Cosmology as a Crossed Product

TL;DR

This paper introduces a novel quantum framework for inflationary cosmology using crossed product algebras, providing a model-independent way to derive inflationary parameters consistent with Planck data.

Contribution

It develops a new formalism based on crossed product algebras to describe inflationary cosmology quantum mechanically, linking algebraic structures to observable fluctuations.

Findings

Derives inflationary parameters (1-n_s) and epsilon consistent with Planck data.

Provides a model-independent characterization of scalar curvature fluctuations.

Extends the algebra of observables in a de Sitter universe to include time evolution as an inner automorphism.

Abstract

We suggest a new conceptual frame where inflationary Cosmology is quantum mechanically described using the Hilbert space representation of the crossed product of the type $III$ factor associated with the algebra of local operators on the de Sitter static patch and the automorphism of translations in the emergent conformal time. In this formal framework scalar curvature fluctuations are determined by the gravitational potential induced by the quantum variance of the generator of the conformal time automorphisms. Choosing as representative, in the Hilbert space of the crossed product, the effective vacuum state for each mode and conformal time, we evaluate the quantum variance and the induced scalar curvature fluctuations. This leads to a model independent characterization of the inflationary parameters (1-n_s) \sim 0.0318 and \epsilon\sim 0.0027 in good agreement with Planck experimental results. In this formal frame inflation extends the algebra of observables in a primordial dS expanding Universe. In this extension time evolution becomes an inner automorphism and the conjugated energy fluctuations manifest as the observable scalar curvature fluctuations.