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Varieties of expanding universe

We apply some general theorems of Hardy and Fowler, which give the asymptotic behaviour of all ultimately monotonic solutions of first- and second-order polynomial differential equations, to the Einstein equations describing expanding universes. We determine the possible forms for the time evolution of the expansion scale factor of expanding universes. We also examine the possible monotonic asymptotes for scalar fields driving slow-roll inflation by applying these methods to the Hamilton - Jacobi equations for the scalar field evolution. The asymptotes that describe inflationary universes are identified with known solutions.