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Astron. Astrophys. 353, 1-9 (2000)

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6. Discussion and summary

In the present model all kinds of singularities (e.g. of the expansion velocity, the expansion rate, or [FORMULA] for [FORMULA] as in the BP-model) are avoided and no necessity arises for employing a theory of quantum gravity. For [FORMULA] the (closed) universe is a tiny micro-universe in a classical static state. Its expansion is triggered by an instability and starts quite slowly at the velocity [FORMULA] and expansion rate [FORMULA]. Only much later it gains appreciably until it becomes exponentially inflating and finally reaches the later expansion rate of an inflationary big bang universe. It is an advantage of the present model that the universe is not "born" with an unexplained and extreme expansion rate like in most big bang models, but that the observed expansion can be explained as the consequence of an instability in its far past.

It should be noted that the dynamical solution describing the departure from the unstable equilibrium obtained in this paper is quite different from the well known Eddington-Lemaître solution of the Einstein-Lemaître equations that describes the departure from Einstein's static solution. While it has been shown by Börner & Ehlers (1988) and Ehlers & Rindler (1989) that for [FORMULA] the latter violates the maximum red-shift constraint, the present model is in good agreement with this constraint and other observations, as was shown in Sect. 5.

The coexistence of a quantum field of energy density [FORMULA] with some sort of primordial relativistic matter and/or radiation is an essential ingredient of the present model that may be critically considered and certainly needs discussion. The fact that a similar coexistence is assumed in big bang models with inflation may be invoked in support, but it may be a weak argument in view of opposing arguments raised by other authors. Priester et al. (1989) emphasize that a quantum vacuum state of the universe is the more natural choice for its primordial stage than a state in which elementary particles already have been present. Usually the origin of the vacuum energy density is assumed to be either the Higgs field that was introduced in elementary particle physics in order to explain the mass of elementary particles through interactions, or some other quantum field or other causes like worm-holes etc. The question is whether a primordial quantum field can exist on its own as a precondition for massive particles to be formed later, or whether it in turn needs these particles for its own existence. When the venerable principle actio = reactio is invoked, the second view appears as the more natural one. If this is accepted, then still another feature of the model becomes plausible. At first glance an extreme fine tuning appears to be needed in order that the condition [FORMULA] is getting satisfied. However, in an equilibrium state equipartition between two interacting ingredients is the only natural constellation, and in this spirit the fact, that equipartition in the equilibrium state results from the the field equations, may even appear as a confirmation.

It is interesting to note that the condition for the existence of an unstable equilibrium state from which the universe evolves through instability is quite contrary for the classical equilibrium of the present model and the quantum equilibrium considered by Starobinsky (1980): in the classical case in addition to the quantum field the presence of particles is necessary while in the quantum case just the absence of particles is required.

In this paper, in agreement with its modern interpretation as the energy density of some quantum field, the cosmological constant is treated as a dynamical variable rather than a geometrical one. However, this treatment is rather crude because [FORMULA] is still kept constant for most of the time - at a very large value during early stages of the universe and at its present low value after the phase transition until today. Many models have been proposed coupling the decay of [FORMULA] to the time evolution of the universe or its scale factor S - a survey is presented in a paper by Overduin & Cooperstock (1998) -, and it should be possible to replace the instant transition assumed in this paper with one of these more refined transitions without major changes in the results. The validity of this assumption was demonstrated in Sect. 4 for one specific model.

A look at the evolution of [FORMULA] used for the calculations of this paper (Fig. 4) shows that the present model could contribute to a solution of the "coincidence problem" (see Zlatev et al. 1999). This is raised by the latest observations according to which [FORMULA] lies in the same range as the matter density [FORMULA] today. The coincidence problem consists in the fact that, according to many models, [FORMULA] and [FORMULA] start at very different values in the early universe and require an extreme fine tuning at that time in order to reach almost equal values at present. In the soft bang model of this paper the universe starts with [FORMULA], during the evolution of the universe there are times at which the two densities temporarily depart from each other, but today they are very close to each other again. It appears that the present model would provide a good starting point for developing a quintessence field with "tracking properties" (see Zlatev et al. 1999) - at least it appears to be in good agreement with the requirements of such a concept.

[FIGURE] Fig. 4. . Ratio [FORMULA] as function of [FORMULA] for the present model.

Conceptually it has been considered as a very satisfying property of big bang models that in them the universe does not have an eternal past but originated from some act of creation. In this sense the eternal past of the present model may appear as a conceptual disadvantage. However, the situation is not as bad as it appears. In physics time is a parameter that is used for ordering changes of states. However, when there are no changes then this order parameter loses its sense. In a very slowly changing situation it may therefore become more useful to consider the changes themselves as the order parameter that represents time instead of using an order parameter ordering no changes. In this sense the lifetime of the universe considered in this paper is not greater than that of big bang models because there has been even a smaller change from the original state to the present.

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© European Southern Observatory (ESO) 2000

Online publication: December 8, 1999