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72
DETAILS FOR THE DYNAMICS OF SUPERCURVATURE MODE
73
Details for the dynamics of supercurvature mode
This part will largely rely on the basis of Ref. [53]. The formulation of ScmDE setup for inhomogeneous dark
energy most importantly relies on and starts from the feature that, the supercurvature mode of an ultralight scalar
field may exists and stays superhorizon with spatial fluctuations on the present universe, which was developed in
Ref. [54]. However, this part may make for a better understanding of this setup. Consider an ultralight scalar field
φ as a canonical free field on the CDL instanton geometry described by the Euclidean spacetime metric
ds2Euc. = a2 (X)(dX 2 + dθ2 + sin2 θdΩ22 ),
(A.1)
whose symmetry of geometry is deformed by the different vacuum states before and after the quantum tunneling
and the associative bubble wall, i.e., the tunneling wall, which is demonstrated in Fig. 17.
𝑋!
−∞
Bubble
Ancestor
Vacuum
Bounce wall at X=X0
(thin-wall approximation)
Figure 17: The figure shows a schematic picture of the CDL tunneling of instanton ψ and the related Euclidean
spacetime. The true vacuum within the bubble nucleated after the CDL tunneling is S3 denoted apparently by
the S1 surface, and the ancestor false vacuum with Swithin the deformed surface of S 2 in the figure. The dashed
boundary of the bubble corresponds to the tunneling wall in the potential V (ψ) with a thin-wall approximation.
The evolution of scale factor a in this spacetime is consequently a piecewise function determined by the deformation, giving a well in the potential term of φ in its equation of motions under certain circumstances of parameters
related to the scale factor a, Hubble rate H in primordial epochs, mass m(φ) of ultralight scalar φ.
After solving for perturbative bounce solution of the eigenfunctions for the mode expansion of φ in the complete
basis in X coordinates, which is performable because the spatial coordinate along X is compact, the reflection and
DETAILS FOR THE DYNAMICS OF SUPERCURVATURE MODE
74
transmission coefficients R(k) and T (k) of the mode functions, associated with the quantum tunneling probability
from ancestor vacuum to the new vacuum can then be obtained. Subsequently, the contribution to correlation
function from the eigenmode functions of field φ on the Lorentzian geometry
ds2Lor. = a2 (η)(−dη 2 + dR2 + sinh2 RdΩ22 ),
(A.2)
where R and η are the time-constant spatial slice and conformal time of the open universe bubble that evolved into
the observable universe today following Ref. [53]. Taking only the contributions from the supercurvature modes
with k = i(1 − ), it is given as (Eq. (4.5) in Ref. [53])
hφ(η, R)φ(η 0 , 0)i(scm) =
−2πi
1 sinh(1 − )R
· Res(i(1 − ))e(1−)(η+η +2˜η1 )
8π 2 a(η)a(η 0 )
sin π
sinh R
(A.3)
where a(η) is the scale factor. Res(i(1 − )) comes from the reflection coefficient R(k) of the eigen functions of
modes at the pole kB = i(1 − ) related to the bound state of energy arising from the pontential well mentioned
previously, whose explicit form is also given in [53].
These bound states, shifted by the tiny but nonzero field mass mφ of φ from kB = i pole with a small quantity
, are vanishing at X → ∞ in the Euclidean coordinates; hence they are normalizable and must be included in
the mode expansion in the complete basis with respect to X coordinates. However, after applying the analytic
continuation across the null infinity between the Euclidean spacetime and the open de Sitter chart of H3 where
our universe resides, these discrete modes became non-normalizable on the spatial slice and stay almost constant
without decaying; the fluctuation scale is much larger than the curvature scale by factor −1 ; hence these modes
are named the supercurvature modes.
Let R be the radial coordinate parametrizing the spatial slice H3 . η˜1 is a phase shift introduced for connecting
the CDL and the open FLRW geometries smoothly expressed as
eη˜1 =
HA
(1 + e2X0 ),
HI
(A.4)
where X0 is related to the size of the bubble (X0 → −∞ corresponds to a small bubble limit). For small , Eq. (A.3)
reduces to Eq. (2.43) with
ϕ(η) =
1/2 HA
c∗
mA
HI
HA
ϕ∗ (η),
(A.5)
DETAILS FOR THE DYNAMICS OF SUPERCURVATURE MODE
75
where c∗ is an O(1) constant (Eq. (5.33) in [53]). ϕ∗ (η) represents the time evolution in the FLRW universe, and,
for instance, in the periods (ii) and (iii) in Sec. V-C of Ref. [53], it is given by
ϕ∗ (η) '
sin m0 t
m0 t
(A.6)
where t is the proper time in the FLRW universe. When m0 ∼
H0 is satisfied, m0 t . 1, hence one obtains
ϕ∗ (η) ' 1; the supercurvature mode is almost “frozen” in its dynamics. If more stringent condition m0 t 1 holds,
then ϕ(η) ' const. behaves extremely close to a cosmological constant.
