*3.2. Baker-Campbell-Hausdorff Decomposition*

Explicit calculations involving the evolution operator derived in the last section turn out to be rather tedious, even for simple initial conditions. This is due to the structure of the exponential in Γˆ † *ξ* and the associated generator, respectively. As we will show in this section, we can perform a generalized Baker-Campbell-Hausdorff (BCH) decomposition of the operator Γˆ *ξ* that brings it into a form that is more suitable for actual practical computations. For this purpose it is of advantage to change the representation and henceforth work in the occupation number basis, that is with the usual ladder operators defined as (where we omit the explicit mentioning of the representation from now on):

$$
\pi\_{\boldsymbol{\theta}}(\hat{\boldsymbol{q}}) = \frac{1}{\sqrt{2\omega\_{0}}} (\hat{A}^{\dagger} + \hat{A}), \quad \pi\_{\boldsymbol{\theta}}(\hat{\boldsymbol{p}}) = i\sqrt{\frac{\omega\_{0}}{2}} (\hat{A}^{\dagger} - \hat{A}), \quad [\hat{A}, \hat{A}^{\dagger}] = \mathbb{1}\_{\mathcal{H}}.\tag{34}
$$

where we set as before *h*¯ = 1 and *m* = 1. Inserting these identities into the generator *<sup>π</sup>q*(G<sup>ˆ</sup>) from (26) and Γˆ *ξ* , we obtain (again without the explicit representation):

$$\begin{split} \mathcal{M}\_{\tilde{\xi}} &= \exp\left\{ \frac{i}{2} \ln(\xi) \left( i(A^{\dagger}A^{\dagger} - \bar{A}A) + \frac{h(\tilde{\xi})}{2\omega\_{0}} (\hat{A}^{\dagger}\hat{A}^{\dagger} + \hat{A}^{\dagger}\hat{A} + \bar{A}A^{\dagger} + \bar{A}A) \right) \right\} \\ &= \exp\left\{ \frac{1}{2} \ln(\tilde{\xi}) \left( \left( 1 + \frac{ih(\tilde{\xi})}{2\omega\_{0}} \right) \hat{A}\hat{A} - \left( 1 - \frac{ih(\tilde{\xi})}{2\omega\_{0}} \right) \hat{A}^{\dagger}\hat{A}^{\dagger} + \frac{ih(\tilde{\xi})}{\omega\_{0}} (\hat{A}^{\dagger}\hat{A} + \frac{1}{2}) \right) \right\} \\ &= \exp\left\{ \overline{\pi}(\tilde{\xi}) \frac{\hat{A}\hat{A}}{2} - a(\tilde{\xi}) \frac{\hat{A}^{\dagger}\hat{A}^{\dagger}}{2} + i\lambda(\tilde{\xi}) \left( \hat{A}^{\dagger}\hat{A} + \frac{1}{2} \right) \right\} \\ &=: \exp\left\{ \overline{\pi}(\tilde{\xi}) \delta\_{-} - a(\tilde{\xi}) \delta\_{+} + i\lambda(\tilde{\xi}) \delta\_{3} \right\}. \end{split} \tag{35}$$

where we made the following redefinitions for later notational convenience:

$$\mathfrak{d}\_{+} := \frac{1}{2} \hat{A}^{\dagger} \hat{A}^{\dagger}, \quad \mathfrak{d}\_{-} := \frac{1}{2} \hat{A} \hat{A}, \quad \mathfrak{d}\_{3} := \hat{A}^{\dagger} \hat{A} + \frac{1}{2}, \quad [\mathfrak{d}\_{3}, \mathfrak{d}\_{\pm}] = \pm 2 \mathfrak{d}\_{\pm}, \quad [\mathfrak{d}\_{-}, \mathfrak{d}\_{+}] = \mathfrak{d}\_{3}. \tag{36}$$

The coefficients are in fact explicitly time-dependent functions *<sup>α</sup>*(*ξ*), *<sup>λ</sup>*(*ξ*), where the time dependency is carried by the solution *ξ* of the Ermakov Equation (5) as we have seen in the discussion of the classical setup. They are defined as:

$$\alpha = \ln(\xi) \left( 1 - \frac{i h(\xi)}{2 \omega\_0} \right), \quad \lambda = \frac{h(\xi)}{2 \omega\_0} \ln(\xi), \quad |a|^2 > \lambda^2 \,\forall \, \xi \colon \mathbb{R} \supseteq \mathbb{I} \to \mathbb{R}. \tag{37}$$

*Universe* **2019**, *5*, 170

After this replacement the resulting expression for Γ ˆ *ξ* takes the form of a generalized, time-dependent squeezing operation. The commutation relations in Equation (36) are those of sl(2, <sup>R</sup>), which was already evident in the classical sector of the theory. It is straightforward to see that the standard Baker-Campbell-Hausdorff decomposition does not work, since the iterated commutator structure leads to infinitely many non-vanishing contributions in the well-known formula. However, a BCH decomposition of SL(2, R) elements has been performed using analytic techniques as shown in Reference [30]. This was done by introducing a parametric rescaling of Γ ˆ *ξ* and allowing a corresponding dependence of the coefficient functions in the decomposition on this parameter. In our case, a rescaling of the original Γ ˆ *ξ* leads to:

$$\hat{\Gamma}\_{\vec{\xi}}(\mu) := \exp\left\{\mu \hat{\mathcal{G}}\right\} = \exp\left\{\mu \left(\overline{a}(\vec{\xi})\mathfrak{t}\_{-} - a(\vec{\xi})\mathfrak{t}\_{+} + i\lambda(\vec{\xi})\mathfrak{t}\_{3}\right)\right\},\tag{38}$$

with an arbitrary rescaling by some parameter *μ* ∈ R. Let us denote the decomposed version of Γ ˆ *ξ* (*μ*) by ˜ Γ ˆ *ξ* (*μ*), with a semicolon representing a parametric dependence:

$$\tilde{\Gamma}\_{\tilde{\xi}}(\mu) = :\exp\left\{\beta\_{+}(\tilde{\xi};\mu)\vartheta\_{+}\right\}\exp\left\{\gamma(\tilde{\xi};\mu)\vartheta\_{3}\right\}\exp\left\{\beta\_{-}(\tilde{\xi};\mu)\vartheta\_{-}\right\}\tag{39}$$

Then we aim at determining the coefficient functions *β*+(*ξ*; *μ*), *γ*(*ξ*; *μ*) and *β*−(*ξ*; *μ*) such that we have Γ ˆ *ξ* (*μ*) = ˜ Γ ˆ *ξ* (*μ*). This rescaling allows us to differentiate Γˆ *ξ* (*μ*) and ˜ Γ ˆ *ξ* (*μ*) with respect to *μ*. Considering this we start with a consistency requirement for Γ ˆ *ξ* (*μ*) and ˜ Γ ˆ *ξ* (*μ*) given by:

$$\left(\frac{\partial}{\partial \mu} \mathbb{\hat{I}\_{\xi}(\mu)}\right) \left(\mathbb{\hat{I}\_{\xi}(\mu)}\right)^{\dagger} = \left(\frac{\partial}{\partial \mu} \mathbb{\hat{I}\_{\xi}(\mu)}\right) \left(\mathbb{\hat{I}\_{\xi}(\mu)}\right)^{\dagger}.\tag{40}$$

In the next step we will omit the arguments of the coefficient functions for the sake of a more compact notation. Explicitly evaluating the differentials and using the unitarity of Γ ˆ *ξ* , we end up with three contributions. A closer look reveals that these contributions contain the adjoint action of Γ ˆ *ξ* onto the three generators of the algebra in (36), which can be easily computed due to the simple structure of their commutators. The linear independence of the generators then leads to a coupled system of differential equations for the coefficient functions:

$$
\pi = \exp\{-2\gamma\} \frac{\partial \beta\_-}{\partial \mu} \tag{41}
$$

$$i\lambda = \frac{\partial \gamma}{\partial \mu} - \beta\_+ \exp\{-2\gamma\} \frac{\partial \beta\_-}{\partial \mu} \tag{42}$$

$$
\hbar \alpha = 2\beta\_+ \frac{\partial \gamma}{\partial \mu} - \frac{\partial \beta\_+}{\partial \mu} - \beta\_+^2 \exp\{-2\gamma\} \frac{\partial \beta\_-}{\partial \mu}.\tag{43}
$$

Performing a number of substitutions, this system of differential equations can be cast into the form of a complex Riccati-type ordinary differential equation, for more details we refer the reader to the explicit computations done in Reference [30]. An appropriate ansatz for this equation yields a solution, subsequent resubstitution then leads to the desired BCH coefficient functions of the normal-ordered decomposition of Γ ˆ *ξ* . A similar procedure can be performed for the normal and anti-normal ordering of both Γ ˆ *ξ* and Γ ˆ † *ξ* , respectively, while we have chosen that *σ*ˆ3 remains in the middle for computational convenience. Although depending on the given initial state <sup>Ψ</sup>(*q*, *<sup>t</sup>*0) at our disposal, the most useful forms of the coefficients (or operator orderings, respectively) regarding computational convenience are given by:

$$\begin{aligned} \delta\_{+}(\mu) &= + \frac{a \operatorname{sh}(\Delta \mu)}{\Delta \operatorname{ch}(\Delta \mu) + i \lambda \operatorname{sh}(\Delta \mu)} & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \pi\_{+}(\mu) = + \frac{a \operatorname{sh}(\Delta \mu)}{\Delta \operatorname{ch}(\Delta \mu) - i \lambda \operatorname{sh}(\Delta \mu)} \\ \delta\_{-}(\mu) &= - \frac{\overline{a} \operatorname{sh}(\Delta \mu)}{\Delta \operatorname{ch}(\Delta \mu) + i \lambda \operatorname{sh}(\Delta \mu)} & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad$$

$$\begin{aligned} \beta\_{+}(\mu) &= -\frac{a \operatorname{sh}(\Delta \mu)}{\Delta \operatorname{ch}(\Delta \mu) - i \lambda \operatorname{sh}(\Delta \mu)} & \varepsilon\_{+}(\mu) &= -\frac{a \operatorname{sh}(\Delta \mu)}{\Delta \operatorname{ch}(\Delta \mu) + i \lambda \operatorname{sh}(\Delta \mu)} \\ \beta\_{-}(\mu) &= +\frac{\overline{a} \operatorname{sh}(\Delta \mu)}{\Delta \operatorname{ch}(\Delta \mu) - i \lambda \operatorname{sh}(\Delta \mu)} & \varepsilon\_{-}(\mu) &= +\frac{\overline{a} \operatorname{sh}(\Delta \mu)}{\Delta \operatorname{ch}(\Delta \mu) + i \lambda \operatorname{sh}(\Delta \mu)} \\ \gamma(\mu) &= -\underbrace{\mathrm{ln}\left(\mathrm{ch}(\Delta \mu) - \frac{i \lambda}{\Delta} \operatorname{sh}(\Delta \mu)\right)}\_{\text{normal ordering of } \mathbb{F}\_{\mathbb{Z}}} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad$$

