On the initial conditions and solutions of the semiclassical Einstein equations in a cosmological scenario

TL;DR

This paper investigates the existence and behavior of solutions to the semiclassical Einstein equations with a quantum scalar field in cosmological spacetimes, showing solutions exist locally and discussing their dependence on quantum states.

Contribution

It provides a rigorous proof of local existence of solutions using an iteration scheme and analyzes the influence of quantum states on the solutions over time.

Findings

Solutions exist on small time intervals.

Quantum states with Hadamard property do not affect local solutions.

End-point behavior in expanding spacetimes is largely independent of initial quantum states.

Abstract

In this paper we shall discuss the backreaction of a massive quantum scalar field on the curvature, the latter treated as a classical field. Furthermore, we shall deal with this problem in the realm of cosmological spacetimes by analyzing the Einstein equations in a semiclassical fashion. More precisely, we shall show that, at least on small intervals of time, solutions for this interacting system exist. This result will be achieved providing an iteration scheme and showing that the series, obtained starting from the massless solution, converges in the appropriate Banach space. The quantum states with good ultraviolet behavior (Hadamard property), used in order to obtain the backreaction, will be completely determined by their form on the initial surface if chosen to be lightlike. Furthermore, on small intervals of time, they do not influence the behavior of the exact solution. On large intervals of time the situation is more complicated but, if the spacetime is expanding, we shall show that the end-point of the evolution does not depend strongly on the quantum state, because, in this limit, the expectation values of the matter fields responsible for the backreaction do not depend on the particular homogeneous Hadamard state at all. Finally, we shall comment on the interpretation of the semiclassical Einstein equations for this kind of problems. Although the fluctuations of the expectation values of pointlike fields diverge, if the spacetime and the quantum state have a large spatial symmetry and if we consider the smeared fields on regions of large spatial volume, they tend to vanish. Assuming this point of view the semiclassical Einstein equations become more reliable.