Von Neumann algebra description of inflationary cosmology

TL;DR

This paper explores the von Neumann algebra framework for inflationary quasi-de Sitter space, revealing its description as a Type II$_ty$ algebra due to unbounded entropy and fluctuations, contrasting with perfect dS space.

Contribution

It introduces a von Neumann algebra description of inflationary quasi-dS space, highlighting the role of energy flux fluctuations and their divergence in the semiclassical limit.

Findings

Quasi-dS space is described by Type II$_ty$ algebra.

Entropy in quasi-dS space is unbounded, unlike perfect dS.

Fluctuations in energy flux diverge as G approaches zero.

Abstract

We study the von Neumann algebra description of the inflationary quasi-de Sitter (dS) space. Unlike perfect dS space, quasi-dS space allows the nonzero energy flux across the horizon, which can be identified with the expectation value of the static time translation generator. Moreover, as a dS isometry associated with the static time translation is spontaneously broken, the fluctuation in time is accumulated, which induces the fluctuation in the energy flux. When the inflationary period is given by $(\epsilon_H H)^{-1}$ where $\epsilon_H$ is the slow-roll parameter measuring the increasing rate of the Hubble radius, both the energy flux and its fluctuation diverge in the $G \to 0$ limit. Taking the fluctuation in the energy flux and that in the observer's energy into account, we argue that the inflationary quasi-dS space is described by Type II$_\infty$ algebra. As the entropy is not bounded from above, this is different from Type II$_1$ description of perfect dS space in which the entropy is maximized by the maximal entanglement. We also show that our result is consistent with the observation that the von Neumann entropy for the density matrix reflecting the fluctuations above is interpreted as the generalized entropy.