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

• Applicant: Professor Dr. Hanno Gottschalk
• Subject Area: Mathematics
• Term: from 2015 to 2018
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 279133405

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.

• DFG Programme: Research Grants