  We study cosmological perturbations in the framework of Loop Quantum
Cosmology, using a hybrid quantization approach and Mukhanov-Sasaki variables.
The formulation in terms of these gauge invariants allows one to clarify the
independence of the results on choices of gauge and facilitates the comparison
with other approaches proposed to deal with cosmological perturbations in the
context of Loop Quantum Theory. A kind of Born-Oppenheimer ansatz is employed
to extract the dynamics of the inhomogeneous perturbations, separating them
from the degrees of freedom of the Friedmann-Robertson-Walker geometry. With
this ansatz, we derive an approximate Schr\"odinger equation for the
cosmological perturbations and study its range of validity. We also prove that,
with an alternate factor ordering, the dynamics deduced for the perturbations
is similar to the one found in the so-called "dressed metric approach", apart
from a possible scaling of the matter field in order to preserve its unitary
evolution in the regime of Quantum Field Theory in a curved background and some
quantization prescription issues. Finally, we obtain the effective equations
that are naturally associated with the Mukhanov-Sasaki variables, both with and
without introducing the Born-Oppenheimer ansatz, and with the different factor
orderings that we have studied.
