**6. Applications**

#### *6.1. Solution of the Ermakov Equation on Quasi-de Sitter Spacetime*

In the following we will derive and investigate a specific solution to the Ermakov equation on a quasi-de Sitter background. This leaves us with the opportunity to simplify this solution to the case of de Sitter, where the solution is known and perform a quantitative comparison of the behavior of *ξ*(*η*) for these two spacetimes. Our starting point is again the Mukhanov-Sasaki equation shown in (58), which will be restated for the reader's convenience:

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

Naturally, in the limit of vanishing slow-roll parameters *ε*, *τ* and *κ* the equation merges into the Mukhanov-Sasaki equation on de Sitter, that is, we find that *ν*2 − 14 → 2 as expected. Next, we will bring the equation into a slightly different but also frequently used form. This is done by multiplying the entire equation by *η*2, which is possible since *η* ranges from −∞ to 0, both obviously excluded. Further, we introduce new functions *<sup>w</sup>*(−*kη*) with *k* = "**k**" that are related to the original mode functions by *<sup>w</sup>*(−*kη*) = <sup>√</sup>*<sup>v</sup>***k**−*<sup>η</sup>* . This leads to the following differential equation for *<sup>w</sup>*(−*kη*):

$$
\lambda \chi^2 \frac{d^2 w(-\chi)}{d\chi^2} + \chi \frac{dw(-\chi)}{d\chi} + \left(\chi^2 - \nu^2\right) w(-\chi) = 0, \quad \chi = k\eta,\tag{82}
$$

This is an advantage because (82) precisely corresponds to the generalized Bessel differential equation with a well-studied framework of solution techniques. Primarily, we are interested in a set of two linearly independent solutions of this equation in order to construct a solution for the Ermakov equation following the path taken in Reference [36]. The most general solution to the Bessel equation is given in terms of Bessel functions *Jν*(−*kη*),*Yν*(−*kη*) of the first and second kind, respectively. These can be rewritten in terms of Hankel functions *H*(1) *ν* (−*kη*), *H*(2) *ν* (−*kη*) of the first and second kind, which are given by:

$$H\_{\nu}^{(1)}(-k\eta) := J\_{\mathcal{V}}(-k\eta) + i\mathcal{Y}\_{\mathcal{V}}(-k\eta), \quad H\_{\nu}^{(2)}(-k\eta) := J\_{\mathcal{V}}(-k\eta) - i\mathcal{Y}\_{\mathcal{V}}(-k\eta). \tag{83}$$

These functions form a linearly independent set of solutions to Equation (82). Introducing constants *α*, *β* ∈ C we can give a general solution to the Mukhanov-Sasaki equation on quasi-de Sitter by means of resubstituting *v***k** = √−*η <sup>w</sup>*(−*kη*) and inserting the previously found basis of solutions for the Bessel equation:

$$v\_{\mathbf{k}}(\eta) = \sqrt{-\eta} \left( \alpha H\_{\nu}^{(1)}(-k\eta) + \beta H\_{\nu}^{(2)}(-k\eta) \right),$$

Now we follow Reference [36] and can start to construct a (unique) solution of the Ermakov equation with either the use of *Jν*(−*kη*),*Yν*(−*kη*) or *H*(1) *ν* (−*kη*), *H*(2) *ν* (−*kη*), respectively. This can be achieved by the following procedure, which can be straightforwardly verified by direct computation. We have

$$\mathcal{G}\_{\mathbf{k}}(\eta) = \sqrt{A\_{\mathbf{k}} u\_{\mathbf{k}}^2 + 2B\_{\mathbf{k}} u\_{\mathbf{k}} v\_{\mathbf{k}} + \mathbb{C}\_{\mathbf{k}} v\_{\mathbf{k}'}^2} \qquad A\_{\mathbf{k}} \mathbb{C}\_{\mathbf{k}} - B\_{\mathbf{k}}^2 = \left\| \mathbf{k} \right\|^2 W(u\_{\mathbf{k}}, v\_{\mathbf{k}})^{-2} \text{.} \tag{84}$$