With the frozen supercurvature modes, we can set, for example, η = 0 to evaluate the its energy density
interpreted as dark energy. In the flat universe limit ΩK 1, the supercurvature modes behave as the dark energy
with the density
8πG m20 ϕ2 (0)
8πG
ρDE '
= H02 ΩΛ .
(A.7)
An additional note in the massless limit → 0 is that, with a further assumption of small-bubble approximation
as X0 → −∞, the well-known result for the coincident-point correlation function in de Sitter spacetime [82]
hφ2 i = ϕ2 (0) =
3 HA
8π mA
(A.8)
can be reproduced.
A.1
Probability distribution functions of ScmDE density
Following the setup of Eq. (A.7), the explicit form of the probability functions of the dark energy density and
the density parameter from ScmDE can be demonstrated. For a normalized probability variable of the field, the
distribution function is given by
1 e2
P (φ(x)) = √ exp − φ (x) .
2π
(A.9)
Note that hφe2 (x)i = 1. Using φ(x),
one may write the scalar field as φ(η, x) = ϕ(η) φ(x),
where ϕ(0) is defined
in Appendix A. One will find the for the supercurvature-mode dark energy, the probability density function its
density is given by
ρDE (x) =
1 2 2
m φ (η0 , x) ≈ m20 ϕ2 (0)φe2 (x).
2 0
(A.10)
DETAILS FOR THE DYNAMICS OF SUPERCURVATURE MODE
76
On the large scales R > Rsc , the spatial variation is significant, however, as long as we consider a region of the
present Hubble horizon, which is much smaller than the scale Rsc , ρDE (x) can be regarded as a probability variable
through φe by Eq. (A.10). Following the conservation of the probability,
dφ(x)P
(φ(x))
= dρDE f (ρDE ),
(A.11)
we define the probability density function of ρDE (x)
f (ρDE ) =
dφ(x)δ(ρ
DE − ρDE (x))P (φ(x)).
(A.12)
It can be analytically calculated as
exp −ρDE /m20 ϕ2 (0)
f (ρDE ) = √
4πm20 ϕ2 (0)
ρDE /m20 ϕ2 (0)
(A.13)
which is plotted in left panel Figire 18. This figure shows the a wide range for the probability distribution of ρDE
at scales larger than the supercurvature scale Rsc even when we fix the parameter as Eq. (3.52).
For scales within the horizon, one can also discuss the probability distribution function of the density parameter
for dark energy defined by
ΩΛ (x) ≡
ΩΛ φe2 (x)
ρDE (x)
ρDE (x) + ρm
1 − ΩΛ + ΩΛ φe2 (x)
(A.14)
where ρm is the dark matter energy density. In a similar way to the case for the dark energy density, we can find
the probability density function of ΩΛ as
f (ΩΛ ) =
dφδ(Ω
Λ − ΩΛ (x))P (φ(x)).
(A.15)
It can be analytically calculated as
f (ΩΛ )
2 2πΩΛ (1 − ΩΛ )
ΩΛ (1 − ΩΛ )
ΩΛ (1 − ΩΛ )
exp −
ΩΛ (1 − ΩΛ )
2ΩΛ (1 − ΩΛ )
(A.16)
The right panel of Fig. 18 plots the function f (ΩΛ ) assuming ΩΛ = 0.7 in Eq. (A.16). f (ΩΛ ) has a peak at a point
of ΩΛ slightly larger than ΩΛ = 0.7, but this figure demonstrates a wide distribution of probability of ΩΛ at scales
larger than the supercurvature scale Rsc .
DETAILS FOR THE DYNAMICS OF SUPERCURVATURE MODE
f(ΩΛ)
1.0
0.8
0.6
0.4
77
0.5
1.5
0.2
0.2
0.4
0.6
0.8
1.0
ΩΛ
Figure 18: The left panel shows the probability density distribution √
f (ρDE ) as a function of ρDE in Eq. (A.13).
The horizontal axis is X = ρDE /m20 ϕ2 (0), and the vertical axis is Y = 4πm20 ϕ2 (0)f (ρDE ). The right panel shows
the probability density distribution f (ΩΛ ) for ΩΛ in Eq. (A.16), with its expectation value fixed as ΩΛ = 0.7.
ISOCURVATURE AND ADIABATIC INITIAL CONDITIONS
78
Isocurvature and adiabatic initial conditions
This appendix is devoted to the explanation of isocurvature initial conditions used for the superhorizon perturbations of φ representing dark energy.
