Evolution of the density parameter in inflationary cosmology reexamined

The evolution of the cosmological density parameter is examined with particular reference to the inflationary phase of the early Universe. The standard treatments of the Friedmann-Lem\ᵃitre equations as a dynamical system are considerably extended, and are generalized to allow the presence of an arbitrary mixture of perfect fluids with different equations of state. Phase-plane diagrams are constructed which show the evolution of the density parameter with the expansion of the Universe in both inflationary and noninflationary cases. More detailed models are constructed from sequences of simple models, and the relation between the results presented here and those previously obtained is shown explicitly, using analytical and graphical methods. The treatment is then extended to deal with the inflationary case directly, namely, when the energy density is dominated by a scalar inflaton field. It is shown in this case that a number of special regions of parameter space can be projected onto plane systems so as to permit a useful phase-plane representation, and some of these cases are displayed graphically.