  We develop a consistent analytic approach to determine the conditions under
which slow roll inflation can arise when the inflaton is the same scalar field
that is responsible for the bounce in Loop Quantum Cosmology (LQC). We find
that the requirement that the energy density of the field is fixed at the
bounce having to match a critical density has important consequences for its
future evolution. For a generic potential with a minimum, we find different
scenarios depending on the initial velocity of the field and whether it begins
life in a kinetic or potential energy dominated regime. For chaotic potentials
that start in a kinetic dominated regime, we find an initial phase of
superinflation independent of the shape of the potential followed by a damping
phase which slows the inflaton down, forcing it to turnaround and naturally
enter a phase of slow-roll inflation. If we begin in a potential energy
dominated regime, then the field undergoes a period where the corrections
present in LQC damp its evolution once again forcing it to turnaround and enter
a phase of slow roll inflation. On the other hand we show for the Starobinsky
potential that inflation never occurs when we begin in a potential dominated
regime. In fact traditional Starobinsky inflation has to start in a kinetic
energy dominated regime, with corresponding tighter constraints on the initial
value of the field for successful inflation than in the conventional case.
Comparing our analytic results to published numerical ones, we find remarkable
agreement especially when we consider the different epochs that are involved.
In particular the values of key observables obtained from the two approaches
are in excellent agreement, opening up the possibility of obtaining analytic
results for the evolution of the density perturbations in these models.
