4.1 Spatially-homogeneous spacetimes

The only spatially-homogeneous spacetimes admitting a compact Cauchy surface are the Bianchi types I, IX and the Kantowski-Sachs model; to allow for cosmological solutions with more general symmetry types, it is enough to replace the condition that the spacetime is spatially homogeneous, with the condition that the universal covering of spacetime is spatially homogeneous. Spacetimes with this property are called locally spatially homogeneous and these include, in addition, the Bianchi types II, III, V, VI0, VII0, and VIII.

One of the first studies on the Einstein–Vlasov system for spatially-homogeneous spacetimes is the work [152] by Rendall. He chooses a Gaussian time coordinate and investigates the maximal range of this time coordinate for solutions evolving from homogeneous data. For Bianchi IX and for Kantowski–Sachs spacetimes he finds that the range is finite and that there is a curvature singularity in both the past and the future time directions. For the other Bianchi types there is a curvature singularity in the past, and to the future spacetime is causally geodesically complete. In particular, strong cosmic censorship holds in these cases.

Although the questions on curvature singularities and geodesic completeness are very important, it is also desirable to have more detailed information on the asymptotic behavior of the solutions, and, in particular, to understand in which situations the choice of matter model is essential for the asymptotics.

In recent years several studies on the Einstein–Vlasov system for spatially locally homogeneous spacetimes have been carried out with the goal to obtain a deeper understanding of the asymptotic structure of the solutions. Roughly, these investigations can be divided into two cases: (i) studies on non-locally rotationally symmetric (non-LRS) Bianchi I models and (ii) studies of LRS Bianchi models.

In case (i) Rendall shows in [153Jump To The Next Citation Point] that solutions converge to dust solutions for late times. Under the additional assumption of small initial data this result is extended by Nungesser [127], who gives the rate of convergence of the involved quantities. In [153] Rendall also raises the question of the existance of solutions with complicated oscillatory behavior towards the initial singularity may exist for Vlasov matter, in contrast to perfect fluid matter. Note that for a perfect fluid the pressure is isotropic, whereas for Vlasov matter the pressure may be anisotropic, and this fact could be sufficient to drastically change the dynamics. This question is answered in [93Jump To The Next Citation Point], where the existence of a heteroclinic network is established as a possible asymptotic state. This implies a complicated oscillating behavior, which differs from the dynamics of perfect fluid solutions. The results in [93] were then put in a more general context by Calogero and Heinzle [46], where quite general anisotropic matter models are considered.

In case (ii) the asymptotic behaviour of solutions has been analyzed in [159Jump To The Next Citation Point, 160Jump To The Next Citation Point, 48Jump To The Next Citation Point, 47Jump To The Next Citation Point]. In [159], the case of massless particles is considered, whereas the massive case is studied in [160Jump To The Next Citation Point]. Both the nature of the initial singularity and the phase of unlimited expansion are analyzed. The main concern in these two works is the behavior of Bianchi models I, II, and III. The authors compare their solutions with the solutions to the corresponding perfect fluid models. A general conclusion is that the choice of matter model is very important since, for all symmetry classes studied, there are differences between the collision-less model and a perfect fluid model, both regarding the initial singularity and the expanding phase. The most striking example is for the Bianchi II models, where they find persistent oscillatory behavior near the singularity, which is quite different from the known behavior of Bianchi type II perfect fluid models. In [160] it is also shown that solutions for massive particles are asymptotic to solutions with massless particles near the initial singularity. For Bianchi I and II, it is also proven that solutions with massive particles are asymptotic to dust solutions at late times. It is conjectured that the same also holds true for Bianchi III. This problem is then settled by Rendall in [157]. The investigation [48] concerns a large class of anisotropic matter models, and, in particular, it is shown that solutions of the Einstein–Vlasov system with massless particles oscillate in the limit towards the past singularity for Bianchi IX models. This result is extended to the massive case in [47].

Before finishing this section we mention two other investigations on homogeneous models with Vlasov matter. In [106] Lee considers the homogeneous spacetimes with a cosmological constant for all Bianchi models except Bianchi type IX. She shows global existence as well as future causal geodesic completeness. She also obtains the time decay of the components of the energy momentum tensor as t → ∞, and she shows that spacetime is asymptotically dust-like. Anguige [28] studies the conformal Einstein–Vlasov system for massless particles, which admit an isotropic singularity. He shows that the Cauchy problem is well posed with data consisting of the limiting density function at the singularity.


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