, BH[S]{} , where \[AandB\] A=- , B=--(+) . By integration over time one can get the expressions for ${\cal R}$ and ${\cal S}$ at late time, which is (in matrix form): (

[c]{} [R]{}\
[S]{}

)=(

[cc]{} 1 & T\_[[R]{}[S]{}]{}\
0 & T\_[[S]{}[S]{}]{}

)(

[c]{} [R]{}\
[S]{}

)\_ , where \[TrsandTss\] T\_[[R]{}[S]{}]{}(t\_,t)&&\^t\_[t\_]{}A(t\_1)T\_[[S]{}[S]{}]{}(t\_,t\_1)H(t\_1)dt\_1 ,\
T\_[[S]{}[S]{}]{}(t\_,t)&&(\^t\_[t\_]{}B(t\_2)(t\_,t\_2)H(t\_2)dt\_2) .

The power spectra at horizon-reentering therefore can be expressed as: \[spectrum2\] [P]{}\_[R]{}=(1+T\_[[R]{}[S]{}]{}\^2)[P]{}\_[[S]{}]{} , [P]{}\_[S]{}=T\_[[S]{}[S]{}]{}\^2[P]{}\_[[S]{}]{} , [C]{}\_[[R]{}[S]{}]{}=T\_[[R]{}[S]{}]{}T\_[[S]{}[S]{}]{}[P]{}\_[[S]{}]{} , assuming that $\cal R$ and $\cal S$ are uncorrelated at Hubble-exit time. Define the rotation angle $\Theta$ such that $\sin\Theta=T_{{\cal R}{\cal S}}/\sqrt{1+T_{{\cal R}{\cal S}}^2}$, and applying the definition of spectral index in Eq. (\[index\]), one can furtherly have: \[index2\] n\_[R]{}&=&n\_[[R]{}]{}-H\_\^[-1]{}(2)\
&=&1-2\_-\_-A\_(2)-2B\_\^2 , where the definition of $T_{{\cal R}{\cal S}}$ has been used.

Pulling back perturbations into Jordan frame
--------------------------------------------

In the previous sections we have calculated the perturbations of the model (\[action\]), including the scalar and tensor perturbations, in its Einstein frame. This is what usually people do to analyse modified gravity theories, for it is much easier to deal with pure field theories in Einstein frame and one doesn’t need to bother with the higher order curvature terms. For the case of pure modified gravity or 