probing classically conformal b-l model with …seminar/pdf_2016_zenki/.../ 60 probing classically...

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Probing classically conformal B-L model with gravitational waves

Ryusuke Jinno (KEK)

Based on arXiv:1604.05035 (by RJ & Masahiro Takimoto)

July 19th, 2016 @ Osaka Univ.

1

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Introduction & Conclusion

2

/ 60

FIRST DECECTION OF GWS

LIGO announcement @ 2016/2/11

- Black hole binary

36M⊙ + 29M⊙ →62M⊙

with 3.0M⊙ radiated in GWs

- Frequency ~ 35 to 250 Hz

- Significance > 5.1σ

3

/ 60



- Black hole binary

36M⊙ + 29M⊙ →62M⊙



- Significance > 5.1σ

The era of

Gravitational-wave astronomy

has come

4

/ 60

Next will come Gravitational-wave “COSMOLOGY”

FROM ASTRONOMYTO COSMOLOGY

KAGRA と DECIGO

原始重力波シンポ (日本物理学会 2014年秋季大会, 2014年9月19日, 佐賀大学) 49

KAGRA (~2017)Ground-based DetectorÆ 高周波数の重力波イベント

目標: 重力波の検出, 天文学

DECIGO (~2030)Space observatoryÆ 低周波数の重力波

目標: 宇宙論的な知見など

From Ando-san’s talk @日本物理学会2014年秋季大会 5

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重力波天文学のロードマップ

原始重力波シンポ (日本物理学会 2014年秋季大会, 2014年9月19日, 佐賀大学) 58

2010

2015

2020

2025

~10 event/yrのイベントレート

地上望遠鏡

KAGRAＡｄ. ＬＩＧＯ

LIGO TAMA

EnhancedLIGO

CLIO

Advanced LIGO KAGRA Advanced

Virgo

VIRGOGEO

ET

より遠くを観測 (10Hz-1kHz)宇宙望遠鏡

0.1mHz-10mHz確実な重力波源

0.1Hz帯宇宙論的な重力波

低周波数帯の観測 (1Hz以下)

LPF

DECIGO

DECIGO

LISA

BBO

LPFDPF

Pre-DECIGOLISA

Note : + Recent progress in “Atomic interferometry”[See papers by Stanford group,

e.g. Dimopoulos et al., PLB ’09]

From Ando-san’s talk @日本物理学会2014年秋季大会 6

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eLISA⌦GWh2

GW frequency [Hz]

100 10410�4

http://rhcole.com/apps/GWplotter/ OR arXiv:1408.0740

LISA DECIGO

BBO

7

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- Space interferometers (LISA, BBO, DECIGO,...) are planned in the future

What kind of cosmology can we search by GWs ?

- Inflationary quantum fluctuations

- Preheating

- Cosmic strings, domain walls

- First-order phase transition

8

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- Electroweak symmetry breaking

- SUSY breaking

- PQ symmetry breaking

- GUT breaking ...

Many particle-physics candidates- Inflationary quantum fluctuations

- Preheating

- Cosmic strings, domain walls

- First-order phase transition

- Space interferometers (LISA, BBO, DECIGO,...) are planned in the future


What kind of cosmology can we search by GWs ?

9

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TAKE-HOME MESSAGE

“Classical conformal” models

can lead to first-order PT

with large amount of gravitational waves

10

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TAKE-HOME MESSAGE

“Classical conformal” models

can lead to phase transition

with large amount of gravitational waves

KEYs1. BIG bubbles

2. ULTRA supercooling in the early universe

11

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TALK PLAN

0. Introduction & Conclusion

1. GW production in cosmic phase transition (General)

2. GWs produced in classicaly conformal B-L model (Model specific)

3. Conclusion

Extra stage. Recent progress in GW calc. from bubble collisions

12

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1. GW production in cosmic phase transition

13

/ 6014

Quantum tunneling

Φ

V

false vacuum true vacuum

released energy

(thermal trap)