with Δ<sup>2</sup> := |*α*|<sup>2</sup> − *λ*<sup>2</sup> and Δ<sup>2</sup> > 0 for all real solutions *ξ*(*t*) of the Ermakov Equation (5). By fixing the parameter *μ* = 1, we recover the unitary transformation we initially started with. Let us note that for an initially time-independent Hamiltonian, the decomposed transformation reproduces the identity operator, as expected. This is due to the fact that in this case *ξ*(*t*) = 1, which in turn leads to a vanishing generator. Furthermore one can explicitly check that the adjoints of the decompositions of Γ ˆ *ξ* and Γ ˆ † *ξ* are the decompositions of the adjoints, which illustrates mutual consistency and conservation of unitarity among the obtained results. To briefly summarize this chapter, we have used analytical techniques to perform a decomposition of the exponentiated generator in (35) into three individual contributions. Due to the fact that we are working with unitary representations of the algebra of non-compact Lie group with mutually non-commuting elements, this result is nontrivial and enables the realization of computations in a compact form. Examples of applications of the Baker-Campbell-Hausdorff decomposition of Γ ˆ *ξ* can be found in Sections 3.3 and 6, respectively.

### *3.3. Time-Dependent Bogoliubov Maps*

In this section we will show that the transformation induced by the operator Γ ˆ *ξ* can be understood as a time-dependent Bogoliubov transformation when applied to the ladder operators. Given the action of Γ ˆ *ξ* on the elementary position and momentum operators, it can naturally be extended to the ladder operators as well. The same applies also to the adjoint action, which is however tedious to evaluate in the original form of the generator. Due to the possibility of decomposing the operator Γ ˆ *ξ* and its adjoint, we can take advantage of the result in the last section and compute the action on *A* ˆ and *A* ˆ † with a normal and anti-normal ordered decomposition, respectively.

Using the commutator structure of the generators in (36), we obtain:

$$\operatorname{Ad}\_{\hat{\Gamma}\_{\xi}}(\hat{A}) = \mathfrak{e}^{-\gamma(\xi)} (\hat{A} - \mathfrak{E}\_{+}(\xi)\hat{A}^{\dagger}),\tag{44}$$

$$\operatorname{Ad}\_{\hat{\mathbb{L}}\_{\vec{\mathbb{L}}}}(\vec{A}^{\dagger}) = e^{\iota(\vec{\xi})} (\vec{A}^{\dagger} + \varepsilon\_{-}(\vec{\xi})\vec{A}),\tag{45}$$

$$\operatorname{Ad}\_{\hat{\mathbb{F}}\_{\tilde{\xi}}^{\star}}(\hat{A}) = \mathfrak{e}^{-\nu(\tilde{\xi})} \left( \hat{A} - \delta\_{+}(\xi)\hat{A}^{\dagger} \right), \tag{46}$$

$$\operatorname{Ad}\_{\hat{\mathbb{L}}^{\dagger}\_{\hat{\mathbb{L}}}}(\hat{A}^{\dagger}) = e^{\varrho(\hat{\mathbb{L}})} (\hat{A}^{\dagger} + \tau\_{-}(\hat{\mathbb{x}})\hat{A}),\tag{47}$$

with the corresponding coefficient functions derived in the preceding section. These functions carry an explicit time dependence via *ξ*(*t*), which is a solution of the Ermakov Equation (5) with the time-dependent frequency of the initial Hamiltonian. In fact, the transformations of the ladder operators in (47) look already close to that of a time-dependent Bogoliubov transformation. Whether this is indeed the case depends on the coefficient functions involved and will be analyzed in the following. For this purpose, let us rewrite the action of Γˆ *ξ* on these operators as a 2 × 2 matrix representation, considering the Equations (44)–(47):

$$
\begin{pmatrix}
\mathcal{g}\left(\underline{\boldsymbol{\xi}},\boldsymbol{\xi}\right) & \mathsf{\widetilde{h}}\left(\underline{\boldsymbol{\xi}},\boldsymbol{\xi}\right) \\
h\left(\underline{\boldsymbol{\xi}},\boldsymbol{\xi}\right) & \overline{\boldsymbol{\mathfrak{g}}}\left(\underline{\boldsymbol{\xi}},\boldsymbol{\xi}\right)
\end{pmatrix}
\begin{pmatrix}
\boldsymbol{A} \\
\boldsymbol{A}^{\dagger}
\end{pmatrix} = \begin{pmatrix}
\mathcal{g}\left(\underline{\boldsymbol{\xi}},\boldsymbol{\xi}\right)\boldsymbol{\mathring{A}} + \mathsf{\widetilde{h}}\left(\underline{\boldsymbol{\xi}},\boldsymbol{\xi}\right)\boldsymbol{\mathring{A}}^{\dagger} \\
h\left(\underline{\boldsymbol{\xi}},\boldsymbol{\xi}\right)\boldsymbol{\mathring{A}} + \overline{\boldsymbol{\mathfrak{g}}}\left(\underline{\boldsymbol{\xi}},\boldsymbol{\xi}\right)\boldsymbol{\mathring{A}}^{\dagger}
\end{pmatrix} := \begin{pmatrix}
\mathcal{B} \\
\mathcal{B}^{\dagger}
\end{pmatrix},\tag{48}
$$

with the additional requirement that if [*A*ˆ, *A*ˆ †] = 1H, then similarly [*B*ˆ, *B*ˆ †] = 1H needs to hold, usually required for a Bogoliubov transformation. In order to achieve this, we need to impose the condition that the determinant of the matrix on the left-hand side of (48) involving *g*(*ξ*, ˙ *ξ*) and *h*(*ξ*, ˙ *ξ*) is equal to one, which amounts to:

$$\left[\left[\mathcal{B},\mathcal{B}^\dagger\right] = \mathbbm{1}\_{\mathcal{H}} \quad \Longleftrightarrow \quad \left|\mathcal{g}\left(\xi',\xi\right)\right|^2 - \left|h(\xi',\xi)\right|^2 = 1.$$

Applying this to the transformation in (44), we get:

$$
\begin{pmatrix}
\mathfrak{g}\left(\xi,\xi\right) & \widetilde{h}\left(\xi,\xi\right) \\
h\left(\xi,\xi\right) & \overline{\mathfrak{z}}\left(\xi,\xi\right)
\end{pmatrix} = \begin{pmatrix}
e^{-\gamma\left(\xi\right)} & -e^{-\gamma\left(\overline{\xi}\right)}\beta\_{+}\left(\xi\right) \\
\end{pmatrix} \quad \Longrightarrow \quad e^{-\left(\gamma+\overline{\gamma}\right)}\left(1-\left|\beta\_{+}\right|^{2}\right) \stackrel{!}{=} 1.
$$

Given the explicit functional form of the BCH coefficients, it can be easily shown that Γˆ *ξ* indeed describes a time-dependent Bogoliubov transformation and the expression above equals one. For all remaining cases this can be also shown using the same method. We are now in a situation where we can formulate the time evolution of *A*ˆ, *A*ˆ † in the Heisenberg picture, using the time-evolution operator *U*ˆ (*<sup>t</sup>*0, *t*). It consists of the aforementioned Bogoliubov map together with an exponential operator involving the Lewis-Riesenfeld invariant or the autonomous Hamiltonian, respectively. The additional exponent also carries the information of the time-rescaling encoded in the function *ξ*(*t*) and hence is sensitive to the underlying spacetime geometry. In the following, we introduce the following notation for the coefficient functions *β*+(*ξ*(*t*)) = *β*+(*ξ*) and *β*+(*ξ*(*<sup>t</sup>*0)) := *β*+(*ξ*0) involved in the decomposition of Γˆ *ξ* and Γˆ *ξ*,0, respectively. Carefully applying *U*ˆ (*<sup>t</sup>*0, *t*) and collecting everything together, we obtain:

$$\begin{split} \dot{A}\_{H}(t\_{0},t) &= \exp\left\{-\left(\gamma(\xi) + i\omega\_{0}\int\_{t\_{0}}^{t} \frac{\mathbf{d}\tau}{\xi^{2}(\tau)} + \nu(\xi\_{0})\right)\right\} \Big(\hat{A} - \delta\_{+}(\xi\_{0})\dot{A}^{\dagger}\Big) \\ &- \exp\left\{-\left(\gamma(\xi) - i\omega\_{0}\int\_{t\_{0}}^{t} \frac{\mathbf{d}\tau}{\xi^{2}(\tau)} + \bar{\nu}(\xi\_{0})\right)\right\} \beta\_{+}(\xi) \Big(\hat{A}^{\dagger} - \delta\_{+}(\xi\_{0})\dot{A}\Big), \end{split} \tag{49}$$

$$\begin{split} \boldsymbol{A}\_{H}^{\dagger}(t\_{0},\boldsymbol{t}) &= \exp\left\{-\left(\boldsymbol{\bar{\gamma}}(\boldsymbol{\xi})-i\boldsymbol{\omega}\_{0}\int\_{t\_{0}}^{t} \frac{\mathbf{d}\boldsymbol{\tau}}{\boldsymbol{\xi}^{2}(\boldsymbol{\tau})} + \boldsymbol{\nu}(\boldsymbol{\xi}\_{0})\right)\right\} \Big(\boldsymbol{A}^{\dagger} - \boldsymbol{\delta}\_{+}(\boldsymbol{\xi}\_{0})\boldsymbol{A}\Big) \\ &- \exp\left\{-\left(\boldsymbol{\bar{\gamma}}(\boldsymbol{\xi})+i\boldsymbol{\omega}\_{0}\int\_{t\_{0}}^{t} \frac{\mathbf{d}\boldsymbol{\tau}}{\boldsymbol{\xi}^{2}(\boldsymbol{\tau})} + \boldsymbol{\nu}(\boldsymbol{\xi}\_{0})\right)\right\} \boldsymbol{\beta}\_{+}(\boldsymbol{\xi}) \Big(\boldsymbol{A} - \boldsymbol{\delta}\_{+}(\boldsymbol{\xi}\_{0})\boldsymbol{A}^{\dagger}\Big). \end{split} \tag{50}$$

Although the expressions (49) and (50) look rather complicated at first, as expected they reduce to *A* ˆ , *A* ˆ † in the limit *t* → *t*0. This can be seen by replacing *t* by *t*0 in the expression above and using the definitions of the Baker-Campbell-Hausdorff coefficients from Section 3.2. One finally observes that all contributions apart from *A* ˆ or *A* ˆ †, respectively, cancel upon inserting the definition of Δ and using the fact that cosh<sup>2</sup>(Δ) − sinh<sup>2</sup>(Δ) = 1. In principle, these results allow to compute expectation values for various initial conditions and investigate the behavior of these operators for the single-mode Mukhanov-Sasaki equation. We will discuss some application of this framework in Section 6, where we consider the derived unitary map for the single-mode Mukhanov-Sasaki equation in the context of quasi-de Sitter spacetimes. Prior to that, in the next section we will discuss whether the results obtained so far can be carried over to field theory, that is whether the obtained unitary map can be extended to the bosonic Fock space.

