chaotic inflation, supersymmetry breaking, and moduli...
TRANSCRIPT
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Chaotic Inflation, Supersymmetry
Breaking, and Moduli Stabilization
Clemens Wieck !
COSMO 2014, Chicago August 25, 2014
!!
Based on 1407.0253 with W. Buchmuller, E. Dudas, L. Heurtier
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Outline
1. Chaotic inflation in supergravity !2. Bounds on supersymmetry breaking !3. Comments on moduli stabilization !4. Conclusion/Outlook
1
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1. Chaotic inflation in supergravity
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Chaotic inflation
2
• Chaotic (large-field) inflation is attractive scenario for
explaining initial conditions of the universe
• Simplest setup: free massive scalar field,
V = m2'2
• Field values trans-Planckian while inflaton mass small,
' ⇠ 10� 15MP , m ⇠ 10�5 MP
,! guarantee smallness of m via approximate shift symmetry
![Page 5: Chaotic Inflation, Supersymmetry Breaking, and Moduli ...cosmo2014.uchicago.edu/depot/talk-wieck-clemens.pdfChaotic Inflation, Supersymmetry Breaking, and Moduli Stabilization Clemens](https://reader030.vdocuments.mx/reader030/viewer/2022040115/5e7d198add826a188f7b4538/html5/thumbnails/5.jpg)
Chaotic inflation in supergravity
• Naive implementation of chaotic inflation with quadratic
potential:
3
W = m�2 , K =1
2
��+ �
�2
• Scalar potential unbounded from below at
,! introduce ‘stabilizer field’ S
' ⇠ O(1),
V = eK⇣KIJDIWDJW � 3|W |2
⌘
⇠ m2'2 � 3m2'4 .
![Page 6: Chaotic Inflation, Supersymmetry Breaking, and Moduli ...cosmo2014.uchicago.edu/depot/talk-wieck-clemens.pdfChaotic Inflation, Supersymmetry Breaking, and Moduli Stabilization Clemens](https://reader030.vdocuments.mx/reader030/viewer/2022040115/5e7d198add826a188f7b4538/html5/thumbnails/6.jpg)
Chaotic inflation in supergravity
• Naive implementation of chaotic inflation with quadratic
potential:
3
W = m�2 , K =1
2
��+ �
�2
• Scalar potential unbounded from below at
,! introduce ‘stabilizer field’ S
' ⇠ O(1),
V = eK⇣KIJDIWDJW � 3|W |2
⌘
⇠ m2'2 � 3m2'4 .
![Page 7: Chaotic Inflation, Supersymmetry Breaking, and Moduli ...cosmo2014.uchicago.edu/depot/talk-wieck-clemens.pdfChaotic Inflation, Supersymmetry Breaking, and Moduli Stabilization Clemens](https://reader030.vdocuments.mx/reader030/viewer/2022040115/5e7d198add826a188f7b4538/html5/thumbnails/7.jpg)
Chaotic inflation in supergravity
• Define supergravity model by
[Kawasaki, Yamaguchi, Yanagida ’00]
4
• CMB observables are predicted to be
W = mS� , K =1
2
��+ �
�2+ |S|2 � ⇠|S|4
,! FS drives inflation while S = 0 stabilized during inflation
ns ' 0.967 and r ' 0.13
with horizon crossing at '? ' 15 ) H ⇠ m'? ⇠ 1014 GeV
• Possible generalization: W = Sf(�)[Kallosh, Linde, Rube ’11]
[Kallosh, Linde ’08]
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Chaotic inflation in supergravity
• Define supergravity model by
[Kawasaki, Yamaguchi, Yanagida ’00]
4
• CMB observables are predicted to be
W = mS� , K =1
2
��+ �
�2+ |S|2 � ⇠|S|4
,! FS drives inflation while S = 0 stabilized during inflation
ns ' 0.967 and r ' 0.13
with horizon crossing at '? ' 15 ) H ⇠ m'? ⇠ 1014 GeV
• Possible generalization: W = Sf(�)[Kallosh, Linde, Rube ’11]
[Kallosh, Linde ’08]
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2. Bounds on supersymmetry breaking
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Minimal setup
• Couple chaotic inflation & Polonyi model purely gravitational,
5
W = mS�+ fX +W0 ,
K =1
2
��+ �
�2+ SS +XX � ⇠1(XX)2 � ⇠2(SS)
2 .
,! ⇠1, ⇠2 needed for stabilization, ⇠1 e.g. from O’Raifeartaigh model
• Minkowski vacuum after inflation:
h�i = hSi = 0 , hXi ⇠ 1
⇠1, m3/2 ' W0 ' fp
3
• Note: if vacuum becomes unstable & c.c. cancellation
must be different
W0 > mp3
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Minimal setup
• Couple chaotic inflation & Polonyi model purely gravitational,
5
W = mS�+ fX +W0 ,
K =1
2
��+ �
�2+ SS +XX � ⇠1(XX)2 � ⇠2(SS)
2 .
