# A possible origin of the $\alpha$-vacuum as the initial state of the
Universe

Pisin<EMAIL_ADDRESS>Kuan-Nan
<EMAIL_ADDRESS>Wei-Chen<EMAIL_ADDRESS>and Dong-
han<EMAIL_ADDRESS>aLeung Center for Cosmology and
Particle Astrophysics, National Taiwan University, Taipei 10617, Taiwan
bDepartment of Physics and Center for Theoretical Sciences, National Taiwan
University, Taipei 10617, Taiwan
cGraduate Institute of Astrophysics, National Taiwan University, Taipei 10617,
Taiwan
dKavli Institute for Particle Astrophysics and Cosmology, SLAC National
Accelerator Laboratory, Stanford University, Stanford, California 94305, USA
eCenter for Cosmological Constant Problem, Extreme Physics Institute, Pusan
National University, Busan 46241, Republic of Korea
fDepartment of Physics, Pusan National University, Busan 46241, Republic of
Korea
gDepartment of Physics Education, Pusan National University, Busan 46241,
Republic of Korea
hResearch Center for Dielectric and Advanced Matter Physics, Pusan National
University, Busan 46241, Republic of Korea

###### Abstract

We investigate the cosmological observables using the Euclidean path integral
approach. Specifically, we study both the no-boundary compact instantons
scenario and the Euclidean wormholes scenario that can induce the creation of
two universes from nothing. It is known that perturbations associated with the
no-boundary scenario can only be consistent with the Bunch-Davies vacuum. Here
we demonstrate that the Euclidean wormholes can allow for a de Sitter
invariant vacuum, the so-called $\alpha$-vacuum state, where the Bunch-Davies
vacuum is a special case. This therefore provides the $\alpha$-vacuum a
geometrical origin. As an aside, we discuss a subtle phase issue when
considering the power spectrum related to $\alpha$-vacuum in the closed
universe framework.

###### Contents

  1. I Introduction
  2. II Essentials of Euclidean cosmology
    1. II.1 Euclidean path integral approach
    2. II.2 Background solutions
    3. II.3 Perturbations
    4. II.4 Wave function of matter perturbation
  3. III Quantum states with Euclidean geometric origins
    1. III.1 Reproduction of the Bunch-Davies state from the no-boundary proposal
      1. III.1.1 Computations from instantons
      2. III.1.2 Dependence on the evaluation time
      3. III.1.3 Scale-invariance of the power spectrum
    2. III.2 Existence of the $\alpha$-vacuum state in Euclidean wormholes
      1. III.2.1 Standard $\alpha$-vacuum in flat de Sitter space
      2. III.2.2 No-boundary vs. Euclidean wormholes: Euclidean and Lorentzian initial conditions
      3. III.2.3 The $n$-dependent phase factor in the Euclidean regime
      4. III.2.4 Agreement between analytic and numerical results
  4. IV Conclusion
  5. A Expectation value of matter perburbation
  6. B Curvature perturbation and cosmological observables
  7. C Equivalence of mode functions

##  I Introduction

While modern cosmology has entered the era of precision cosmology, the
inflationary cosmology Starobinsky:1980te  is currently the most-favored one
amongst all existing early universe theories. In an inflationary scenario,
quantum fluctuations of the inflaton field were amplified and eventually
became the seeds for the large-scale structure formation and the observed
cosmic microwave background (CMB) anisotropies. For instance, the latest CMB
anisotropy measurement from Planck Planck:2018vyg  shows a nearly scale-
invariant scalar power spectrum with the non-measurements of the primordial
non-Gaussianities and of the isocurvature perturbations. These results,
together with other measurements, constrain the possible inflationary models
and currently are in favor of a simple inflationary model with a Starobinsky-
type potential, in which a plateau-like potential generates the quasi-de
Sitter phase during the slow-roll period Martin:2013tda .

Besides the difference in the actions of the inflaton fields, to compute the
theoretical expectations of an inflation model, we also need to assume a
specific quantum vacuum state for the scalar perturbations. Since the de
Sitter space serves as a good approximation of the exponentially expanding
period of inflation, the choice of vacuum is commonly reduced to the
discussion of the choice of vacuum in a de Sitter space Mukhanov:2007zz . In
this sense, the most natural one is to choose the Bunch-Davies vacuum
Bunch:1978yq .

However, interestingly, this is not the unique de Sitter invariant vacuum. If
we introduce a kind of Bogoliubov transformation from the Bunch-Davies vacuum,
we will obtain a new vacuum that is still de Sitter invariant. This continuous
class of vacua is known as the $\alpha$-vacuum Chernikov:1968zm ;
Mottola:1984ar ; Allen:1985ux ; Bousso:2001mw , where $\alpha$ is a parameter
that describes a specific vacuum and the Bunch-Davies vacuum is just a special
case of the $\alpha$-vacuum.

In inflationary cosmology, the surge of the consideration of $\alpha$-vacuum
was triggered by the discussion of the trans-Planckian effect on the
predictions of inflation for cosmological fluctuation spectrum
Brandenberger:1999sw , where Danielsson suggested that the cutoff of our
ignorance of the trans-Planckian physics can be encoded in the choice of
initial conditions for the field modes when they start out at cutoff energy
scale Danielsson:2002kx  (see also Ref. Shankaranarayanan:2002ax ). As a
result, the correction to the cosmological fluctuation spectrum can be of the
order of $\mathcal{O}(H/\Lambda)$, where $H$ is the Hubble parameter when the
physical momentum of the mode is on a par with the fixed cutoff scale
$\Lambda$, which in the most modest case is set to be the Planck scale
$M_{Pl}$. While in a pure de Sitter background, the above-mentioned Hubble
parameter is a constant for different $k$-modes, in a realistic slow-rolling
inflationary model, the slowly changing Hubble parameter induces an
oscillatory behavior on the power spectrum, which was first quantitatively
estimated in Ref. Easther:2002xe . However, later study shows this modulation
on the power spectrum might be too small to be measured Easther:2004vq . On
the other hand, the $\alpha$-vacuum creates distinct features in the non-
Gaussianities Holman:2007na , so the existence of the $\alpha$-vacuum might be
either supported or strongly limited by future observations once the
primordial non-Gaussianities are measured.

Next, one should understand that Danielsson’s method is only an effective
approach, and the correction to the power spectrum can be smaller depending on
different theories considered for quantum gravity. For instance, before the
model-independent effective approach to the trans-Planckian effect on the
cosmological fluctuation spectrum, studies were carried out in a model-
dependent way on the modification of dispersion relations or the non-
commutative geometry from string theory. Meanwhile, models from quantum
cosmology, for instance, the no-boundary proposal (NBP) Hartle:1983ai  and the
tunneling proposal Vilenkin:1982de  were not considered in the discussion of
the trans-Planckian effect. This is, in fact, not so surprising since in both
models, the Bunch Davies vacuum is the only consistent choice of the vacuum
state for the matter perturbations Laflamme:1987mx ; Vachaspati:1988as . Thus,
if future observations support the $\alpha$-vacuum, the above two popular
models from the quantum cosmology frameworks will be disfavored if no new
mechanism generating deviation from the Bunch-Davies vacuum can be introduced
in a consistent way. Then, a natural question one can ask is, do there exist
models from quantum cosmology that support the $\alpha$-vacuum for the
fluctuations?

We find that the answer to this interesting question is positive by
considering the Euclidean wormhole scenario. However, we report that deviation
from the Bunch-Davies vacuum is exponentially suppressed in the Euclidean
regime, so the imprint on the cosmological observables might be small without
fine-tuning. Another result worth mentioning is that since the Euclidean
wormhole, like the NBP, is formulated in a closed spacetime, we find a non-
trivial phase issue for the mode solutions once we consider the mixing of the
positive and negative frequency modes. To our best knowledge, this issue has
never been raised since the modulation of the power spectrum by the
$\alpha$-vacuum is always carried out in a spatially flat universe framework.
Only after identifying the correct phase factor, can the power spectrum
approach to the spatial flat result in the large $n$ limit. We analytically
demonstrate the origin of this $n$-dependent phase issue by using a massless
scalar field.

This paper is organized as follows. In Sec. II, we give a brief introduction
to the Euclidean wormhole and summarize our formalism to define and compute
the power spectrum of the cosmological perturbations. In Sec. III, we compute
the power spectrum and compare the results between that of compact instantons
and Euclidean wormholes. The reason for including the $n$-dependent phase
factor for the initial condition is discussed. Finally, in Sec. IV, we
summarize our results and discuss possible future research topics.

##  II Essentials of Euclidean cosmology

In Euclidean quantum cosmology, one deals with the Euclidean path integral,
which is a solution to the Wheeler-DeWitt equation at the formal level.
Therefore, in principle, this path integral can describe non-perturbative
aspects of the Universe. The Euclidean path integral is the integration
between the in-state and the out-state Hartle:1983ai . Using this approach,
Hartle and Hawking provided a wave function, where there is no dependence on
the initial boundary; there exists the final boundary dependence only in the
wave function. This proposal of the boundary condition of the Universe is
known as the no-boundary proposal because there is no initial boundary.
Interestingly, according to Halliwell and Hawking Halliwell:1984eu , the
compact instantons of the no-boundary proposal conform with the Bunch-Davies
state for density perturbations Bunch:1978yq .

