We examine the four dimensional path integral for Euclidean quantum gravity in the context of the EPRL-FK spin foam model. The state sum is restricted to certain symmetric configurations which resembles the geometry of a flat homogeneous and isotropic universe. The vertex structure is specially chosen so that a basic concept of expansion and contraction of the lattice universe is allowed.We compute the asymptotic form of the spin foam state sum in the symmetry restricted setting, and recover a Regge-type action, as well as an explicit form of the Hessian matrix, which captures quantum corrections. We investigate the action in the three cases of vacuum, a cosmological constant, and coupled to dust, and find that in all cases, the corresponding FRW dynamics is recovered in the limit of large lattices. While this work demonstrates a large intersection with computations done in the context of cosmological modelling with Regge Calculus, it is ultimately a setup for treating curved geometries in the renormalization of the EPRL-FK spin foam model. *

We study the renormalization group flow of the Euclidean Engle-Pereira-Rovelli-Livine and Freidel-Krasnov (EPRL-FK) spin foam model in the large-j-limit. The vertex amplitude is deformed to include a cosmological constant term. The state sum is reduced to describe a foliated spacetime whose spatial slices are flat, isotropic and homogeneous. The model admits a non-vanishing extrinsic curvature whereas the scale factor can expand or contract at successive time steps. The reduction of degrees of freedom allows a numerical evaluation of certain geometric observables on coarser and finer discretizations. Their comparison defines the renormalization group (RG) flow of the model in the parameters (α, Λ, G). We first consider the projection of the RG flow along the α direction, which shows a UV-attractive fixed point. Then, we extend our analysis to two-and three-dimensional parameter spaces. Most notably, we find the indications of a fixed point in the (α, Λ, G) space showing one repulsive and two attractive directions. arXiv:1804.00023v3 [gr-qc]

In this article, we consider an ad-hoc deformation of the EPRL model for quantum gravity by a cosmological constant term. This sort of deformation has been first introduced by Han for the case of the 4-simplex. In this article, we generalise the deformation to the case of arbitrary vertices, and compute its large-j-asymptotics. We show that, if the boundary data corresponds to a 4d polyhedron P , then the asymptotic formula gives the usual Regge action plus a cosmological constant term. We pay particular attention to the determinant of the Hessian matrix, and show that it can be related to the one of the undeformed vertex.