No quasi-stable scalaron lump forms after $R^2$ inflation

In the Einstein frame picture of Starobinky's $R^2$ inflation model, cosmic inflation is driven by a slowly rolling inflaton field, called scalaron, and followed by a coherently oscillating scalaron phase. Since the scalaron oscillates excessively many times in its potential, which has a quadratic minimum and is a little shallower than quadratic on the positive side, it may fragment into long-living localized objects, called oscillons or I-balls, due to nonlinear growth of fluctuations before reheating of the universe. We show that while parametric self-resonances amplify scalaron fluctuations in the Minkowski background, the growth cannot overcome the decay due to expansion in the Friedmann background after $R^2$ inflation. By taking into account back-reaction from the metric of spacetime, modes that are larger than a critical scale are indeed amplified and become non-decaying. However, those non-decaying modes are not growing enough to form spatially localized lumps of the scalaron. Thus, reheating processes are unaltered by oscillons/I-balls and they proceed through perturbative decay of the scalaron as studied in the original work.