  We study the imprint that certain quantization ambiguities may leave in
effective regimes of the hybrid loop quantum description of cosmological
perturbations. More specifically, in the case of scalar perturbations we
investigate how to reconstruct the Mukhanov-Sasaki field in the effective
regime of Loop Quantum Cosmology, taking as starting point for the quantization
a canonical formulation in terms of other perturbative gauge invariants that
possess different dynamics. This formulation of the quantum theory, in terms of
variables other than the Mukhanov-Sasaki ones, is crucial to arrive at a
quantum Hamiltonian with a good behavior, elluding the problems with ill
defined Hamiltonian operators typical of quantum field theories. In the
reconstruction of the Mukhanov-Sasaki field, we ask that the effective
Mukhanov-Sasaki equations adopt a similar form and display the same Hamiltonian
structure as the classical ones, a property that has been widely assumed in
Loop Quantum Cosmology studies over the last decade. This condition actually
restricts the freedom inherent to certain quantization ambiguities. Once these
ambiguities are removed, the reconstruction of the Mukhanov-Sasaki field
naturally identifies a set of positive-frequency solutions to the effective
equations, and hence a choice of initial conditions for the perturbations. Our
analysis constitutes an important and necessary test of the robustness of
standard effective descriptions in Loop Quantum Cosmology, along with their
observational predictions on the primordial power spectrum, taking into account
that they should be the consequence of a more fundamental quantum theory with a
well-defined Hamiltonian, in the spirit of Dirac's long-standing ideas.
