Perturbations of Single-field Inflation in Modified Gravity Theory
  In this paper, we study the case of single field inflation within the
framework of modified gravity theory where the gravity part has an arbitrary
form $f(R)$. Via a conformal transformation, this case can be transformed into
its Einstein frame where it looks like a two-field inflation model. However,
due to the existence of the isocurvature modes in such a
multi-degree-of-freedom (m.d.o.f.) system, the (curvature) perturbations are
not equivalent in two frames, so in despite of its convenience, it is illegal
to treat the perturbations in its Einstein frame as the "real" ones as we
always do for pure $f(R)$ theory or single field with nonminimal coupling. Here
by pulling the results of curvature perturbations back into its original Jordan
frame, we show explicitly the power spectrum and spectral index of the
perturbations in the Jordan frame, as well as how it differs from the Einstein
frame. We also fit our results with the newest Planck data. Since there are
large parameter space in these models, we show that it is easy to fit the data
very well.
