Cosmology as a Crossed Product

TL;DR

This work recasts inflationary cosmology within an algebraic framework, describing local observables as type $III$ factors and using the crossed-product construction to convert modular (outer) time into inner time. Inflation is modeled as a state in the crossed-product Hilbert space $L^2(R^+,{rak H}^{dS})$, with scalar curvature fluctuations driven by the quantum variance of the conformal-time generator, yielding model-independent predictions. The approach delivers concrete inflationary parameters, notably a scalar spectral tilt $(1-n_s) \,\approx\;0.0328$ and a slow-roll parameter $\epsilon\approx 0.0027$, in good accord with Planck data, without invoking a specified inflaton potential. It also highlights emergent causality and gravity-regulated entanglement as integral features, offering a principled link between quantum gravity effects and observable cosmological fluctuations with potential connections to causal diamonds and holographic notions of entropy.

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 ε\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.