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.Comment: 4-page review for Scientifica Act