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 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:
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