where *u***k**, *v***k** are two linearly independent solutions of the Ermakov equation and *<sup>W</sup>*(*<sup>u</sup>***k**, *<sup>v</sup>***k**) denotes the Wronskian determinant. As an additional 'initial' condition next to the Wronskian, we impose the well-definedness of the solution in the limit of past conformal infinity where for each mode the Mukhanov-Sasaki equation reduces to an harmonic oscillator with constant frequency. The function *ξ***k** should also solve the Ermakov equation in this limiting case of constant frequency. Consequently, we can insert the linear independent solutions (83) into formula in (84) and investigate the behavior for *η* → <sup>−</sup>∞. Analyzing the asymptotic behavior of the Bessel functions (and correspondingly Hankel functions) according to Reference [37], we get:

$$H\_{\nu}^{(1)}(-k\eta) \sim \sqrt{-\frac{2}{\pi k \eta}} \exp\left\{-i\left(k\eta + \frac{\pi}{4}(2\nu+1)\right)\right\} \qquad \text{for} \qquad |\eta| \gg 1,$$

$$H\_{\nu}^{(2)}(-k\eta) \sim \sqrt{-\frac{2}{\pi k \eta}} \exp\left\{+i\left(k\eta + \frac{\pi}{4}(2\nu+1)\right)\right\} \qquad \text{for} \qquad |\eta| \gg 1.$$

We realize that in this limit the summand under the square root in (84) is only well-defined for vanishing coefficients *A***k**, *C***k** such that only the mixed term remains. In order to determine the coefficient *B***k** we need to find an expression for the Wronskian determinant of Hankel functions, which is non-trivial to obtain in a straightforward manner. However, we know that the Wronskian of solutions of the harmonic oscillator equation is constant in time and we have an relation for the asymptotic behavior of the Hankel functions. Given this we have

$$\mathcal{W}(\sqrt{-\eta}H\_{\nu}^{(1)}(-k\eta), \sqrt{-\eta}H\_{\nu}^{(2)}(-k\eta)) = -\eta \, \mathcal{W}(H\_{\nu}^{(1)}(-k\eta), H\_{\nu}^{(2)}(-k\eta)).\tag{85}$$

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

As the next step, let us rewrite the derivative with respect to conformal time of *W*(*H*(1) *ν* (−*kη*), *H*(2) *ν* (−*kη*)) in terms of a differential equation by the use of its anti-symmetry and the Bessel differential Equation (82) obeyed by *H*(1) *ν* , *H*(2) *ν* :

$$\mathcal{W}'(H\_{\nu}^{(1)}, H\_{\nu}^{(2)}) = -\frac{1}{\eta} \mathcal{W}(H\_{\nu}^{(1)}, H\_{\nu}^{(2)}) \quad \Longrightarrow \quad \mathcal{W}(H\_{\nu}^{(1)}(-k\eta), H\_{\nu}^{(2)}(-k\eta)) \propto \frac{D\_{\mathbf{k}}}{\eta},\tag{86}$$

where we allowed for that the constant *D***k** can vary for each mode. Note that the proportionality of the Wronskian of the Hankel functions in (86) is in accordance with the fact that it is conserved on solutions of the Mukhanov-Sasaki equation, as seen in Equation (85). Finally, after insertion of the asymptotic behavior of the Hankel functions, we find:

$$\mathcal{W}\left(H\_{\nu}^{(1)}(-k\eta), H\_{\nu}^{(2)}(-k\eta)\right) \sim \frac{4i}{\pi\eta} \quad \text{for} \quad |\eta| \gg 1 \quad \implies \quad D\_{\mathbf{k}} = \frac{4i}{\pi}.$$

Note that the Wronskian is purely imaginary, which is expected due to the negative sign of the *B*2**k** term in the condition presented in the second equation in (84) for the coefficients. If we had chosen a different route and had taken Bessel instead of Hankel functions, we would have to choose *B***k** = 0 for consistency and with a corresponding purely real Wronskian determinant. As a final result, we can determine *B***k**:

$$\mathcal{W}(\sqrt{-\eta}H\_{\upsilon}^{(1)}(-k\eta), \sqrt{-\eta}H\_{\upsilon}^{(2)}(-k\eta)) = -\eta \frac{4i}{\pi\eta} = -\frac{4i}{\pi} = \text{const} \implies \quad B\_{\mathbf{k}} = -\frac{k\pi}{4} \tag{87}$$

