probing classically conformal b-l model with …seminar/pdf_2016_zenki/.../ 60 probing classically...
TRANSCRIPT
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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.
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Introduction & Conclusion
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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
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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
The era of
Gravitational-wave astronomy
has come
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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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Next will come Gravitational-wave “COSMOLOGY”
FROM ASTRONOMYTO COSMOLOGY
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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FROM ASTRONOMYTO COSMOLOGY
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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Next will come Gravitational-wave “COSMOLOGY”
FROM ASTRONOMYTO COSMOLOGY
- 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
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FROM ASTRONOMYTO COSMOLOGY
- 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
Next will come Gravitational-wave “COSMOLOGY”
What kind of cosmology can we search by GWs ?
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TAKE-HOME MESSAGE
“Classical conformal” models
can lead to first-order PT
with large amount of gravitational waves
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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
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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
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1. GW production in cosmic phase transition
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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
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Quantum tunneling
Φ
V
Bubble formation & GW production
x3
false vacuum true vacuum
released energy
(thermal trap)
← ~ T →
Field space Position space
true
true
true
2. Bubbles expand
3. PT completes when bubbles collide
1. Bubbles nucleate
ROUGH SKETCH
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Quantum tunneling
Φ
V
Bubble formation & GW production
GWs
x3
false vacuum true vacuum
released energy
(thermal trap)
← ~ T →Released energy localized around walls sources GWs
⇤hij ⇠ GT (wall)ij⇤hij ⇠ GT (wall)ij
true
Field space Position space
true
true
ROUGH SKETCHds
2 = �dt
2 + a
2(t)(�ij + 2hij)dxidx
j
Def. of GWs
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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 :
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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 :
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NO scattering = NO information lossbecause of Planck-suppressed interactions
ROUGH SKETCH
present
Redshift
GWs
tΓ ~ H 4
After production, GWs evolve just by redshifting
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Then, what makes LARGE GWs?
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KEY 1BIG bubbles
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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
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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
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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
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KEY 2ULTRA supercooling
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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”)
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ULTRA supercooling makes LARGE GWs
false
true
wall
friction
pressure
scalar+plasma dynamics
potential
falsetrue
- Roughly speaking,
determines bubble-wall behavior
Before explaining why, ...
↵ ⌘ ✏⇤/⇢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( )
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- (R) Runaway case,Kinetic & gradient of the scalar fieldcarries energy
- (T) Terminal veolcity case
Φ
r
true
false
Φ
r
true
false
Before explaining why, ...
reaches terminal vel. ( < c )
reaches speed of light ( c )
ULTRA supercooling makes LARGE GWs
↵ � 1( )
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- (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
Before explaining why, ...
reaches terminal vel. ( < c )
reaches speed of light ( c )
ULTRA supercooling makes LARGE GWs
↵ � 1( )
Heated-up plasma
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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
↵ ⌘ ✏⇤/⇢radiation
small αlarge α
wall
rad
wall
rad8<
:{
sourceGWs
sourceGWs
ULTRA supercooling makes LARGE GWs
↵ � 1( )
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Let’s see some equations
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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)
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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)
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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)
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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 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)
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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 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)
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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 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)
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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 > 5.1σO(1)%
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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 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)
BIG bubbles
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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 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)
ULTRA supercooling
↵ ⌘ ✏⇤/⇢radiation
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Summary so far
Two keys to make LARGE GWs
- BIG bubbles
- ULTRA supercooling
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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
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Finite temperature potential
�
M
g2B�LT2�2
- As temperature changes, ...
T
V��4(� < 0)
�/T
V/T 4g2B�LT
2�2 ��4(� < 0)
POTENTIAL BEHAVIOR
- Thermal mass + Quartic V ⇠ g2B�LT2�2
+ �(max(T,�))�4
T is the only mass scale
relevant to tunneling
All dimensionful quantities
normalized by T
48
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Finite temperature potential
POTENTIAL BEHAVIOR
- Thermal mass + Quartic V ⇠ g2B�LT2�2
+ �(max(T,�))�4
- As temperature changes, ...
