Towards spinfoam cosmology

We compute the transition amplitude between coherent quantum states of geometry peaked on homogeneous-isotropic metrics. We work in the context of pure gravity without matter, we use the holomorphic representations of loop quantum gravity and the Kaminski-Kisielowski-Lewandowski generalization of the new vertex, and work at first order in the vertex expansion, second order in the graph (multipole) expansion, and first order in ${\mathrm{\text{volume}}}^{-1}$. We show that the resulting amplitude is in the kernel of a differential operator whose classical limit is the canonical Hamiltonian of a Friedmann-Robertson-Walker cosmology. This result is an indication that the dynamics of loop quantum gravity defined by the new vertex reproduces the gravity part of the Friedmann equation in the appropriate limit.