Expansions in spin foam cosmology

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 factorization 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.