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.