological constant to zero, but such attempts typically involve the implicit use of conformal invariance, and fail when the symmetry is broken [1]. This leaves quantum gravity as a tempting place to look for an explanation. Perhaps the most intriguing proposal to date has been Coleman’s wormhole model [3], in which topological fluctuations of spacetime induce effective nonlocal interactions that smear Λ into a probabilistic distribution peaked sharply at zero. The proposal presented in this paper is similar in spirit to Coleman’s, but different in detail: I consider a different set of topologies, with metrics that (unlike Coleman’s) are exact saddle points of the functional integral, and I interpret the resulting partition function rather differently. In particular, I will argue that a rapidly growing density of topologies may drive the cosmological constant to zero, as processes that could increase |Λ| instead merely produce more complicated “spacetime foam.” 1. The Euclidean Gravitational Partition Function I shall work in Euclidean quantum gravity, that is, quantum gravity “analytically continued” to Riemannian (positive-definite) metrics, since this seems to be the most natural setting in which to consider fluctuations of spacetime topology. The partition function for the volume canonical ensemble is [4, 5]