The goal of this paper is to analyze the effects of the matter fields in the evolution of the Bianchi IX cosmology close to the singularity. Although the dynamics of this model is very involved, asymptotically, as the singularity is approached, it can be well approximated as a succession of Bianchi I periods connected by quick bounces against potential walls. Moreover, in such limit, matter fields (excluding stiff matter) are known to be subdominant with respect to the anisotropies. Therefore, by performing an expansion around small volumes, and assuming that the matter contribution in this regime can be described as a barotropic perfect fluid with a linear equation of state, we obtain an approximate analytic solution of the dynamics. Then, we explicitly compute the form of the transition (Kasner) map that relates the pre- and postbounce Bianchi I periods including the leading matter effects. As an important conclusion, we observe that generically the presence of matter leads to a postbounce velocity with a lower deflection angle than in the vacuum case, and thus effectively increases the convex curvature of the potential walls. This effect may have important consequences in the chaotic nature of general relativity near spacelike singularities.

中文翻译：

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物质存在下 Bianchi IX 的渐近：近似 Kasner 图


本文的目标是分析物质场在靠近奇点的比安奇九世宇宙学演化中的影响。尽管这个模型的动力学非常复杂，但随着奇点的接近，它可以很好地近似为一连串的比安奇 I 周期，这些周期通过与潜在壁的快速反弹连接起来。此外，在这样的极限下，已知物质场（不包括硬质）在各向异性方面是次为主的。因此，通过围绕小体积进行展开，并假设该状态下的物质贡献可以描述为具有线性状态方程的气压完美流体，我们得到了动力学的近似解析解。然后，我们显式计算过渡 （Kasner） 图的形式，该映射与反弹前和反弹后的 Bianchi I 周期相关，包括主要物质效应。作为一个重要的结论，我们观察到，通常物质的存在会导致后反弹速度的偏转角比真空情况下的偏转角小，从而有效地增加了电位壁的凸曲率。这种效应可能会对广义相对论在类空间奇点附近的混沌性质产生重要影响。