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.

Figure 1

The two ways to glue discs to the edges. On the left, the boundary of the disc is identified with the whole figure eight; on the right, two discs are glued to the two triangles individually.

Figure 2

The one-vertex spin foams and their associated KKL spin networks. The dashed lines indicate that the face bounded by the edges it connects does not shrink to a point.

Figure 3

Our notation for the boundary state space.

Figure 4

${\Gamma }^5$, the complete graph on five vertices. The vertices are labeled by $j$, $k=1\dots 5$; each edge is labeled by the end vertices and carries a spin $j_{kl}$, and each vertex carries an intertwiner ${\iota }_k$.