Abstract
We analyse spatially homogenous cosmological models of locally rotationally symmetric Bianchi type III with a massive scalar field as matter model. Our main result concerns the future asymptotics of these spacetimes and gives the dominant time behaviour of the metric and the scalar field for all solutions for late times. This metric is forever expanding in all directions, however, in one spatial direction only at a logarithmic rate, while at a power-law rate in the other two. Although the energy density goes to zero, it is matter dominated in the sense that the metric components differ qualitatively from the corresponding vacuum future asymptotics. Our results rely on a conjecture for which we give strong analytical and numerical support. For this we apply methods from the theory of averaging in nonlinear dynamical systems. This allows us to control the oscillations entering the system through the scalar field by the Klein–Gordon equation in a perturbative approach.
Highlights
Homogenous cosmologies have been investigated in vacuum [21] and with various matter models such as perfect fluids [8, 25], collisionless kinetic gases (Vlasov matter) [4, 18, 19] magnetic fields or scalar fields [8, 19], in particular with a focus on their asymptotic dynamics and stability
We present our main results in Subsection 5.4 which give the dominant future asymptotic behaviour of locally rotationally symmetric (LRS) Bianchi III Einstein-KleinGordon solutions; cf theorems 1 and 2
Our main results give the behaviour of LRS Bianchi III Einstein-Klein-Gordon solutions for large times in terms of the shear variable and energy density (Theorem 1) and in terms of the metric components (Theorem 2)
Summary
Homogenous cosmologies have been investigated in vacuum [21] and with various matter models such as perfect fluids [8, 25], collisionless kinetic gases (Vlasov matter) [4, 18, 19] (recently [5, 9, 13]) magnetic fields or scalar fields [8, 19] (recently [10, 12, 27]), in particular with a focus on their asymptotic dynamics and stability. We give the dominant late time behaviour of the Klein-Gordon field (Theorem 2).The premises of the main theorems contain the assumption that a conjecture holds (Conjecture 1) For the latter we give strong analytical and numerical support. We mention that one of our motivations to consider LRS Bianchi III Einstein-Klein-Gordon cosmologies was the result of [17] which concerns EinsteinVlasov cosmologies with massive particles within the same symmetry class. Under the assumption that the conjecture holds, we derive our main results in Section 5: the future asymptotics of LRS Bianchi III Einstein-Klein-Gordon cosmologies; cf theorems 1 and 2.
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