N ON - INTEGRABILITY OF THE AXISYMMETRIC B IANCHI IX COSMOLOGICAL MODEL VIA D IFFERENTIAL G ALOIS T HEORY Primitivo ACOSTA-HUMÁNEZ Universidad Simón Bolívar, Barranquilla - Colombia Instituto Superior de Formación Docente Salomé Ureña Santiago de los Caballeros - Dominican Republic primitivo.acosta-humanez@isfodosu.edu.do Juan J. MORALES-RUIZ Depto. de Matemática Aplicada Universidad Politécnica de Madrid Madrid, Spain juan.morales-ruiz@upm.es Teresinha J. STUCHI Instituto de Física Universidade Federale de Rio de Janeiro Rio de Janeiro, Brazil tstuchi@if.ufjr.br ABSTRACT We investigate the integrability of an anisotropic universe with matter and cosmological constant formulated as Bianchi IX models. The presence of the cosmological constant causes the existence of a critical point in the finite part of the phase space. The separatrix associated to this Einstein’s static universe is entirely contained in an invariant isotropic plane forgetting the singularity at the origin. This invariant plane of isotropy is an integrable sub-space of the Taub type. In this paper we analyse the differential Galois group of the second order variational equations to this plane in order to apply the integrability theorem of the second author with Ramis and Simó. The main result is that the model is non-integrable by meromorphic functions. Keywords Bianchi IX models · cosmological constant · differential Galois theory · integrability · Poincaré sections. MSC 2010. 34M25; 35A22; 35C05; 92D25 Introduction Belinskii, Khalatnikov and Lifishitz [8] started the question of chaotic behaviour of general Bianchi IX models in Relativistic Cosmology. The interest in the chaoticity (or not) of Bianchi IX models has been mainly focused on the Mixmaster case. Vacuum Bianchi IX models with three scale factors was taken by Misner [17]). Also [9, 10] studied the three dimensional Bianchi IX model from the real dynamics point of view. The absence of an invariant (or topological) characterisation of chaos in the model, i.e., standard chaotic indicators such as Lyapunov exponents being coordinate dependent and therefore questionable [14] ,[12]. For discussions of the issue of chaotic dynamics on these models we refer to the works of Barrow [7]. Cornish and Levin [13] proposed to quantify chaos in the Mixmaster universe by calculating the dimensions of fractal basin boundaries in initial-conditions sets for the full dynamics. Calzeta [12] and El Hasi proved integrability under adiabatic approximation, but the tori breaked up without the adiabatic terms. We recall that the conjunction of a cosmological constant and anisotropy implies in the existence of a critical point, the Einstein universe, in the finite region of phase space. This point is linearly a saddle center that implies the existence of a centre manifold tangent to the linear periodic orbits around E. From this centre manifold the stable and unstable manifolds which are topologically cylinders carry the dynamics away and to the inflationary region. This was the object of the work of H.P. Oliveira et al. (see [22]) for the axisymmetric Bianchi IX model. This is the model we study in this paper. This problem was numerically shown non integrable; these cylinders, when the full dynamics is considered, arXiv:2102.10108v1 [math.DS] 19 Feb 2021