The Semiclassical Einstein Equations in Cosmological Spacetimes

The semiclassical Einstein equation describes the backreaction of quantum matter fields on classical background spacetimes. This conference proceeding reviews some recent results obtained by N. Pinamonti, D. Siemssen and the author [Ann. Henri Poincaré 22 , 3965–4015, 2021] on the initial-value problem of the semiclassical Einstein equation coupled to a quantum, massive, scalar field with arbitrary coupling to the scalar curvature in cosmological spacetimes. The central issue of the problem arises from the fact that the linearized expectation value of the renormalized stress-energy tensor of the quantum matter field hides a nonlocal contribution depending on the highest derivative. This is encoded in an unbounded, tame operator which lose derivatives, and thus it prevents a direct analysis of the dynamical equation. The system can nevertheless be reformulated as an inverse problem, allowing one to isolate and invert the highest-derivative contribution. In this form, existence and uniqueness of solutions can be established using Banach fixed-point methods. This proceeding is based on the talk given by the author at the IQSA2025 Intermediate conference (Tropea, Italy).