← ~ T →

Bubble formation & GW production

false

x3

true

true

true

ROUGH SKETCH

Field space Position space

1. Bubbles nucleate

/ 6015

Quantum tunneling

Φ

V


x3


released energy

(thermal trap)

← ~ T →


true

true

true

2. Bubbles expand

3. PT completes when bubbles collide

1. Bubbles nucleate

ROUGH SKETCH

/ 6016

Quantum tunneling

Φ

V


GWs

x3


released energy

(thermal trap)

← ~ T →Released energy localized around walls sources GWs

⇤hij ⇠ GT (wall)ij⇤hij ⇠ GT (wall)ij

true


true

true

ROUGH SKETCHds

2 = �dt

2 + a

2(t)(�ij + 2hij)dxidx

j

Def. of GWs

/ 6017

Bubble formation & GW prod. occur and complete when...

Γ ~ H4 Γ : Bubble nucleation rate per unit time & vol.

H : Hubble parameter

ROUGH SKETCH

because finds bubbles when

Γ × vol × t ~ 1

H -1H -3

is satisfiedH-3, vol ~H-1radius ~

: Hubble horizon

c

H ⌘ a

aHubble parameter :

/ 6018

Bubble formation & GW prod. occur and complete when...

Γ ~ H4 Γ : Bubble nucleation rate per unit time & vol.

H : Hubble parameter

ROUGH SKETCH

because finds bubbles when

Γ × vol × t ~ 1

H -1H -3

is satisfied

cGWs

H-3, vol ~H-1radius ~

: Hubble horizon

H ⌘ a

aHubble parameter :

/ 6019

NO scattering = NO information lossbecause of Planck-suppressed interactions

ROUGH SKETCH

present

Redshift

GWs

tΓ ~ H 4

After production, GWs evolve just by redshifting

Then, what makes LARGE GWs?

20

KEY 1BIG bubbles

21

/ 6022

Why ?

BIG bubbles

small bubbles

⌦GW

⌘ 1

⇢tot

d⇢GW

d ln k: Hubble radius

at the transition

BIG bubbles produce LARGE GWs

→ LONG time for GW sourcing

→ LONG time from nucleation to PT completion

: GW amplitude

BIG bubbles

GW frequency

/ 6023

Typical bubble size ~ β

- Taylor expansion β(t - t )Γ ~ Γ e* *Taylor exp.

around t*

t * : typical transition time(when Γ ~ H )4

-1- Γ changes significantly with timescale β

(because many bubbles start to nucleate here and there after this time)

How to make BIG bubbles

- Then, bubbles can expand only for t ~ t + (a few) * *-1β

t = t* →t = t + (a few)*t = t +*→

-1β -1β

-1

/ 6024

Typical bubble size ~ β

- Taylor expansion β(t - t )Γ ~ Γ e* *Taylor exp.

around t*

t * : typical transition time(when Γ ~ H )4

-1- Γ changes significantly with timescale β

(because many bubbles start to nucleate here and there after this time)

How to make BIG bubbles

- Then, bubbles can expand only for t ~ t + (a few) * *-1β

t = t* →t = t + (a few)*t = t +*→

-1β -1β

-1

BIG bubblesll

SMALL βll

SLOWLY changing nucleation rate

KEY 2ULTRA supercooling

25

/ 6026

false

true

wall

friction

pressure

scalar+plasma dynamics

ULTRA supercooling makes LARGE GWs

Before explaining why, ...

scalar field & plasma

- In thermal PT, two main players :

- Let’s see qualitative classification of bubble-wall behavior

- Walls (where the scalar field value changes)

want to expand (“pressure”) / but are pushed back by plasma (“friction”)

/ 6027


false

true

wall

friction

pressure

scalar+plasma dynamics

potential

falsetrue

- Roughly speaking,

determines bubble-wall behavior


↵ ⌘ ✏⇤/⇢radiation

(R) Runaway case

(T) Terminal velocity case

Wall velocity approaches Energy dominated by

speed of light (c)

terminal velocity (< c)

scalar field

plasma bulk motion

↵ & O(10�2)

↵ . O(10�2)

Note : is model dependent↵ . O(10�2)

