Lattice-refining loop quantum cosmology and inflation

We study the importance of lattice refinement in achieving a successful inflationary era. We solve, in the continuum limit, the second order difference equation governing the quantum evolution in loop quantum cosmology, assuming both a fixed and a dynamically varying lattice in a suitable refinement model. We thus impose a constraint on the potential of a scalar field, so that the continuum approximation is not broken. Considering that such a scalar field could play the role of the inflaton, we obtain a second constraint on the inflationary potential so that there is consistency with the cosmic microwave background data on large angular scales. For a $m^2{\varphi }^2/2$ inflationary model, we combine the two constraints on the inflaton potential to impose an upper limit on $m$, which is severely fine-tuned in the case of a fixed lattice. We thus conclude that lattice refinement is necessary to achieve a natural inflationary model.