from the above solution, one obtains the Bunch-Davies vacuum solution: Q\^()Q\^s()  deep inside the horizon as usual, while in the $k\rightarrow 0$ limit, one gets \[largescale\] Q\_\^Q\_\^s\~Hk\^[-]{}  at the Hubble-exit time. Here $\tau\equiv\int a^{-1}(t)dt$ is the conformal time.

The inflationary observables are usually expressed in terms of power spectra and correlation functions. One could define the power spectrum for $Q^\sigma$ and $Q^s$ as: Q\^[m]{}Q\^n=\_m(k)\^[(3)]{}(-\^)\_[mn]{} , (m,n=,s)  where $\delta_{mn}$ denotes that there are no correlations between $Q^\sigma$ and $Q^s$ until the Hubble-exit. This is actually reasonable, since inside the horizon, the correlations of the variables are determined by the communication relations of their production-annihilation operators, which should commute because they are independent degrees of freedom. Although they have coupling, it should be subdominant inside the horizon where $|k\tau|\gg 1$. Considering the relation between ${\cal R}$, ${\cal S}$ and $Q^\sigma$, $Q^s$ (Eq. (\[curvature\]) and the sentences thereafter), one can have \_[R]{}=\_ , [P]{}\_[S]{}=\_[s]{} , and from the results (\[largescale\]), one has: \[spectrum1\] [P]{}\_[[R]{}]{}\_[[S]{}]{}=  at the Hubble-exit time. Here we denote the values of variables at Hubble-exit time by a star in the subscript. The spectral index of scalar spectrum at Hubble-exit time is then evaluated as \[index\] n\_[[R]{}]{}=1-2\_-\_ .

One can also 