Isotropization of Bianchi class A models with curvature for a minimally coupled scalar tensor theory

Abstract

We look for necessary isotropization conditions of Bianchi class A models with curvature in the presence of a massive and minimally coupled scalar field when a function ℓ of the scalar field tends to a constant, diverges monotonically or with sufficiently small oscillations. Isotropization leads the metric functions to tend to a power or exponential law of the proper time t and the potential, respectively, to vanish as t−2 or to a constant. Moreover, isotropization always requires late-time accelerated expansion and flatness of the universe.

Isotropization of Bianchi class A models with a minimally coupled scalar field and a perfect fluid

“…When isotropy occurs with k → 0, we have thus Ω m → 0, U ∝ p φ − ρ φ > ρ m and the results are the same as in [15] where no perfect fluid is present: Considering the limit near isotropy of the Hamiltonian equation forφ rewritten with the normalised variables (see appendice), we deduce that the scalar field asymptotically behaves as the limit of the solution foṙ…”

“…As instance, when we look for x ± asymptotical behaviours, we need to calculate exp( ℓ 2 dΩ). As explains in the section 2.2.2, we have shown in [15] that near the isotropic state, ℓ 2 tends to a constant ℓ 0 (vanishing or not). Then, in our calculation, we have assumed that asymptotically when Ω → −∞, exp( ℓ 2 dΩ) → exp(ℓ 2 0 Ω) but this is true only if ℓ 2 tends sufficiently fast to its constant equilibrium value.…”

“…In this paper, we will study the class 1 isotropisation. In the two next subsections, we briefly recall the results we obtained in [15] without a perfect fluid and then discuss the assumptions we have made related to their stability.…”

Bianchi type IX asymptotical behaviours with a massive scalar field: chaos strikes back

“…The third one has been studied in [18,19,20,21] from the Bianchi models isotropisation point of view. In [12], the case with U = m 2 φ 2 was studied.…”

Scalar fields properties for flat galactic rotation curves

“…The limit (18) is doubly important. Firstly, in (Fay 2001(Fay , 2003Fay & Luminet 2003) it has been shown that a necessary condition for isotropisation of Bianchi models was ψU 2 ψ ωU 2 → 2 , being a constant in a close interval depending on the presence of curvature and perfect fluid. Consequently, galactic scalar field properties for large r could match a cosmological scalar field present in the entire Universe which would allow for its isotropisation.…”