paper, we calculated the power spectrum and spectral index of the curvature perturbations in Jordan frame, and showed their difference between those in Einstein frame. In the solo-degree-of-freedom system however, the quantities such as power spectrum and spectral indexes are given the same via computations in Jordan and Einstein frame, and one can directly use the results he gets in Einstein frame as his final results.

The main reason that causes the difference of the curvature perturbations in the two frames of our model is that, as a m.d.o.f. system, isocurvature perturbations will be generated, and the isocurvature perturbations in Einstein frame will contribute to the curvature ones in Jordan frame. In other words, although mathematically it has no problem to do such a transformation, physically we are concerning different quantities. So the difference of curvature perturbations between two frames are real, not only from conformal transformations. One can check that when the isocurvature fraction of the perturbations approaches zero, adiabatic perturbations in the two frames will again coincide. Our results are consistent with that in Ref. [@Gong:2011qe], though the analysis are different. For more general systems of m.d.o.f., it is straightforward to prove that the conclusions are the same.