�/T
V/T 4g2B�LT
2�2 ��4(� < 0)
�
M
g2B�LT2�2
V��4(� < 0)
T
49
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Finite temperature potential
POTENTIAL BEHAVIOR
- Thermal mass + Quartic V ⇠ g2B�LT2�2
+ �(max(T,�))�4
�
M
g2B�LT2�2
- As temperature changes, ...
�/T
V/T 4g2B�LT
2�2 ��4(� < 0)
V��4(� < 0)
T
50
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Finite temperature potential
POTENTIAL BEHAVIOR
- Thermal mass + Quartic V ⇠ g2B�LT2�2
+ �(max(T,�))�4
- As temperature changes, ...
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)
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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
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Transition occurs at very low temperature
- Phase transition occurs when →�H4 ⇠ 1 S ⇠ 100
(� ⇠ 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
This small gradient
gives small β (prev. slide)
This small T/M gives large α(Need to wait until
couplings run significantly!)
55
KEY 2 : ULTRA supercooling
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Transition occurs at very low temperature
- Phase transition occurs when →�H4 ⇠ 1 S ⇠ 100
(� ⇠ 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
This small gradient
gives small β (prev. slide)
This small T/M gives large α(Need to wait until
couplings run significantly!)Transition succeeds
Transition fails
Note :
In the actual calc.,
we use more
sophisticated cond.
56
KEY 2 : ULTRA supercooling
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Transition occurs at very low temperature
- Phase transition occurs when →�H4 ⇠ 1 S ⇠ 100
(� ⇠ 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
This small gradient
gives small β (prev. slide)
This small T/M gives large α(Need to wait until
couplings run significantly!)Transition succeeds
Transition fails
Note :
In the actual calc.,
we use more
sophisticated cond.
ULTRA supercoolingoccurs in this model
57
KEY 2 : ULTRA supercooling
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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
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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
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Backup
61
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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
label for transition time
:*
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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)
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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]
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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)
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Estimation of the transition rate
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”)
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BOUNCE CALCULATION
Estimation of the transition rate
- 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, )
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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
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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
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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)
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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
→
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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
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GW spectrum from bubble collisions is
WHAT WE DID
Exactlydetermined by analytic calculation
in the same setup as in numerical simulations
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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 )*
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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
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
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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
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
✏⇤
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
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ANALYTIC CALCULATIONHOW IS IT POSSIBLE ?
The essence :
Ensemble average
hTij
(tx
,x)Tkl
(ty
,y)iens
GW spectrum is determined by
ll
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ANALYTIC CALCULATIONHOW IS IT POSSIBLE ?
The essence : GW spectrum is determined by
Why ?
hTij
(tx
,x)Tkl
(ty
,y)iens
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ANALYTIC CALCULATIONHOW IS IT POSSIBLE ?
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]
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ANALYTIC CALCULATIONHOW IS IT POSSIBLE ?
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
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ANALYTIC CALCULATIONHOW IS IT POSSIBLE ?
(tx
,x)
(ty,y)
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×
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
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ANALYTIC CALCULATIONHOW IS IT POSSIBLE ?
(tx
,x)
(ty,y)
nucleation point
Estimation of the ensemble average
- 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
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ANALYTIC CALCULATIONHOW IS IT POSSIBLE ?
(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
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ANALYTIC CALCULATIONHOW IS IT POSSIBLE ?
- 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)
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ANALYTIC CALCULATIONHOW IS IT POSSIBLE ?
single
double
Final expression
contains many polynomials, exponentials, and Bessel functions, but just that
�(k/�) ⌘ 3
8⇡G
�2⇢tot
2✏2⇤⌦
GW
(k)
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ANALYTIC CALCULATIONHOW IS IT POSSIBLE ?
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)
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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)
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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]