Vertex expansion for the Bianchi type I model

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 “coloring” of the spin foams. Furthermore, the isotropic expansion is recovered by integrating out the anisotropies.

Figure 1

The Bianchi I model can be written in terms of three parameters $(n,x_1,x_2)$. $n$ dictates the volume, while $x_1$ and $x_2$ are analogous to the spin labels of LQG and are represented here as two edges. Note that this is not intended to be a true LQG graph, rather it is a pictorial description of the degrees of freedom that our states have.

Figure 2

The history of the spin network state i.e. the spin foam analogue of our vertex expansion, with $M=1$ is represented pictorially. The initial state has volume $n$ and anisotropies $x_i$. The state then “evolves,” keeping $n$ constant but allowing the anisotropies to vary. Eventually there is a transition to a new volume, $\overline{n}$. The anisotropy before and after the transition are $y_i$ and ${\overline{x}}_i$, respectively. The final state has labels $\overline{n}$ and ${\overline{y}}_i$. The dashed lines indicate that the “spin” labels ($x_i$) evolve along each of the constant volume pieces. The amplitude for such process, in the notation of Eq. (3.19), is given by $A(\overline{\mathbf{y}},\overline{n},\overline{\mathbf{x}},\mathbf{y},n,\mathbf{x})$.