the scope of this paper.

It is also straightforward to get the tensor spectrum $\tilde{\cal P}_{\cal T}$ in Jordan frame of our model from Eq. (\[tensor\]). After taking conformal tranformations (\[relation\]) and keeping only leading order, the result will be \[tensorspectrumJF\] \_[T]{}= , which coincides with ${\cal P}_{\cal T}$. This is because since tensor degrees of freedom of perturbations are decoupled from the scalar degrees of freedom, and in such a system there is only one tensor degree of freedom, so the tensor modes of the two frames are equivelant, and the tensor spectra in the two frames differ only by a conformal transformation. Eqs. (\[differenceR\])-(\[tensorspectrumJF\]) are our main results in this context.

Fitting with the PLANCK data
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In the previous section, we derived the perturbation generated by a “modified gravity+single scalar" system, and after calculating in Einstein frame, we pull it back to its original Jordan frame, which can be used to fit the data. Actually, one can see that the results are controlled by a series of parameters, namely $\tilde{H}_\ast$, $\tilde{\epsilon}_\ast$, $\tilde{\eta}_\ast$, $\tilde{\omega}_\ast$, $\tilde{z}_\ast$, $F_\ast$ and ${\cal C}$, while ${\cal C}$ contains information after Hubble-exits, namely $A$, $B$ and $N_\ast$. Therefore, even we take explicit forms of 