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I am currently reading Mukhanov's and Winitzki's book with title: "Introduction to Quantum Fields and Classical Backgrounds".

At chapter 6, they argue that in a FRW (Freedman - Robertson - Walker) spacetime, a defined vacuum state $|0_1 >$ at one instance of (conformal) time $\eta_1$ cannot be regarded as a vaccum in another instance of (conformal) time $\eta_2$. Hence, one needs different creation and annihilation operators $(\alpha^\pm , \beta^\pm)$ for each vacuum state so that:
\begin{equation} \alpha^- |0_1> = 0, \quad \beta^- |0_2> = 0 \end{equation}

Now, since our spacetime is not Minkowski spacetime, the usual Klein Gordon equation is not the equation of motion and as a result our fields are not expanded as plane waves. In contrast, they are expanded as a series of mode functions of conformal time $u_k (\eta)$, which are the solutions of the proper equation of motion. For a scalar field $\chi$, in FRW spacetime, we have: \begin{equation} \chi (\eta, \vec x) = \int \frac{d^3k}{(2 \pi)^3} u_k (\eta) \alpha_{\vec k}^- e^{i \vec k \vec x} + u*_k (\eta) \alpha_{\vec k}^+ e^{-i \vec k \vec x} \end{equation}

Now, since there are two different creation and annihilation operators and the requirement that each expansion has to give back the same fields, we can argue that we should expect a second set of mode functions $v_k (\eta)$ which will be paired with $\beta^\pm$ so that:
\begin{equation} \chi (\eta, \vec x) = \int \frac{d^3k}{(2 \pi)^3} v_k (\eta) \beta_{\vec k}^- e^{i \vec k \vec x} + v*_k (\eta) \beta_{\vec k}^+ e^{-i \vec k \vec x} \end{equation}

Additionally, the two mode functions are connected with the Bogolyobov transformations.

Then, in chapter 7, the authors take the example of the De sitter spacetime and solve the differential equation which gives back the Hankel Solutions.

Finally, here is my question.

If in De sitter spacetime, for an initial time $\eta_1$ we know that the mode functions are the Hankel functions, how can we find different mode functions which would correspond to a different time $\eta_2$?

Since, the spacetime will be the same for both times.

Basically, I am trying to determine the second type of mode functions that would be paired with a second set of creation and annihilation operators and connect them with the initial Hankel functions via a Bogolyobov transformation.

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