### **4. Implementing the Time-Dependent Canonical Transformation as a Unitary Map on the Bosonic Fock Space**

For the reason that we were able to construct a unitary map for the toy model of the single-model Mukhanov-Sasaki Hamiltonian, the next obvious step is to aim at a unitary implementation of the time evolution operator *U* ˆ (*<sup>t</sup>*0, *t*) on the full Fock space F. Since every mode of the Mukhanov-Sasaki equation is a time-dependent harmonic oscillator, we need to treat every mode separately and with a different frequency, depending on the absolute value of **k**. Hence it is natural to equip the solution of the Ermakov equation, which also differs from mode to mode for precisely this reason, with a corresponding mode label, which in turn carries over to the time-dependent Bogoliubov transformation Γ ˆ *ξ* . In the conventional formalism, the Mukhanov-Sasaki Hamiltonian and the mode expansion of the Mukhanov-Sasaki variable and its conjugate momentum are of the form:

$$\hat{H}(\eta) = \frac{1}{2} \int \mathrm{d}^3 \mathbf{x} \left( \mathfrak{H}\_v^2(\eta, \mathbf{x}) + \left( \partial\_\mathbf{a} \mathfrak{H}(\eta, \mathbf{x}) \right) \left( \partial^a \mathfrak{H}(\eta, \mathbf{x}) \right) - \frac{z''(\eta)}{z(\eta)} \hat{v}^2(\eta, \mathbf{x}) \right), \tag{51}$$

$$\psi(\eta, \mathbf{x}) = \int \frac{\mathbf{d}^3 k}{(2\pi)^3} \left( v\_\mathbf{k}(\eta) \hbar\_\mathbf{k} \exp\{i \mathbf{k} \cdot \mathbf{x}\} + \overline{v}\_\mathbf{k}(\eta) \hbar\_\mathbf{k}^\dagger \exp\{-i \mathbf{k} \cdot \mathbf{x}\} \right), \tag{52}$$

$$i\hbar\_v(\boldsymbol{\eta}, \mathbf{x}) = \int \frac{d^3k}{(2\pi)^3} \left(\partial\_0 \boldsymbol{\upsilon}\_{\mathbf{k}}(\boldsymbol{\eta}) \hbar\_{\mathbf{k}} \exp\{i\mathbf{k}\cdot\mathbf{x}\} + \partial\_0 \overline{\boldsymbol{\upsilon}}\_{\mathbf{k}}(\boldsymbol{\eta}) \hbar\_{\mathbf{k}}^\dagger \exp\{-i\mathbf{k}\cdot\mathbf{x}\}\right),\tag{53}$$

with *∂*0 denoting a derivative with respect to conformal time *η*, *a*(*η*) is the scale factor, H is the conformal Hubble function and *φ*¯ stands for the homogeneous and isotropic part of the inflaton scalar field. Given the canonical commutator [*v*<sup>ˆ</sup>(*η*, **<sup>x</sup>**), *π*ˆ *<sup>v</sup>*(*η*, **y**)] = *<sup>i</sup>δ*(3)(**<sup>x</sup>**, **y**) H together with the mode expansion of *<sup>v</sup>*<sup>ˆ</sup>(*η*, **x**) and *π*ˆ *<sup>v</sup>*(*η*, **x**) as well as the following choice for the Wronskian

$$\mathcal{W}(\upsilon\_{\mathbf{k}'}\overline{\upsilon}\_{\mathbf{k}}) := \upsilon\_{\mathbf{k}}\overline{\upsilon}\_{\mathbf{k}}' - \upsilon\_{\mathbf{k}}'\overline{\upsilon}\_{\mathbf{k}} = i,\tag{54}$$

the corresponding annihilation and creation operators satisfy the commutator algebra

$$[\pounds\_{\mathbf{k}\prime} \pounds\_{\mathbf{m}}^{\dagger}] = (2\pi)^3 \delta^{(3)}(\mathbf{k}\prime \mathbf{m}) \mathbb{I}\_{\mathcal{H}\prime}$$

where all remaining commutators vanish. Compared to the one-particle case we obtain an additional factor of (<sup>2</sup>*π*)<sup>3</sup> here, which in principle needs to be considered when deriving the corresponding Bogoliubov coefficients in (44)–(47) for the field theory case. In order to avoid to include appropriate powers of 2 *π* in the derivation of the Bogoliubov coefficients, as an intermediate step we rescale the creation and annihilation operators such that they satisfy a commutator algebra that involves just the *δ*-function. This yields:

$$
\hat{A}\_{\mathbf{k}} := (2\pi)^{-\frac{3}{2}} \hbar\_{\mathbf{k}\prime} \quad \hat{A}\_{\mathbf{k}}^{\dagger} := (2\pi)^{-\frac{3}{2}} \hbar\_{\mathbf{k}\prime}^{\dagger} \quad [\hat{A}\_{\mathbf{k}\prime} \hat{A}\_{\mathbf{m}}^{\dagger}] = \delta^{(3)}(\mathbf{k}, \mathbf{m}) \, \mathbf{1}\_{\hat{\mathbb{M}}}.\tag{55}
$$

Note that we consider a quantization of the inflaton perturbation in the context of quantum field theory on a curved background, where the background quantities are considered as external quantities and we thus neglect any backreaction effects.

Now we can let the Bogoliubov transformation act on the rescaled operators *<sup>A</sup>*<sup>ˆ</sup>**k**, *A*ˆ † **k** and all results obtained in the previous Section 3.3 can be easily carried over to the field theoretic case, where the Bogoliubov transformation maps *<sup>A</sup>*<sup>ˆ</sup>**k**, *A*ˆ † **k** to a new set of creation and annihilation operators *B*ˆ **k**, *B*ˆ † **k** that fulfill the same rescaled commutation relation. Due to the linearity of the Bogoliubov transformation, the rescaling affects both sides of the equation and thus can be easily removed and the standard algebra we started with is restored. In the field theory the generator in the exponential of Γˆ **k** *ξ* is smeared with the Baker-Campbell-Hausdorff coefficient functions that act as the smearing functions. The action on the operators *<sup>A</sup>*<sup>ˆ</sup>**k**, *A*ˆ † **k** is then diagonal, because at each order of the iterated commutator, the to be found Dirac distributions can be absorbed into the integral involved due to the smearing. Hence, the generalization of Bogoliubov coefficients we obtained in the one-particle case in (48) to the field theory case just consists of equipping them with a mode label. The questions that still needs to answered is whether the so defined extension of Γˆ **k** *ξ* to Fock space describes a unitary map on the latter. Fortunately, there exists a criterion whether a given Bogoliubov transformation can be unitarily implemented on Fock space, called the *Shale-Stinespring* condition. A review on the Shale-Stinespring condition with a sketched proof can be for example found in Reference [31]. The theorem essentially states that the anti-linear part of the Bogoliubov transformation under consideration needs to be a Hilbert-Schmidt operator. In our case, this condition carries over to the product of the off-diagonal coefficients in Equation (48) being bounded when integrated over all of R3:

$$\int\_{\mathbb{R}^3} \mathrm{d}^3 k \, h\_{\mathbf{k}}(\underline{\mathfrak{x}}, \underline{\mathfrak{x}}') \overline{h}\_{\mathbf{k}}(\underline{\mathfrak{x}}, \underline{\mathfrak{x}}') < \infty, \quad \overline{h}\_{\mathbf{k}}(\underline{\mathfrak{x}}, \underline{\mathfrak{x}}') = -\exp\left\{-\gamma(\underline{\mathfrak{x}}\_{\mathbf{k}})\right\} \beta\_{+}(\underline{\mathfrak{x}}\_{\mathbf{k}}),\tag{56}$$

where *<sup>γ</sup>*(*ξ***k**) and *β*+(*ξ***k**) are the Baker-Campbell-Hausdorff coefficients from the decomposition in Section 3.2, *ξ***k**(*t*) is the mode-dependent solution of the Ermakov equation and *ξ* **k** is the derivative with respect to conformal time. At this point the advantage of rescaling the operator algebra becomes evident, since we can copy our results from previous computations of the Bogoliubov coefficients. A discussion on the initial conditions regarding the solutions *ξ***k**(*t*) can be found in Section 6 below, which will provide the basis for the investigations of the finiteness of the integral over the anti-linear part of Γˆ **k** *ξ* . Explicitly inserting the coefficients while still keeping *ξ***k**(*η*) in the arguments and considering the rescaled operators such that they satisfy the same commutator algebra as in Section 3.3 leads to:

$$\left|h\_{\mathbf{k}}(\mathbf{\tilde{t}}',\mathbf{\tilde{t}}')\right|^{2} = \left(\operatorname{ch}^{2}(\Delta\_{\mathbf{k}}) + \frac{\lambda\_{\mathbf{k}}^{2}}{\Delta\_{\mathbf{k}}^{2}}\operatorname{sh}^{2}(\Delta\_{\mathbf{k}})\right)\frac{|a\_{\mathbf{k}}|^{2}\operatorname{sh}^{2}(\Delta\_{\mathbf{k}})}{\Delta\_{\mathbf{k}}^{2}\operatorname{ch}^{2}(\Delta\_{\mathbf{k}}) + \lambda\_{\mathbf{k}}^{2}\operatorname{sh}^{2}(\Delta\_{\mathbf{k}})} = \frac{|a\_{\mathbf{k}}|^{2}\operatorname{sh}^{2}(\Delta\_{\mathbf{k}})}{|a\_{\mathbf{k}}|^{2} - \lambda\_{\mathbf{k}}^{2}}.$$

