second term in $f(\tilde{R})$.

Since the parameter space is still large for a global fitting, In Fig. \[tt\] we take three cases of parameter choice for each example, and plot the TT power spectrum as well as compare them to the Planck data points. Although these three cases are randomly chosen and are different from each other, they are all consistent with the data, showing that it is quite easy to have our model fit the observations. Moreover, with proper choices, one of our cases tends to give a large tensor/scalar ratio $r$, which can contribute to TT power spectrum at large scales, and has possiblities to be justified by near future data. We present the parameter choices as well as the result such as $A_s$, $n_s$ and $r$ for each case in the caption of Fig. \[tt\]. The newest PLANCK constraints on these quantities are $\ln(10^{10}A_s)\approx3.089\pm0.036$ and $n_s\approx0.9655\pm0.0062$ at $1\sigma$ confidence level, where PLANCK+LowP data are used [@Planck:2015xua].

![The plot of TT spectrum from the examples in this section. The green and orange points and error bars are Planck data for low-$l$ and high-$l$, respectively. Upper panel: the $f(\tilde R)=\xi \tilde R^n$ case, where we choose $(\xi,n)$ as $(1,1.1)$ 