On the Degree of Generality of Inflation in Friedmann Cosmological Models with a Massive Scalar Field

Abstract

Friedmann cosmological models with a massive minimally coupled scalar field are investigated by numerical methods. It is indicated that the majority of the solutions undergo the inflationary stage in flat and open models. In a closed model the ratio of the number of solutions without adequate inflation to the number of the total solutions are obtained numerically. I ) One oL the questions concerning this model is a generality of the inflationary solutions. We investigate this problem studying classical solutions of a homogeneous model with a massive scalar field. Among all solutions in this model some are inflationary and the others are not. If it is found that a fraction of the inflationary solutions (the ratio of the number of the inflationary solutions to the total number of solutions in some measure) is large enough, it becomes very reasonable to take such solutions for the model of our universe. For the flat Friedmann-Robertson-Walker (FRW) models, this problem was treated in the paper by Belinsky, Grishchuk, Zel'dovich and Khalatnikov. 2 ) Analyz­ ing the behavior of trajectories in the phase space of this dynamical system, they have shown that a fraction of the non-inflationary solution is the order of m/mp, mp and m being the Planck mass and a mass of the scalar field respectively. In fact, in order for the density fluctuation generated during the inflation to be small, the ratio m/mp is restricted to be smalIsuch as m/mp~ 1O-5~ 10- 6 • 4 ) Therefore, at least for the flat FRW model, the overwhelming majority of the solutions are found to be inflationary. N ow, as for the open and closed FRW models, the problem was treated by the analytic method in the paper by Belinsky and Khalatnikov. 5 ) For the open model, the result is not so different from the flat model. However, for the closed model, the result is quite different: Due to the curvature effect, the expansion tends to turn into collapse. In the previous work, it was shown that the fra<;:tion of the non-inflationary solution is about 1/4 in some measure. In this paper, we investigate the evolution of the universe.modelsby an.explicit numerical computation of the phase space trajectories and have confirmed the.results of the previous work. The phase space for open and closed models is three dimen­ sional and the trajectories start from the two dimensional surface called Quantum