Scalaron from $R^2$-gravity as a Heavy Field
  We study a model of inflation in which a scalar field $\chi$ is non-minimally
coupled to Starobinsky's $R^2$ gravity. After transforming it to the Einstein
frame, a new scalar field, the scalaron $\phi$, will appear and couple to
$\chi$ with a nontrivial field metric, while $\chi$ acquires a positive mass
via the non-minimal coupling. Initially inflation occurs along the $\phi$
direction with $\chi$ trapped near its origin by this induced mass. After
$\phi$ crosses a critical value, it starts rolling down rapidly and proceeds
todamped oscillations around an effective local minimum determined by the value
of $\chi$, while inflation still continues, driven by the $\chi$ field at this
second stage where the effect of the non-minimal coupling becomes negligible.
The presence of the damped oscillations during the transition from the first to
second stage of inflation causes enhancement and oscillation features in the
power spectrum of the curvature perturbation. Assuming that the oscillations
may be treated perturbatively, we calculate these features by using the $\delta
N$ formalism, and discuss its observational implications to large scale CMB
anomalies or primordial black hole formation, depending on the scale of the
features.
