Dark Energy with Large-scale Inhomogeneities

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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

...