The Cosmological Semiclassical Einstein Equation as an Infinite-Dimensional Dynamical System

TL;DR

This paper establishes a new framework for solving the semiclassical Einstein equation in cosmology, accommodating full renormalization freedom and analyzing the evolution of quantum states as an infinite-dimensional dynamical system.

Contribution

It introduces a comprehensive approach that handles full renormalization freedom and develops a novel infinite-dimensional dynamical system formulation for the SCE.

Findings

Existence of maximal and global solutions for vacuum-like states.

Local solutions established for thermal-like states.

The equations do not exhibit Minkowski solution instability.

Abstract

We develop a comprehensive framework in which the existence of solutions to the semiclassical Einstein equation (SCE) in cosmological spacetimes is shown. Different from previous work on this subject, we do not restrict to the conformally coupled scalar field and we admit the full renormalization freedom. Based on a regularization procedure, which utilizes homogeneous distributions and is equivalent to Hadamard point-splitting, we obtain a reformulation of the evolution of the quantum state as an infinite-dimensional dynamical system with mathematical features that are distinct from the standard theory of infinite-dimensional dynamical systems (e.g., unbounded evolution operators). Nevertheless, applying methods closely related to Ovsyannikov's method, we show existence of maximal/global solutions to the SCE for vacuum-like states, and of local solutions for thermal-like states. Our equations do not show the instability of the Minkowski solution described by other authors.