\[einstein\] S\_[E]{}&=&d\^4x  , where \[phi\] &=&- , b()= ,\
V(,)&=&\^[-4]{}(U()+[V]{}()) .

From the above one can see, via conformal transformation one can transform our model (\[action\]) into a minimal coupling two-field inflation model. The equation of motion for such two-field models contains two second-order differential equations, but no higher-order derivatives involving, thus is easier to be solved. That is a very important reason why people like to do the calculations in Einstein frame rather than directly in the original Jordan frame. For the system (\[einstein\]), we would like to first analyze the background dynamics, and define some slow-roll parameters for later use. From (\[einstein\]), The energy density and the pressure in Einstein frame are: &=&\^2+e\^[2b()]{}\^2+V(,) ,\
P&=&\^2+e\^[2b()]{}\^2-V(,) , where dot denotes derivative w.r.t. cosmic time in Einstein frame $t$. By varying the action w.r.t. $\phi$ and $\chi$, we can obtain the equations of motion for both fields: \[eom\] +3H+V\_-b\_e\^[2b()]{}\^2&=&0 ,\
+3H+2b\_+e\^[-2b]{}V\_&=&0 . Finally, the Friedmann equations read: \[friedmann\] H\^2= , =-(+P) . Hereafter we take the unit such that $\kappa^2=1$. The various slow-roll parameters can be defined as: \[slowroll\] \_= , \_=e\^[2b]{} , =- , which should be much smaller than unity during inflation, and satisfy the relation: $\epsilon_\phi+\epsilon_\chi=\epsilon$. Under 