Vertex Expansion for the Bianchi I model

TL;DR

This paper develops a perturbative expansion for Bianchi I loop quantum cosmology models, revealing a spin foam-like structure with anisotropic degrees of freedom acting as colorings, and recovers isotropic models by integrating out anisotropies.

Contribution

It extends the spin foam analogy to anisotropic Bianchi I models and demonstrates how isotropic models are recovered from anisotropic expansions.

Findings

Expansion expressed as a series of fixed-transition amplitudes

Anisotropic degrees of freedom act as spin foam colorings

Isotropic models recovered by integrating out anisotropies

Abstract

A perturbative expansion of Loop Quantum Cosmological transitions amplitudes of Bianchi I models is performed. Following the procedure outlined in [1,2] for isotropic models, it is shown that the resulting expansion can be written in the form of a series of amplitudes each with a fixed number of transitions mimicking a spin foam expansion. This analogy is more complete than in the isotropic case, since there are now the additional anisotropic degrees of freedom which play the role of `colouring' of the spin foams. Furthermore, the isotropic expansion is recovered by integrating out the anisotropies.