5 Conclusions and Outlook

To the extent that the presence or absence of a boundary constitutes a topological property of a manifold, we may refer to the idea underlying the whole discussion in this paper as topological symmetry breaking. The effect of this new mechanism on the inflationary-axion scenario is apparent in our formulation of Bubble Dynamics, and was illustrated in four as well as in two spacetime dimensions. Indeed, the action functional of Bubble Dynamics can be defined in any number of dimensions as a generalization of the Einstein-Maxwell action for the dynamics of point charges on a Riemannian manifold. As a matter of fact, gravity plays no special role in it: even though we have formulated the model for a bubble in $(3+1)$ Minkowski spacetime, it can be extended to a generic $p$-brane embedded in a target space with $D=p+2$ dimensions. In four dimensions and under the assumption of spherical symmetry, the field equations of bubble dynamics are integrable [15]: the net physical result of the $A_{\mu\nu\rho}$ coupling to the membrane degree of freedom is the nucleation of a bubble whose boundary separates two vacuum phases characterized by two effective and distinct cosmological constants, one inside and one outside the bubble ¹¹.
The three-index representation of the cosmological field is the key to the whole formulation of topological symmetry breaking and self-consistent generation of mass. That field, we have argued, represents the ultimate source of energy in the bubble universe. But how does matter manage to `` bootstrap '' itself into existence out of that source of latent energy? Here is where the difference between the conventional ``cosmological constant'' and the cosmological field comes into play: the original cosmological constant introduced by Einstein plays a somewhat passive role, in that it is ``frozen'' within the Hilbert action of general relativity; the cosmological field, on the contrary, even though it represents a non dynamical gauge field, will interact with gravity and combine with a bubble single degree of freedom thereby acquiring mass as a consequence of topological symmetry breaking. In fact, there are some similarities with the Higgs mechanism which may help to clarify the boundary mechanism of mass production. For instance, the extended gauge symmetry (76), (77) represents the end result of a process that begins with the violation of gauge invariance due to the presence of a boundary. Then, the Kalb-Ramond field $B_{\mu\nu}$, prescribed by the Stueckelberg procedure, represents massless, spinless particles that play the role of Goldstone bosons. However, while in the usual Higgs mechanism the spin content of the gauge field is the same before and after the appearance of mass, a new effect occurs when the gauge field is $A_{\mu\nu\rho}$: when massless, $A_{\mu\nu\rho}$ carries no degrees of freedom, while equation (27) describes massive spin-0 particles in a representation which is dual to the familiar Proca representation of massive, spin-1 particles [24]. In the light of this formal analogy with the Higgs mechanism, axions are interpreted as massless spin-0 Goldstone bosons represented by a Kalb-Ramond field while the physical spectrum consists of massive spin-0 particles represented by the $\widehat A_{\mu\nu\rho}$-field. In this sense, topological symmetry breaking by the boundary has the same effect as the breaking of the Peccei-Quinn symmetry in the local standard model of particle physics.

We do not have at present a fully fledged quantum theory of bubble-dynamics even though we have taken several steps on the way to that formulation [7], [25]. However, if bubble dynamics in two dimensions is any guide, one can anticipate the main features of the quantum theory: as the volume of the bubble universe increases exponentially during the inflationary phase, so does the total (volume) energy of the interior ``De Sitter vacuum''. At least classically. Quantum mechanically there is a competitive effect which is best understood in terms of an analogous effect in (1+1)-dimensions. As we have argued in the previous section, the``volume'' within a one-dimensional bubble is the linear distance between the two end-point charges. As the distance increases, so does the potential energy between them. Quantum mechanically, however, it is energetically more favorable to polarize the vacuum through the process of pair creation [26], which we interpret as the nucleation of secondary bubbles out of the vacuum enclosed by the original bubble. The net physical result of this mechanism is the production of massive spin-0 particles [14]. The same mechanism can be lifted to $(3+1)$-dimensions and reinterpreted in the cosmological context: the $A_{\mu\nu\rho}$ field shares the same properties of the gauge potential $A_\mu$ in two dimensions and polarizes the vacuum via the formation of secondary bubbles. Consider now a spherical bubble and focus on the radial evolution alone. The intersection of any diameter with the bubble surface evolves precisely as a particle-anti particle pair in $(1+1)$-dimensions. However, since there is no preferred direction, the mechanism operates on concentric shells inside the original bubble. Remarkably, the final result is again the production of massive pseudoscalar particles in the bubble universe. However, while in two dimensions Goldstone bosons do not exist, in $(3+1)$-dimensions they do exist and have a direct bearing on the axion mass problem. Born out of the darkness of the cosmic vacuum, axions were invisible to begin with and remain invisible to the extent that they are `` eaten up '' by the cosmological field. According to this interpretation, one of the possible forms of dark matter in the universe, besides MACHO's, emerges as the necessary end product of a process, driven by the cosmic vacuum energy, according to which gauge invariance and vacuum decay conspire to extract massive particles out of the cosmological field of dark energy.