single field nonminimal coupled with gravity, the curvature perturbation are invariant in both frames, i.e. $\tilde{\cal R}={\cal R}$ [@Makino:1991sg], and one can directly use the Einstein frame results to compare with the observations. For our case, however, it is no longer the case. To see this, let’s pull the results we get in Einstein frame back into the Jordan frame and see how different they are. From the conformal transformation (\[conformal\]), we have: \[relation\] &=&\_0\^[-1]{} a , d=\_0\^[-1]{} dt ,\
&=&H\_0-\_0=H\_0-\_0\^/\_0 , where $\Omega=\sqrt{\varphi}=\sqrt{F}$.

It is useful to define the Jordan-frame-based slow-varying parameter: && , \_ , - ,\
&& , , where one could find from Eq. (\[freom2\]) that first three of these parameters have the relation: =-+\_+[O]{}(\^2) , while we have $\epsilon_\phi\simeq 3\tilde{\omega}^2$ from Eqs. (\[phi\]) and (\[slowroll\]). Furthermore, one could get that the slow-roll parameters in the two frames are related as (up to leading order): =+\_+3\^2 , = .

Furthermore, from perturbed metric (\[perturbmetric\]), one can have =- , =+ , and the curvature perturbation in Jordan frame is: =- . From this expression, one can see that the curvature perturbation is not conformal invariant any more. To see this more clearly, one can calculate the 