,! ⇠1, ⇠2 needed for stabilization, ⇠1 e.g. from O’Raifeartaigh model
• Minkowski vacuum after inflation:
h�i = hSi = 0 , hXi ⇠ 1
⇠1, m3/2 ' W0 ' fp
3
• Note: if vacuum becomes unstable & c.c. cancellation
must be different
W0 > mp3
![Page 12: Chaotic Inflation, Supersymmetry Breaking, and Moduli ...cosmo2014.uchicago.edu/depot/talk-wieck-clemens.pdfChaotic Inflation, Supersymmetry Breaking, and Moduli Stabilization Clemens](https://reader030.vdocuments.mx/reader030/viewer/2022040115/5e7d198add826a188f7b4538/html5/thumbnails/12.jpg)
Minimal setup - during inflation
• During inflation, all fields heavy except inflaton
6
' =p2 Im�
• But, is shifted due to supersymmetry breaking� =p2 ImS
,! integrate out heavy � consistently:
• Obtain effective inflaton potential,
V (') =1
2m2'2
✓1� 4f2
f2 + 3m2 + 6m2'2⇠2
◆
� ' � 2fm'
f2 + 3m2 + 6m2'2⇠2
) potential becomes too steep for large f !
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Minimal setup - during inflation
• During inflation, all fields heavy except inflaton
6
' =p2 Im�
• But, is shifted due to supersymmetry breaking� =p2 ImS
,! integrate out heavy � consistently:
• Obtain effective inflaton potential,
V (') =1
2m2'2
✓1� 4f2
f2 + 3m2 + 6m2'2⇠2
◆
� ' � 2fm'
f2 + 3m2 + 6m2'2⇠2
) potential becomes too steep for large f !
![Page 14: Chaotic Inflation, Supersymmetry Breaking, and Moduli ...cosmo2014.uchicago.edu/depot/talk-wieck-clemens.pdfChaotic Inflation, Supersymmetry Breaking, and Moduli Stabilization Clemens](https://reader030.vdocuments.mx/reader030/viewer/2022040115/5e7d198add826a188f7b4538/html5/thumbnails/14.jpg)
Minimal setup - CMB observables
7
0 2 4 6 8 10 120.1
0.15
0.2
0.25
0.3
0.35
0.4
f · 105
r
0 2 4 6 8 10 120.84
0.86
0.88
0.9
0.92
0.94
0.96
0.98
1
f · 105
ns
) bound from observations:
m3/2 . 10�4 MP ⇠ H
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Non-minimal setup I
8
• Introduce additional interaction to superpotential, stabilizes S
W = mS�+MX�+ fX +W0 ,
K =1
2
��+ �
�2+ SS +XX � ⇠1(XX)2 .
• Perform similar analysis as in minimal setup, integrate out ImS
) V (') =1
2(1 + �2)m2'2
⇣1� F (f,', �)
⌘
• has similar effect as before, but even more
destructive due to absence of
F (f,', �)⇠2
, with � = Mm
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Non-minimal setup I
8
• Introduce additional interaction to superpotential, stabilizes S
W = mS�+MX�+ fX +W0 ,
K =1
2
��+ �
�2+ SS +XX � ⇠1(XX)2 .
