Triangulated loop quantum cosmology: Bianchi IX universe and inhomogeneous perturbations

We develop the triangulated version of loop quantum cosmology, recently introduced in the literature. We focus on the dipole cosmology, where space is a three-sphere and the triangulation is formed by two tetrahedra. We show that the discrete fiducial connection has a simple and appealing geometrical interpretation and we correct the ansatz on the relation between the model variables and the Friedmann-Robertson-Walker scale factor. The modified ansatz leads to the convergence of the Hamiltonian constraint to the continuum one. We then ask which degrees of freedom are captured by this model. We show that the model is rich enough to describe the (anisotropic) Bianchi IX universe, and give the explicit relation between the Bianchi IX variables and the variables of the model. We discuss the possibility of using this path in order to define the quantization of the Bianchi IX universe. The model contains more degrees of freedom than Bianchi IX, and therefore captures some inhomogeneous degrees of freedom as well. Inhomogeneous degrees of freedom can be expanded in representations of the $SU(2)$ Bianchi IX isometry group, and the dipole model captures the lowest integer representation of these, connected to hyperspherical harmonic of angular momentum $j=1$.