perturbations, thus the system will not be in an adiabatic state. The equation of motion of the system (\[action\]) can be written as: \[freom1\] 3\^2&=&(f+3F\^+3F\^)+\^2 ,\
\[freom2\]-2\^&=&(F\^-F\^)+\^2 , where $\tilde{H}$ is the Hubble parameter, $F$ is defined as $F\equiv\partial f/\partial\tilde{R}$, and $\rho_s$ and $P_s$ are energy density and pressure of the scalar part, respectively. Prime denotes derivative with $\tilde{t}$, the cosmic time in Jordan frame. One can also define the “effective" energy density and pressure for the whole system, which are &=&(f+3F\^+3F\^)+ ,\
&=&-(f+F\^+5F\^)+ , satisfying the Friedmann equations $3\tilde{H}^2=\kappa^2\tilde\rho$ and $-2\tilde{H}^\prime=\kappa^2(\tilde{\rho}+\tilde{P})$.

As is well known, the system (\[action\]) can be written into a Brans-Dicke form by simple field redefinition. Define $\varphi\equiv F$, $U(\varphi(\tilde{R}))=F\tilde{R}-f(\tilde{R})$, the action (\[action\]) becomes: S\_[BD]{}=d\^4  , which looks like two scalars, but still with one of them nonminimally coupled to gravity. The nonminimal coupling can be removed by further transformation, however, by the price of modifying the scaling of space-time, namely conformal transformation. To do this, we need to define a new metric, which we call the metric in Einstein frame. The metric in Einstein frame connects to the original metric (the metric in Jordan frame) as: \[conformal\] g\_=\^2\_  where $\Omega\equiv\sqrt{\varphi}$. Therefore by manipulation we have: 