This no-boundary proposal is a very elegant and reasonable proposal for the
boundary condition of the wave function, but this requires the separation of
the Euclidean manifold between the initial and the final hypersurfaces. In
other words, the Euclidean manifold for the final state must be regular and
compact, such as the poles of the sphere. However, it is not necessarily true
in general Chen:2017qeh . If one considers a kind of excited quantum state
Robles-Perez:2010uvf  or non-trivial dynamics of the matter field Chen:2017qeh
, the regular and compact boundary conditions of the instantons are not
allowed. In a cosmological context, the next candidate instanton, which is
regular but non-compact, is the Euclidean wormholes that connect from the
initial boundary to the final boundary Hawking:1988ae . There might be several
ways to interpret this Euclidean wormhole; either tunneling from the initial
boundary to the final boundary or tunneling of two universes from nothing
Chen:2016ask . In this paper, we will mainly focus on the latter
interpretation.

###  II.1 Euclidean path integral approach

The Euclidean path integral is a solution to the Wheeler-DeWitt equation at
the formal level. Therefore, in principle, this path integral can describe
non-perturbative aspects of the Universe. The Euclidean path integral is the
integration between the in-state and the out-state as follows Hartle:1983ai :

$\displaystyle\Psi\left[h_{\mathrm{f}},\phi_{\mathrm{f}};h_{\mathrm{i}},\phi_{\mathrm{i}}\right]=\int\mathcal{D}g_{\mu\nu}\mathcal{D}\phi\;e^{-S_{\mathrm{E}}\left[g_{\mu\nu},\phi\right]},$
(1)

where $h_{\mathrm{i,f}}$ are the 3-metrics at the initial and the
hypersurfaces, $\phi_{\mathrm{i,f}}$ are the field values of the matter field
$\phi$ at the initial and the final hypersurfaces, respectively, and
$S_{\mathrm{E}}$ is the Euclidean action of the metric $g_{\mu\nu}$ and the
matter field $\phi$. This integration is to sum over all geometries and field
configurations that connect from $|h_{\mathrm{i}},\phi_{\mathrm{i}}\rangle$ to
$|h_{\mathrm{f}},\phi_{\mathrm{f}}\rangle$.

From this formal wave function, one can use the steepest-descent approximation
for practical computations. One can approximate this integration as a sum of
on-shell solutions, or so-called instantons:

$\displaystyle\Psi\left[h_{\mathrm{f}},\phi_{\mathrm{f}};h_{\mathrm{i}},\phi_{\mathrm{i}}\right]\simeq\sum
e^{-S^{\mathrm{instanton}}_{\mathrm{E}}}.$ (2)

In several examples, the instanton of the in-state
$|h_{\mathrm{i}},\phi_{\mathrm{i}}\rangle$ and the instanton of the out-state
$|h_{\mathrm{f}},\phi_{\mathrm{f}}\rangle$ can be independent of each other.
In this case, one can present this as follows:

$\displaystyle\Psi\left[h_{\mathrm{f}},\phi_{\mathrm{f}};h_{\mathrm{i}},\phi_{\mathrm{i}}\right]\simeq
e^{-\left[S^{\mathrm{instanton}}_{\mathrm{E}}(\mathrm{final})-S^{\mathrm{instanton}}_{\mathrm{E}}(\mathrm{initial})\right]},$
(3)

where $S^{\mathrm{instanton}}_{\mathrm{E}}(\mathrm{initial})$ and
$S^{\mathrm{instanton}}_{\mathrm{E}}(\mathrm{final})$ are the Euclidean
actions for the initial and final instantons, respectively. If this kind of
separation of instantons between the initial and the final states is possible,
it is not surprising to write the wave function only for the final state:

$\displaystyle\Psi\left[h_{\mathrm{f}},\phi_{\mathrm{f}}\right]=\int\mathcal{D}g_{\mu\nu}\mathcal{D}\phi\;e^{-S_{\mathrm{E}}\left[g_{\mu\nu},\phi\right]}\simeq
e^{-S^{\mathrm{instanton}}_{\mathrm{E}}(\mathrm{final})}.$ (4)

In this case, there is no dependence on the initial boundary; there exists the
final boundary dependence only in the wave function. This proposal of the
boundary condition of the Universe is known as the no-boundary proposal
because there is no initial boundary.

On the other hand, if we do not exclude the initial boundary, we need to see
the generic Euclidean path integral in the following form:

$\displaystyle\Psi\left[h_{\mu\nu}^{(2)},\phi_{0}^{(2)};h_{\mu\nu}^{(1)},\phi_{0}^{(1)}\right]=\int\mathcal{D}g_{\mu\nu}\mathcal{D}\mathcal{\phi}\;e^{-S_{\mathrm{E}}[g_{\mu\nu},\phi]},$
(5)

where $g_{\mu\nu}$ is the metric, $\phi$ is a matter field, and we sum over
all regular Euclidean geometries that have boundary values $\partial
g_{\mu\nu}=h_{\mu\nu}^{(1)}\cup h_{\mu\nu}^{(2)}$ and
$\partial\phi=\phi_{0}^{(1)}\cup\phi_{0}^{(2)}$.

Figure 1: Interpretation of Euclidean solutions with two boundaries.
$\mathrm{E}\cup\mathrm{E}^{\prime}$ corresponds the Euclidean manifold that
connects two Lorentzian manifolds $\mathrm{L}$ and $\mathrm{L}^{\prime}$. One
can interpret either there exists one arrow or two arrows of time.

Since this path integral has two boundaries, there can be two interpretations
(Fig. 1):

  * –

1\. One arrow of time: A universe with boundary values $\partial
g_{\mu\nu}=h_{\mu\nu}^{(1)}$, $\partial\phi=\phi_{0}^{(1)}$ tunnels to
boundary values $\partial g_{\mu\nu}=h_{\mu\nu}^{(2)}$,
$\partial\phi=\phi_{0}^{(2)}$, or vice versa.

  * –

2\. Two arrows of time: Two universes with boundary values $\partial
g_{\mu\nu}=h_{\mu\nu}^{(1,2)}$ and $\partial\phi=\phi_{0}^{(1,2)}$ are created
from nothing.

In this paper, we will follow the second interpretation Chen:2016ask ;
Chen:2017qeh .

###  II.2 Background solutions

As a simple model, let us consider a Euclidean action consisting of the
standard Einstein-Hilbert term and the kinetic and potential energies of the
inflaton:

$\displaystyle
S_{\mathrm{E}}=\int\sqrt{+g}d^{4}x\left[-\frac{R}{16\pi}+\frac{1}{2}\left(\partial_{\mu}\Phi\right)^{2}+U(\Phi)\right],$
(6)

where $\Phi$ is the scalar inflaton with the potential $U(\Phi)$. In addition,
we assume the background metric to be a homogeneous, isotropic, and closed
universe:

$\displaystyle
ds_{\mathrm{E}}^{2}=\sigma^{2}\left(d\tau^{2}+a^{2}(\tau)d\Omega_{3}^{2}\right),$
(7)

where $d\Omega_{3}^{2}=d\chi^{2}+\sin^{2}\chi(d\theta^{2}+\sin^{2}\theta
d\varphi^{2})$ (with angle variables $\chi$, $\theta$, and $\varphi$) and
$\sigma^{2}=8\pi U_{0}/3$ with a constant $U_{0}$.

After redefining $\phi=\sqrt{4\pi/3}\Phi$ and $V=U/U_{0}$, one can derive the
equations of motion for the background geometry and the inflaton:

$\displaystyle\dot{a}^{2}-1+a^{2}\left(-\dot{\phi}^{2}+V\left(\phi\right)\right)$
$\displaystyle=$ $\displaystyle 0,$ (8)
$\displaystyle\ddot{a}+2a\dot{\phi}^{2}+aV$ $\displaystyle=$ $\displaystyle
0,$ (9)
$\displaystyle\ddot{\phi}+3\frac{\dot{a}}{a}\dot{\phi}-\frac{1}{2}\frac{dV}{d\phi}$
$\displaystyle=$ $\displaystyle 0,$ (10)

where dots denote differentiation for the Euclidean time $\tau$.

##### Compact instantons

In the no-boundary proposal, one requires the regular and compact condition of
the manifold. So, for example, one can present the compactness condition such
that $a(0)=0$. Due to the regularity of the solution, this requires
$\dot{a}(0)=1$ and $\dot{\phi}(0)=0$.

As a simple demonstration, let us consider $U(\Phi)=\lambda^{2}U_{0}$ and
$\Phi$ is a constant. Then, the solution becomes

$\displaystyle a=\lambda^{-1}\sin(\lambda\tau).$ (11)

The location for the Wick-rotation is at $\tau=\pi/2\lambda$. After the Wick-
rotation to the Lorentzian time $\tau\rightarrow\pi/2\lambda+it$, the result
becomes

$\displaystyle a=\lambda^{-1}\cosh(\lambda t).$ (12)

##### Euclidean wormholes

In Euclidean gravity, there are several models of Euclidean wormholes
Chen:2016ask ; Chen:2017qeh .

  * (1)

Conformal gravity: From the Wheeler-DeWitt equation of Einstein gravity with a
conformally coupled scalar field, one can introduce the separation of
variables for the entire wave function Hartle:1983ai ; Robles-Perez:2010uvf .
Thanks to this separation of variables, one can specify the energy eigenvalues
$E_{n}=n+1/2$, where $n=0,1,2,3,...$ and $a$ should satisfy

$\displaystyle\dot{a}^{2}=1-\frac{\Lambda}{3}a^{2}-\frac{2E_{n}}{a^{2}},$ (13)

where $\Lambda$ is the cosmological constant. The on-shell solution of this
equation has two turning points: $a_{\mathrm{min}}^{2}\simeq 2E_{n}$ and
$a_{\mathrm{max}}^{2}\simeq 3/\Lambda$, where the former corresponds to the
throat of the Euclidean wormhole and the latter determines the Wick-rotation
point.

