4 The Three Stepping Stones of `Dark Matter' Production:
Formulation of the Mechanism

In order to place our previous discussion in the right perspective and partly to justify the more technical approach in the following subsections, let us consider the inflationary idea that the early phase of the exponential expansion of the universe inflated a microscopic volume of space to a size much larger that the presently observable part of the universe; this idea can be formulated within the framework of General Relativity as a special case of ``Classical Bubble Dynamics'' (CBD), i.e., the study of the evolution of a vacuum bubble in the presence of gravity [10]. In our own formulation of CBD, inflation is driven by a gauge field $A_{\mu\nu\rho}(x)$ which is equivalent to a cosmological constant [11], and the boundary effects in CBD, completely similar to those discussed in the previous section, constitute the precise mechanism which extracts dark matter from the self-energy of $A_{\mu\nu\rho}$.
In short, how does that process take place? The following properties of $A_{\mu\nu\rho}$ constitute the crux of the boundary mechanism in the inflation-axion scenario:
a) When massless, $A_{\mu\nu\rho}$ represents `` dark stuff '' by definition, since in (3+1)-dimensions $A_{\mu\nu\rho}$ does not possess radiative degrees of freedom. In fact, the field strength $F_{\mu\nu\rho\sigma}\equiv\nabla_{[ \mu} A_{\nu\rho\sigma]}= \partial_{[ \mu} A_{\nu\rho\sigma ]}$, as a solution of the classical field equation, is simply a constant disguised as a gauge field. This property, even though peculiar, is not new in field theory: it is shared by all $d$-potential forms in $(d+1)$-spacetime dimensions. For instance in two dimensions, $F_{\mu\nu}=\partial_{[ \mu}A_{\nu]}=\epsilon_{\mu\nu} \Lambda$ while in four dimensions, $F_{\mu\nu\rho\sigma}=\epsilon_{\mu\nu\rho\sigma} f$, and $f$ represents a constant background field in both cases by virtue of the field equations. What is then the meaning of ``$f$''? As a gauge field, $A_{\mu\nu\rho}$ is endowed with an energy momentum tensor and thus it couples to gravity [15]: the resulting equations are Einstein's equations with the cosmological term $\Lambda=4 \pi G f^2$. For this reason we call $A_{\mu\nu\rho}$ the ``cosmological field''. This alternative interpretation of the cosmological constant can be traced back to Ref. [15] and its application to the inflationary scenario in Ref. [11]; it will be discussed in more detail in the following subsection.
b) If the cosmological field acquires a mass, then it describes massive pseudoscalar particles, in contrast with the usual Higgs mechanism. Indeed, in the massive case the free field equation for $A_{\mu\nu\rho}$

$$\partial_\lambda \partial^{ [ \lambda} A^{\mu\nu\rho ]}+m^2 A^{\mu\nu\rho}=0 , \quad \Longrightarrow \partial_\mu A^{\mu\nu\rho}=0$$ (27)

imposes the divergence free constraint on the four components of $A_{\mu\nu\rho}$ leaving only one propagating degree of freedom. In other words, the introduction of a mass term ``excites'' a dynamical (pseudoscalar) particle of matter out of the cosmological energy background.
c) Evidently, the transition from case a) (massless, non dynamical field) to case b) (massive propagating particles) requires a physical mechanism for its enactment. Here is where the idea of topological symmetry breaking and the concomitant rearrangement of gauge symmetry come into play. We hasten to say here, and expand our discussion in the following subsection, that the cosmological field $A_{\mu\nu\rho}$ does not interact directly with the ordinary matter fields that represent point-like particles. Rather, $A_{\mu\nu\rho}$ is the ``gauge partner'' of relativistic closed membranes, or bubbles, in the sense that it mediates the interaction between surface elements according to the same general principle of gauge invariance which dictates the coupling of point charges to vector gauge bosons, or the coupling of Kalb-Ramond potentials to elementary string-like objects. Clearly, this type of coupling to relativistic membranes as fundamental extended objects, is a crucial assumption of the whole mechanism of mass generation advocated in this paper.