*Universe* **2019**, *5*, 170

Given the former definition of *α*, *λ* and Δ we consider the extension of these quantities to the multi-mode case given by Δ**k** = 4<sup>|</sup>*<sup>α</sup>***k**|<sup>2</sup> − *<sup>λ</sup>*2**k**. Inserting the explicit form of *α* and *λ* from Equation (37) we end up with:

$$
\Delta\_{\mathbf{k}}^{2} = |a\_{\mathbf{k}}|^{2} - \lambda\_{\mathbf{k}}^{2} = \left| \ln(\boldsymbol{\zeta}\_{\mathbf{k}}) \left( 1 - \frac{i\boldsymbol{h}(\boldsymbol{\zeta}\_{\mathbf{k}})}{2\left(\boldsymbol{\omega}\_{\mathbf{k}}^{(0)}\right)^{2}} \right) \right|^{2} - \left( \frac{\boldsymbol{h}(\boldsymbol{\zeta}\_{\mathbf{k}})}{2\left(\boldsymbol{\omega}\_{\mathbf{k}}^{(0)}\right)^{2}} \ln(\boldsymbol{\zeta}\_{\mathbf{k}}) \right)^{2} = \ln^{2}(\boldsymbol{\zeta}\_{\mathbf{k}}).
$$

Note that ln<sup>2</sup>(*ξ***k**) > 0 for all modes **k** ∈ R<sup>3</sup> with "**k**" = 0 and all conformal times *η* ∈ R− \ {0}, from which we can conclude that Δ**k** = ln(*ξ***k**) since we already know that Δ**k** > 0 holds. Explicitly substituting the definitions of *αk*, *λk* and Δ*k* into *vk*(*ξ*, ˙*ξ*), we arrive at the following integral for the de Sitter case with *ξ*(*η*) as derived in the succeeding Section 6:

$$\int\_{\mathbb{R}^3} \mathbf{d}^3 k \, |h\_{\mathbf{k}}(\mathbf{j}, \mathbf{\xi}')|^2 = \int\_{\mathbb{R}^3} \mathbf{d}^3 k \left( 1 + \frac{1}{\left(\omega\_{\mathbf{k}}^{(0)}\right)^2} \left[ \frac{\mathfrak{f}\_{\mathbf{k}}(\eta)\mathfrak{f}\_{\mathbf{k}}^{\prime}(\eta)}{1 - \mathfrak{f}\_{\mathbf{k}}^2(\eta)} \right]^2 \right) \left( \frac{1}{2} \frac{\mathfrak{f}\_{\mathbf{k}}^2(\eta) - 1}{\mathfrak{f}\_{\mathbf{k}}(\eta)} \right)^2$$

$$= \frac{1}{4} \int\_{\mathbb{R}^3} \mathbf{d}^3 k \left[ \left( \frac{\mathfrak{f}\_{\mathbf{k}}^2(\eta) - 1}{\mathfrak{f}\_{\mathbf{k}}(\eta)} \right)^2 + \left( \frac{\mathfrak{f}\_{\mathbf{k}}^{\prime}(\eta)}{\omega\_{\mathbf{k}}^{(0)}} \right)^2 \right]$$

$$= \frac{1}{4} \int\_{\mathbb{R}^3} \mathbf{d}^3 k \left[ \left( \frac{(k\eta)^2}{1 + (k\eta)^2} \frac{1}{(k\eta)^4} \right) + \frac{1}{k^2} \left( \frac{(k\eta)^2}{1 + (k\eta)^2} \frac{1}{k^4 \eta^6} \right) \right].$$

This expression allows us to consider a simple power-counting procedure of the individual contributions. For large *k*, the first term behaves as *k*−<sup>4</sup> whereas the second term decays as *k*−6, so there is no divergence in the ultraviolet. For small *k* we observe the first term to be proportional to *k*−<sup>2</sup> and the second contribution to *k*−4, which leads to an infrared divergence of the latter, which in turn shows that the integral above is not finite. Consequently, the Shale-Stinespring condition is not satisfied in our case and the Bogoliubov transformation Γ ˆ *ξ* cannot be unitarily implemented on Fock space by simply extending the toy model of the single-mode case to the multi-mode case due to 'infinite particle production' between mutually different vacuum states. Interestingly, there is no issue with the ultraviolet here but just in the infrared sector, showing that next to the large *k* behavior one also needs to check whether there are occurring singularities in the infrared, as they can equally add a diverging contributions to the number operator expectation value with respect to different vacua. It is not obvious to us that this aspect has been considered in the recent work of Reference [15], where a similar Bogoliubov transformation is used on Fock space. As can be seen form our analysis, the behavior of the BCH coefficients is different for small *k* than it is for large *k*, hence it is not obvious that even if the Bogoliubov coefficients are finite for large *k* this is a sufficient check in order to conclude that the Bogoliubov transformation under consideration can be unitarily implemented on Fock space.

As discussed before at the end of Section 3.1, the most common transformation in the literature in the context of the Lewis-Riesenfeld invariant is the one where the time-dependent Hamiltonian is mapped to the Hamiltonian of a harmonic oscillator with frequency *ω*(0) *k* = 1. For this reason we also analyze what happens to the Shale-Stinespring condition if we do not require the time-independent frequency to be just *ω*(0) **k** = *k* but unity instead. This changes the solution *ξ***k**(*η*) by an additional factor of *k*− 12 if we impose similar initial conditions, that is *ξ*(*sq*) **k** (*η*) = *k*− 12 *ξ***k**(*η*) and leads to the residual squeezing transformation in Γ ˆ **k** *ξ* in the limit of past conformal infinity already mentioned in Section 3.1. Considering this modification in *ξsq***k** compared to *ξ***<sup>k</sup>**, we can also analyze whether the Shale-Stinespring conditions is satisfied here. We have:

$$\begin{split} \int\_{\mathbb{R}^3} \mathbf{d}^3 k \, |h\_{\mathbf{k}}(\xi^{(sq)}, \mathfrak{f}\_{\mathbf{k}}^{(sq)})|^2 &= \frac{1}{4} \int\_{\mathbb{R}^3} \mathbf{d}^3 k \left[ \left( \frac{(\mathfrak{f}\_{\mathbf{k}}^{(sq)}(\eta))^2 - 1}{\mathfrak{f}\_{\mathbf{k}}^{(sq)}(\eta)} \right)^2 + \left( \mathfrak{f}\_{\mathbf{k}}^{(sq)}(\eta) \right)^2 \right] \\ &= \frac{1}{4} \int\_{\mathbb{R}^3} \mathbf{d}^3 k \left[ \left( \mathfrak{f}\_{\mathbf{k}}^{(sq)} \right)^{-2} \left( \frac{1}{k^3 \eta^2} - \frac{k - 1}{k} \right)^2 + \frac{1}{k} \left( \frac{(k\eta)^2}{1 + (k\eta)^2} \frac{1}{k^4 \eta^6} \right) \right] \\ &= \frac{1}{4} \int\_{\mathbb{R}^3} \mathbf{d}^3 k \left[ k \left( \frac{(k\eta)^2}{1 + (k\eta)^2} \right) \left( \frac{1}{k^3 \eta^2} - \frac{k - 1}{k} \right)^2 + \frac{1}{k} \left( \frac{(k\eta)^2}{1 + (k\eta)^2} \frac{1}{k^4 \eta^6} \right) \right]. \end{split}$$

We apply a similar power counting to the two terms involved in the last line separately. For small *k* the second summand in that line decays as *k*−3, whereas it is proportional to *k*−<sup>5</sup> in the limit of large *k*, which yields a finite contribution in the ultraviolet and an infrared divergence. Similarly, in the small *k* region at lowest order, the first summand behaves as *k*−<sup>3</sup> and thus is divergent in the infrared. Furthermore, it increases linearly in *k* in the large *k* limit causing a divergence in the ultraviolet. As a consequence, also the transformation associated with *ξ*(*sq*) **k** is not unitarily implementable on Fock space, just as it was the case with *ξ***<sup>k</sup>**. However, there is a subtle distinction between the two cases. For *ξ***k** we found that the infrared modes lead to a divergence, whereas this time both small and large values of "**k**" are problematic. Let us understand a bit more in detail why it is expected that the infrared modes can be problematic in the case of the map corresponding to *ξ***<sup>k</sup>**. This map transforms the time-dependent harmonic oscillator Hamiltonian with frequency *<sup>ω</sup>*2**k**(*η*) = *k*2 − *z*(*η*) *z*(*η*) into the Hamiltonian of the harmonic oscillator with constant frequency *ω*(0) **k** = *k*. Hence, for **k** = 0 the latter corresponds to the Hamiltonian of a free particle because here the frequency just vanishes. This aspect has not been carefully taken into account in the map constructed so far. Therefore, in the next section we will discuss how the map constructed up to now could be modified for the low "**k**" modes such that the infrared singularity can be avoided. For the map corresponding to *ξ***<sup>k</sup>**, this attempt is possible because the problematic behavior of these modes constitutes only a compact domain in the space of modes, in contrast to the additional ultraviolet divergence involved in the map associated with *ξ*(*sq*) **k** . For this purpose, we will introduce the so-called Arnold transformation that has been used already in Reference [16] at the quantum level, which is designed to perform a mapping to the free particle Hamiltonian.