Due to the requirement that the transformation induced by Γ*ξ* should be unitary, we need *B***k** to be chosen such that the final solution *ξ***k**(*η*) is real, which is always possible in this case due to the involved squares:

$$\mathcal{J}\_{\mathbf{k}}(\eta) = \sqrt{-\frac{k\pi\eta}{2}H\_{\nu}^{(1)}(-k\eta)H\_{\nu}^{(2)}(-k\eta)} = \sqrt{-\frac{k\pi\eta}{2}\left(\left(I\_{\nu}(-k\eta)\right)^{2} + \left(\mathcal{Y}\_{\nu}(-k\eta)\right)^{2}\right)}\tag{88}$$

Another important aspect is the correct limit at past conformal infinity, which we can immediately deduce from the asymptotic forms of the Hankel functions above. This suggests that for each Fourier mode *ξ***k**(*η*) solves the Ermakov equation in the case where the Mukhanov-Sasaki frequency becomes a constant *ω*(0) **k** := lim*η*→−∞ *<sup>ω</sup>***k**(*η*) = "**k**", that is:

$$\lim\_{\eta \to -\infty} \zeta\_\mathbf{k}(\eta) = \lim\_{\eta \to -\infty} \sqrt{-\frac{k \pi \eta}{2} \frac{2}{\pi k |\eta|}} = \lim\_{\eta \to -\infty} \sqrt{-\text{sgn}(\eta)} = 1.$$

At this point we still need to investigate whether given the solution *ξk*(*η*) on quasi-de Sitter we can rediscover the solution for de Sitter in the case of vanishing slow-roll parameters. For this purpose, we consider the half-integer expressions for the Bessel functions:

$$J\_{n+\frac{1}{2}}(\mathbf{x}) = (-1)^n \sqrt{\frac{2}{\pi}} x^{n+\frac{1}{2}} \left(\frac{\mathbf{d}}{\mathbf{x} \mathbf{d} \mathbf{x}}\right)^n \frac{\sin(\mathbf{x})}{\mathbf{x}} \qquad \forall \ n \in \mathbb{N},$$

$$\mathcal{Y}\_{n+\frac{1}{2}}(\mathbf{x}) = (-1)^{n+1} \sqrt{\frac{2}{\pi}} x^{n+\frac{1}{2}} \left(\frac{\mathbf{d}}{\mathbf{x} \mathbf{d} \mathbf{x}}\right)^n \frac{\cos(\mathbf{x})}{\mathbf{x}} \quad \forall \ n \in \mathbb{N}.$$

Form this we obtain an expression for *ξ***k**(*η*) on de Sitter where *ν* = 3/2:

$$\mathcal{E}\_{\mathbf{k}}^{(d\mathbf{S})} = \sqrt{-\frac{k\pi\eta}{2} \left( \left( \sqrt{\frac{2}{\pi}} \frac{k\eta\cos(k\eta) - \sin(k\eta)}{(-k\eta)^{\frac{3}{2}}} \right)^2 + \left( \sqrt{\frac{2}{\pi}} \frac{k\eta\sin(k\eta) + \cos(k\eta)}{(-k\eta)^{\frac{3}{2}}} \right)^2 \right)} = \sqrt{1 + \frac{1}{(k\eta)^2}} \mathcal{E}\_{\mathbf{k}}$$

which of course retains the same limit at past conformal infinity as the more complicated solution for non-vanishing slow-roll parameters. The solution *ξ* (*dS*) **k** (*η*) can be obtained in full analogy to the procedure outlined above using the well-known solution for the Mukhanov-Sasaki frequency on de Sitter. Depending on the particular choice of basis for the space of solutions, one needs to eliminate either of the coefficients in (84) due to the required well-definedness of the limiting case |*η*| → ∞. The outcome precisely corresponds to *ξ* (*dS*) **k**(*η*) found in the limit above.