↵ � 1( )

/ 6028

- (R) Runaway case,Kinetic & gradient of the scalar fieldcarries energy

- (T) Terminal veolcity case

Φ

r

true

false

Φ

r

true

false


reaches terminal vel. ( < c )

reaches speed of light ( c )


↵ � 1( )

/ 6029

- (R) Runaway case,Kinetic & gradient of the scalar fieldcarries energy

- (T) Terminal veolcity case

Φ

r

true

false

Φ

r

true

false

Bulk motion of plasma fluid around walls


reaches terminal vel. ( < c )

reaches speed of light ( c )


↵ � 1( )

Heated-up plasma

/ 6030

Why’s ULTRA supercooling favorable ?

- Focus on (R) Runaway case, since this occurs in our model

- In this case, large α (ULTRA supercooling) makes LARGE GWs

Rough sketch

simply because walls source GWs


small αlarge α

wall

rad

wall

rad8<

:{

sourceGWs

sourceGWs


↵ � 1( )

Let’s see some equations

31

/ 6032

GWs from bubble collisions

⌦GW

(t) ⌘ 1

⇢tot

(t)

d⇢GW

(t)

d ln k

GW power spectrum (def.)

GW energy densitydecomposed into each lnk

normalized by total (= wall & radiation) energy density

⌦GW(t, k)

/ 6033


⌦GW

(t) ⌘ 1

⇢tot

(t)

d⇢GW

(t)

d ln k




⌦GW(t, k)

/ 6034


⌦GW

(t) ⌘ 1

⇢tot

(t)

d⇢GW

(t)

d ln k




⌦GW(t, k)

/ 6035


⌦GW

(t) ⌘ 1

⇢tot

(t)

d⇢GW

(t)

d ln k




GW power spectrum from bubble collisions

⌦GW,peak(t⇤) ⇠ O(10�2)

✓�

H⇤

◆�2 ✓↵

1 + ↵

◆2

maximal fraction which goes into GWs

label for transition time

:*# of bubbles in one Hubble radius( )

-2 wall energy

total energy( )

2

⌦GW(t, k)

/ 6036


⌦GW

(t) ⌘ 1

⇢tot

(t)

d⇢GW

(t)

d ln k





⌦GW,peak(t⇤) ⇠ O(10�2)

✓�

H⇤

◆�2 ✓↵

1 + ↵

◆2




-2 wall energy

total energy( )

2

⌦GW(t, k)

/ 6037


⌦GW

(t) ⌘ 1

⇢tot

(t)

d⇢GW

(t)

d ln k





⌦GW,peak(t⇤) ⇠ O(10�2)

✓�

H⇤

◆�2 ✓↵

1 + ↵

◆2




-2 wall energy

total energy( )

2

⌦GW(t, k)

/ 6038



- Black hole binary

36M⊙ + 29M⊙ →62M⊙



- Significance > 5.1σO(1)%

/ 6039


⌦GW

(t) ⌘ 1

⇢tot

(t)

d⇢GW

(t)

d ln k





⌦GW,peak(t⇤) ⇠ O(10�2)

✓�

H⇤

◆�2 ✓↵

1 + ↵

◆2




-2 wall energy

total energy( )

2

⌦GW(t, k)

BIG bubbles

/ 6040


⌦GW

(t) ⌘ 1

⇢tot

(t)

d⇢GW

(t)

d ln k





⌦GW,peak(t⇤) ⇠ O(10�2)

✓�

H⇤

◆�2 ✓↵

1 + ↵

◆2




-2 wall energy

total energy( )

2

⌦GW(t, k)

ULTRA supercooling


/ 6041

Summary so far

Two keys to make LARGE GWs

- BIG bubbles

- ULTRA supercooling

/ 60

2. GW production in classically conformal B-L model

42

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CLASSICALLY CONFORMAL

What is “classically conformal” ?

