solution in Einstein frame. In Section III, we study the perturbation theory of our model in detail in the Einstein frame and then pull back to the Jordan frame. One can see that the results of the two frames will be quite different. Section IV is devoted to the constraints on this model by use of PLANCK data. Finally, we conclude with a discussion in Section V. Note that, we will work with the reduced Planck mass, $M_{p} = 1/\sqrt{8\pi G}$, where $G$ is the gravitational constant, and adopt the mostly-plus metric sign convention $(-,+,+,+)$.

The model
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We start with the action: \[action\] S=d\^4   with $\kappa^2=8\pi G$, where $f(\tilde{R})$ is an arbitrary function of the Ricci curvature $\tilde{R}$, and $\mathcal{L}_s=-\tilde{\nabla}_\mu\chi\tilde{\nabla}^\mu\chi/2-{\cal V}(\chi)$ is the lagrangian for the matter field $\chi$. For later convenience, we refer variables with tilde to those in Jordan frame while their correspondence without tilde are in Einstein frame. This is just a simple example of modified gravity accompanied with normal matter, but a big difference from pure modified gravity is that it contains more than one degree of freedom, where at large scales, the curvature perturbation is no longer a conserved variable but sourced by isocurvature 