The adiabatic (curvature) perturbations are defined as those inherited from the initial perturbations of some
decayed background (e.g., some scalar field), leaving the relative particle number density nX and nY of different
species (e.g., matter and radiation) unchanged
δR ≡
δnY
δnX
∝ δt,
nX
nY
(B.1)
or
nX
nY
= 0,
(B.2)
where the entropy related to the EoS of the system of X and Y is conserved. The perturbations in form of this
kind perturbs the energy density related to particle number density in the system, inducing spatial curvature Φ
(specifically, 4k 2 Φ/a2 in Fourier representation in wavenumber k) related to the energy density enclosed in the
spacetime at a fixed time, hence are sometimes called the curvature perturbations as well.
However, it is also reasonable to consider initial perturbations that are orthogonal to the adiabatic perturbations,
resulting in perturbations to the entropy, not changing the total energy density of the system but changing the
(local) EoS of the system, hence
δnY
δnX
6=
nX
nY
(B.3)
or
δS ≡
δnY
δnX
nX
nY
(B.4)
which is the definition of entropy perturbation. The isocurvature perturbations are defined as orthogonal to
hence independent from the perturbations to energy density and curvature by adiabatic perturbations discussed
previously, hence is called the isocurvature perturbations as well.
ISOCURVATURE AND ADIABATIC INITIAL CONDITIONS
79
The Boltzmann equations for the first moment (monopole) of the perturbations to photon distribution and
matter distribution are [56, 83]
˙ 0 + Φ˙ = 0,
(B.5)
δ˙ + 3Φ˙ = 0,
(B.6)
respectively.
From the previous definition of isocurvature perturbations, consequently the isocurvature initial condition reads
iso
iso
Θiso
0 (0) = −Φ (0) = 0 = Ψ (0),
(B.7)
δ iso (0) + 3Φiso (0) = 0.
(B.8)
Eq. (B.8) corresponds to the initial conditions for Eq. (3.66) and Eq. (4.195). From Eq. (B.5) one obtains
iso
iso
iso
Θiso
0 (ηd ) = −Φ (ηd ) + Φ (0) + Θ0 (0),
(B.9)
hence inserting Eq. (B.7) one will see for the isocurvature initial condition
iso
iso
Θiso
0 (ηd ) = −Φ (ηd ) = Ψ (ηd ).
(B.10)
Then one can evaluate the ISW effect contribution from the isocurvature modes as
[Θ0 + Ψ]iso (η) ∼ 2Ψiso .
(B.11)
The prefactor 2 here is nontrivial, and can referred to in Eq. (3.75). On the contrary, for the adiabatic perturbations,
the prefactor is 1/3 in matter dominant epoch.
MULTIPOLE EXPANSION MATRICES
80
Multipole Expansion Matrices
(m)
The matrices Pi
utilized in the definitions of the perturbations in Sec. 3 are simply written as
(m=1)
Pi
3
4π
(m=2)
Pi
3
4π
(m=3)
Pi
3
4π
(C.1)
,
,
(C.2)
,
(C.3)
MULTIPOLE EXPANSION MATRICES
(m)
while Pij
81
are traceless matrices related to the multipole expansion of the perturbations, listed as following
(m=1)
Pij
15
16π
(m=2)
Pij
15
16π
(m=3)
Pij
15
16π
(m=4)
Pij
15
16π
(m=5)
Pij
(C.5)
,
(C.6)
−1
,
(C.4)
,
−1
15
0
16π
,
(C.7)
−1
.
(C.8)
Eq. (3.78)–(3.80) are by virtue of the multipole expansion of the inhomogeneous perturbations under the real
spherical harmonics in the space up to ` = 2, the quadrupole component, with the ` = 0 component standing for
the homogeneous background as the monopole.
Using θ and ϕ to denote the polar angle and azimuthal angle in the spherical coordinates respectively, taking
MULTIPOLE EXPANSION MATRICES
82
spatial basis
x1 = χ sin θ cos ϕ,
x2 = χ sin θ sin ϕ,
x3 = χ cos θ,
(C.9)
the relation between these matrices and the spherical harmonics can be understood as
(m)
(m) i
(C.10)
(m)
(m)
(C.11)
Y`=1 (θ, ϕ) ≡ Pi
x /χ,
Y`=2 (θ, ϕ) ≡ Pij xi xj /χ2 ,
with integer m ∈ [1, 2` + 1] instead of m ∈ [−`, `], corresponding to the three matrices for ` = 1 and five matrices
for ` = 2 previously.
