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
Outline
1. Chaotic inflation in supergravity !2. Bounds on supersymmetry breaking !3. Comments on moduli stabilization !4. Conclusion/Outlook
1
1. Chaotic inflation in supergravity
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
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 .
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 .
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]
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]
2. Bounds on supersymmetry breaking
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
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
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 !
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 !
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
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
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
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
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
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
3. Comments on moduli stabilization
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
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
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
4. Conclusion / Outlook
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
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