#### *Proposal of a Modified Map for the Infrared Modes: The Arnold Transformation*

In the previous section we have observed that the map Γ ˆ *ξ* that maps the time-dependent harmonic oscillator system onto the system of a harmonic oscillator with constant frequency *ω*(0) **k** = *k* with *k* := "**k**" is not a unitary operator on Fock space due to an infrared divergence that occurs in the off-diagonal trace of the Bogoliubov coefficients for the infrared modes. Given the fact that no ultraviolet singularities arise, the strategy we will follow in this section is to consider a modification of the map induced by Γ ˆ *ξ* for a finite spherical neighbourhood 1 "**<sup>k</sup>**" > 0 of the infrared modes in such a way that no infrared singularities occur. As mentioned above, the natural target Hamiltonian we should map to in the case of the zero mode is the Hamiltonian of a free particle. At the classical level this so-called Arnold transformation [32] was introduced in order to transform a generic second-order differential equation, that physically describes a driven harmonic oscillator with time-dependent friction coefficient and time-dependent frequency, into the differential equation corresponding to the motion of a free particle. Its implementation as a unitary map at the quantum level has been investigated for instance in Reference [16]. From our approach we can make an immediate connection to this formalism by going back to Equation (15) that has played an important role in deriving our classical transformation. Now if we aimed at mapping the original time-dependent Hamiltonian *<sup>H</sup>*(*η*) onto the free particle Hamiltonian, *ξ*(*η*) would need to satisfy the harmonic equation of motion with the

frequency *<sup>ω</sup>***k**(*η*) for each mode instead of the Ermakov equation, as it was presented in our case before. As already discussed in Reference [16], one can recover the Ermakov equation when considering three physical systems, a time-dependent and a time-independent harmonic oscillator together with the free particle. Then one constructs the two Arnold transformations that relate the time-dependent and the time-independent harmonic oscillator to the free particle. From combining one of these Arnold transformation with the inverse of the second one, one obtains a map that relates the systems of the time-dependent harmonic oscillator with the time-independent one via a time-rescaling. For more details regarding this aspect we refer the reader to the presentation in Reference [16]. In order to be able to discuss the approach from Reference [16] and ours in parallel, we will denote the time-rescaling function associated with the Arnold transformation by Θ**k** (*η*). The non-zero "**<sup>k</sup>**" modes lead to the well-known solutions of the Mukhanov-Sasaki equation for finite "**k**", whereas for the **k** = 0 mode we need to find appropriate solutions. By construction, Θ**k** (*η*) satisfies the Ermakov equation with vanishing *ω*(0) **k** , that is the time-dependent harmonic oscillator equation of the associated mode, given by:

$$
\Theta\_{\mathbf{k}\_\varepsilon}''(\eta) + \omega\_{\mathbf{k}\_\varepsilon}^2(\eta)\Theta\_{\mathbf{k}\_\varepsilon}(\eta) = 0,
$$

where we are interested in the case where *<sup>ω</sup>***k** (*η*) is determined by the Mukhanov-Sasaki equation. If we compare the time rescaling in Equation (11) with the one given in Reference [16], we obtain an exact agreemen<sup>t</sup> if we take into account that the Wronskians of two solutions of the time-dependent and time-independent harmonic oscillator, respectively, are constant and can be chosen to be identical. Since in our work the physical system under consideration is described by a time-dependent harmonic oscillator (that is, the Mukhanov-Sasaki equation), let us first consider this equation for arbitrary modes **k** and prior to any gauge-fixing:

$$v\_{\mathbf{k}}''(\eta) + \left(\|\mathbf{k}\|\|^2 - \frac{z''(\eta)}{z(\eta)}\right) v\_{\mathbf{k}}(\eta) = 0,\tag{57}$$

,

For the particular case of a quasi-de Sitter spacetime, the explicit form of this equation can be given in terms of the so-called slow-roll parameters. The Friedmann equations together with the Klein-Gordon equation describing the dynamics of the background scalar field *φ* on a slow-rolling quasi-de Sitter background can be used to rewrite the Mukhanov-Sasaki equation in a convenient way. For this purpose we define a set of three slow-roll parameters *ε*, *τ* and *κ*, that describe the fractional change of *H* ˙ per Hubble time, the fractional chance of *ε* per Hubble time as well as the fractional change of *τ* per Hubble time, respectively:

$$
\varepsilon = -\frac{\dot{H}}{H^2}, \quad \tau = \frac{\dot{\varepsilon}}{\varepsilon H'}, \quad \kappa = \frac{\tau}{\tau H}
$$

where a dot denotes a derivative with respect to cosmological time in these expressions. Inserting the Klein-Gordon equation and using the Friedmann equations along with an assumed subdominance of the second-order derivatives of *φ*, we can rewrite *z*(*η*) and its derivatives in terms of *ε*, *τ* and *κ*. By truncating the resulting expressions after the first order in the slow-roll parameters, the time-dependent part of the frequency *<sup>ω</sup>k*(*η*) becomes:

$$z = \frac{a^2 \varepsilon}{4\pi \prime}, \quad \frac{z'}{z} = \mathcal{H} \left( 1 + \frac{\tau}{2} \right), \quad \frac{z''}{z} = \frac{1}{\eta^2} (1 + \varepsilon)^2 \left( 2 - \varepsilon + \frac{3\tau}{2} \right) \approx \frac{4\nu^2 - 1}{4\eta^2}, \quad \nu = \frac{3}{2} + \varepsilon + \frac{\tau}{2}.$$

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Thus, the Mukhanov-Sasaki equation up to first order in the slow-roll parameters for a quasi-de Sitter background reads:

$$v\_{\mathbf{k}}^{\prime\prime}(\eta) + \left(\left\|\mathbf{k}\right\|^2 - \frac{4\nu^2 - 1}{4\eta^2}\right) v\_{\mathbf{k}}(\eta) = 0. \tag{58}$$

Given the equation above we can read off the time-dependent frequency that we considered for the time-dependent harmonic oscillator in our single-mode toy model approach. This is also precisely the equation that <sup>Θ</sup>(*η*) needs to satisfy for a given but finite "**<sup>k</sup>**". Let us emphasize that the solutions to Equation (58) need to be computed separately for vanishing and non-vanishing "**k**", respectively. The real-valued solutions for "**k**" > 0 are given by the Bessel functions of first and second kind, for details the reader is referred to Section 6. If we consider the limiting case of **k** = 0 in the context of the quantum Arnold transformation, we obtain the following linear differential equation with time-dependent coefficients for the rescaling function Θ**k** (*η*), omitting the label for the zero mode:

$$\Theta''(\eta) - \frac{4\nu^2 - 1}{4\eta^2} \Theta(\eta) = 0 \qquad \Longleftrightarrow \qquad \eta^2 \Theta''(\eta) - \left(\nu^2 - \frac{1}{4}\right) \Theta(\eta) = 0 \qquad \text{for} \qquad \|\mathbf{k}\| = 0.$$

This differential equation with time-dependent coefficients can be transformed into an equation with constant coefficients, which then again can be solved by means of the substitution *y* = ln(|*η*|) and an exponential ansatz of this new variable incorporating the dependence on the effective slow-roll parameter *ν*. The general solution of this differential equation is given by:

$$\Theta(\eta) = \mathfrak{c}\_1 |\eta|^r + \mathfrak{c}\_2 |\eta|^s, \quad \text{with} \quad r, \mathfrak{s} = \frac{1}{2} \left( 1 \pm \sqrt{4\nu^2} \right) \quad \text{for} \quad \nu^2 > 0 \tag{59}$$

From this solution we readily obtain two linearly independent solutions Θ1, Θ2 that can be used to construct the Arnold transformation for the **k** = 0 mode. Note that the differential equation above can be also solved for *ν* = 0 or *ν*2 < 0, respectively. However, according to the parameter space of the slow-roll parameters in Reference [33], this range is not physically reasonable and hence we only use the result for strictly positive, real-valued slow-roll parameters. Due to the range of conformal time (*η* ∈ R− \ {0}), the relevant solution here is the growing branch proportional to |*η*| *s* with *s* < 0, since the decreasing branch diverges in the limit of past conformal infinity. This then coincides with the choice of the final time-rescaling transformation suggested in Reference [16]. We reconsider the form of the transformation Γˆ *ξ* and insert the corresponding solution for Θ**k** (*η*) to obtain an analogous transformation Γˆ <sup>Θ</sup>**k** by means of which we can transform the Schrödinger equation similar to Equation (31) and according to:

$$i\frac{\partial}{\partial t}\Psi(q,t) = \frac{1}{2}\Big(\pi\_q(\not p)^2 + \omega\_{\mathbf{k}\_\ell}^2(t)\pi\_q(\not q)^2\Big)\Psi(q,t) \quad \Longleftrightarrow \quad \left(\frac{1}{2\Theta\_{\mathbf{k}\_\ell}^2}\pi\_q(\not p)^2 - i\frac{\partial}{\partial t}\right)\Gamma\_{\Theta\_{\mathbf{k}\_\ell}}\Psi(q,t) = 0.$$

This means that Γˆ <sup>Θ</sup>**k** maps the time-dependent Hamiltonian for the **k** mode into the time-independent Hamiltonian of the free particle modulo a rescaling of the momentum operator. It is important to emphasize that this unitary map can only be performed at the level of the full Schrödinger equation, as otherwise the spectrum of the two related operators would have to be equivalent, which is clearly not the case for the time-dependent harmonic oscillator and the free particle. It is the time-derivative in the Schrödinger equation that is crucial for removing the term proportional to *<sup>π</sup>q*(*q*<sup>ˆ</sup>)<sup>2</sup> altogether. Furthermore we would like to stress that the time rescaling function Θ**k** is different for "**<sup>k</sup>**" > 0 and "**<sup>k</sup>**" = 0, respectively. In the first case, depending on the imposed initial conditions, it is given by the Bessel functions of first and second kind *Jν*(<sup>−</sup> **<sup>k</sup>***η*) and *<sup>Y</sup>ν*(<sup>−</sup> **<sup>k</sup>***η*). In the latter case, Θ**k** corresponds to the above power-law solution. Unfortunately, there are some drawbacks of the implementation of the quantum Arnold transformation with the help of Γˆ <sup>Θ</sup>**k**. Firstly, the limit of past

conformal infinity is not well-defined in terms of the generator Gˆ as depicted in (26), for neither of the two cases. This especially means that we do not ge<sup>t</sup> an asymptotic identity map for an already free particle (i.e., the "**<sup>k</sup>**" = 0 case in the limit of past conformal infinity) as we do with the initially time-independent harmonic oscillator in the case of the original transformation Γˆ *ξ* . Secondly, the attempt to relate the free particle with a Lewis-Riesenfeld type invariant does not work as smoothly as in the case of the previous map. If we construct a similar invariant in this case here for the initially time-dependent oscillator Hamiltonian, it can be trivially factorized and has the following form:

$$\mathbb{I}\_{LR} = \frac{1}{2} \mathbb{I}\_{\Theta\_{\mathbf{k}}}^{\star} \pi\_q(\boldsymbol{\uprho})^2 \mathbb{I}\_{\Theta\_{\mathbf{k}}} = \frac{1}{2} \left( \Theta\_{\mathbf{k}\_i} \pi\_q(\boldsymbol{\uprho}) - \Theta\_{\mathbf{k}\_i}' \pi\_q(\boldsymbol{\uprho}) \right)^2 = \mathbb{I}\_{\mathbf{k}\_i}^{\dagger} \mathbb{A}\_{\mathbf{k}\_{i'}} \quad \mathbb{A}\_{\mathbf{k}\_{i'}} = \frac{i}{\sqrt{2}} \left( \Theta\_{\mathbf{k}\_i} \pi\_q(\boldsymbol{\uprho}) - \Theta\_{\mathbf{k}\_i}' \pi\_q(\boldsymbol{\uprho}) \right). \tag{60}$$

