  In scenarios of physical interest in Loop Quantum Cosmology, with a
preinflationary epoch where the kinetic energy of the inflaton dominates, the
analytic study of the dynamics of the primordial fluctuations has been carried
out by neglecting the inflaton potential in those stages of the evolution. In
this work we develop approximations to investigate the influence of the
potential as the period of kinetic dominance gives way to the inflationary
regime, treating the potential as a perturbation. Specifically, we study how
the potential modifies the effective mass that dictates the dynamics of the
scalar perturbations in the preinflationary epochs, within the framework of the
so-called hybrid prescription for Loop Quantum Cosmology. Moreover, we motivate
and model a transition period that connects the kinetically dominated regime
with inflation, allowing us to study the interval of times where the
contribution of the potential is no longer negligible but an inflationary
description is not yet valid. Finally, we include the main modifications coming
from a slow-roll correction to a purely de Sitter evolution of the
perturbations during inflation. We analytically solve the dynamics of the
perturbations in each of these different epochs of cosmological evolution,
starting from initial conditions fixed by the criterion of asymptotic
Hamiltonian diagonalization. This enables us to compute and quantitatively
analyze the primordial power spectrum in a specific case, using a quadratic
inflaton potential.
