  We study the compatibility of the quantum homogeneitiy and isotropy
hypothesis (QHIH), proposed by Ashtekar and Gupt to restrict the choice of
vacuum state for the cosmological perturbations in Loop Quantum Cosmology
(LQC), with the requirement that the selected vacuum should lead to a power
spectrum that does not oscillate. We inspect in close detail the procedure that
these authors followed to construct a set of states satisfying the QHIH, and
how a preferred vacuum can be determined within this set. We find a step that
is not univocally specified in this procedure, in relation with the replacement
of the set of states that was originally allowed by the QHIH with an
alternative set that is more manageable. In fact, the first of these sets does
not contain the state that has been used in most of the implementations of the
QHIH to the analysis of the power spectrum of the perturbations in LQC. We
focus our attention on the original set picked out by the QHIH and investigate
whether some of its elements may display a non-oscillatory behavior. We show
that, to the extent to which the techniques used in this paper apply, this
possibility is feasible. Thus, the two aforementioned criteria for the physical
restriction of the vacuum state in LQC are compatible with each other and not
exclusive.