• Perform similar analysis as in minimal setup, integrate out ImS
) V (') =1
2(1 + �2)m2'2
⇣1� F (f,', �)
⌘
• has similar effect as before, but even more
destructive due to absence of
F (f,', �)⇠2
, with � = Mm
![Page 17: Chaotic Inflation, Supersymmetry Breaking, and Moduli ...cosmo2014.uchicago.edu/depot/talk-wieck-clemens.pdfChaotic Inflation, Supersymmetry Breaking, and Moduli Stabilization Clemens](https://reader030.vdocuments.mx/reader030/viewer/2022040115/5e7d198add826a188f7b4538/html5/thumbnails/17.jpg)
Non-minimal setup I - CMB observables
9
0 1 2 3 4 5 60
0.1
0.2
0.3
0.4
0.5
f · 105
r
� = 1� = 1.5� = 2� = 2.5� = 3� = 5� = 100
0 1 2 3 4 5 60.9
0.95
1
1.05
1.1
f · 105ns
� = 1� = 1.5� = 2� = 2.5� = 3� = 5� = 100
• Observational bounds more stringent by factor of 10
,! reintroduce ⇠2-term to obtain similar result as in minimal setup
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Non-minimal setup II - O’R model
10
• Consider O’Raifeartaigh model with one shift-symmetric
‘O’Raifearton‘ and the other identified with the stabilizer field,
W = mS�+ �XS2 + fX +W0 ,
,! inflation driven by FS , supersymmetry breaking via FX
• But: mixed terms of the form arise which induce
tachyonic modes with
V � m'XS
m2tach ⇠ �H
,! cure via quartic term in K with unrealistically large ⇠1
K =1
2
��+ �
�2+ SS +XX
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Non-minimal setup II - O’R model
10
• Consider O’Raifeartaigh model with one shift-symmetric
‘O’Raifearton‘ and the other identified with the stabilizer field,
W = mS�+ �XS2 + fX +W0 ,
,! inflation driven by FS , supersymmetry breaking via FX
• But: mixed terms of the form arise which induce
tachyonic modes with
V � m'XS
m2tach ⇠ �H
,! cure via quartic term in K with unrealistically large ⇠1
K =1
2
��+ �
�2+ SS +XX
![Page 20: Chaotic Inflation, Supersymmetry Breaking, and Moduli ...cosmo2014.uchicago.edu/depot/talk-wieck-clemens.pdfChaotic Inflation, Supersymmetry Breaking, and Moduli Stabilization Clemens](https://reader030.vdocuments.mx/reader030/viewer/2022040115/5e7d198add826a188f7b4538/html5/thumbnails/20.jpg)
3. Comments on moduli stabilization
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Chaotic inflation without stabilizer
11
• Remember: chaotic inflation without stabilizer unbounded from
below
• Idea: can stabilized modulus help via no-scale cancellation of
term?
�3W 2
W = Wmod
(⇢) +1
2m�2 + fX +W
0
V = eK⇢(⇢+ ⇢)2
3|@⇢W |2 � (⇢+ ⇢)(@⇢WW + @⇢WW ) +K↵↵D↵WD↵W
�
)
,! naively looks like cancellation successful
K = �3 log (⇢+ ⇢) +1
2
��+
¯��2
+X ¯X � ⇠1(X ¯X)
2
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Chaotic inflation without stabilizer
11
• Remember: chaotic inflation without stabilizer unbounded from
below
• Idea: can stabilized modulus help via no-scale cancellation of
term?
�3W 2
W = Wmod
(⇢) +1
2m�2 + fX +W
0
V = eK⇢(⇢+ ⇢)2
3|@⇢W |2 � (⇢+ ⇢)(@⇢WW + @⇢WW ) +K↵↵D↵WD↵W
�
)
,! naively looks like cancellation successful
K = �3 log (⇢+ ⇢) +1
2
��+
¯��2
+X ¯X � ⇠1(X ¯X)
2
![Page 23: Chaotic Inflation, Supersymmetry Breaking, and Moduli ...cosmo2014.uchicago.edu/depot/talk-wieck-clemens.pdfChaotic Inflation, Supersymmetry Breaking, and Moduli Stabilization Clemens](https://reader030.vdocuments.mx/reader030/viewer/2022040115/5e7d198add826a188f7b4538/html5/thumbnails/23.jpg)
Chaotic inflation without stabilizer
12
• But: modulus (with ) must be properly integrated out,
find during inflation for supersymmetric stabilization
m⇢ > H
⇢min = ⇢0 +Winfp2⇢0 m⇢
+O(H2/m2⇢)
[Buchmüller, Wieck, Winkler ’14]
,! e↵ective inflaton potential again unbounded from below
• Modulus completely decouples from inflation in limit of infinite
mass
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4. Conclusion / Outlook
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Conclusion
13
• Models with renormalizable coupling between inflation and
supersymmetry breaking hard to construct and constrained
• In simplest, decoupled setups gravitino mass bounded from
above roughly by the inflaton mass
• Removing stabilizer means potential is unbounded from below
,! Modulus stabilized supersymmetrically above Hubble scale
can not cure problem
![Page 26: Chaotic Inflation, Supersymmetry Breaking, and Moduli ...cosmo2014.uchicago.edu/depot/talk-wieck-clemens.pdfChaotic Inflation, Supersymmetry Breaking, and Moduli Stabilization Clemens](https://reader030.vdocuments.mx/reader030/viewer/2022040115/5e7d198add826a188f7b4538/html5/thumbnails/26.jpg)
Outlook
14
• Different constraints on gravitino mass in models with more
complicated Kahler potentials?
• What happens to unboundedness problem in scenarios with
non-supersymmetrically stabilized modulus, e.g. LVS, Kahler
uplifting?
,! Modulus may not decouple completely
,! Large F⇢ may yield approximate no-scale cancellation
to cure unboundedness