General covariant transition amplitudes in quantum cosmology

The path-integral approach to cosmology consists in the computation of transition amplitudes between states of the quantum geometry of the universe. In the past, the concrete computation of these transitions amplitudes has been performed in a perturbative regime, breaking the full general covariance of the theory. Here I present how it is possible to define a general covariant path integral in quantum cosmology, by relying on the most recent results of the canonical and covariant formulations of Loop Quantum Gravity. I present two strategies that have been implemented. The first starts from the full Spinfoam theory, i.e. the path-integral framework for Loop Quantum Gravity, and defines a cosmological system. This is not obtained from symmetry reduced variables that are successively quantized, but directly considering the approximations that are characteristic of the full theory. The Spinfoam Cosmology obtained in this way includes quantum fluctuations beyond standard perturbation theory. The second strategy exploits the Hamiltonian constraint of Loop Quantum Cosmology, that is exponentiated in the formal expression of the usual path integral. The result is a general covariant path integral, that reproduces the form of the amplitude in the full Spinfoam theory. Therefore, this procedure connects the canonical and the covariant formalisms.