We also plot the TT spectrum according to the Jordan frame results we’ve got and compare it to the PLANCK data, with two simple examples. In the first one $f(R)$ takes the form of $R^n$, while the other is the famous Starobinsky model, $f(R)=R+\xi R^2$. From the plot one can see that, since in both examples the parameter space is quite large, it is very easy to have the theoretical plots of the spectrum consistent with the data points. We also explicitly gave the spectrum amplitude and spectral indexes of the two examples respectively, which is well within the constraints given by the PLANCK data. Thus it also indicates that the model itself is interesting and deserves further investigations in the future.

Acknowledgements
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We thank Yun-Song Piao, Antonio De Felice and Jonathan White for useful discussions. T.Q. is supported by the Open Innovation Fund of Key Laboratory of Quark and Lepton Physics (MOE), Central China Normal University (No.:QLPL2014P01), and J.X. is supported by the National Youth Thousand Talents Program.

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