  * (2)

String-inspired theory: Euclidean wormholes are also possible in the axion-
induced model Giddings:1987cg  given by the Euclidean action

$\displaystyle S_{E}=\int
d^{4}x\sqrt{+g}\left(-\frac{R}{16\pi}+\frac{1}{2}G_{IJ}(\phi)\nabla^{\mu}\phi^{I}\nabla_{\mu}\phi^{J}\right),$
(14)

where $I,J$ denotes different species of scalar fields and $G_{IJ}$ can have
negative signs in Euclidean signatures. Thanks to the negative-sign
contributions, $a$ should satisfy

$\displaystyle\dot{a}^{2}=1-\frac{C}{a^{4}},$ (15)

where $C$ is a positive constant. This allows a turning point at
$a_{\mathrm{min}}^{2}=\sqrt{C}$, which corresponds to the throat of an
asymptotically flat Euclidean wormhole. Further investigations of Euclidean
wormholes in the anti-de Sitter space are in Maldacena:2004rf ; Arkani-
Hamed:2007cpn .

  * (3)

Kinetic effects of a scalar field: For simplicity, let us consider a free
scalar field with $U(\Phi)=\lambda^{2}U_{0}$ case, where $\lambda$ is a
rescaled cosmological constant. Then, the scalar field satisfies

$\displaystyle\frac{d\phi}{d\tau}=i\frac{A}{a^{3}},$ (16)

where $A$ is a constant. Hence,

$\dot{a}^{2}=1-\lambda^{2}a^{2}-\frac{A^{2}}{a^{4}}$ (17)

should be satisfied in the Euclidean domain. This allows two turning points:
$a_{\mathrm{min}}^{2}\simeq|A|$ and $a_{\mathrm{max}}^{2}\simeq\lambda^{-2}$
for real-valued $A$ Chen:2016ask ; Chen:2017qeh . The former corresponds to
the throat of an asymptotically de Sitter Euclidean wormhole and the latter is
the Wick-rotation point. Note that this kind of Euclidean wormhole can be
generalized to the models with a non-trivial potential with inflationary
scenario Chen:2019cmw  or even in the anti-de Sitter background Kang:2017jmq .

In summary, Euclidean wormholes can be described by the following equation of
motion:

$\dot{a}^{2}=1-Ba^{2}-\frac{C}{a^{n}},$ (18)

where $B$ and $C$ are positive constants and $n$ is a positive number. Here,
$B$, $C$, and $n$ depend on a specific model. In this paper, we will focus on
the case that $n=4$, but our formalism does not depend on a specific model of
the Euclidean wormhole. $C$ determines the throat size of the wormhole
$a_{\mathrm{min}}^{2}\simeq C^{2/n}$ and $B$ determines the Wick-rotation
point $a_{\mathrm{max}}^{2}\simeq 1/B$. Note that $C=0$ provides a compact
instanton according to the no-boundary proposal. Fig. 2 shows an example of
the Euclidean wormhole solution as well as the compact instanton following the
model in Chen:2016ask ; Chen:2017qeh . Here, one can notice that the Euclidean
wormhole is specified by the model parameters $a_{\mathrm{min}}$ and
$a_{\mathrm{max}}$.

Figure 2: The black curve is the real part and the blue curve is the imaginary
part of $a(\tau)$, where we assume an Euclidean wormhole solution following
the model in Chen:2016ask ; Chen:2017qeh  with $a_{\mathrm{min}}\simeq
0.200854$ and $a_{\mathrm{max}}\simeq 1.54488$, while the red dashed curve is
$a(\tau)$ of a compact instanton with the same $a_{\mathrm{max}}\simeq
1.5451$.

###  II.3 Perturbations

In cosmology, in addition to the background geometry and the inflaton field,
cosmological perturbations are also important since they are essential for the
structure formation. In this section, we introduce the perturbations to the
minisuperspace metric following Halliwell-Hawking prescription
Halliwell:1984eu :

$\displaystyle
ds^{2}=\sigma^{2}\left[-\left(N^{2}-N_{i}N^{i}\right)dz^{2}+2N_{i}dx^{i}dz+h_{ij}dx^{i}dx^{j}\right],$
(19)

where

$\displaystyle N$ $\displaystyle=$ $\displaystyle
N_{0}\left[1+\frac{1}{\sqrt{6}}\sum_{nlm}g_{nlm}Q_{nlm}\right],$ (20)
$\displaystyle N_{i}$ $\displaystyle=$ $\displaystyle
a\sum_{nlm}\left[\frac{1}{\sqrt{6}}k_{nlm}\left(P_{i}\right)_{nlm}+\sqrt{2}j^{o}_{nlm}\left(S^{o}_{i}\right)_{nlm}+\sqrt{2}j^{e}_{nlm}\left(S^{e}_{i}\right)_{nlm}\right],$
(21) $\displaystyle h_{ij}$ $\displaystyle=$ $\displaystyle
a^{2}\left[\gamma_{ij}+\sum_{nlm}\left(\frac{\sqrt{6}}{3}a_{nlm}Q_{nlm}\gamma_{ij}+\sqrt{6}b_{nlm}\left(P_{ij}\right)_{nlm}+\sqrt{2}c^{o}_{nlm}\left(S_{ij}^{o}\right)_{nlm}+\sqrt{2}c^{e}_{nlm}\left(S_{ij}^{e}\right)_{nlm}\right.\right.$
(22)
$\displaystyle\left.\left.+2d^{o}_{nlm}\left(G_{ij}^{o}\right)_{nlm}+2d^{e}_{nlm}\left(G_{ij}^{e}\right)_{nlm}\right)\right],$
$\displaystyle\Phi$ $\displaystyle=$
$\displaystyle\sqrt{\frac{3}{4\pi}}\phi+\sqrt{\frac{3\pi}{2}}\sum_{nlm}f_{nlm}Q_{nlm},$
(23)

and $\gamma_{ij}dx^{i}dx^{j}=d\Omega_{3}^{2}$. Here, $\\{a_{n\ell m},b_{n\ell
m},c_{n\ell m},d_{n\ell m},g_{n\ell m},k_{n\ell m},j_{n\ell m},f_{n\ell m}\\}$
are perturbations beyond the minisuperspace, the superscript $o\;(e)$ denotes
odd (even) parity, $Q_{n\ell m}$ is the scalar harmonic on a 3-sphere of the
unit radius, $\\{(P_{i})_{n\ell m},(S_{i})_{n\ell m}\\}$ are vector harmonics,
and $\\{(P_{ij})_{n\ell m},(S_{ij})_{n\ell m},(G_{ij})_{n\ell m}\\}$ are
tensor harmonics.

As usual, $z$ is real-valued in the metric. The Euclidean time is defined by
$d\tau^{2}=-N_{0}^{2}dz^{2}$ for an imaginary-valued $N_{0}$, while the
Lorentzian time is defined by $dt^{2}=N_{0}^{2}dz^{2}$ for a real-valued
$N_{0}$. Therefore, the conventional Wick-rotation corresponds to taking
$d\tau\rightarrow idt$, which casts Eqs. (8), (9), and (10) into the
Lorentzian form:

$\displaystyle{\dot{a}}^{2}+1-a^{2}\left({\dot{\phi}}^{2}+V\left(\phi\right)\right)$
$\displaystyle=$ $\displaystyle 0,$ (24)
$\displaystyle{\ddot{a}}+2a{\dot{\phi}}^{2}-aV$ $\displaystyle=$
$\displaystyle 0,$ (25)
$\displaystyle{\ddot{\phi}}+3\frac{\dot{a}}{a}{\dot{\phi}}+\frac{1}{2}\frac{dV}{d\phi}$
$\displaystyle=$ $\displaystyle 0,$ (26)

where dots now denote differentiation with respect to the Lorentzian time $t$.
In addition, in the slow-roll limit, one can approximate the equation for the
matter perturbation as follows Halliwell:1984eu :

$\displaystyle\ddot{f}_{nlm}+3\frac{\dot{a}}{a}\dot{f}_{nlm}+\left(\frac{1}{2}\frac{d^{2}V}{d\phi^{2}}+\frac{n^{2}-1}{a^{2}}\right)f_{nlm}\simeq
0.$ (27)

###  II.4 Wave function of matter perturbation

Cosmological observables are related to the expectation values; as in quantum
mechanics, these expectation values are determined by the wave functions. In
quantum cosmology, the wave function of the universe can be factorized as:

$\displaystyle\Psi\left[a,\phi,\delta\phi\right]=\Psi_{\mathrm{bg}}\left[a,\phi\right]\prod_{nlm}\psi_{nlm}\left[f_{nlm};a,\phi\right],$
(28)

where $\Psi_{\mathrm{bg}}$ is the wave function of the background fields $a$
and $\phi$, and $\psi_{nlm}$ is the wave function of the matter perturbation
${f}_{nlm}$ with metric determined by the background fields $(a,\phi)$. The
matter wave function $\psi_{nlm}$ can be evaluated from the Euclidean path
integral:

$\displaystyle\psi_{nlm}=\int\mathcal{D}f_{nlm}\text{exp}\left[-\int^{\tau_{\mathrm{f}}}_{\tau_{\mathrm{i}}}d\tau
L_{nlm}\right]\simeq
A_{nlm}\text{exp}\left[-\int^{\tau_{\mathrm{f}}}_{\tau_{\mathrm{i}}}d\tau
L_{nlm}\right],$ (29)

where $A_{nlm}$ is a normalization factor, $L_{nlm}$ is the Euclidean
Lagrangian of the matter perturbation, and the steepest-descent approximation
is applied in the last step. In the Euclidean wormhole considerations, we
regard that $\tau_{\mathrm{i}}$ corresponds to the wormhole throat
($a_{\mathrm{min}}$) and $\tau_{\mathrm{f}}$ corresponds the endpoint of the
integral that can be chosen arbitrarily. We will evaluate the wave function
after the horizon crossing time and check whether our result is independent of
the time choice or not. Also, our result must be consistent with the standard
result in quantum field theory. After integrating the Euclidean action, one
obtains

$\displaystyle\psi_{nlm}\simeq
A_{nlm}\exp\left[\left.-\frac{1}{2}a^{3}f_{nlm}\dot{f}_{nlm}\right|^{\tau_{\mathrm{f}}}_{\tau_{\mathrm{i}}}\right]=B_{nlm}\exp\left[-\left.\frac{1}{2}a^{3}f_{nlm}\dot{f}_{nlm}\right|_{\tau_{\mathrm{f}}}\right],$
(30)

where $B_{nlm}$ is a normalization constant that absorbs the constant
contribution of the Euclidean action at $\tau_{\mathrm{i}}$.

