  In this paper, we systematically study the evolution of the Universe in the
framework of a modified loop quantum cosmological model (mLQC-I) with various
inflationary potentials, including chaotic, Starobinsky, generalized
Starobinsky, polynomials of the first and second kinds, generalized T- models
and natural inflation. In all these models, the big bang singularity is
represented by a quantum bounce, and the evolution of the Universe both before
and after the bounce is universal and weakly depends on the inflationary
potentials, as long as the evolution is dominated by the kinetic energy of the
inflaton at the bounce. In particular, the evolution in the pre-bounce region
can be universally divided into three different phases: pre-bouncing,
pre-transition, and pre-de Sitter. The pre-bouncing phase occurs immediately
before the quantum bounce, during which the evolution of the Universe is
dominated by the kinetic energy of the inflaton. Thus, the equation of state of
the inflaton is about one, w = 1. Soon, the inflation potential takes over, so
w rapidly falls from one to negative one. This pre-transition phase is very
short and quickly turns into the pre-de Sitter phase, whereby the effective
cosmological constant with a Planck size takes over and dominates the rest of
the contracting phase. In the entire pre-bounce regime, the evolution of the
expansion factor and the inflaton can be approximated by analytical solutions,
which are universal and independent of the inflation potentials.