### *6.2. Eigenstates of the Lewis-Riesenfeld Invariant*

As a test scenario for the formalism outlined in this work we construct and analyze the explicitly time-dependent eigenstates of the Lewis-Riesenfeld invariant. This will happen at the level of a quantum mechanical toy-model and serve the purpose of exhibiting the mathematical convenience of the formalism as well as the (squeezing) properties of the unitary transformation obtained in the context of the Lewis-Riesenfeld invariant. These eigenstates can be easily found by applying the previously obtained (inverse) Bogoliubov transformation Γˆ † *ξ* to the defining property of the vacuum, that is *A*ˆ |0 = 0. We obtain:

$$
\Gamma\_{\xi}^{\dagger} A \Gamma\_{\xi} \Gamma\_{\xi}^{\dagger} |0\rangle = \operatorname{Ad}\_{\Gamma\_{\xi}^{\dagger}}(\vec{A}) \Gamma\_{\xi}^{\dagger} |0\rangle = \varepsilon^{-\nu(\xi)} \left(\vec{A} - \delta\_{+}(\xi) \vec{A}^{\dagger}\right) \Gamma\_{\xi}^{\dagger} |0\rangle =: \mathcal{B} \Gamma\_{\xi}^{\dagger} |0\rangle = 0,
$$

with the BCH coefficients *ν*(*ξ*) and *δ*+(*ξ*) determined in Section 3.2. That is, the vacuum state of the Bogoliubov transformed annihilation operator *B*ˆ corresponds to the unitarily transformed initial vacuum state. Recall that Γˆ *ξ* was capable of relating the time-independent Hamiltonian *H*ˆ 0 and the Lewis-Riesenfeld invariant ˆ*ILR* via the adjoint action, in other words, the Lewis-Riesenfeld invariant factorizes in terms of *B*ˆ, *B*ˆ †. Reexpressing the operators above in position representation we end up with a first-order differential equation for the transformed vacuum state. Understandably, this equation contains explicitly time-dependent coefficients due to the explicit time dependence of the Bogoliubov transformation. We obtain the following solution for the ground state <sup>Ψ</sup>0(*q*, *η*):

$$\Psi\_0(q,\eta) = \left(\frac{\omega\_0}{\pi \,\hat{\xi}^2(\eta)}\right)^{\frac{1}{4}} \exp\left\{ \left(\frac{i}{2}\frac{\hat{\xi}'(\eta)}{\hat{\xi}(\eta)} - \frac{\omega\_0}{2\,\hat{\xi}^2(\eta)}\right) q^2 \right\},\tag{89}$$

where *ξ* (*η*) denotes the derivative with respect to conformal time, we again used that the mass *m* = 1 here and conveniently have set *h*¯ = 1 as before. The first excited state can be obtained from AdΓ<sup>ˆ</sup> † *ξ*(*A*ˆ †)Γ<sup>ˆ</sup> † *ξ* |0 = *B*ˆ †Γ<sup>ˆ</sup> † *ξ* |0 and is found to be:

$$\Psi\_1(q,\eta) = \left(\frac{\omega\_0}{\pi \, \xi^2(\eta)}\right)^{\frac{1}{4}} \sqrt{\frac{2\omega\_0}{\xi^2(\eta)}} q \exp\left\{ \left(\frac{i}{2} \frac{\xi^\nu(\eta)}{\xi(\eta)} - \frac{\omega\_0}{2\, \xi^2(\eta)}\right) q^2 \right\}.\tag{90}$$

Note that for a time-independent frequency *ω*(*η*) = *ω*0 the solution merges into the standard quantum harmonic oscillator since ˆ*ILR* and *H*ˆ 0 coincide in this limit by construction due to *ξ*(*η*) = *ξ*0 = 1. The details of the underlying spacetime, that is, what determines the values of the various slow-roll parameters enters through the solution *ξ*(*η*) of the Ermakov equation, which is sensitive to the background via the Mukhanov-Sasaki frequency *ω*(*η*) and consequently through the real index *ν* of the Hankel and Bessel functions in the final solution in Equation (88). The plots in Figure 2 display the absolute squares of the solutions in Equations (89) and (90), respectively, at two different conformal times.