This quantity has for example been already obtained in Reference [17] as a quantum invariant based on orthogonal functions in a similar context. It is immediate that the above factorization is in this sense pathological, as one can immediately see that the occurring operators *<sup>a</sup>*<sup>ˆ</sup>**k** , *a*ˆ† **k** can not be interpreted as ladder operators due to [*a*<sup>ˆ</sup>**k** , *a*ˆ† **k** ] = 0, since they only differ by a global sign. This can be also seen by looking at the original invariant (24) which has an additional term proportional to *q*ˆ <sup>2</sup>*ξ*−<sup>2</sup> that is absent in the case of the Arnold transformation for **k** by construction, simply because of the lack of a term proportional to *<sup>π</sup>q*(*q*<sup>ˆ</sup>) in the transformed free particle Hamiltonian. Nevertheless, the transformation Γˆ <sup>Θ</sup>**k** is unitary for all finite times *η* and all considered modes. However, due to the non-preservation of the commutator structure between *<sup>a</sup>*<sup>ˆ</sup>**k** and *a*ˆ† **k** , it is not a Bogoliubov transformation, hence it does not qualify as an infrared continuation of the map Γˆ *ξ* used throughout this work.

In summary, it was not possible to find a transformation similar to Γˆ *ξ* for the infrared modes. Regarding predictions in inflationary comsology, we are naturally interested in the large *k* modes, which are properly implementable in the context of our symplectic transformation. Hence, as an alternative to the proposed maps for the infrared, we sugges<sup>t</sup> the identity map as a proper choice, that is: 

$$\Gamma\_{\vec{\xi}} := \begin{cases} \exp\left(-i\left[\int\_{V\_{\mathsf{c}}} \mathbf{d}^{3} k \, \mathcal{G}(\xi\_{\mathsf{k}}, \xi\_{\mathsf{k}})\_{\prime} \cdot \right] \right) & \text{for} \quad ||\mathbf{k}|| > ||\mathbf{k}\_{\mathsf{c}}||, \\ \mathbf{1}\_{\mathcal{H}} & \text{for} \quad ||\mathbf{k}|| \le ||\mathbf{k}\_{\mathsf{c}}||, \end{cases} \tag{61}$$

where *V* := {**k** ∈ R<sup>3</sup> : "**k**" > "**<sup>k</sup>**"} is the smearing domain and Gˆ **k** denotes the mode-dependent generator of the Bogoliubov transformation depicted in Equation (26) and especially in (35) in terms of annihilation and creation operators, respectively. This is possible and well-defined since the occurring coefficients in the generator are smooth for "**k**" > "**<sup>k</sup>**" and moreover lie in *<sup>L</sup>*<sup>1</sup>(*<sup>V</sup>*, d3*k*) as can be checked by explicit integration. The reasons for choosing the identity map are twofold. Firstly, this trivially constitutes a Bogoliubov transformation with the off-diagonal coefficients in (48) vanishing, rendering the Shale-Stinespring integral finite, thus allowing for unitary implementability of Γˆ **k** *ξ* on Fock space. Secondly, the functions multiplying the off-diagonal elements in the Mukhanov-Sasaki Hamiltonian remain unchanged compared to the standard case, which means that they can be neglected for sufficiently early times. This is due to the fact that the effective friction term in the equation of motion for these functions is subdominant in this regime. For details the reader is referred to the discussion in the succeeding section.

### **5. Relation of the Lewis-Riesenfeld Invariant Approach to the Bunch-Davies Vacuum and Adiabatic Vacua**

In the context of the results of the previous sections, it is a natural question whether there exists a relation of the mode functions obtained in the framework of the formalism in this work and the ones obtained in the standard approach in cosmology. As we will show by taking time-rescaling transformation into account, we can relate the solutions *ξ***k** of the Ermakov equation to the mode

functions associated with the Bunch-Davies vacuum and other adiabatic vacua. For this purpose we consider the following form of the Mukhanov-Sasaki mode function

$$w\_{\mathbf{k}}(\eta) = N\_{\mathbf{k}} \xi\_{\mathbf{k}}(\eta) \exp\left\{-i\omega\_{\mathbf{k}}^{(0)} \int^{\eta} \frac{\mathbf{d}\tau}{\xi\_{\mathbf{k}}^{2}(\tau)}\right\},\tag{62}$$

corresponding to a polar representation of the complex mode *vk* into a real function *ξ***k** and a complex phase that was in a similar form already mentioned in Reference [15]. *N***k** is time-independent for each mode, *ξ***k**(*η*) remains arbitrary at this point and *ω*(0) **k** can take the values *k* or 1 depending on the choice of map that is considered. We want to show that the Mukhanov-Sasaki equation

$$
\omega\_{\mathbf{k}}^{\prime\prime}(\eta) + \omega\_{\mathbf{k}}^2(\eta)\upsilon\_{\mathbf{k}}(\eta) = 0 \tag{63}
$$

expressed in terms of the polar representation exactly coincides with the Ermakov equation. Starting from this polar representation of the mode functions we compute the second derivative and reinsert it into the Mukhanov-Sasaki equation to obtain:

$$v\_{\mathbf{k}}^{\prime\prime}(\eta) = N\_{\mathbf{k}} \exp\left\{-i\omega\_{\mathbf{k}}^{(0)} \int^{\eta} \frac{\mathbf{d}\tau}{\xi\_{\mathbf{k}}^{2}(\tau)}\right\} \Big(\mathfrak{J}\_{\mathbf{k}}^{\prime\prime} - i\omega\_{\mathbf{k}}^{(0)} \frac{\mathfrak{J}\_{\mathbf{k}}^{\prime}}{\xi\_{\mathbf{k}}^{2}} + i\omega\_{\mathbf{k}}^{(0)} \frac{\mathfrak{J}\_{\mathbf{k}}^{\prime}}{\xi\_{\mathbf{k}}^{2}} - \frac{(\omega\_{\mathbf{k}}^{(0)})^{2}}{\xi\_{\mathbf{k}}^{3}}\Big). \tag{64}$$

We realize that summands involving *ξ* **k** cancel each other and the Mukhanov-Sasaki equation can be rewritten as:

$$
\xi\_{\mathbf{k}}^{\prime\prime} + \omega\_{\mathbf{k}}^2(\eta)\xi\_{\mathbf{k}} - \frac{(\omega\_{\mathbf{k}}^{(0)})^2}{\xi\_{\mathbf{k}}^3} = 0 \quad \Longleftrightarrow \quad v\_{\mathbf{k}}^{\prime\prime}(\eta) + \omega\_{\mathbf{k}}^2(\eta)v\_{\mathbf{k}}(\eta) = 0. \tag{65}
$$

That is, we recover the Ermakov equation for the radial part of the polar representation in Equation (62). The polar representation of the mode functions can also be obtained if we consider how the Fourier modes transform under the time-dependent canonical transformation that relates the time-dependent and time-independent harmonic oscillator. The mode functions in the system of the harmonic oscillator written as a function of conformal time are given by *<sup>u</sup>***k**(*η*) = *N***k** exp − *<sup>i</sup>ω*(0) **k** *η dτ ξ*2 **k**(*τ*) . Here *<sup>u</sup>***k**(*T*) satisfies the standard harmonic oscillator differential equation with respect to the time variable *T*. Using that for each mode we have *T* **k** = *ξ*−<sup>2</sup> **k** , one can easily derive the corresponding differential equation that *<sup>u</sup>***k**(*η*) fulfills with respect to conformal time *η*. Now the time-dependent canonical transformation rescales the spatial coordinate by *ξ*−<sup>1</sup> **k** . Considering this as well as the fact that the mode *<sup>u</sup>***k**(*η*) depends on *k* only, the corresponding Fourier mode after the transformation is given by *<sup>v</sup>***k**(*η*) = *ξ***k***u***k**(*η*), yielding again the polar representation of the Fourier mode shown in (62), where we used how the Fourier transform changes under a scaling of the coordinates.

A second way to obtain this result is via the explicit form of the Bogoliubov transformation associated with the time-dependent canonical transformation. We denote the time-dependent annihilation and creation operators of the harmonic oscillator system by ˆ *b***k**(*T*) = *<sup>u</sup>***k**(*T*)<sup>ˆ</sup> *b***k** and ˆ *b*† **k**(*T*) = *<sup>u</sup>***k**(*T*)<sup>ˆ</sup> *b*† **k** respectively, where the time-dependent annihilation and creation operators satisfy the Heisenberg equation associated with the Hamiltonian of the harmonic oscillator. Once more considering the relation between *T* and *η* for each mode, we can also understand ˆ *b***k**(*η*) and ˆ *b*† **k**(*η*) as operator-valued functions of conformal time *η*. The mode expansion in the system of the time-dependent harmonic oscillator can be written in terms of time-dependent annihilation and creation operators *<sup>a</sup>*<sup>ˆ</sup>**k**(*η*) = *<sup>v</sup>***k**(*η*)*a*<sup>ˆ</sup>**k** and *a*ˆ† **k**(*η*) = *<sup>v</sup>***k**(*η*)*a*ˆ† **k** which both satisfy the Heisenberg equation associated to the Mukhanov-Sasaki Hamiltonian. As shown in Section 4, the time-dependent canonical

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transformation corresponds to a time-dependent Bogoliubov map at the quantum level. In the notation of the last section, this relates the two sets of annihilation and creation operators as follows3:

$$
\hat{a}\_{\mathbf{k}}(\eta) = \mathcal{g}\_{\mathbf{k}}(\underline{\mathfrak{z}}, \underline{\mathfrak{z}}')\hat{b}\_{\mathbf{k}}(\eta) + \overline{h}\_{\mathbf{k}}(\underline{\mathfrak{z}}, \underline{\mathfrak{z}}')\hat{b}\_{\mathbf{k}}^{\dagger}(\eta), \quad \hat{a}\_{\mathbf{k}}^{\dagger}(\eta) = \overline{\mathfrak{g}}\_{\mathbf{k}}(\underline{\mathfrak{z}}, \underline{\mathfrak{z}}')\hat{b}\_{\mathbf{k}}^{\dagger}(\eta) + h\_{\mathbf{k}}(\underline{\mathfrak{z}}, \underline{\mathfrak{z}}')\hat{b}\_{\mathbf{k}}(\eta).
$$

