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