$f(R)$ to break the degeneracy of some of the parameters, there are still large parameter space that can easily make our model consistent with the data. Here for illumination we just show two simple examples. In the first example $f(\tilde{R})$ is taken to be of the form [@DeFelice:2010aj] f()=\^n , F=n\^[n-1]{}=n\^[n-1]{} , and $\tilde\omega_\ast$ and $\tilde{z}_\ast$ can be simply expressed in terms of $\tilde{\epsilon}_\ast$, $\tilde{\eta}_\ast$ as \_(1-n)\_ , \_(1-n)\_ , and Eqs. (\[scalarspectrumJF\]) and (\[indexJF\]) becomes \_&=&[C]{} ,\
\_[R]{}&=&1+2(n-2)\_+(+n)\_+  with &=&1+{\[1+e\^[BN\_]{}(nB-B-1)\]\
&&+(B-1)e\^[BN\_]{}}\^2 ,\
[C]{}\^&=&2H\_e\^[BN\_]{}{\[1+e\^[BN\_]{}(nB-B-1)\]\
&&+(B-1)e\^[BN\_]{}}\
&& , One can check that it returns to the results of standard inflation when $\xi=n=1$.

Another example is the well-known Starobinsky model [@Starobinsky:1980te]: f()=+\^2 , F=1+2=1+12\^2(2-) , and $\tilde\omega_\ast$ and $\tilde{z}_\ast$ can be simply expressed in terms of $\tilde{\epsilon}_\ast$, $\tilde{\eta}_\ast$ as \_-\_ , \_=+\_ , and Eqs. (\[scalarspectrumJF\]) and (\[indexJF\]) becomes \_&=&[C]{} ,\
\_[R]{}&=&1-2\_-2\_+  with &=&1+{+e\^[BN\_]{}}\^2 ,\
[C]{}\^&=&2H\_e\^[BN\_]{}{+e\^[BN\_]{}}\
&& , Interestingly, we find that in this case, although expressed using slow-roll parameters in Jordan frame, the expressions of $\tilde{n}_{\cal R}$ has the same form of that in Einstein frame, namely Eq. (\[index\]), except for the ${\cal C}$ term. This holds for any value of $\xi$, and only depend on the quadratic scaling of the 