On the Expansions in Spin Foam Cosmology

TL;DR

This paper analyzes the expansions in spin foam cosmology, highlighting the contributions of complex amplitudes at the one vertex level, and discusses the geometric and topological aspects of these expansions.

Contribution

It provides a detailed examination of the contributions at various levels of the spin foam expansion and discusses the necessary conditions for controlling these expansions.

Findings

Complex amplitudes contribute at the one vertex level.

Factorisation of amplitudes has a topological interpretation.

Truncating dynamics is necessary for consistent approximations.

Abstract

We discuss the expansions used in spin foam cosmology. We point out that already at the one vertex level arbitrarily complicated amplitudes contribute, and discuss the geometric asymptotics of the five simplest ones. We discuss what type of consistency conditions would be required to control the expansion. We show that the factorisation of the amplitude originally considered is best interpreted in topological terms. We then consider the next higher term in the graph expansion. We demonstrate the tension between the truncation to small graphs and going to the homogeneous sector, and conclude that it is necessary to truncate the dynamics as well.