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

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3. Classification of inflationary solutions with [FORMULA] const and k = 1

In this section it is shown how the inflationary branch of the soft bang model presented in this paper fits into the framework of general inflationary solutions with [FORMULA] = const (or [FORMULA] const equivalently) for [FORMULA]. With [FORMULA], [FORMULA] and [FORMULA] Eq. (5) becomes

[EQUATION]

with

[EQUATION]

1. Case [FORMULA]

This is the matter free case of the IR-model or the BP-model resp. According to (18), [FORMULA] is restricted to values [FORMULA] with [FORMULA], since [FORMULA]. From (19) one obtains

[EQUATION]

with [FORMULA] for [FORMULA] according to (18). In the solution of the BP-model, [FORMULA] comes from [FORMULA] for [FORMULA], decreases until a minimum value [FORMULA] is reached, and then turns around to increasing values. This solution is obtained from (18) for [FORMULA].

On the other hand, the solution of the IR-model is obtained for [FORMULA], starting at [FORMULA] with [FORMULA] and [FORMULA].

According to (20) [FORMULA] for

[EQUATION]

and thus the following classification is obtained:

[EQUATION]

2. Case [FORMULA]

Fig. 2 shows the potential [FORMULA] with [FORMULA] for three different kinds of solution together with its shape for [FORMULA]. For [FORMULA] it has a maximum

[EQUATION]

a) For [FORMULA] or

[EQUATION]

there is an unstable equilibrium point at [FORMULA], and the soft bang solution of this paper is obtained. The curve given by (23) is shown in Fig. 3.

[FIGURE] Fig. 2. Potential [FORMULA] for the cases [FORMULA] and [FORMULA]. In the case [FORMULA], curve a) applies for[FORMULA], b) for [FORMULA] and c) for [FORMULA].

[FIGURE] Fig. 3. Diagram [FORMULA] versus [FORMULA] with location of the different kinds of inflation solutions with [FORMULA]. The curve soft bang corresponds to (23) and is the location of soft bang solutions, the upper bound of the shaded area corresponds to (25). BP-model solutions are located on the [FORMULA] axis in the range from 0 to [FORMULA], IR-model solutions in the range [FORMULA]. In the shaded area generalized big bounce solutions are obtained, in the area above it and below the soft bang curve generalized IR-solutions. The region above the soft bang curve is the location of big bang solutions.

b) For [FORMULA] or [FORMULA], big bounce solutions of the same type as in the BP-model are possible as well as solutions of the type used in the IR-model, the difference being that a matter and/or radiation density [FORMULA] coexists with the vacuum energy density [FORMULA]. Since for all solutions [FORMULA], it follows that [FORMULA]. Now, [FORMULA] is restricted to [FORMULA] (see Fig. 2) with

[EQUATION]

obtained from [FORMULA], and therefore

[EQUATION]

Generalized big bounce solutions of the type considered in the BP-model are obtained for [FORMULA], and generalized solutions of the IR-model type for [FORMULA]. The boundary between the two is given by [FORMULA] which with (24) leads to the boundary equation

[EQUATION]

represented in Fig. 3 by the upper boundary of the shaded area.

c) For [FORMULA] or [FORMULA] finally, big bang solutions with [FORMULA] are obtained with [FORMULA] for [FORMULA].

Fig. 3 shows where the different kinds of inflation solutions with [FORMULA] are located in a [FORMULA] versus [FORMULA] diagram, the definitions [FORMULA], [FORMULA] and [FORMULA] being used. The boundary curves (21) and (25) don't intersect but only touch at

[EQUATION]

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© European Southern Observatory (ESO) 2000

Online publication: December 8, 1999
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