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

Abstract

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. Working on a level surface H = h with h = 0 and h = − 1 4λ and using the Morales-Ramis-Simo "higher variational" theory ([11]), we prove that the hamiltonian system cannot be integrable for particular values of λ among the exceptional values and that it is completely integrable in two special cases (λ = Λ = −m 2 and λ = Λ = −m 2 3 ). We conjecture that there is no other case of complete integrability and give detailed arguments towards this.Keywords: Hamiltonian systems; Integrability; Morales-Ramis-Simo Theorem; Computer algebra I. THE PROBLEMThe Friedmann-Robertson-Walker model ([7]) is a model which explains some features of the observed universe at the present time. 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 formwhere ν is the time; a(ν) is the scale factor, which means that if a distance is measured as d 0 today, then at any other instant, it is d 0 times a; k = 0, 1, −1 is the curvature of space-time and dΩ 2 is the distance element on a two-sphere. The dynamics is described by Raychauduri equation and Klein-Gordon equation ([7]) which derive from the Hamil-After the canonical transformation of variables p a = ip a and a = −ia (suggested from [4]), and assuming k = 1, we get the Hamiltonian:where a is the scale factor of the universe; φ is the scalar field with self-coupling constant λ and with mass m (which we will assume to be non zero); Λ is the cosmological constant. • to show that the variational equation, computed along a given particular solution of the hamiltonian system, can be seen as a direct sum of two Lamé equations (when λ = 0);• to find the result of non-integrability already proved in [5] when λ = − 2m 2 (n + 1)(n + 2), n ∈ IN or Λ = − 2m 2 (N + 1)(N + 2), N ∈ IN using a criterion of non integrability ([2, 3]) deduced from Morales-Ramis theorem ([10]);• to prove that the system is not completely integrable in the particular case

Global integrability of cosmological scalar fields

“…The second part of the above theorem can be strengthened to meromorphic first integrals, although not for all values of the parameters, as described in [12].…”

“…Later, Yoshida's results were sharpened by Morales-Ruiz and Ramis [49], and used by the present authors in [44] to obtain countable families of possibly integrable cases with some restrictions on λ and Λ. Also recently, more conditions for integrability have been given in [12], although only for a non-zero spatial curvature k and a generic value of energy, that is, when the particular solution is a non-degenerate elliptic function defined on a non-zero energy level.Our work shows, that the conjecture of that paper is in fact correct -as shown in Section 4 -the conformal system is only integrable in two cases (with the above assumptions). We go further than that and show that for a generic energy value, a spatially flat (k = 0) the universe is only integrable in four cases.…”

Non-integrability of the Anisotropic Stormer Problem and the Isosceles Three-Body Problem

“…In particular a generalization of this theory to higher order variational equations (developed quite recently by Morales-Ruiz, Ramis and Simó [24,25]) has already given results on the non-integrability of a case of the Henon-Heiles system [24,25], natural Hamiltonian systems with homogeneous potential of degree 3 or 4 [20,22], the motion of a satellite in circular orbit [6] and the Friedmann-Robertson-Walker cosmological model [7]. In our work this generalization was also needed in order to prove the nonintegrability of one case of the reduced ASP.…”

Periodic orbits and non-integrability in a cosmological scalar field

“…We have one parametric family of periodic solutions generated by the periodic solution (6) with r * = − 2h(m 2 +λ) 2m 2 +λ+Λ , ρ * = 2h(m 2 +Λ) 2m 2 +λ+Λ , α * = 0. Families (IV): Either h > 0 and (λ, Λ) ∈ R 2 ∪ R 4 , or h < 0 and (λ, Λ) ∈ Ω 2 ∪ Ω 4 .…”

“…Our results are on the non-integrability in the sense of Liouville-Arnold for any second first integral of class C 1 . In [6] and [8] also the problem of non-existence of any additional meromorphic first integral in the Hamiltonian system (5) was considered.…”