Linear Stability of Semiclassical Theories of Gravity

The linearization of semiclassical theories of gravity is investi- gated in a toy model, consisting of a quantum scalar field in interaction with a second classical scalar field which plays the role of a classical back- ground. This toy model mimics also the evolution induced by semiclassical Einstein equations, such as the one which describes the early universe in the cosmological case. The equations governing the dynamics of linear perturbations around simple exact solutions of this toy model are ana- lyzed by constructing the corresponding retarded fundamental solutions, and by discussing the corresponding initial value problem. It is shown that, if the quantum field which drives the back-reaction to the classical background is massive, then there are choices of the renormalization pa- rameters for which the linear perturbations with compact spatial support decay polynomially in time for large times, thus indicating stability of the underlying semiclassical solution.