  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.