We must regard $\psi_{nlm}$ as a functional of $f_{nlm}$ and hence we need to
regard $f_{nlm}$ as an operator $\hat{f}_{nlm}$. Although we know the value of
the wave function, within the constraint of the value, there are infinitely
many possibilities to choose the functional dependence between $\hat{f}_{nlm}$
and $\psi_{nlm}$. To provide a probability distribution that is consistent
with the results of the quantum field theory, one needs to consider the
quantum mode expansion:

$\displaystyle\hat{f}_{nlm}=f_{nlm}\hat{a}_{nlm}+f^{*}_{nlm}\hat{a}^{\dagger}_{nlm},$
(31)

where $a_{nlm}$ and $a_{nlm}^{\dagger}$ are annihilation and creation
operators of a certain vacuum, respectively. If we impose the condition that
the wave function is annihilated by the operator $a$, we obtain the result
Laflamme:1987mx :

$\displaystyle\psi_{nlm}\left[\hat{f}_{nlm}\right]=C_{nlm}\exp\left[-\left.\left(\frac{a^{3}\dot{f}_{nlm}}{2f_{nlm}}\right)\right|_{\tau_{\mathrm{f}}}\hat{f}_{nlm}^{2}\right],$
(32)

and hence, the wave function has a Gaussian-like form. This is a kind of
vacua, but this is not guaranteed whether this vacuum is the Bunch-Davies
state or not. Indeed, the properties of the vacuum are related to the
background manifold. We will investigate this problem later.

Equivalently, if we Wick-rotate the wave function to the Lorentzian time $t$,
we present as follows:

$\displaystyle\psi_{nlm}\left[\hat{f}_{nlm}\right]=C_{nlm}\exp\left[i\left.\left(\frac{a^{3}{\dot{f}}_{nlm}}{2f_{nlm}}\right)\right|_{t_{\mathrm{f}}}\hat{f}_{nlm}^{2}\right].$
(33)

By defining $f_{nlm}=v_{nlm}/a$ and $d\eta=dt/a$, the Lorentzian equations of
motion of $v_{nlm}(\eta)$ is then

$\displaystyle{v}^{\prime\prime}_{nlm}=-\left[n^{2}-1-\frac{a^{\prime\prime}}{a}+\frac{a^{2}}{2}\frac{d^{2}V}{d\phi^{2}}\right]v_{nlm},$
(34)

where ′ denotes differentiation with respect to $\eta$.

After some computations (see Appendix), we obtain the following formula:

$\displaystyle\mathcal{P}(n)=2\pi^{2}P(n)=n\left(n^{2}-1\right)\left\langle\left|\hat{f}_{n}\right|^{2}\right\rangle=\frac{n\left(n^{2}-1\right)}{2a^{2}\mathrm{Re}\left[-i\frac{v^{\prime}_{n}}{v_{n}}\right]},$
(35)

where ′ denotes a differentiation with respect to $\eta$. Here, $v_{n}=af_{n}$
and $P(n)$ has the conventional normalization.

##  III Quantum states with Euclidean geometric origins

Now we are ready to compute the power spectrum of a scalar field. We will
compare our result from the Euclidean path integral approach with the results
from the quantum field theory in the flat space limit, and we will further ask
the physical meaning as well as their consistency.

###  III.1 Reproduction of the Bunch-Davies state from the no-boundary
proposal

First, let us compute the power spectrum of the compact instantons (no-
boundary proposal).

####  III.1.1 Computations from instantons

In the no-boundary proposal, the Euclidean scale factor is
$a(\tau)=\lambda^{-1}\sin(\lambda\tau)$, and the perturbation mode
$f_{n}(\tau)$ satisfies the equation of motion:

$\displaystyle\ddot{f}_{n}(\tau)=-3\frac{\dot{a}}{a}\dot{f}_{n}+\left(\frac{n^{2}-1}{a^{2}}+\mu^{2}\right)f_{n},$
(36)

where $\lambda$ is a cosmological constant and $\mu$ is the inflaton mass.
After we solve the equation, we obtain the solution:

$\displaystyle
f_{n}(\tau)=D_{n}(\zeta-\zeta^{2})^{(n-1)/2}{}_{2}F_{1}\big{(}n-\nu,n+\nu+1;n+1;\zeta\big{)},$
(37)

where

$\displaystyle\nu=-\frac{1}{2}+\sqrt{\frac{9}{4}-\frac{\mu^{2}}{\lambda^{2}}},\quad\zeta=\frac{1-\cos(\lambda\tau)}{2},$
(38)

$D_{n}$ is a normalization constant, and ${}_{2}F_{1}$ is the hypergeometric
function Laflamme:1987mx . This solution satisfies the Euclidean initial
conditions: $\dot{f}_{1}(\tau=0)=0$ and $f_{n\geq 2}(\tau=0)=0$, as required
by the regularity condition in the no-boundary proposal implied by Eq. (36).
By performing analytic continuation $\tau\rightarrow\pi/2\lambda+it$, one
obtains the corresponding Lorentzian solution. It is known that this solution
picks out the Bunch-Davies vacuum state in a closed universe Halliwell:1989myn
.

In our numerical treatment, we set the Euclidean initial conditions as follows
to make the equation of motion regular Chen:2017aes :

$\displaystyle
f_{n}(\tau_{i})=e^{-i\delta_{n}}\frac{1}{2}\epsilon\tau_{i}^{2},\quad\dot{f}_{n}(\tau_{i})=e^{-i\delta_{n}}\epsilon\tau_{i},$
(39)

where $\delta_{n}$ is an arbitrary phase that can be mode-dependent. Here,
$\tau_{i}\ll 1$ and $\epsilon$ is an arbitrary parameter. (In our work, we
set555The reason for this choice will be discussed in the following sections.
$\delta_{n}=n\pi/2,\epsilon=\tau_{i}=10^{-3}$.) Fig. 3 (left) shows that the
scalar power spectrum obtained based on our numerical treatment agrees well
with that obtained according to the above analytic formula.

####  III.1.2 Dependence on the evaluation time

In general, the power spectrum is both mode-dependent and time-dependent.
However, in certain situations, the time dependence only results in a trivial
amplitude shift. For example, in a spatially flat Lorentzian universe with
power-law $(a=a_{0}t^{p})$ inflation, the positive frequency Bunch-Davies
perturbation mode $U_{k}(\tilde{\eta})/a(\tilde{\eta})$, which now has a
continuous mode $k$ instead of the discrete one $n$ and the conformal time
$\tilde{\eta}\in(-\infty,0^{-})$, is given by

$\displaystyle
U_{k}(\tilde{\eta})=\sqrt{\frac{-\pi\tilde{\eta}}{4}}i^{\nu+1/2}H_{\nu}^{(1)}(-k\tilde{\eta}),$
(40)

where $H_{\nu}^{(1)}$ is the Hankel function and $\nu=3/2+1/(p-1)$. In this
case, the Hubble parameter $H(t)=\dot{a}(t)/a(t)$ is time-dependent:

$\displaystyle
H(y)y^{-\frac{\epsilon}{1-\epsilon}}=\frac{1}{\epsilon}\big{(}\frac{y}{k\epsilon}\big{)}^{\frac{\epsilon}{1-\epsilon}}y^{-\frac{\epsilon}{1-\epsilon}}=H_{*}=const.,$
(41)

where $y=k/aH=-k(1-\epsilon)\tilde{\eta}$, $\epsilon=1/p$, and $H_{*}$ here is
the Hubble parameter evaluated at the horizon-crossing:
$y_{*}=k/a(t_{*})H(t_{*})=1$, which is a constant.

For $\epsilon=0$, which corresponds to the massless de Sitter inflation in a
flat universe, $H=H_{*}$. The corresponding scalar power spectrum
$P(k)=k^{3}|U_{k}|^{2}/(2\pi^{2}a^{2})$ for a given mode $k$ evaluated at any
moment in time $y$ is

$\displaystyle
P(k)=\frac{H^{2}}{4\pi^{2}}(1+y^{2})=\frac{H_{*}^{2}}{4\pi^{2}}(1+y^{2}).$
(42)

Therefore, if we choose the horizon-crossing time, $y=1$, we obtain
$P(k)=H_{*}^{2}/2\pi^{2}$; if we choose the super-horizon limit, $y\ll 1$, we
obtain $P(k)=H_{*}^{2}/4\pi^{2}$, which is the standard result for matter
perturbations in the Bunch-Davies state of a spatially flat, massless de
Sitter universe. In this case, the power spectra at the horizon crossing time
and at the super-horizon limit simply differ by a trivial factor of 2
Kinney:2005vj .

####  III.1.3 Scale-invariance of the power spectrum

In the no-boundary proposal, the Hubble parameter $H(t)=\lambda\tanh(\lambda
t)$ is time-dependent, and, e.g., it has the horizon-crossing value
$H(t_{n})=\lambda n/\sqrt{n^{2}+1}$, which is mode-dependent. By setting
$\lambda=H_{*}$, where $H_{*}$ is the horizon-crossing value for the Hubble
parameter in a flat universe, $H(t_{n})=H_{*}n/\sqrt{n^{2}+1}$. Therefore, for
large modes $n\gg 1$, the curvature term $``1"$ is negligible, which results
in $H(t_{n})\simeq H_{*}$ becomes approximately mode-independent, and this
recovers the scale-invariance of the spectrum.