- Classically no mass scale & violation of scale invariance by quantum effect

- Naturalness problem

Motivation

(Coleman-Weinberg mechanism)

�

V0 ⇠ �(�)�4

M

Rough sketch

� < 0 � > 0

↓

[Bardeen ‘95]

produces the EW scale|�|2|H|2

43

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Gauge & matter content

- Gauge :

- Matter :

Gauge couping gB�L

(equivalently, )↵B�L = g2B�L/4⇡

CLASSICALLY CONFORMAL B-L MODEL

[Iso et. al., ‘09]

44

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Zero temperature potential

� < 0�

V0 ⇠ �(�)�4

M

- Quartic terms + No mass terms

+ no mass terms

- Scale is induced by the running of (determined by )gB�L

�

V0 ⇠ �(�)�4

~ M

�

� > 0

(“classically no-scale” assumption)

POTENTIAL BEHAVIOR

tree

45

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Zero temperature potential

� < 0�

V0 ⇠ �(�)�4

M

- Scale is induced by the running of (determined by )gB�L

�

V0 ⇠ �(�)�4

~ M

�

� > 0

POTENTIAL BEHAVIOR

46

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Finite temperature potential

- Thermal mass + Quartic V ⇠ g2B�LT2�2

+ �(max(T,�))�4

�

V

M

g2B�LT2�2

thermal mass

��4(� < 0)

POTENTIAL BEHAVIOR

47

/ 60


�

M

g2B�LT2�2

- As temperature changes, ...

T

V��4(� < 0)

�/T

V/T 4g2B�LT

2�2 ��4(� < 0)

POTENTIAL BEHAVIOR


+ �(max(T,�))�4

T is the only mass scale

relevant to tunneling

All dimensionful quantities

normalized by T

48

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POTENTIAL BEHAVIOR


+ �(max(T,�))�4


�/T

V/T 4g2B�LT

2�2 ��4(� < 0)

�

M

g2B�LT2�2

V��4(� < 0)

T

49

/ 60


POTENTIAL BEHAVIOR


+ �(max(T,�))�4

�

M

g2B�LT2�2


�/T

V/T 4g2B�LT

2�2 ��4(� < 0)

V��4(� < 0)

T

50

/ 60


POTENTIAL BEHAVIOR


+ �(max(T,�))�4


potential structure at the origin SLOWLY changes (~ beta function)

�

M

g2B�LT2�2

V��4(� < 0)

51

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Γ ~ O(T ) e -S /T34 dimensionless

Note : Usually does not hold since m/T (m : mass scale of the potential) enters

g2B�LT2�2 ��4(� < 0)

Nucleation rate is calculated with “bounce method”

S /T depends only on couplings, not on T (since it’s dimensionless!)3

S3/T ⇠ 10gB�L

|�|

Nucleation rate Γ changes SLOWLY (with beta function)

Key 1 : BIG bubbles(slowly-changing nucl. rate)

52

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S /T depends only on couplings, not on T (since it’s dimensionless!)3

Γ ~ O(T ) e -S /T34 dimensionless

Note : Usually does not hold since m/T (m : mass scale of the potential) enters

g2B�LT2�2 ��4(� < 0)

Nucleation rate is calculated with “bounce method”

S3/T ⇠ 10gB�L

|�|

Nucleation rate Γ changes SLOWLY (with beta function)

SLOWLY changing nucleation ratell

BIG bubblesll

LARGE GWs

53

Key 1 : BIG bubbles(slowly-changing nucl. rate)

/ 60

Transition occurs at very low temperature

- Phase transition occurs when →�H4 ⇠ 1 S ⇠ 100

KEY 2 : ULTRA supercooling

(� ⇠ O(T 4)e�S3/T ⇠ M4e�S3/T+4 ln(T/M) ⌘ M4e�S)(� ⇠ O(T 4)e�S3/T ⇠ M4e�S3/T+4 ln(T/M) ⌘ M4e�S)

�

M

g2B�LT2�2

V

↵B�L = 0.016↵B�L = 0.008S

T/M

54

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�

M

g2B�LT2�2

V

↵B�L = 0.016↵B�L = 0.008S

T/M

This small gradient

gives small β (prev. slide)

This small T/M gives large α(Need to wait until

couplings run significantly!)