The explicit form of these coefficients is given by:

$$g\_{\mathbf{k}}(\mathbf{\tilde{s}}',\mathbf{\tilde{s}}') = \frac{1}{2} \left(\mathbf{\tilde{s}}\_{\mathbf{k}} + \frac{1}{\mathbf{\tilde{s}}\_{\mathbf{k}}}\right) + \frac{i}{2\omega\_{\mathbf{k}}^{(0)}}\mathbf{\tilde{s}}', \qquad h\_{\mathbf{k}}(\mathbf{\tilde{s}}',\mathbf{\tilde{s}}') = \frac{1}{2} \left(\mathbf{\tilde{s}}\_{\mathbf{k}} - \frac{1}{\mathbf{\tilde{s}}\_{\mathbf{k}}}\right) - \frac{i}{2\omega\_{\mathbf{k}}^{(0)}}\mathbf{\tilde{s}}'.\tag{66}$$

Given this time-dependent Bogoliubov map, the Fourier modes in the two systems are related via

$$\boldsymbol{w}\_{\mathbf{k}}(\boldsymbol{\eta}) = \left(g\_{\mathbf{k}}(\boldsymbol{\xi}, \boldsymbol{\xi}^{\prime}) + h\_{\mathbf{k}}(\boldsymbol{\xi}, \boldsymbol{\xi}^{\prime})\right) \\ \boldsymbol{u}\_{\mathbf{k}} = \boldsymbol{\xi}\_{\mathbf{k}} \\ \boldsymbol{u}\_{\mathbf{k}}(\boldsymbol{\eta}) = N\_{\mathbf{k}} \boldsymbol{\xi}\_{\mathbf{k}} \exp\left\{-i\omega\_{\mathbf{k}}^{(0)} \int^{\eta} \frac{d\boldsymbol{\tau}}{\boldsymbol{\xi}\_{\mathbf{k}}^{2}(\boldsymbol{\tau})}\right\}. \tag{67}$$

Hence, we again recover the polar representation of the Fourier mode. At this point we did not ye<sup>t</sup> clarify the purpose of the *N***k**, which is intricately connected with the commutator algebra of annihilation and creation operators as we will see. Recall the well-known (off-diagonal) form of the Mukhanov-Sasaki Hamiltonian if we insert the mode expansions into the Hamiltonian density:

$$H = \int \frac{\mathbf{d}^3 k}{(2\pi)^3} \left[ F\_\mathbf{k}(\eta) \mathfrak{A}\_\mathbf{k} \mathfrak{A}\_{-\mathbf{k}} + F\_\mathbf{k}(\eta) \mathfrak{A}\_\mathbf{k}^\dagger \mathfrak{A}\_{-\mathbf{k}}^\dagger + E\_\mathbf{k}(\eta) \left( 2 \mathfrak{A}\_\mathbf{k}^\dagger \mathfrak{A}\_\mathbf{k} + (2\pi)^3 \delta^{(3)}(0) \right) \right],\tag{68}$$

where we used the isotropy of the mode functions due to the high degree of symmetry of the spacetime, the invariance of the measure under reflection and the following definitions:

$$F\_{\mathbf{k}}(\eta) := \left(v\_{\mathbf{k}}'\right)^2 + \omega\_{\mathbf{k}}^2(\eta)v\_{\mathbf{k}'}^2 \quad E\_{\mathbf{k}}(\eta) := v\_{\mathbf{k}}'\overline{v}\_{\mathbf{k}}' + \omega\_{\mathbf{k}}^2(\eta)v\_{\mathbf{k}}\overline{v}\_{\mathbf{k}}.\tag{69}$$

Regarding the normalization of the mode functions *v***k**, we can transfer this condition to the polar representation given in Equation (62) by just inserting the definition into the Wronskian. This removes the dependence on *ξ***k** completely and we can explicitly give a relation between *N***k** and the Wronskian of the original mode functions:

$$\mathcal{W}(\upsilon\_{\mathbf{k}}, \overline{\upsilon}\_{\mathbf{k}}) = 2i\omega\_{\mathbf{k}}^{(0)} N\_{\mathbf{k}}^2. \tag{70}$$

This is not surprising upon closer inspection. Recall that the *ω*(0) **k** in the Ermakov equation corresponds to the time-independent frequency in the transformed Schrödinger equation. We conveniently chose to map the Mukhanov-Sasaki frequency into just the **k**-dependent part, completely removing the time dependence. This has the effect that Γ ˆ *ξ* becomes the identity transformation for the case of an initially time-independent oscillator, whereas we would obtain a residual squeezing if we mapped every mode to unity. This freedom of choice is reflected in the explicit form of the normalization constant *N***k**, which depends on the choice of the oscillator frequency in the target system in order to preserve the normalization of the mode functions and hence the standard commutator algebra of annihilation and creation operators. Given the Mukhanov-Sasaki Hamiltonian in the form of annihilation and creation operators in (68), we can discuss the assumptions for the initial condition regarding the Fourier modes associated with the Bunch-Davies vacuum and

<sup>3</sup> Note that the roles of *<sup>a</sup>*<sup>ˆ</sup>**k**, *a*ˆ†**k** and ˆ*b***k**, ˆ*b*†**k** are interchanged in comparison to the one-particle case considered in (48) for notational convenience, whereas the coefficients are named analogously. Here the first set of operators belongs to the Mukhanov-Sasaki Hamiltonian, whereas the second set is associated to the time-independent harmonic oscillator. In contrast, in (48) the operators *B* ˆ , *B* ˆ † belong to the time-dependent system, whereas *A*ˆ, *A*ˆ † are associated with the time-independent harmonic oscillator.

the ones obtained in our work and compare them, consequently. First, we rewrite the Fourier mode associated to the Bunch-Davies vacuum given by

$$v\_{\mathbf{k}}^{\rm BD}(\eta) = \frac{1}{\sqrt{2k}} \left( 1 - \frac{i}{k\eta} \right) e^{-i\mathbf{k}\eta} \tag{71}$$

in the polar representation as shown in (62) for our general solution. This yields:

$$v\_{\mathbf{k}}^{\rm BD}(\eta) = i|v\_{\mathbf{k}}^{\rm BD}| \exp\left\{-i\omega\_{\mathbf{k}}^{(0)} \int^{\eta} \frac{\mathbf{d}\tau}{\xi\_{\mathbf{k}}^{2}(\tau)}\right\} = \frac{i}{\sqrt{2k}} \sqrt{1 + \frac{1}{(k\eta)^{2}}} \exp\left\{-i\omega\_{\mathbf{k}}^{(0)} \int^{\eta} \frac{\mathbf{d}\tau}{\xi\_{\mathbf{k}}^{2}(\tau)}\right\},\tag{72}$$

which corresponds exactly to the *ξ*(*sq*) **k** that we obtained from the Ermakov equation by requiring appropriate initial conditions for *ξ***k** which carry over to initial conditions on the Fourier mode and the additional factor *i* comes from the phase of (71) compared to the one arising from the integral.

As far as the Hamiltonian diagonalization (HD) of the Mukhanov-Sasaki Hamiltonian is concerned, one diagonalizes this Hamiltonian instantaneously at some time *η*0 which requires the coefficients *F***k** and *F***k** to vanish at *η*0. In addition it can be shown that the state satisfying this requirement also minimizes the energy at that time *η*0, so that requiring both does not yield to further conditions on the state. If the requirement *F***k** = 0 and the normalization of the Wronskian *<sup>W</sup>*(*<sup>v</sup>***k**, *<sup>v</sup>***k**) to *<sup>W</sup>*(*<sup>v</sup>***k**, *<sup>v</sup>***k**) = *i* is satisfied, we choose the following initial conditions for the model:

$$\operatorname{HD}\left(\mathbf{I}\right)\quad \left|v\_{\mathbf{k}}\right|\left(\eta\_{0}\right) = \frac{1}{\sqrt{2\omega\_{\mathbf{k}}(\eta\_{0})}}\quad \text{and}\quad \operatorname{HD}\left(\mathbf{II}\right)\quad v\_{\mathbf{k}}'(\eta\_{0}) = -i\omega\_{\mathbf{k}}(\eta\_{0})v\_{\mathbf{k}}(\eta\_{0}).\tag{73}$$

If we consider the specefic choice *η*0 → −∞ in this context we exactly end up with the initial conditions usually chosen to obtain the Bunch-Davies vacuum:

$$\text{BD}\left(\text{I}\right) \quad |v\_{\mathbf{k}}| ( - \infty) = \frac{1}{\sqrt{2k}} \quad \text{and} \quad \text{BD}\left(\text{II}\right) \quad v\_{\mathbf{k}}'( - \infty) = -ikv\_{\mathbf{k}}( - \infty) \tag{74}$$

where we used that *ωk* = *k* at *η*0 → <sup>−</sup>∞, meaning that the modes become the standard Minkowski modes in this limit. Looking closer into the condition *F***k** = 0 we can rewrite this non-linear differential equations as:

$$F\_{\mathbf{k}} = 0 \quad \Longleftrightarrow \quad v\_{\mathbf{k}}'(\eta) \left( v\_{\mathbf{k}}'' - \left( \frac{\omega\_{\mathbf{k}}'(\eta)}{\omega\_{\mathbf{k}}(\eta)} \right) v\_{\mathbf{k}}'(\eta) + \omega\_{\mathbf{k}}^2(\eta) v\_{\mathbf{k}}(\eta) \right) = 0. \tag{75}$$

We realize that *F***k** = 0 at all times *η* requires that *vk* satisfies a differential equations that looks like the Mukhanov-Sasaki equation but with an additional friction term included. For a constant frequency *ω***k** the friction term vanishes, which for the case of de Sitter where *<sup>ω</sup>*2**k**(*η*) = *k*2 − 2*η*2 is given in the limit of large *k*. For de Sitter the friction coefficient reads *<sup>ω</sup>***k** *<sup>ω</sup>***k** = 2*η* 1 (*kη*)<sup>2</sup>−<sup>2</sup> and thus, depending on the values of *k* and *η* it will not always be negligible, which is the reason why in the case of Bunch-Davies one can only achieve an instantaneous Hamiltonian diagonalization. This is due to the fact that *vk* satisfies the Mukhanov-Sasaki equation and at the same time needs to fulfill *F***k** = 0, generally being in conflict already for the simple case of a de Sitter universe. Note that in our work the Hamiltonian diagonalization of the Mukhanov-Sasaki Hamiltonian can be obtained for each instant in time and is not obtained by setting *<sup>F</sup>***k**(*η*) equal to zero but by a time-dependent unitary transformation that involves also a time-rescaling. Now since we fixed our initial condition in the limit *η* → −∞ and, as we will show below, the solution we obtained satisfies the differential equation for adiabatic vacua