On the other hand, as far as the time evolution of the comoving Hubble radius
$(1/aH)$ is concerned, the spatial curvature of the Universe can cause
observable differences around the onset of inflation Bonga:2016iuf . Since
perturbation modes with small $n$ (long wavelength) leave the horizon earlier
than modes with large $n$ (short wavelength), these small $n$ modes can carry
imprints of the curvature before the curvature effect is diluted away by the
rapid expansion of the Universe. As a result, the power spectrum in the no-
boundary proposal is suppressed at small $n$ due to the positive spatial
curvature, while the spectrum is always scale-invariant in the absence of
curvature.

The above discussions of the scalar power spectrum of the no-boundary
instantons are summarized in Fig. 3 (right):

  * – 1.

The power spectrum is also time-dependent, where the power spectra at the
horizon crossing time and at the super-horizon limit differ by a factor of “2”
Kinney:2005vj  in the large $n$ limit.

  * – 2.

The scale-invariance is recovered at large $n$.

  * – 3.

The power spectrum is suppressed at small $n$ which was observed in
Chen:2017aes ; Yeom:2017ikw ; Chen:2019mbu .

Now we conclude that our numerical treatment is consistent with the analytic
solution as well as various theoretical expectations.

Figure 3:  Left: The scalar power spectrum based on our numerical treatment
agrees well with analytic results evaluated at different time $y=n/aH$. Right:
In the no-boundary proposal, scale-invariant behavior is recovered at large
$n$, while there is suppression at small $n$. The invariant values are
indicated by the horizontal dashed lines.

###  III.2 Existence of the $\alpha$-vacuum state in Euclidean wormholes

In the previous subsection, we reported that our numerical computations are
consistent with analytic solutions as well as previously known theoretical
expectations. Now, using our technique, we will extend the computations to the
Euclidean wormholes.

Now one can ask what is the crucial difference between the no-boundary
proposal and the Euclidean wormholes. As mentioned in the previous subsection,
the no-boundary proposal imposes initial conditions on the matter
perturbation. However, if we relax the no-boundary proposal, e.g., considering
the Euclidean wormhole solution, we will have more freedom to choose the
initial conditions which may allow the possibility of going beyond the usual
Bunch-Davies state Chernikov:1968zm ; Mottola:1984ar ; Allen:1985ux ;
Bousso:2001mw .

Interestingly, in our setup, we already chose operators for a vacuum
condition. Therefore, it is very reasonable to conclude that the Euclidean
wormholes may explain a kind of vacuum state which goes beyond the Bunch-
Davies state. If we further impose the de Sitter invariance condition, we will
show that the Euclidean wormholes allow the $\alpha$-vacua for matter
perturbations.

####  III.2.1 Standard $\alpha$-vacuum in flat de Sitter space

Standard discussions of $\alpha$-vacuum involve a mode-independent
$\alpha$-parameter with $\mathrm{Re}(\alpha)<0$ and a massive scalar field in
a spatially flat inflationary universe. $\alpha$-vacuum is de Sitter invariant
and its corresponding mode function is given by the Mottola-Allen
transformation Mottola:1984ar ; Allen:1985ux :

$\displaystyle
u_{k}\left(\tilde{\eta}\right)=A_{k}U_{k}\left(\tilde{\eta}\right)+B_{k}U^{*}_{k}\left(\tilde{\eta}\right),$
(43)

where

$\displaystyle\left|A_{k}\right|^{2}-\left|B_{k}\right|^{2}=1$ (44)

is the normalization condition, $\tilde{\eta}\in(-\infty,0^{-})$,

$\displaystyle
U_{k}\left(\tilde{\eta}\right)=\sqrt{\frac{-\pi\tilde{\eta}}{4}}i^{\nu+1/2}H_{\nu}^{(1)}\left(-k\tilde{\eta}\right),$
(45)

and

$\displaystyle\nu=\sqrt{\frac{9}{4}-\frac{\mu^{2}}{H_{*}^{2}}}.$ (46)

Here, $U_{k}$ is the standard positive frequency Bunch-Davies mode and
$(A_{k},B_{k})=N_{\alpha}(1,e^{\alpha})$ are the Bogoliubov coefficients with
$N_{\alpha}=(1-e^{\alpha+\alpha^{*}})^{-1/2}$ Bousso:2001mw .

Note that the existence of the $\alpha$-parameter modifies the scalar power
spectrum. For example, in the massless case $\mu=0$ and in the super-Hubble
limit $y=k/\dot{a}(t)\ll 1$,

$\displaystyle\left|u_{k}\left(y\ll 1\right)\right|^{2}=\left|U_{k}(y\ll
1)\right|^{2}\left(\frac{1+e^{\alpha+\alpha^{*}}-2\mathrm{Re}(e^{\alpha})}{1-e^{\alpha+\alpha^{*}}}\right),$
(47)

where $|U_{k}(y\ll 1)/a|^{2}=H_{*}^{2}/2k^{3}$ and
$a=e^{H_{*}t}=-1/H_{*}\tilde{\eta}$. Since the scalar power spectrum is
proportional to $|u_{k}/a|^{2}$, the mode-independent $\alpha$-parameter
results in a mode-independent amplitude shift.

####  III.2.2 No-boundary vs. Euclidean wormholes: Euclidean and Lorentzian
initial conditions

In the no-boundary proposal, the requirements of compactness and regularity at
the South Pole uniquely pick out the Euclidean growing mode, and this leads to
a Bunch-Davies mode. To allow the $\alpha$-vacuum, the other linearly
independent solution, which is an Euclidean decaying mode, must be allowed.
However, the Euclidean decaying mode diverges at the South Pole and breaks the
regularity condition of the no-boundary instanton. This shows that the no-
boundary proposal does not admit an $\alpha$-vacuum state.

On the other hand, in the Euclidean wormhole scenario, since
$a(\tau=0)=a_{\mathrm{min}}\neq 0$, there would be no divergence for the two
linearly independent mode functions, and they can both survive to construct
the $\alpha$-vacuum mode:

$\displaystyle v_{n}=N_{\alpha}\left(V_{n}+e^{\alpha^{*}}V^{*}_{n}\right)$
(48)

in the Lorentzian domain, where $V_{n}$ ($V^{*}_{n}$) is approximately the
negative (positive) frequency Bunch-Davies mode, which will correspond to a
growing (decaying) mode in the Euclidean domain, and $\alpha^{*}$ is the
complex conjugate of $\alpha$.

Let us denote the Euclidean growing (decaying) mode corresponding to $V_{n}/a$
$(V^{*}_{n}/a)$ by $g_{n}$ $(d_{n})$. Then, Lorentzian initial conditions of
$v_{n}$ at $t=0$ are determined by the Euclidean modes evaluated at the Wick-
rotation point $\tau=X$ by

$\displaystyle v_{n}$ $\displaystyle=$ $\displaystyle
N_{\alpha}a_{\mathrm{max}}\left(g_{n}+e^{\alpha^{*}}g^{*}_{n}\right),$ (49)
$\displaystyle v^{\prime}_{n}$ $\displaystyle=$ $\displaystyle
iN_{\alpha}a_{\mathrm{max}}^{2}\left(\dot{g}_{n}-e^{\alpha^{*}}\dot{g^{*}}_{n}\right),$
(50)

where ′ is the derivation with respect to $\eta\equiv
2\arctan[\tanh\big{(}\lambda t/2\big{)}]$, $a(\tau=X)=a_{\mathrm{max}}$, and
we have made use of the fact that

$\displaystyle d_{n}(X)$ $\displaystyle=$ $\displaystyle g^{*}_{n}(X),$ (51)
$\displaystyle\dot{d}_{n}(X)$ $\displaystyle=$
$\displaystyle-\dot{g^{*}}_{n}(X),$ (52)

which are necessary for the analyticity. This shows that, for a wide range in
the $\alpha$-parameter space, i.e., $|\mathrm{Re}(\alpha)|\geq 1$, $v_{n}$ can
be approximately dominated by a single Bunch-Davies component, if $g_{n}(X)$
and $\dot{g}_{n}(X)$ have the same order of magnitudes as their complex
conjugates, due to the exponentially hierarchy between two modes by the factor
$e^{\mathrm{Re}(\alpha)}$. One can observe that if $\mathrm{Re}(g_{n})$
monotonically increases forward in $\tau$, then $\mathrm{Re}(d_{n})$
monotonically decreases (see Fig. 4). This means that, even when
$|\mathrm{Re}(\alpha)|\geq 1$, it is still possible that $v_{n}$ in the
Euclidean domain has different Euclidean initial conditions compared to the
no-boundary proposal, depending on the value of $e^{\alpha^{*}}d_{n}(0)$.

To summarize, the Euclidean growing and decaying modes are exponentially
increasing and decreasing in the Euclidean domain. In the no-boundary
proposal, since the decaying mode diverges at $\tau=0$, one is forced to
choose $\mathrm{Re}(\alpha)\rightarrow-\infty$, which uniquely picks out the
Bunch-Davies vacuum. However, it is fair to say that Euclidean wormholes allow
the adjustment of the $\alpha$-parameter to allow non-trivial $\alpha$-vacua
by superposing the growing and decaying modes, since the two modes are
regular.

Figure 4:  Perturbation modes ($n=3$) in the Euclidean regime with
$\delta_{n}=0$. The left is the Euclidean growing mode and the right is the
Euclidean decaying mode. These two Euclidean modes can be significantly
different at small $\tau$, but they can have identical values at the Wick
rotation point $\tau=X$.

