Global integrability of cosmological scalar fields

Abstract

We investigate the Liouvillian integrability of Hamiltonian systems describing a universe filled with a scalar field (possibly complex). The tool used is the differential Galois group approach, as introduced by Morales-Ruiz and Ramis. The main result is that the generic systems with minimal coupling are non-integrable, although there still exist some values of parameters for which integrability remains undecided; the conformally coupled systems are only integrable in four known cases. We also draw a connection with the chaos present in such cosmological models, and the issues of the integrability restricted to the real domain.fields [23], as has chaotic dynamics [48].The first of our results is that minimally coupled fields are not integrable in the generic case. There are however special families of the system's parameters which leave the question open. We give the appropriate conditions in the concluding section.There are several physical reasons to study more than just minimally coupled fields. Early works on chaotic inflation found the coupling constant ξ small or negative [30] but some argue [26] that the paradigm of inflation should be generalised to the case with non-zero coupling constant which should not be fine-tuned close to zero, and WMAP observations seem to indicate non-negligible ξ.The coupling could be generated by quantum corrections [10,29], or from the renormalisation of the Klein-Gordon equation as described in [16]. The coupling constant should be fixed by particle physics of the matter composing the scalar field, for example the way ξ = 1/6 was found in the large N approximation to the Nambu-Jona-Lasimo model in [34]. Non-minimally coupled fields are also interesting in the context of description of the dark energy for which the ratio between the pressure and the energy density is less than −1. Such matter is called a phantom matter, and cannot be achieved by standard scalar fields [27].Conformally coupled fields were subject to more rigorous integrability analysis, as opposed to minimally coupled ones, thanks to the natural form of their Hamiltonian. As will be shown in the next section, the kinetic part is of natural form, albeit indefinite, and the potential is polynomial (in the case of real fields).Chaos has been studied in such fields by means of Lyapunov exponents, perturbative approach, breaking up of the KAM tori [17,11]. Also the Painlevé property [35] was employed as an indicator of the system's integrability.Ziglin proved that the system given by (20) is not meromorphically integrable when Λ = λ = 0 and k = 1 [67]. His methods were also used by Yoshida to homogeneous potentials which is the case for the system when k = 0 [61,62,63,64]. 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...

Periodic orbits and non-integrability in a cosmological scalar field

“…For a good introduction to the cosmological model here studied we suggest to the reader to look at the paper of Maciejewski et al [14] and references therein for a detailed deduction and implications about the importance of this model.…”

“…As was saw in [14] the general action (2) for conformally coupled scalar fields must includes the following part where the additional coupling to gravity through the Ricci scalar R, and a quartic potential term with constant λ are considered. After some algebraic manipulations, assuming that the constant angular momentum is null, and under the use of convenient variables the Hamiltonian associated to the action (3) assumes the form (4) H = H(q 1 , q 2 , p 1 , p 2 ) = 1 2 (−p 2 1 + p 2 2 ) + 1 2 k(−q 2 1 + q 2 2 ) + m 2 q 2 1 q 2 2 + 1 4 Λq 4 1 + λq 4 2 , with k ∈ {−1, 0, 1} K = k|K| is associated to the index of curvature of the space), λ, Λ, m 2 ∈ R. Notice that the kinetic part is of natural form, albeit indefinite, and the potential associated is a polynomial of degree four.…”

“…As usual the dot denotes derivative with respect to the independent variable t ∈ R, the time. According to [14] we name (5) the conformal coupled scalar field Hamiltonian systems with three parameters.…”

“…In [14] it is proved that the above two cases are the only meromorphically integrable cases of the Hamiltonian system (5) when m = 0. In fact, in order to prove the nonexistence of an additional meromorphic first integral on a non-zero Hamiltonian level the authors applied the Morales-Ramis theory (see [19] or [20]).…”

On the integrability of the $5$-dimensional Lorenz system for the gravity-wave activity

“…More precisely, we want to know what is the maximal number of functionally independent either global analytic or Darboux first integrals that system (1) can exhibit?. This question has been considered for many other differential equations and other classes of first integrals not necessarily analytic or Darboux; see for instance [6,7,10] and the references therein.…”