Application of the Abel equation of the 1st kind to inflation analysis of non-exactly solvable cosmological models

Abstract

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.

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“…It is also a strategy adopted in a recent paper by James Lidsey [2], where the author pursued an ambitious goal: to establish a new method of constructing general solutions for the universe that undergoes an inflationary expansion. The premise of the slow-rolling assumption is simple, for it assumes (correctly, as has been demonstrated in [7]) that the for the major part of the inflationary phase the superpotential W (φ) ≈ V (φ), that is,…”

“…Such a representation ends up being a versatile tool, particularly for analyzing the inflationary dynamics; for example, it is easy to see that for V ≥ 0 the inflation condition a/a > 0 transforms to the condition |y| > √ 3 (cf. [7]). The problem with this method is, of course, the fact that even in the form (2) the cosmological equations remain, generally speaking, nonintegrable (the list of all known integrable cases in [8] has only 11 entries, and 7 of them contains no external parameters at all)-which implies non-integrability of the corresponding Friedmann equations.…”

“…also solves (7). Now, since F (φ) = e λφ serves as an obvious special solution to (7), "enhancing" it by means of (8) yields a whole new family of solutions of (7), and therefore, of (5) (due to (6)):…”

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“…The key to the next step would be to introduce the desired power-law relationship, except we do it not for potential V (φ), but for superpotential W (φ) instead. Then, letting W (φ) = A 2 φ n and integrating (11) we end up with:…”

“…With the "band of opportunity" (2) being extremely narrow, the potential V (φ) inside of it remains practically constant, and one can estimate that V (φ) ∼ V (0) + V (0)φ. Therefore, ifφ decreases in time, one is free to use the slow-roll approximation [9]- [11] to get an estimate on the time t * which measures how long it will take from the current time t 0 till the beginning of the collapse. In particular, ifφ 0 =φ(t = t 0 ) and V (0) = 0, then t * ∼ t D (φ 0 /m p ) 2 , where t D ∼ ρ D −1/2 , and so we end up with t * ∼ 10 12 yr.…”