Many-nodes/many-links spinfoam: the homogeneous and isotropic case

TL;DR

This paper computes a Lorentzian spinfoam vertex amplitude for regular graphs with many nodes and links, demonstrating that in the large volume limit, all such amplitudes support the Friedmann equation, relevant for quantum cosmology.

Contribution

It provides a general computation of the spinfoam vertex amplitude for arbitrary regular graphs in the homogeneous and isotropic case, extending previous models.

Findings

All amplitudes have the same support in the large j limit.

The amplitudes yield the Friedmann equation in the semiclassical limit.

Quantum corrections in spinfoam cosmology are due to relaxing the large j limit, not graph refinement.

Abstract

I compute the Lorentzian EPRL/FK/KKL spinfoam vertex amplitude 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 compete graph on 5 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.