Gauge invariance in loop quantum cosmology: Hamilton-Jacobi and Mukhanov-Sasaki equations for scalar perturbations

Gauge invariance of scalar perturbations is studied together with the associated equations of motion. Extending the methods developed in Hamiltonian general relativity, the Hamilton-Jacobi equation is investigated into the details in the framework of loop quantum gravity. The gauge-invariant observables are built, and their equations of motions are reviewed both in the Hamiltonian and Lagrangian approaches. This method is applied to scalar perturbations with either holonomy or inverse-volume corrections.