On the non-integrability of a class of hamiltonian cosmological models

Abstract

The method of Morales and Ramis determines whether a given Hamiltonian system is non-integrable. We apply this method to Friedmann Robertson Walker models with a self-interacting scalar fi eld and cosmological constant. It is shown that, with the exception of a set of measure zero, these models are non-integrable. Our results complement those of Helmi and Vucetich who used the Painlevé property to fi nd integrable models within this class.

About the non-integrability in the Friedmann-Robertson-Walker cosmological model

“…, N ∈ IN Furthermore, numerical evidence of integrability is illustrated in [5] when λ = Λ = −m = −1 and when λ = Λ = − m 2 3 . This paper is a continuation of [5].…”

“…Although it does not describe the real universe in an essential way because of too many symmetries, it remains a fundamental model. We consider the Friedmann-Robertson-Walker (FRW) universe ( [5]) where the metric takes the form…”

“…The dynamics is described by Raychauduri equation and Klein-Gordon equation ( [7]) which derive from the Hamiltonian ( [5], [7])…”

“…In [5], Coehlo, Skea and Stuchi use the Morales-Ramis theory and the Kovacic algorithm (which solves second order linear differential equations) to show that the FRW model is not completely integrable except eventually when λ and Λ satisfy the conditions…”

“…

We study the non integrability of the Friedmann-Robertson-Walker cosmological model, in continuation of the work [5] of Coehlo, Skea and Stuchi. Using Morales-Ramis theorem ([10]) and applying a practical nonintegrability criterion deduced from it, we find that the system is not completely integrable for almost all values of the parameters λ and Λ, which was already proved by the authors of [5] applying Kovacic's algorithm.

…”