Application of the Abel Equation of the 1st kindto an inflation analysis of non-exactly solvable cosmological modelsIn this paper we revisit the relationship between the Einstein-Friedman and the Abel equations to demonstrate how it might be applied to the inflationary analysis in a flat Friedman universe filled with a real-valued scalar field. The analysis is performed for three distinct cases of polynomial potentials. As a result of a numeric integration of Abel equation, the necessary and sufficient conditions for both slow-rolling and inflation proper are estimated with respect to the initial value of the field. In addition, the relationship between the slow-rolling condition and the inflation is ascertained.

In a recent article by J. Lidsey, an interesting link has been established between the Kortewegde Vries (KdV) equation and the equations describing cosmological inflation. Due to this link, the author was able to devise rather a simple way of construction of inflationary solutions. We here demonstrate that the technique developed therein is but a consequence of a linearizability of the original cosmological equations. Furthermore, we show the required linearized equation to be nothing else but a Schr¨odinger equation. To emphasize the importance of this fact, we provide the reader with a way to use this relationship for a construction of not just one but an entire family of exact solutions. In conclusion, we discuss the general possibility to involve the KdV equation involvement in cosmological dynamics and provide a particular example where this equation might arise.