####  III.2.3 The $n$-dependent phase factor in the Euclidean regime

Now we ask how to adjust the initial condition of scalar perturbations at
$\tau=0$. In this subsection, we discuss the detailed reason why we have
chosen a specific form Eq. (39).

Recall that we have included a non-trivial phase $\delta_{n}=n\pi/2$ for the
Euclidean initial conditions in Eq. (39). In the following, we will
demonstrate that there is a non-trivial $n$-dependent phase factor in the mode
function for closed universes which is consistent with the flat space limit.
As a result, we need to include this phase factor $\delta_{n}$ for the
Euclidean initial conditions to have the correct power spectrum for a general
$\alpha$-vacuum.

In the following, we utilize a massless spectator scalar field in a pure de
Sitter background to analytically show why this phase factor emerges in the
closed universe setting666We are aware that the mixing of positive and
negative frequency modes for the massless scalar field cannot be called an
alpha vacuum since the corresponding vacuum is not invariant under the full
continuous de Sitter symmetry. Nevertheless, this phase issue exists both in
the massive and massless cases, and the analytic result is much easier to
derive in the massless case.. Before we enter the main discussion, we first
quickly review the parallel scenario in the flat slicing de Sitter space and
introduce a more useful variable that will play a crucial role in the closed
universe case later.

Firstly, the mode equation of a massless spectator scalar field in flat
slicing de Sitter space is given by Baumann:2009ds ; Baumann:2018muz

$u_{k}^{\prime\prime}(\tilde{\eta})+\left(k^{2}-\frac{2}{\tilde{\eta}^{2}}\right)u_{k}(\tilde{\eta})=0,$
(53)

where $k$ is the comoving wave number and $\tilde{\eta}$ is the conformal time
runs from $-\infty$ to $0^{-}$. This differential equation admits the general
solution

$u_{k}(\tilde{\eta})=A_{k}\frac{1}{\sqrt{2k}}\left(1-\frac{i}{k\tilde{\eta}}\right)e^{-ik\tilde{\eta}}+B_{k}\frac{1}{\sqrt{2k}}\left(1+\frac{i}{k\tilde{\eta}}\right)e^{ik\tilde{\eta}},$
(54)

where the coefficients are chosen to follow the relation
$|A_{k}|^{2}-|B_{k}|^{2}=1$ to have the creation and annihilation operators
satisfying the required commutation relation.

When the wavelength of a mode is much smaller compared to the Hubble radius,
the curvature of the spacetime has little effect on it. In the flat slicing de
Sitter space, it happens for all modes in the infinite past
$\tilde{\eta}\rightarrow-\infty$, where the mode solutions reduce to the form
of the usual positive mode solution in a Minkowski spacetime

$\lim_{\tilde{\eta}\rightarrow\\-\infty}u_{k}(\tilde{\eta})=\frac{1}{\sqrt{2k}}e^{-ik\tilde{\eta}}.$
(55)

Thus, modes in their sub-horizon limit offer the natural choice of
coefficients: $A_{k}=1$ and $B_{k}=0$, which leads to the mode solution

$u_{k}(\tilde{\eta})=\frac{1}{\sqrt{2k}}\left(1-\frac{i}{k\tilde{\eta}}\right)e^{-ik\tilde{\eta}}.$
(56)

That is, we choose only the positive frequency mode function.

Next, to make the comparison of the wavelength of a mode and the
characteristic length scale of the expanding background more explicitly, we
can consider the following variable

$\tilde{y}\equiv\frac{k}{aH}=-k\tilde{\eta}\sim\frac{R_{H}}{l_{phy}(k)},$ (57)

where $R_{H}$ is the physical Hubble radius, and $l_{phy}(k)$ is the physical
length of a mode specified by the comoving wave number $k$. Using Eq. (57),
one can rewrite Eq. (53) into

$u_{k}^{\prime\prime}(\tilde{y})+\left(1-\frac{2}{\tilde{y}^{2}}\right)u_{k}(\tilde{y})=0,$
(58)

and the positive frequency mode function Eq. (56) into

$u_{k}(\tilde{y})=\frac{1}{\sqrt{2k}}\left(1+\frac{i}{\tilde{y}}\right)e^{i\tilde{y}}.$
(59)

Notice that the condition of sub-horizon limit, where we previously determined
the coefficients $A_{k}$ and $B_{k}$, becomes $\tilde{y}\gg 1$, and under this
limit, Eq. (58) reduces to

$u_{k}^{\prime\prime}(\tilde{y})+u_{k}(\tilde{y})=0,$ (60)

for which the positive frequency mode solution is

$u_{k}=\frac{1}{\sqrt{2k}}e^{i\tilde{y}}=\frac{1}{\sqrt{2k}}e^{-ik\tilde{\eta}}.$
(61)

We shall refer to Eq. (61) as the standard sub-horizon limit (for the massive
case, it is called the Bunch-Davies limit) in which the mode solution is of a
plane-wave form both in $\tilde{\eta}$ and $\tilde{y}$. Meanwhile, the super-
horizon limit, where we evaluate the power spectrum of the scalar field, is
simply given by the limit $\tilde{y}\ll 1$. We now see how these relations
play out in the closed de Sitter space.

Returning to the case of a closed universe, the equation of motion for the
perturbation mode $f_{n}=v_{n}/a=u^{*}_{n}/a$,777Following the usual
convention, we denote the positive frequency mode by $u$ and the negative
frequency mode by $v$. They are simply the complex conjugate of each other.
Hence, $u$ and $v^{*}$ define the same vacuum state. Eq. (36), can be cast
into the following massless Lorentzian form:

$u^{\prime\prime}_{n}+\left[n^{2}-1+\frac{1}{2}\left(\cos
2\eta-3\right)\sec^{2}\eta\right]u_{n}=0,$ (62)

upon the standard analytic continuation and transforming the comoving time
into the conventional conformal time: $\eta=2\arctan[\tanh\big{(}\lambda
t/2\big{)}]$. In this convention, $0\leq\eta<\pi/2$. Similar to Eq. (57), we
can define the variable

$y=\frac{n}{aH}=\frac{n}{\tan\eta}\sim\frac{R_{H}}{l_{phy}(n)},$ (63)

and use it to rewrite Eq. (62) into

$\left(1+\frac{y^{2}}{n^{2}}\right)\left[(n^{2}+y^{2})u^{\prime\prime}_{n}(y)+2yu^{\prime}_{n}(y)\right]+\left[\frac{n^{2}}{y^{2}}(y^{2}-2)-2\right]u_{n}=0.$
(64)

This admits the positive frequency no-boundary solution Invariant vacuum 888In
appendix C, we proved the equivalence between this solution and the massless
limit of Eq.(37).:

$u_{n}=\frac{n}{\sqrt{2(n+1)n(n-1)}}\left(1+\frac{i}{y}\right)\exp\left[-in\arctan\frac{n}{y}\right],$
(65)

where $n\geq 2$. Now, comparing Eq. (58) with Eq. (64), we see that one major
difference between them is that the $k$ is removed from the differential
equation Eq. (58) but $n$ still exists in Eq. (64). As a result, there are
three special regimes in the closed de Sitter scenario instead of two as in
the flat de Sitter case, i.e., $y\ll 1$ and $y\gg 1$. Putting these three
special regimes according to the time order, they are $y\gg n\gg 1$, $n\gg
y\gg 1$, and $n\gg 1\gg y$.

One can check that the mode equation Eq. (64) reduces to the same form of the
sub-horizon and super-horizon limits of Eq. (58) in the regimes $n\gg y\gg 1$
and $n\gg 1\gg y$ as

$\begin{split}u_{n}^{\prime\prime}(y)+u_{n}(y)=0\quad&when\quad n\gg y\gg
1,\\\ u_{n}^{\prime\prime}(y)-\frac{2}{y^{2}}u_{n}(y)=0\quad&when\quad n\gg
1\gg y,\end{split}$ (66)

respectively, wherein the second line we have assumed
$|n^{2}u_{n}^{\prime\prime}|\gg|yu_{n}^{\prime}|$, which can be checked by the
solution whether it is consistent. Now, the crucial point is this: does the
positive frequency solution Eq. (65) also reduce to the similar forms in these
two regimes as those in the flat de Sitter case?

