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

3. Classification of inflationary solutions with 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 = const (or const equivalently) for . With , and Eq. (5) becomes

with

1. Case

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

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

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

According to (20) for

and thus the following classification is obtained:

2. Case

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

a) For or

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

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

Fig. 3. Diagram versus with location of the different kinds of inflation solutions with . 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 axis in the range from 0 to , IR-model solutions in the range . 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 or , 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 coexists with the vacuum energy density . Since for all solutions , it follows that . Now, is restricted to (see Fig. 2) with

obtained from , and therefore

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

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

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

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

© European Southern Observatory (ESO) 2000

Online publication: December 8, 1999
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