Dynamics of Anisotropic Universes

We present a general study of the dynamical properties of Anisotropic Bianchi Universes in the context of Einstein General Relativity. Integrability results using Kovalevskaya exponents are reported and connected to general knowledge about Bianchi dynamics. Finally, dynamics toward singularity in Bianchi type viii and ix universes are showed to be equivalent in some precise sence. 1 Homogeneous Universe and Bianchi models Considering the usual synchronous frame of General Relativity 1 ds = gμν dx μ dx = g̃ij dx i dx − dt. A Universe is said homogeneous when there exists an isometry group wich preserves the infinitesimal spacial lenght dl = g̃ij dx i dx A characterization of the isometry group is possible writing structure constants C c ab = ( ∂ie c j − ∂jei ) ea e i b (1) where dx = ej dy j . Constants C c ab are tensorial, low components are antisymetrics, and follow the Jacobi rule : C e ab C d ec + C e bcC d ea + C e caC d eb = 0 (2) Decomposing C c ab = εabd N dc + δ b Aa − δ a Ab , where N is a symetric tensor, one can show that equivalence classes of homogeneous universes are equivalence classes of N with NAb = 0. Without less of generality, the symetry of N ab allows us to write N =   n1 0 0 0 n2 0 0 0 n3   and Ab = [a, 0, 0] (3) without contrary indications greek indexes run from 0 to 3, latin indexes from 1 to 3, metric signature is (−,+,+,+), ε and δ are respectively the completely antisymetric LeviCevita tensor and the Kronecker symbol 1 Models split then into Class A with a = 0 and Class B with a 6= 0 and can be arranged in the well known Bianchi models n1 n2 n3 a Model 0 is a triple eigenvalue of N 0 0 0 0 BI 0 0 0 ∀ BV 0 is a double eigenvalue of N 1 0 0 0 BII 0 1 0 ∀ BIV 0 is a simple eigenvalue of N 1 1 0 0 BVIIo 0 1 1 ∀ BVIIa 1 −1 0 0 BVIo 0 1 −1 6= 1 BVIa 0 1 −1 1 BIII 0 is not an eigenvalue of N 1 1 1 0 BIX 1 1 −1 0 BVIII 2 Einstein Equations Following [1], one writes ds = γ (τ)ωiωj−N2 (τ) dτ with γ (τ) = diag [ e1, e2, e3 ] and dt = N (τ) dτ . The so called invariant differential forms basis ω are linear combinations of dxi with exponential or trigonometric coefficient in xi (See [1]). Finally, τ and N (τ) are respectivelly the conformal time and the lapse function. 2.1 BKL Formalism This formalism was introduced in the 70’s by [3]. Filling Universe by a barotropic fluid with pressure P and energy density ρ such that P = (Γ− 1)ρ, taking N (τ) = V 2 = e123 for the lapse function, some algebra then gives from Einstein equations, the equations for the dynamics of Bianchi Universes