To answer this question, we use the following approximation in the two limits:

$\begin{split}\arctan\frac{n}{y}\simeq\frac{n}{y}+\mathcal{O}\left(\frac{n^{3}}{y^{3}}\right)\quad&when\quad\frac{n}{y}\ll
1,\\\
\arctan\frac{n}{y}\simeq\frac{\pi}{2}-\frac{y}{n}+\mathcal{O}\left(\frac{y^{3}}{n^{3}}\right)\quad&when\quad\frac{y}{n}\ll
1.\end{split}$ (67)

Then we have Eq. (65) reducing to

$\begin{split}u_{n}(y)\sim\frac{1}{\sqrt{2n}}e^{-in^{2}/y}\quad&when\quad y\gg
n\gg 1,\\\ u_{n}(y)\sim\frac{1}{\sqrt{2n}}e^{-in\pi/2}e^{iy}\quad&when\quad
n\gg y\gg 1,\\\
u_{n}(y)\sim\frac{1}{\sqrt{2n}}e^{-in\pi/2}\frac{i}{y}e^{iy}\sim\frac{1}{\sqrt{2n}}e^{-in\pi/2}\frac{i}{y}\quad&when\quad
n\gg 1\gg y,\end{split}$ (68)

where the approximation $e^{iy}\sim 1$ is used to have the final form of
$u_{n}(y)$ in regime $n\gg 1\gg y$ satisfying the super-horizon limit of the
mode equation given in Eq. (66). We then see that the mode solution contains
an extra $n$-dependent phase factor $e^{-in\pi/2}$ in the regimes we are
interested in. To remove this extra factor in the regimes with $n\gg 1\gg y$,
we multiply Eq. (65) by an overall phase factor $e^{in\pi/2}$ to have

$\tilde{u}_{n}(y)=e^{in\pi/2}u_{n}(y)=e^{in\pi/2}\frac{n}{\sqrt{2(n+1)n(n-1)}}\left(1+\frac{i}{y}\right)\exp\left[-in\arctan\frac{n}{y}\right],$
(69)

and conclude that $\tilde{u}_{n}(y)$ should be the correct form of the
positive frequency mode solution for a massless spectator field in the closed
de Sitter space. Meanwhile, we can easily see that the multiplication of
$e^{in\pi/2}$ creates an $n$-dependent phase factor to the regimes $y\gg n\gg
1$. That is,

$\tilde{u}_{n}(y)\sim e^{in\pi/2}\frac{1}{\sqrt{2n}}e^{-in^{2}/y}\quad
when\quad y\gg n\gg 1.$ (70)

As we mentioned earlier, this is the regime corresponding to the earliest
time, and especially this extra phase in Eq. (69) exists after analytic
continuation to the Euclidean regime. This is the reason why we employed
$\delta_{n}=n\pi/2$ for the initial conditions Eq. (39).

####  III.2.4 Agreement between analytic and numerical results

Now, based on these initial conditions for the $\alpha$-vacuum, we report on
the numerical results of the scalar power spectrum.

##### Bunch-Davies vacuum

The left of Fig. 5 shows the scalar power spectrum $P(n)$ at different times
with the Euclidean matter perturbation $f_{n}$ satisfying the initial
conditions Eq. (39) in the presence of an Euclidean wormhole. Interestingly,
the right of Fig. 5 shows that the power spectrum of the Euclidean wormhole is
approximately identical to that of the no-boundary proposal. As indicated in
Fig. 6, the perturbation modes $f_{n}$ satisfying Eq. (39) are in general
different in different Euclidean background geometries, especially at small
$\tau$. However, near the Wick-rotation point, the quantity
$\dot{f}_{n}/f_{n}$ (and therefore the Lorentzian initial condition) becomes
approximately identical. This illustrates that a certain extreme of the
results of Euclidean wormholes consistently corresponds to the result of the
no-boundary proposal.

#####  $\alpha$-vacuum

In the last subsection, we considered the Euclidean wormhole power spectrum in
which the Euclidean perturbation mode $f_{n}=v_{n}/a$ satisfies the initial
conditions given in Eq. (39). This spectrum approximates to the no-boundary
(Bunch-Davies) spectrum. Hence, let us denote this solution as $f_{n}=g_{n}$
and $v_{n}=V_{n}$, where $g_{n}$ is the growing mode satisfying Eq. (39) and
the equalities refer to picking out a single Bunch-Davies component among the
$\alpha$-vacuum expression, Eq. (48). The scalar power spectrum, in this case,
is then, according to Eq. (35),

$\displaystyle
P(n)=\frac{n(n^{2}-1)}{4\pi^{2}\mathrm{Re}\big{[}-i\frac{v^{\prime}_{n}}{v_{n}}\big{]}}=\frac{n(n^{2}-1)}{4\pi^{2}\mathrm{Re}\big{[}-i\frac{V^{\prime}_{n}}{V_{n}}\big{]}}\simeq\frac{\lambda^{2}}{4\pi^{2}}\frac{n^{2}(1+y^{2})}{n^{2}+y^{2}},$
(71)

which is plotted in Fig. 5.

On the other hand, according to Eq. (48), the $\alpha$-vacuum mode leads to
the new power spectrum:

$\displaystyle
P(n)=S_{\alpha}\left(\frac{n(n^{2}-1)}{4\pi^{2}\mathrm{Re}\left[-i\frac{V^{\prime}_{n}}{V_{n}}\right]}\right),$
(72)

where the shifting function

$\displaystyle
S_{\alpha}=N_{\alpha}^{2}\left(1+e^{\alpha+\alpha^{*}}+\frac{2\mathrm{Re}[e^{\alpha}(V_{n})^{2}]}{|V_{n}|^{2}}\right)$
(73)

is the modification to the original Bunch-Davies power spectrum when the
$\alpha$-vacuum state is considered. This result is consistent with the
results of the flat space limit.

Although we do not have an analytic solution for the perturbation mode in the
presence of a Euclidean wormhole, it is still possible to comprehend several
features in the spectrum by utilizing existing analytic no-boundary Bunch-
Davies solutions, since, as previously shown in Fig. 3, Euclidean wormhole (in
our discussion) and the no-boundary proposal have similar spectra when Bunch-
Davies state is considered. Hence, multiplying Eq. (65) by the phase factor
$e^{in\pi/2}$ and taking its complex conjugate, $V_{n}=e^{-in\pi/2}u^{*}_{n}$,
we obtain, for $\alpha\in\mathbb{R}$ and mode-independent,

$\displaystyle\frac{\mathrm{Re}[e^{\alpha}(V_{n})^{2}]}{|V_{n}|^{2}}\simeq-e^{\alpha}\sin(2y)\simeq-e^{\alpha}\sin(2),\quad$
$\displaystyle when\quad n\gg y=1,$ (74)
$\displaystyle\frac{\mathrm{Re}[e^{\alpha}(V_{n})^{2}]}{|V_{n}|^{2}}\simeq-e^{\alpha}\cos(2y)\simeq-e^{\alpha},\quad$
$\displaystyle when\quad n\gg 1\gg y.$ (75)

This indicates that there will be an overall suppression in the spectrum at
horizon-crossing or in the super-horizon regime, which justifies the results
shown in Fig. 7. This is qualitatively consistent with the conventional
suppression that occurs in a flat de Sitter universe with a mode-independent
$\alpha$-parameter Danielsson:2002kx ; Easther:2002xe ; Easther:2004vq .

Figure 5:  Left: Scalar power spectrum $P(n)$ in the presence of an Euclidean
wormhole in a closed universe with a flat inflationary potential. In the
presence of a positive spatial curvature, there is a suppression at small $n$.
At large $n$, the spectrum recovers the scale-invariant behavior. Right:
Comparison between the spectra for the no-boundary proposal and the Euclidean
wormhole scenario with perturbation mode satisfying identical Euclidean
initial conditions Eq. (39).

Figure 6:  The ratio $\dot{f}_{n}/f_{n}$ at $\tau=0.01$ (left) and at the
Wick-rotation point $\tau=X$ (right) in the Euclidean domain for $f_{n}$
satisfying the initial conditions Eq. (39) in the no-boundary proposal (black)
and the Euclidean wormhole scenario (blue).

Figure 7:  Scalar power spectrum $P(n)$ in the presence of an Euclidean
wormhole in a closed universe with a sufficiently flat inflationary potential
at $y=1$ (left) and $y=0.01$ (right). On the one hand, at large $n$, the
spectrum becomes scale-invariant; at small $n$, the spectrum is suppressed by
the curvature. On the other hand, a mode-independent $\alpha$-parameter
results in an overall suppression.

To summarize, we could obtain the scalar power spectrum of the $\alpha$-vacuum
from the perturbations of Euclidean wormholes, where the results are
consistent with the analytic results of the flat space limit (large $n$
limit).

##  IV Conclusion

In this paper, we followed the Euclidean path integral approach and
investigated the cosmological observables under the Euclidean wormholes
geometry, which is a very natural extension of the no-boundary compact
instantons one.

We obtained the power spectrum formula that can be applied to the Euclidean
wormholes case. To define the power spectrum, we need to provide a vacuum
condition, where there is an ambiguity as to whether the vacuum is necessarily
the Bunch-Davies type or the de Sitter invariant vacuum, the so-called
$\alpha$-vacuum. Indeed, if we impose the compact instantons ansatz, the
Euclidean vacuum uniquely points to the Bunch-Davies vacuum, whereas under a
different geometry, i.e., Euclidean wormholes, a different type of vacuum, the
$\alpha$-vacuum state, is allowed. This is not too surprising, because the
regularity condition at the bottleneck of the wormhole is not required. This
renders the quantum states of the universe a geometric origin.

We note, however, that a wide range of $\alpha$-parameter spaces would provide
a vacuum state that is indistinguishable from the Bunch-Davies state. The
simple reason is that one of the two linearly independent modes, i.e., the
exponentially decreasing (in Euclidean time) mode, is suppressed by a factor
of $e^{\mathrm{Re}(\alpha)}$; hence, in practice, only one mode can survive,
and that sufficiently explains the dominance of the Bunch-Davies vacuum.

Nevertheless, the typical $\alpha$-vacuum that is distinguishable from the
Bunch-Davies state remains possible, even though the parameter space is
somewhat narrow. In this sense, from the perspective of the Euclidean path
integral approach, the existence of the $\alpha$-vacuum is a consequence of
the different geometry of the Euclidean wormholes from that of the no-boundary
compact instantons.

Our ultimate goal is to embed this scenario into a realistic inflation model.
We have already obtained a formula that includes curvature perturbations,
where we constructed a background wormhole solution that is consistent with
the inflation model. A more ambitious goal is to find out whether there exists
any other observable, such as primordial gravitational waves, that can either
confirm or falsify our scenario by future observations.

Last but not least, if our universe was indeed originated from the Euclidean
wormholes, then one can naturally interpret that there exist two back-to-back
universes that are created from nothing. Then, these two universes must be
entangled with each other. Can the entanglement entropy between the two
universes be estimated? Furthermore, if it is the case, then our universe
would not be a pure state. What then would be the implications in the
theoretical and observational perspectives? We leave these topics for future
research efforts.