*Universe* **2019**, *5*, 170

without any approximation, it is very natural that our initial conditions at *η*0 = − ∞ are given by the following:

$$\text{LR}\begin{pmatrix}\text{I}\end{pmatrix}\quad |v\_{\mathbf{k}}(\eta\_{0})| = |N\_{\mathbf{k}}\mathbb{I}\_{\mathbf{k}}(\eta\_{0})| = \frac{1}{\sqrt{2k}}\quad\text{and}\quad\text{LR}\begin{pmatrix}\text{II}\end{pmatrix}\quad v\_{\mathbf{k}}'(\eta\_{0}) = -i\omega\_{\mathbf{k}}^{(0)}v\_{\mathbf{k}}(\eta\_{0}),\tag{76}$$

where we used again the same normalization of the Wronskian for the condition LR (II) and that lim *η*0→− ∞ *ξ***k**(*η*0) = 1. Thus the initial conditions obtained here coincide with the initial conditions one chooses for adiabatic vacua to any order as well as the ones chosen for the Bunch-Davies vacuum where we fix them in the large *k* limit and for *η*0 → − ∞. However, in our work the latter was necessary in order that the unitary operator that implements the Bogoliubov transformation (see Equation (35)) becomes the identity operator for an already time-independent harmonic oscillator and is hence considerably natural. Now let us discuss how the results obtained in our work are related to the notion of adiabatic vacua. In the framework of adiabatic vacua one uses the following ansatz for the mode functions: 

$$w\_{\mathbf{k}}(\eta) = \frac{1}{\sqrt{W\_{\mathbf{k}}}} \exp\left\{-i \int d\vec{\eta} \,\mathcal{W}\_{\mathbf{k}}(\vec{\eta})\right\},\tag{77}$$

where *<sup>W</sup>***k**(*η*) is defined through the following differential equation

$$\mathcal{W}\_{\mathbf{k}}^{2}(\eta) = \omega\_{\mathbf{k}}^{2}(\eta) - \frac{1}{2} \left( \frac{\mathcal{W}\_{\mathbf{k}}^{\prime\prime}(\eta)}{\mathcal{W}\_{\mathbf{k}}(\eta)} - \frac{3}{2} \left( \frac{\mathcal{W}\_{\mathbf{k}}^{\prime}(\eta)}{\mathcal{W}\_{\mathbf{k}}(\eta)} \right)^{2} \right), \tag{78}$$

where *<sup>ω</sup>***k**(*η*) is the time-dependent frequency, so in our case the one involved in the Mukhanov-Sasaki Hamiltonian. If we compare the ansatz in (77) with the form of the solution for the Mukhanov-Sasaki equation in (62), we realize that we can map the two expression for *v***k** into each other by the substitution *ξ***k** := *ω*(0) **k** 1 2 *W*<sup>−</sup> 1 2 **k** , where we choose *ω*(0) **k** = *k* and *ω*(0) **k** = 1, respectively to consider the case where the Mukhanov-Sasaki Hamiltonian is mapped to the harmonic oscillator with frequency *k* and 1, respectively. As shown above, rewritten in terms of *ξ***k** the Mukhanov-Sasaki equation merges into the Ermakov equation. Hence, if we express the Ermakov equation in terms of *W***k** we can rewrite the Mukhanov-Sasaki equation in terms of *W***k**. For this we consider the second derivative *ξ* **k** expressed in terms of *W***k**. We obtain: 

$$\tilde{\varsigma}\_{\mathbf{k}}^{\prime\prime} = -\frac{\sqrt{\omega\_{\mathbf{k}}^{(0)}}}{2} \left( \frac{\mathcal{W}\_{\mathbf{k}}^{\prime\prime}(\boldsymbol{\eta})}{\mathcal{W}\_{\mathbf{k}}^{\prime\prime}(\boldsymbol{\eta})} - \frac{3}{2} \frac{(\mathcal{W}\_{\mathbf{k}}^{\prime}(\boldsymbol{\eta}))^2}{\mathcal{W}\_{\mathbf{k}}^{\frac{5}{2}}(\boldsymbol{\eta})} \right).$$

Reinserting this back into the Ermakov equation yields:

$$-\frac{\sqrt{\omega\_{\mathbf{k}}^{(0)}}}{2}\left(\frac{\mathcal{W}\_{\mathbf{k}}^{\prime\prime}(\boldsymbol{\eta})}{\mathcal{W}\_{\mathbf{k}}^{\frac{3}{2}}(\boldsymbol{\eta})}-\frac{3}{2}\frac{(\mathcal{W}\_{\mathbf{k}}^{\prime}(\boldsymbol{\eta}))^{2}}{\mathcal{W}\_{\mathbf{k}}^{\frac{5}{2}}(\boldsymbol{\eta})}\right)+\omega\_{\mathbf{k}}^{2}(\boldsymbol{\eta})\frac{\sqrt{\omega\_{\mathbf{k}}^{(0)}}}{\mathcal{W}\_{\mathbf{k}}^{\frac{1}{2}}(\boldsymbol{\eta})}-\frac{(\omega\_{\mathbf{k}}^{(0)})^{2}}{(\omega\_{\mathbf{k}}^{(0)})^{\frac{3}{2}}}\mathcal{W}\_{\mathbf{k}}^{\frac{3}{2}}(\boldsymbol{\eta})=0.\tag{79}$$

Multiplying the entire equation by *ω*(0) **k** − 1 2 *W* 1 2 **k** we end up with:

$$\mathcal{W}\_{\mathbf{k}}^2 = \omega\_{\mathbf{k}}^2 - \frac{1}{2} \left( \frac{\mathcal{W}\_{\mathbf{k}}^{\prime\prime}(\eta)}{\mathcal{W}\_{\mathbf{k}}(\eta)} - \frac{3}{2} \left( \frac{\mathcal{W}\_{\mathbf{k}}^{\prime}(\eta)}{\mathcal{W}\_{\mathbf{k}}(\eta)} \right)^2 \right), \tag{80}$$

and this agrees precisely with the defining differential equation for *W***k** in (78). The adiabatic condition required for the modes in this context carries over to a condition on the large **k** behavior of the function *ξ***<sup>k</sup>**, being a solution of the Ermakov equation. As usual for adiabatic vacua, they do depend on the chosen extension to the infraed sector. In the formalism presented in this work this arbitrariness is encoded in the choice of how the unitary transformation is modified for the modes **k** with ||**k**|| ≤


$$w\_{\mathbf{k}}(\eta) = \frac{1}{\sqrt{2k}} \sqrt{1 + \frac{1}{(k\eta)^2}} e^{-ik\eta} e^{i\arctan(k\eta)}.\tag{81}$$

In case the solution for *W***k** cannot be determined in a simple manner, one uses a WKB approximation for the integral involved in the adiabatic ansatz in (77), yielding adiabatic vacua of a certain order at which the expansion is truncated, see for instance [34,35] for applications. However, since we have determined an analytical solution for the Ermakov equation for *ω*(0) **k** = *k* we did not ge<sup>t</sup> an approximate solution for *W***k** up to some adiabatic order and obtained the full solution for *W***k**. This way of relating the two formalisms also provides the possibility to have a very clear interpretation of the Fourier mode associated with the Bunch-Davies vacuum in the Lewis-Riesenfeld invariant formalism. Now comparing the phase factors of Fourier modes associated with the Bunch-Davies vacuum with the ones obtained from the ansatz for the adiabatic vacua in (77), we realize the following: The Fourier modes we obtain from the Lewis-Riesenfeld invariant formalism, that agree with the conventional one, can be understood as an adiabatic vacuum of non-linear adiabatic order, that is without any truncation, using the relation between the Ermakov equation and the defining differential equation for adiabatic vacua. Considering the solution in (81) in the limit *kη* 1, we realize that these modes merge into the standard Minkowski modes up to an irrelevant phase and thus satisfy the adiabatic condition. Note that we have chosen the normalization of the Wronskian in such a way that the final mode functions *v***k** agree, regardless of whether we chose the map that relates the MS system with a harmonic oscillator to have frequency *ω*(0) *k* = *k* or *ω*(0) *k* = 1, respectively. However, our analysis shows that on Fock space, the map that intertwines between the harmonic oscillator with *ω*(0) *k* = 1 and the Mukhanov-Sasaki equation cannot be implemented unitarily due to ultraviolet divergences and thus the latter choice cannot be obtained in a natural way in the Lewis-Riesenfeld formalism. For the reason that the solution in (81) was obtained from a unitary transformation that maps the Mukhanov-Sasaki Hamiltonian into the harmonic oscillator Hamiltonian for all modes **k** with "**k**" > "**<sup>k</sup>**", we can interpret this adiabatic vacuum as the natural one associated to this unitary transformation.

We summarize these results of the last two sections in Figure 1 below. We have seen that we can obtain a solution of the Mukhanov-Sasaki equation at the level of the mode functions (and find the associated vacuum) by means of the solution of the Ermakov equation *ξ***k**(*η*) combined together with a time-dependent phase that corresponds to the time rescaling from the classical theory in Equation (11). In our formalism we have the freedom of choosing the target frequency *ω*(0) **k** as we map our Hamiltonian, where we considered two different choices in this work here. One natural choice is to just remove the time dependence and keep the time-independent **k**<sup>2</sup> term in the frequency, which gives a transformation that is implementable for all but the infrared modes, where one can choose to modify the map appropriately as has been discussed above. It is in this sense natural to do so, since in the limit at past conformal infinity, this transformation is the identity as one would expect. Contrary to that, mapping all frequencies to unity results in a residual squeezing at very early times and most importantly in an ultraviolet divergence in the integral of the Shale-Stinespring condition. Using our results it can be shown that the non-squeezed adiabatic vacua are unitarily inequivalent to the generalized Bunch-Davies vacuum because the time-independent squeezing map that relates a harmonic oscillator with frequency *ω*(0) **k** = *k* to the one with frequency *ω*(0) **k** = 1 cannot be implemented as a unitary operator on Fock space.

**Figure 1.** Graphical summary of the two different maps analyzed on Fock space.
