New formalism to define vacuum states for scalar fields in curved space-times

TL;DR

This paper introduces a novel geometric formalism for defining vacuum states of scalar fields in curved space-times, ensuring stability and minimal particle creation even in non-adiabatic regimes.

Contribution

It presents a new geometrical approach mapping phase space dynamics to hyperbolic space, enabling the definition of stable vacua beyond traditional adiabatic methods.

Findings

Stable vacuum states are identified in both adiabatic and non-adiabatic regimes.

The formalism reproduces the Bunch-Davies vacuum in de Sitter space.

A new vacuum state is proposed for cosmological bouncing models.

Abstract

The problem of finding a vacuum definition for a single quantum field in curved space-times is discussed under a new geometrical perspective. The phase space dynamics of the quantum field modes are mapped to curves in a 2-dimensional hyperbolic metric space, in which distances between neighbor points are shown to be proportional to the Bogoliubov coefficients associated with their corresponding mode solutions in phase space. The vacuum state for each mode is then defined as the unique trajectory from which all mapped phase space solutions move within thin annular regions around it. This property implies the stability of the vacuum state: solutions evolved from a point in this trajectory stay close to it as both evolve, and the particle creation is therefore minimized. The new approach is applied to the well-known cases of the time-independent dynamics, where the solutions draw circles around this curve, and when the adiabatic approximation is valid. The analysis is then extended to time-dependent cases in which the adiabatic approximation is not applicable, in the super-Hubble or low-frequency regimes. It is shown that stability trajectories can also be found in these situations, and stable quantum vacua can be obtained. This new formalism is applied to two situations: de Sitter space, where the Bunch-Davies vacuum is obtained in a completely different manner through an analysis in the super-Hubble regime, and in the context of cosmological bouncing models, in which the contracting phase is dominated by a cosmological constant in the asymptotic past. A new vacuum state for cosmological perturbations is proposed in this situation.