Considering the explicit form of the generator of the Bogoliubov transformation Γˆ *ξ* , we realize that it represents a generalized squeezing operator with explicitly time-dependent coefficient functions. These coefficient functions on the other hand are sensitive to the background spacetime via the Ermakov equation and consequently the Mukhanov-Sasaki frequency *<sup>ω</sup>***k**(*η*) involved in Equation (58). In this way it is expected that the eigenstates of the Lewis-Riesenfeld invariant, which are, up to

a phase, eigenstates of the single-mode time-dependent Mukhanov-Sasaki Hamiltonian, show a time-dependent spread which approaches the time-independent case for very large absolute values of conformal time |*η*| 1, that is, close to the Big Bang.

**Figure 2.** Single-mode probability densities |<sup>Ψ</sup>0(*q*, *η*)|<sup>2</sup> (upper line) and |<sup>Ψ</sup>1(*q*, *η*)|<sup>2</sup> (lower line) according to the solutions in (89) and (90) on quasi-de Sitter for three different values of the effective slow-roll parameter *ν* from Equation (58), including de Sitter with *ν* = 3/2 at two different conformal times and with *ω*0 = *k* = 1. The used slow-roll parameters are to be understood as an example, consider Reference [33] for the allowed parameter space and constraints on them according to the Planck mission.

A comparison with the work in Reference [38] bears a strong resemblance to the eigenstate of the Lewis-Riesenfeld invariant found there, however let us compare the results from Reference [38] and ours in more detail. Firstly, the derivation in Reference [38] is performed in cosmological time, whereas we have made the transition to conformal time beforehand, so the explicit occurrences of the scale factor are absent in our work. Secondly, when having a closer look at the Ermakov equation in Reference [38] it becomes evident that in the context of the canonical transformation Γ*ξ* , the time-independent frequency of the Hamiltonian *H*0 is unity. As a consequence, this means for the case of field theory that every mode with *<sup>ω</sup>***k**(*η*) would be mapped to exactly the *same* frequency *ω*(0) **k** = 1, which modifies the solution of the Ermakov equation by an additional *k*− 12 , leading to an ultraviolet divergence in the Shale-Stinespring condition (56) and diminishing the ability to implement it by a separate treatment of the infrared modes with "**k**" ≤ "**<sup>k</sup>**". Thirdly, the authors in Reference [38]

claim that the creation- and annihilation operators that decompose the time *dependent* Hamiltonian are related to the ones associated with the Lewis-Riesenfeld invariant via a Bogoliubov transformation. According to our analysis, while this is true, their relation is more subtle: As it is true that *H*(*t*) can be mapped into *ILR* in the classical theory by means of an extended symplectic map (which in a sense corresponds to a Bogoliubov transformation quantum mechanically), this might not be straightforwardly implementable in the quantum theory even on the one-particle Hilbert space. It can be implemented if and only if the time-rescaling function is chosen such that *ξ*(*t*)−<sup>2</sup> has an analytic anti-derivative, which is for example the case on a de Sitter background. The exponential sandwiched between Γ ˆ † *ξ* and Γ ˆ *ξ*,0 in (33) can then be rewritten as the exponential of an analytic function in the time *operator*, conjugate to the momentum operator *p*<sup>ˆ</sup>*t*. This is the reason why we perform a reduced phase space quantization of Dirac observables, where this problem is absent, rather than Dirac quantization. Consequently, the transformation Γ ˆ *ξ* acts as a one-parameter (i.e., time-dependent) family of unitary transformations on the reduced (physical) phase space. This transformation is suitable for transforming the Hamiltonian within the Schrödinger equation into the independent one *H* ˆ 0, which again can be related to the invariant ˆ *ILR* by means of a time-dependent Bogoliubov transformation.