55


/ 60




�

M

g2B�LT2�2

V

↵B�L = 0.016↵B�L = 0.008S

T/M

This small gradient



couplings run significantly!)Transition succeeds

Transition fails

Note :

In the actual calc.,

we use more

sophisticated cond.

56


/ 60




�

M

g2B�LT2�2

V

↵B�L = 0.016↵B�L = 0.008S

T/M

This small gradient



couplings run significantly!)Transition succeeds

Transition fails

Note :

In the actual calc.,

we use more

sophisticated cond.

ULTRA supercoolingoccurs in this model

57


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Peak frequency & amplitude of the GW spectrum

: Above this → Landau poles below Mp: Below this → Successful PT does not occur: Left to this → Excluded by Z’ mass constraint

RESULT : LARGE GWS

(at present)[Hz]

M ⌘ h�i↵B�L : Gauge coupling at scaleM

58

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Detectability in the future

: eLISA: LISA: DECIGO: BBO

(Regions below dashed lines are detectable)

RESULT : LARGE GWS

59

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CONCLUSION

The era of GW cosmology will come

“No mass scale” at the classical levelKey

Two keys to produce LARGE GWs

- BIG bubbles / - ULTRA supercooling

Classically conformal models satisfy these conditions

and may be tested in future experiments

60

/ 60

Backup

61

/ 6062

GW FREQUENCY

Just redshift strating from transition time

fpeak ⇠ �

H⇤

T⇤108GeV

[Hz]

Hubble-size wave at T = 10^8GeVis stretched to 1Hz by redshift

# of bubbles in one Hubble radius


:*

/ 6063

CONDITION FOR SUCCESSFUL PT

We require transition occur in ~ CMB patch simultaneously

- Survival prob.

- I function behaves like →

- We require I > 100 for T → 0

(“no transition patch” should be

very rare)

/ 6064

THIN-WALL & ENVELOPE

How good are thin-wall & envelope approximations？

- In (R) Runaway case,

envelope approx. → almost justified

thin-wall approx. → justified

Φ

r

energy is stored as kinetic & gradient of the scalar field

no energy sourcingfor collided walls(just the remaining scalar field oscillation)

(N.B. sound wave-enhancement of (T) Terminal velocity case)

- (R) Runaway case is the one where large GW amplitude is expected

n

[Kosowski et al. ‘92]

/ 6065

Estimation of the transition rate

BOUNCE CALCULATION

- Transition rate can be calculated

Φ

V

S ~ 3

Zd3r

✓1

2�0(r)2 + V (�)

◆ll

V0

Vthermal

(temp.-independent)

(temp.-dependent)

~ T

from potential shape

How can we determine

Φ as a function of r ?

Γ ~ O(T ) e4 -S / T3 (in our setup)

/ 6066


BOUNCE CALCULATION

- Transition rate can be calculated

Φ

V

Γ ~ O(T ) e4 -S / T3

S ~ 3

Zd3r

✓1

2�0(r)2 + V (�)

◆

from potential shape

Φ

-V

r = infinity

r = 0

with

Φ : solution of �00(r) +2

r�0(r)� dV

d�= 0

(so-called “bounce”)

/ 6067

BOUNCE CALCULATION


- Then we can calculate β

since it is just Taylor expansion coefficient

Γ ~ Γ e*β(t - t )*

� ' d(S3/T )

dt' H

d(S3/T )

d lnT

(= how fast transition rate changes, )

/ 6068

ROUGH ESTIMATION OF GW AMP.

10!4 0.001 0.01 0.1 1 10 100f!Hz"

10!13

10!11

10!9

10!7

"GW

BBODECIGOLISAeLISA

Detector sensitivities

: eLISA: LISA: DECIGO: BBO- Present GW spectrum

fpeak ⇠ �

H⇤

T⇤108GeV

[Hz]

duration time

h2⌦GW,peak ⇠ O(10�2)O(10�5)

✓�

H⇤

◆2 ✓↵

1 + ↵

◆2

~radiation fraction today~quadrupole factor

cf. SM with → �/H ⇠ O(105),

↵ ⇠ O(0.001)mH ⇠ 10 GeV, �/H ⇠ O(105),

↵ ⇠ O(0.001)

-

Rough estimation of GW amplitude

/ 60

Analytic GW spectrum from bubble collisions

Ryusuke Jinno (KEK)

Based on arXiv:1605.01403 (by RJ & Masahiro Takimoto)

July 19th, 2016 @ Osaka Univ.

