1 Introduction

The modern paradigms of physics are the standard Big Bang model of cosmology and the standard $SU(3)_{\rm C}\otimes SU(2)_{\rm L}\otimes U(1)_{\rm Y}$ model of the strong and electroweak interactions. During the past two decades both models have been refined with the addition of two key ingredients: inflation on the cosmological side [1] and axions as pseudo Goldstone bosons associated with the spontaneous breakdown of the Peccei-Quinn symmetry in particle physics [2]. Inflation requires the existence of dark matter and axions have long been candidates for cold dark matter. A further refinement of the standard models stems from a recent analysis of the cosmic microwave background [3] added to the data from high-redshift supernova observations [4]. Together, they seem to support the idea that the universe is flat and is currently expanding at an accelerated rate⁴. As a result, dark matter and the cosmological constant [5], or some form of dark energy, have become the essential components of the new inflationary scenario⁵. In this paper we wish to suggest that those two components are connected, in a rather fundamental way, by a new mechanism of symmetry re-arrangement that requires the creation of dark matter.
Cold dark matter, in axionic form, could be detected in an experiment capable of probing masses in the range $10^{-6}\sim 10^{-3}$eV. Where does that mass come from? The general consensus is that it comes from the quantum anomaly which violates the chiral $U(1)_{\rm PQ}$ symmetry, thereby evading Goldstone's theorem. However, the chiral anomaly is just one of at least two possible loopholes by which the existence of a Goldstone boson can be avoided. The second loophole is the Higgs mechanism. In its conventional formulation, the Higgs mechanism essentially converts a gauge field, i.e., a massless spin-1 field, into a massive vector field while preserving the value of the spin as well as gauge invariance, thereby ensuring the renormalizability of the theory. Even though this is the mechanism that generates the mass of all known elementary particles within the Standard Model of particle physics, it is clearly unsuitable to describe the cosmological situation envisaged above, namely the conversion of the constant vacuum energy into particles of matter. Thus we are led to ask: How does one connect a ``constant energy background'' (non dynamical by definition) into material particles that are invisible but dynamical? Our suggestion, in a nutshell, is as follows: first, turn the cosmological constant into a non dynamical gauge field, i.e., a gauge field with zero degrees of freedom; second, extract from that gauge field a massive spin-0 field according to the time honored procedure of symmetry breaking followed by restoration of gauge invariance. As we shall see, the new ingredients of that old procedure are: relativistic extended objects (membranes) and their gauge partners (antisymmetric tensor gauge fields).
The topological nature of the new mechanism and its mathematical formulation were discussed in a recent article in connection with the broad issue of electric-magnetic duality of p-branes [7]. As stated above, we are presently interested in applying the notion of ``topological symmetry breaking'' to the new inflationary scenario. Accordingly, we have organized the paper as follows:
In Section(2) we introduce the concept of topological symmetry breaking. In Section(3) we discuss the case of ``electrodynamics in two dimensions'', reinterpreted as ``bubble-dynamics in two dimensions'', as the simplest framework in which topological symmetry breaking can be implemented together with the generation of mass. In Section(4), the details of the new mechanism are illustrated in four spacetime dimensions. There we first outline the three main steps leading to the creation of dark matter, and then discuss each step in a separate subsection. Instrumental to the overall mechanism is the false vacuum decay rate through bubble nucleation from the vacuum which we calculate in real spacetime and compare with the corresponding computation in Euclidean space. Section(5) concludes the paper with a summary of our discussion and a commentary on the applicability of the new mass generation mechanism to the inflationary cosmological scenario.