Notice that the traceless property for the matrices is in correspondence to the conclusion that the large-scale
modes make no source term contribution additional to the scalar modes as its Laplacian vanishes
(m) i
∆(3) Ψ = ∇2 Ψ = Ψ1(m) ∇2 Pi
(m)
x + Ψ2(m) ∇2 Pij xi xj
(m)
= 0 + TrPij Ψ2(m) ∇2 χ2
= 0.
(C.12)
SOME USEFUL TRANSFORMATION RELATIONS
83
Some Useful Transformation Relations
In this appendix, some useful relations to help transform equations quickly between forms as functions of t˜, a, or
η are provided. As the dimensionless quantities was defined in Eqs. (4.146) and (4.165),
t˜ = H0 t,
e = H/H0 ,
with
H=
as a usual convention. Hence, recalling
1 da
a dt
(D.1)
is the derivative with respect to a and overdot ˙ indicates that with
respect to η, for arbitrary function A one has
∂A
H ∂A
∂A
e 0,
=a
= aHA
∂t
∂t
0 ∂a
(D.2)
e ∂A
∂A
∂A
1 ∂A
= A.
H0 ∂t
aH0 ∂η
aH ∂η
∂ t˜
(D.3)
as well as
These will help to transform equations quickly. Consequently, one sees
∂2A
e ∂
= aH
∂a
∂ t˜2
∂A
aH
∂a
e 2 A00 + (a2 H
eH
e 0 + aH
e 2 )A0
= a2 H
(D.4)
and
1 1 ∂a
1 ∂a
= H/H0 = H.
a ∂t
H0 a ∂t
(D.5)
Finally it is worth noting that a universal relation widely used reads
A˙ = a2 HA0 .
(D.6)
DETAILS OF THE CORRELATION FUNCTION FOR RANDOM FIELD φ
84
Details of the correlation function for random field φ
The expectation value in (3.76) can be decomposed into products of two-point functions by using the Wick-theorem
in Eq. (3.48):
h(φ2 (X) − φ2 (0))(φ2 (X 0 ) − φ2 (00 ))i = 2 hφ(X)φ(X 0 )i2 − hφ(X)φ(00 )i2 − hφ(0)φ(X 0 )i2 + hφ(0)φ(00 )i2 .
(E.1)
Here, X, X 0 , 0, 00 are used to denote (η, χ, γ), (η 0 , χ0 , γ 0 ), (η, 0, γ), and (η 0 , 0, γ 0 ), respectively for clarity and simplicity of writing. Then, using the two-point correlation function given in Eq. (2.43) (c.f. Eq. (3.49)), one finds
that Eq. (E.1) can be evaluated as
h(φ2 (X) − φ2 (0))(φ2 (X 0 ) − φ2 (00 ))i
sinh2 (1 − )R
sinh2 (1 − )R1
sinh2 (1 − )R2
= 2ϕ2 (η)ϕ2 (η 0 )
(1 − )2 sinh2 R (1 − )2 sinh2 R1
(1 − )2 sinh2 R2
= −4ϕ2 (η)ϕ2 (η 0 ) (R coth R − R1 coth R1 − R2 coth R2 + 1) + O(2 )
2 0
R − R1 − R2 +
−R + R1 + R2 + O R
' −4 × ϕ (η)ϕ (η )
45
(E.2)
−Kχi for i = 1, 2. The schematic relation of R, R1 , and R2 is presented in Fig. 2. In the expansion
of coth(Ri ), one understands that R1 = −Kχ1 1 and R2 = −Kχ2 1.
where Ri =
Using the relation of Eq. (2.44), one can verify the mathematical property of term in the bracket that
R2 − R12 − R22 +
−R4 + R14 + R24
45
2 2 2 3
' − R1 R2 1 −
R1 + R2 cos ψ − R1 R2
cos ψ −
15
15
2 2 2 3
' − R1 R2 cos ψ − R1 R2
cos2 ψ −
15
Substituting Eqs. (E.2) and (E.3) into Eq. (3.76), Eq. (4.125) can be obtained.
(E.3)
FREQUENTLY USED ABBREVIATIONS
85
Frequently used abbreviations
Abbreviations
Full spellings
ALPs
Axion-Like Particles
BAO
Baryon Acoustic Oscillations
CCP
Cosmological Constant Problem
CDL
Coleman-De Luccia
CPL
Chevallier-Polarski-Linder
CDM
Cold Dark Matter
CMB
Cosmic Microwave Background
EoS
Equation of State
EMT
Energy-Momentum Tensor
FLRW
Friedmann-Lemaitre-Robertson-Walker
GR
General Relativity
ISW effect
Integrated Sachs-Wolfe effect
LSS
Large-Scale Structure
RSD
Redshift Space Distortions
ScmDE
Supercurvature-mode Dark Energy
SPT
Standard Perturbation Theory
...