## Acknowledgment

PC was supported by Taiwan’s National Science and Technology Council (NSTC)
under project number 110-2112-M-002-031, and by the Leung Center for Cosmology
and Particle Astrophysics (LeCosPA), National Taiwan University. KL
appreciates the Elite Doctoral Scholarship jointly provided by National Taiwan
University (NTU) and Taiwan’s National Science and Technology Council (NSTC).
DY and WL were supported by the National Research Foundation of Korea (Grant
No.: 2021R1C1C1008622, 2021R1A4A5031460).

##  Appendix A Expectation value of matter perburbation

In this appendix, we compute the expectation values of observables which are
important to compare with cosmological observations Chen:2017aes ;
Yeom:2017ikw ; Chen:2019mbu .

Since the wave function for perturbation $\psi_{nlm}$ satisfies the
Schrodinger equation, it can be normalized by

$\displaystyle\int df_{nlm}\psi^{*}_{nlm}\psi_{nlm}=1$ (76)

in the Lorentzian domain. Therefore, we obtain the normalization factor:

$\displaystyle\sqrt{\pi}|C_{nlm}|^{2}=\sqrt{\mathrm{Re}\left[a^{2}\frac{\dot{f}_{nlm}}{f_{nlm}}\right]}.$
(77)

From this, we can derive the two-point function:

$\displaystyle\left\langle\left|\hat{f}_{nlm}\right|^{2}\right\rangle=\int
df_{nlm}f_{nlm}^{2}\left|\psi_{nlm}\right|^{2}=\left.\frac{1}{2a^{3}\mathrm{Re}\left[\frac{\dot{f}_{nlm}}{f_{nlm}}\right]}\right|_{t_{\mathrm{f}}}.$
(78)

Now we need to sum over all modes:

$\displaystyle\left\langle\delta\hat{\phi}^{2}\right\rangle=2\pi^{2}\sum_{nlm}\sum_{n^{\prime}l^{\prime}m^{\prime}}\left\langle\hat{f}^{\dagger}_{nlm}\hat{f}_{n^{\prime}l^{\prime}m^{\prime}}\right\rangle
Q_{nlm}^{*}Q_{n^{\prime}l^{\prime}m^{\prime}}.$ (79)

By taking the average over a 3-sphere, one obtains

$\displaystyle\left\langle\delta\hat{\phi}^{2}\right\rangle_{S_{3}}\equiv\frac{\int
d\Omega\left\langle\delta\hat{\phi}^{2}\right\rangle}{\int
d\Omega}=\sum_{nlm}\left\langle\left|\hat{f}_{nlm}\right|^{2}\right\rangle=\sum_{n}n^{2}\left\langle\left|\hat{f}_{n}\right|^{2}\right\rangle,$
(80)

where $d\Omega=\sin^{2}\chi\sin\theta d\chi d\theta d\varphi$ and relying on
the relations

$\displaystyle\int d\Omega~{}Q_{nlm}Q_{n^{\prime}l^{\prime}m^{\prime}}$
$\displaystyle=$
$\displaystyle\delta_{nn^{\prime}}\delta_{ll^{\prime}}\delta_{mm^{\prime}},$
(81) $\displaystyle\sum_{lm}1=\sum_{l=0}^{n-1}\sum_{m=-l}^{l}1$
$\displaystyle=$ $\displaystyle\sum_{l=0}^{n-1}(2l+1)=n^{2}.$ (82)

Therefore, the power spectrum of the scalar field is

$\displaystyle\left\langle\delta\hat{\phi}^{2}\right\rangle_{S_{3}}=\sum_{n}\frac{n\mathcal{P}(n)}{n^{2}-1}=2\pi^{2}\sum_{n}\frac{nP(n)}{n^{2}-1},$
(83)

and hence,

$\displaystyle\mathcal{P}(n)=n\left(n^{2}-1\right)\left\langle\left|\hat{f}_{n}\right|^{2}\right\rangle=\frac{n\left(n^{2}-1\right)}{2a^{2}\mathrm{Re}\left[-i\frac{v^{\prime}_{n}}{v_{n}}\right]},$
(84)

where ′ denotes a differentiation with respect to $\eta$. Here, $v_{n}=af_{n}$
and $P(n)$ has the conventional normalization.

##  Appendix B Curvature perturbation and cosmological observables

Up to now, what we have considered is the matter perturbation. To obtain the
perturbations of gauge-independent observables, we consider the dimensionless
and gauge-invariant comoving curvature perturbations using the flat-slicing
gauge Peter:2013avv :

$\displaystyle\mathcal{R}(t,\chi,\theta,\varphi)=\frac{H}{\dot{\phi}}\delta\phi=\sqrt{2}\pi\sum_{nlm}\mathcal{R}_{n}(t)Q_{nlm}(\chi,\theta,\varphi).$
(85)

From this, the curvature 2-point function is then

$\displaystyle\left\langle\hat{\mathcal{R}}^{\dagger}\hat{\mathcal{R}}\right\rangle_{S_{3}}=\sum_{n}\frac{n\mathcal{P}_{\mathcal{R}}(n)}{n^{2}-1}=\left|\frac{H}{\dot{\phi}}\right|^{2}\sum_{n}n^{2}\left\langle\left|\hat{f}_{n}^{2}\right|\right\rangle.$
(86)

From this, we obtain the primordial comoving curvature power spectrum

$\displaystyle\mathcal{P}_{\mathcal{R}}(n)=\left|\frac{H}{\dot{\phi}}\right|^{2}\mathcal{P}(n).$
(87)

In this paper, since we will only be considering a sufficiently flat
inflationary potential, it does not make sense to discuss the curvature power
spectrum in any more detail, as the inflation never ends and $\dot{\phi}$
would lead to a divergence in the curvature power spectrum. Nevertheless, it
is worth mentioning that the presence of an Euclidean wormhole of our type,
i.e., kinetic effects of a scalar field, should induce a pre-inflationary
phase that emerges in the curvature power spectrum. This may have interesting
effects on small modes $n$ in the power spectrum. For example, the right of
Fig. 8 shows that the Lorentzian inflation, instead of being a constant in a
flat potential, has a period of non-trivial evolution in the presence of a
Euclidean wormhole. The duration of this pre-inflationary phase is related to
the wormhole throat $a_{\text{min}}^{2}\simeq A>0$ by $\dot{\phi}=-A/a^{3}$.
Therefore, the larger the wormhole throat, the more steep and longer the pre-
inflationary phase lasts. It would be interesting to study this pre-
inflationary effect (see also Chen:2017aes ; Yeom:2017ikw ; Chen:2019mbu ) in
detail elsewhere with a more realistic inflationary potential that allows the
Euclidean wormholes (e.g., Chen:2019cmw ) with which the investigation of the
curvature power spectrum is meaningful.

Figure 8:  A demonstration of the time evolution of the inflaton field $\phi$
in the Euclidean ($\tau$) and Lorentzian ($t$) domain, where the Euclidean
wormhole follows the model in Chen:2016ask ; Chen:2017qeh . The black curves
are the real part and the blue curves are the imaginary part of the inflaton
field $\phi$.

##  Appendix C Equivalence of mode functions

By using the identity for the hypergeometric function,

${}_{2}F_{1}(n-1,n+2;n+1;\frac{1-\cos(\lambda\tau)}{2})=[\cos(\lambda\tau/2)]^{-2n}\big{(}\frac{n+\cos(\lambda\tau)}{n+1}\big{)},$
(88)

the massless limit of the regular solution Eq. (37) is simplified to

$f_{n}=\frac{\lambda[\cos(\lambda\tau/2)]^{-2n}[\sin(\lambda\tau)]^{n-1}[n+\cos(\lambda\tau)]}{2^{n}\sqrt{2(n+1)n(n-1)}}.$
(89)

Upon analytic continuation, $\tau\rightarrow\pi/2\lambda+it$, the mode
function in the Lorentzian domain is found out to be

$f_{n}=\frac{\lambda[\cosh(\lambda t)]^{n-1}[1-i\sinh(\lambda
t)]^{-n}[n-i\sinh(\lambda t)]}{\sqrt{2(n+1)n(n-1)}},$ (90)

where we have normalized the mode function such that it has the Wronskian:
$f_{n}\dot{f}_{n}^{*}-f_{n}^{*}\dot{f}_{n}=-i/(\lambda^{-1}\cosh(\lambda
t))^{3}$.

In terms of $y=n/aH$, $t=\lambda^{-1}\text{arcsinh}(n/y)$, the mode function
becomes

$v_{n}=af_{n}=\frac{n}{\sqrt{2(n+1)n(n-1)}}\big{(}1-\frac{i}{y}\big{)}\big{(}1+\frac{n^{2}}{y^{2}}\big{)}^{n/2}\big{(}\frac{iy}{n}\big{)}^{n}\big{(}1+\frac{iy}{n}\big{)}^{-n}.$
(91)

To simplify this, by defining

$e^{inX}\equiv\big{(}1+\frac{n^{2}}{y^{2}}\big{)}^{n/2}\big{(}\frac{iy}{n}\big{)}^{n}\big{(}1+\frac{iy}{n}\big{)}^{-n},$
(92)

taking logarithm on both sides, and using the identity:
$\arctan(x)=\frac{i}{2}\log\big{(}\frac{1-ix}{1+ix}\big{)}$, one finds
$X=\arctan(n/y)$. Therefore, one proves that

$v_{n}=af_{n}=\frac{n}{\sqrt{2(n+1)n(n-1)}}\big{(}1-\frac{i}{y}\big{)}e^{in\arctan(n/y)}$
(93)

is nothing but the complex conjugate of Eq. (65).

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