69

EXTRA STAGE

/ 6070

Same as the previous talk

Let’s focus on (R) Runaway case.

Then, the main GW source is bubble wall collisions.

MOTIVATIONS

Note : In (T) Terminal vel. case, dominant source is said to be

sound waves & turbulence (these occurs due to plasma bulk motion)

/ 6071

We must fix theoretical prediction for GW spectrum

GW spectrum from bubble collisions is usually calculated

by NUMERICAL SIMULATION with some reasonable approx’s

WHAT PEOPLE HAVE USUALLY DONE

/ f�1 ? f�2 ?

∝frequency

ΩGW

Statistical error

fall-off？

[Huber et al., ‘08]

computer

→

/ 6072

We must fix theoretical prediction for GW spectrum

GW spectrum from bubble collisions is usually calculated

by NUMERICAL SIMULATION with some reasonable approx’s

WHAT PEOPLE HAVE USUALLY DONE

/ f�1 ? f�2 ?

∝frequency

ΩGW

Statistical error

fall-off？

[Huber et al., ‘08]

computer

→Statistical error&

Uncertain high-frequency fall-off

/ 6073

GW spectrum from bubble collisions is

WHAT WE DID

Exactlydetermined by analytic calculation

in the same setup as in numerical simulations

/ 6074

INGREDIENTSTO CALCULATE GW SPECTRUM

Definition of GWs : ds

2 = �dt

2 + a

2(t)(�ij + 2hij)dxidx

j

Tij is determined by

- Bubble nucelation rate ( )

- Energy-momentum profile around nucleated bubbles

⇤hij = 8⇡GKij,klTkl

projection to tensor mode

Energy-momentum tensor (from bubble walls)

We need thisPropagation of GWs :

Γ = Γ e*β(t - t )*

/ 6075

APPROX’S USUALLY ADOPTED

All energy is assumed to be condensed in a thin surface of the wall.

false

true thin wall

Thin-wall Envelope

lB

Thin-wall & envelope approximations

[Kosowsky, Turner, Watkins, PRD45 (’92)]

Collided walls are neglectedCollided walls are neglected

Fraction of the released energy is localized ata thin surface of the bubble


rB(t)

✏⇤ : fraction of localized at the wall

: released energy density

: bubble radius

✏⇤

Tij(t,x) = · 4⇡3rB(t)

3✏⇤ ·1

4⇡rB(t)2lB· vivj

/ 6076

APPROX’S USUALLY ADOPTED

All energy is assumed to be condensed in a thin surface of the wall.

false

true thin wall

Thin-wall Envelope

lB

Thin-wall & envelope approximations

[Kosowsky, Turner, Watkins, PRD45 (’92)]

Collided walls are neglectedCollided walls are neglected



rB(t)

✏⇤ : fraction of localized at the wall

: released energy density

: bubble radius

✏⇤

for bubble wall region with width lB

Tij(t,x) = · 4⇡3rB(t)

3✏⇤ ·1

4⇡rB(t)2lB· vivj

Numerical simulation has been donewith these (reasonable) approx’s

/ 6077

ANALYTIC CALCULATIONHOW IS IT POSSIBLE ?

