  In this paper, we first provide a brief review of the effective dynamics of
two recently well-studied models of modified loop quantum cosmologies (mLQCs),
which arise from different regularizations of the Hamiltonian constraint and
show the robustness of a generic resolution of the big bang singularity,
replaced by a quantum bounce due to non-perturbative Planck scale effects. As
in loop quantum cosmology (LQC), in these modified models the slow-roll
inflation happens generically. We consider the cosmological perturbations
following the dressed and hybrid approaches and clarify some subtle issues
regarding the ambiguity of the extension of the effective potential of the
scalar perturbations across the quantum bounce, and the choice of initial
conditions. Both of the modified regularizations yield primordial power spectra
that are consistent with current observations for the Starobinsky potential
within the framework of either the dressed or the hybrid approach. But
differences in primordial power spectra are identified among the mLQCs and LQC.
In addition, for mLQC-I, striking differences arise between the dressed and
hybrid approaches in the infrared and oscillatory regimes. While the
differences between the two modified models can be attributed to differences in
the Planck scale physics, the permissible choices of the initial conditions and
the differences between the two perturbation approaches have been reported for
the first time. All these differences, due to either the different
regularizations or the different perturbation approaches in principle can be
observed in terms of non-Gaussianities.
