Inflation in loop quantum cosmology

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 the future evolution of the field. Initially we consider the case of a generic potential that has a minimum, and we find different scenarios depending on the initial velocity of the field and whether it begins life in a kinetic energy or a potential energy dominated part of its energy density. 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 that slows the inflaton down, forcing it to turn around and naturally enter a phase of slow-roll inflation. If we begin in a potential energy dominated regime, then the inflaton undergoes a period where the corrections present in LQC damp its evolution, once again forcing the field to turn around and enter a phase of slow-roll inflation. On the other hand, we show for Starobinsky inflation that inflation never occurs when we begin in a potential energy dominated regime. What we would normally call potential dominated in traditional Starobinsky inflation where the field lives in its plateau regime; for the case of LQC this corresponds to being in a kinetic energy dominated regime. The requirement that damping slows the field down sufficiently to enter slow roll places tighter constraints on the initial value of the field for successful inflation than in the conventional case. Comparing our analytic results to the 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 our analytical and the published numerical solutions are in excellent agreement, opening up the possibility of using our results to obtain analytic results for the evolution of the density perturbations in these models.