Astron. Astrophys. 353, 1-9 (2000) 3. Classification of inflationary solutions with const and k = 1In 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 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 . 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 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.
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 |