The essence :

Ensemble average

hTij

(tx

,x)Tkl

(ty

,y)iens

GW spectrum is determined by

ll

/ 6078


The essence : GW spectrum is determined by

Why ?

hTij

(tx

,x)Tkl

(ty

,y)iens

/ 6079


The essence : GW spectrum is determined by

- Why ?

hTij

(tx

,x)Tkl

(ty

,y)iens

Formal solution of EOM : ⇤h ⇠ T →

Note : indices omitted below

Energy density of GWs (~ GW spectrum) :

h ⇠Z t

dt0 Green(t, t0)T (t0)

⇢GW(t) ⇠ h ˙h2iens8⇡G

⇠Z

t

dtx

Zt

dty

cos(k(tx

� ty

))hTT iens

same asmassless scalar field

substitute the formal solution

Note : ensemble averagebecause of the stochasticityof the bubbles

[Caprini et al. ‘08]

/ 6080


Estimation of the ensemble average

- Trivial from def. of ensemble average

hT (tx

,x)T (ty

,y)iens

hT (tx

,x)T (ty

,y)iens = ⌃ Probability for ≠ 0

Value of

in that case×

0

B@

1

CA

0

B@

1

CA

T (tx

,x)T (ty

,y)T (t

x

,x)T (ty

,y)

0

B@

1

CA

0

B@

1

CA

(P) Probability part (V) Value part

/ 6081


(tx

,x)

(ty,y)


- Trivial from def. of ensemble average

hT (tx

,x)T (ty

,y)iens

hT (tx

,x)T (ty

,y)iens = ⌃ Probability for ≠ 0

Value of

in that case×

ll

Probability that bubble walls are passing through(t

x

,x)&(ty

,y)

0

B@

1

CA

0

B@

1

CA

0

B@

1

CA

0

B@

1

CA

T (tx

,x)T (ty

,y)T (t

x

,x)T (ty

,y)

0

B@

1

CA

0

B@

1

CA

(P) Probability part (V) Value part

81

/ 6082


(tx

,x)

(ty,y)

nucleation point


- 2 exclusive possibilities for to be nonzero

hT (tx

,x)T (ty

,y)iens

T (tx

,x)T (ty

,y)

1. single nucleation point

2. double nucleation points

(tx

,x)

(ty,y)

82

/ 6083


(tx

,x)

(ty,y)

nucleation point

Estimation of the ensemble average hT (tx

,x)T (ty

,y)iens

1. single nucleation point

2. double nucleation points

(tx

,x)

(ty,y)

- 2 exclusive possibilities for to be nonzeroT (tx

,x)T (ty

,y)

83

/ 6084


- Single nucleation-point contribution

(V) Value part(P) Probability part

Estimation of the ensemble average hT (tx

,x)T (ty

,y)iens

T (tnucl, tx, ty,⌦nucl)(nx

)i

(nx

)j

(ny

)k

(ny

)l

Summation over nucleation points

×

(tx

,x)

(ty,y)Sum over

nucleation points

hTij

(tx

,x)Tkl

(ty

,y)i(s) =Z

dtnucl P (tx

, ty

, |x� y|)�(tnucl)

/ 6085


single

double

Final expression

contains many polynomials, exponentials, and Bessel functions, but just that

�(k/�) ⌘ 3

8⇡G

�2⇢tot

2✏2⇤⌦

GW

(k)

/ 6086


Result

Peak amplitudedetermined

confirmed / f�1 ? f�2 ?

- Consistent with numerical simulation within factor ~2

⌦GW

(k) ⌘ 1

⇢tot

d⇢GW

d ln k

�(k/�) ⌘ 3

8⇡G

�2⇢tot

2✏2⇤⌦

GW

(k)

/ 6087

FUTURE APPLICATIONS

Inclusion of non-envelope part is possible

- Calculation beyond envelope (More exact GW spectrum from bubble coll.)

- Cross check for sound-wave enhancement of GWs (Recent hot topic)

/ 6088

SOUND-WAVE ENHANCEMENT ?

Some authors argue, with numerical simulations, ...

- In (T) Terminal vel. case, bulk motion of plasma sources GWs

as sound waves, for very very very very long time after bubbles collide

However, correlation <T T> seems to drop after such a long time

I expect our method makes clear whether this enhancement exists

- Because, they’re just saying that localized energy structure sources GWs...

[see e.g. Hindmarsh et al., 1304.2433 / 1504.03291]

probing classically conformal b-l model with …seminar/pdf_2016_zenki/.../ 60 probing classically...

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