Many-node/many-link spinfoam: the homogeneous and isotropic case

I compute the Lorentzian EPRL/FK/KKL spinfoam vertex amplitude at the first order for regular graphs, with an arbitrary number of links and nodes, and coherent states peaked on a homogeneous and isotropic geometry. This form of the amplitude can be applied for example to a dipole with an arbitrary number of links or to the 4-simplex given by the complete graph on five nodes. All the resulting amplitudes have the same support, independently of the graph used, in the large-j (large-volume) limit. This implies that they all yield the Friedmann equation: I show this in the presence of the cosmological constant. This result indicates that in the semiclassical limit, quantum corrections in spinfoam cosmology do not come from just refining the graph, but rather from relaxing the large-j limit.