Solutions and stability of the semi-classical Einstein equation on Friedman-Robertson-Walker spacetimes - a phase space approach

本文提出用半经典爱因斯坦方程（SCE）方法研究量子化、相对论性物质与非自相互作用量子场之间的耦合。对于宇宙学弗里德曼-罗伯逊-沃克（FRW）时空，我们将提供这个方程作为（潜在隐式的）无限维量子场模式动力系统的初值公式。我们打算从共形耦合开始给出相空间的数学上严格的描述，其中对于特定初始条件集解的整体存在性已经建立，然后继续到一般耦合。特别地，我们将通过先验估计和Faedeo Galerkin方法研究一般耦合的局部解和全局解的存在性及其稳定性。对上述解的稳定性分析，特别强调无限维动力系统中的稳定和不稳定流形，将应用于动力学的不动点，如闵可夫斯基时空和接近时空奇点的渐近缩放行为（大爆炸）和晚时间（宇宙的命运）。我们还将研究非极大对称时空的推广。此外，将开发一个与SCE的物理和数学性质内在联系的导出方程组的数值求解器。

It is proposed to study the coupling of quantized, relativistic matter modeled as a non- selfinteracting quantum field to the gravitational field treated non-quantum mechanically following the semi-classical Einstein equation (SCE) approach. We will provide an initial value formulation for this equation as a (potentially implicit) infinite-dimensional dynamical system of quantum field modes for the case of cosmological Friedman-Robertson-Walker (FRW) spacetimes. We intend to give a mathematically rigorous description of the phase space starting with the conformally coupled case where global existence of solutions has already been estab- lished for specific sets of initial conditions and proceeding to general couplings. In particular, we will investigate existence of local and global solutions and their stability for general coupling via a priori estimates and the Faedeo Galerkin method. A stability analysis of the afore mentioned solutions, with a special emphasis on stable and unstable manifolds in infinite dimensional dynamical systems, will be applied to fixed points of the dynamics like Minkowski spacetime and asymptotic scaling behaviour close to spacetime singularities (big bang) and late times (fate of the universe). We will also study the generalization to non maximally symmetric space-times. Also, a numerical solver for the derived system of equations will be developed that is intrinsically linked to the physical and mathematical nature of the SCE.