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Page 1: Conformal Description of Starobinsky Model and Beyond

Conformal Description of Starobinsky Model and

BeyondShi Pi

Asia Pacific Center for Theoretical Physics Based on arXiv: 1404.2560 with Yifu Cai and Jinn-ouk Gong

Page 2: Conformal Description of Starobinsky Model and Beyond

Outline• Introduction: Starobinsky model and predictions

• Starobinsky-like model 1: decimal index

• Starobinsky-like model 2: T-models

• Beyond Staobinsky: dynamical index

• Summary

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Page 3: Conformal Description of Starobinsky Model and Beyond

Set up• We begin with (Kallosh et al 1306.5220)

!

• This Lagrangian bears two symmetries:

• 1) local conformal symmetry under

!

• 2) additional SO(1,1) symmetry in field space.

3

L =p�g

1

2@µ�@

µ�+�2

12R(g)� 1

2@µ�@

µ�� �2

12R(g)� �

4(�2 � �2)2

�.

gµ⌫

! e�2�(x)gµ⌫

,� ! e�(x)� ,� ! e�(x)�

Page 4: Conformal Description of Starobinsky Model and Beyond

Fix the gauge• The field χ is called conformon. It is not a real d.o.f. but

can be removed by gauge fixing.

• A gauge also respect the SO(1,1) symmetry is a hyperbola asymptotes to “light cone”.

!

• Or parameterisation

!

• The Higgs-like potential turn to be a pure constant 9λ.

4

�2 � �2 = 6 .

� =

p6 cosh

✓'p6

◆, � =

p6 sinh

✓'p6

◆.

Page 5: Conformal Description of Starobinsky Model and Beyond

Starobinsky model• If we revise the potential a little bit, Starobinsky

model is recovered.

!

• After we take the gauge fixing, it becomes

5

L =p�g

R

12

��2 � �2

�+

1

2@µ�@µ�� 1

2@µ�@µ�� �

4�2 (�� �)2

�. (1)

L =p�g

R

2� 1

2@µ'@µ'� 9

4�e�4'/

p6⇣1� e2'/

p6⌘2

�. (1)

Page 6: Conformal Description of Starobinsky Model and Beyond

Starobinsky model• Starobinsky 1980

!

• After the conformal transformation it goes in the Einstein frame with potential (Barrow et. al. 1988)

!

• This can be described by the conformal inv two-field theory.

L =p�g

✓R+

R2

6M2

◆.

V (�) =3M2

4

⇣1� e�

p2/3�

⌘2.

de Felice & Tsujikawa 2010

Page 7: Conformal Description of Starobinsky Model and Beyond

1� ns =2

N, r =

12

N2.

1� ns =2

N, r =

8

N.

Page 8: Conformal Description of Starobinsky Model and Beyond

Outline• Introduction: Starobinsky model and predictions

• Starobinsky-like model 1: decimal index

• Starobinsky-like model 2: T-models

• Beyond Staobinsky: dynamical index

• Summary

8

Page 9: Conformal Description of Starobinsky Model and Beyond

Extension• We can see that the essential part of the conformal

description is

• 1. Preserve the conformal symmetry;

• 2. Inflation happens near the SO(1,1) symmetry.

• Try to preserve these properties, and see the possible extension of Starobinsky-like model.

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Page 10: Conformal Description of Starobinsky Model and Beyond

Extension• The most general theory that respect the conformal

symmetry is

!

• The potential does not respect the SO(1,1) symmetry. But it is invariant under the local conformal transformation.

• Define z=φ/χ for future convenience.

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L =p�g

R

12

��2 � �2

�+

1

2@µ�@µ�� 1

2@µ�@µ�� 1

36�4f(�/�)

�.

Page 11: Conformal Description of Starobinsky Model and Beyond

Slow-roll parameter• The first slow-roll parameter

!

• Inflation occurs at

• z->0. Then |f’/f| must be small.

!

• z->1. Then f’/f has a first order pole at z=1:

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����f 0

f

���� ⌧ 1

f 0

f=

1� z+ bounded function.

✏1 = 12

⇣V,'V

⌘2= 1

12

h4z + (1� z2) f

0

f

i

Page 12: Conformal Description of Starobinsky Model and Beyond

Residue

• How to interpret the residue β?

• After integration we have

!

• When β=-2, it is equivalent to Starobinsky model.

12

V (�) =3M2

4

⇣1� e�

p2/3�

⌘2.

f(z) = (1� z)��

Page 13: Conformal Description of Starobinsky Model and Beyond

Residue• When β!=-2, it is (recover the conformon)

!

• equivalent to a fractal index of Ricci scalar (Cai, Gong and SP, 1404.2560).

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L =p�g [R+ ↵Rn] ,

n =4 + �

3 + �

L =p�g

R

12

��2 � �2

�+

1

2@µ�@µ�� 1

2@µ�@µ�� �

4�4+�(�� �)��

�,

Page 14: Conformal Description of Starobinsky Model and Beyond

Residue• There are some

constraints based on recent observations on primordial B-modes.

• 2-sigma Planck data (without running): 1.988<β<2.055

• 2-sigma BICEP2 data: 1.806<β<1.848 (Codello et al 1404.3558; Chakravaty et al 1405.1321) β=2 β=1.96

β=1.90

β=1.834

Page 15: Conformal Description of Starobinsky Model and Beyond

Outline• Introduction: Starobinsky model and predictions

• Starobinsky-like model 1: decimal index

• Starobinsky-like model 2: T-models

• Beyond Staobinsky: dynamical index

• Summary

15

Page 16: Conformal Description of Starobinsky Model and Beyond

Other Extensions• Other poles (relatively far from z=1) are possible.

!

• A typical example is the T-model.

!

• The poles at z=-1 and z=0 can never be reached.

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f 0

f=

1� z+

X

i

ciz � zi

+ smooth function.

f 0

f=

2

z � 1+

2

z + 1� 2n

z.

Page 17: Conformal Description of Starobinsky Model and Beyond

T-model

• In the conformal description, T-model is

!

• F is an arbitrary function.

• F(χ/φ) breaks the SO(1,1) symmetry unless F(χ/φ)=constant.

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L =p�g

hR12

��2 � �2

�+ 1

2@µ�@µ�� 1

2@µ�@µ�� 1

36F (�/�)��2 � �2

�2i

Page 18: Conformal Description of Starobinsky Model and Beyond

T-model• After choosing the gauge, it reduces to

!

• The hyper tangent can stretch the potential to make almost any potential flat enough around

• Inflation happens when

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L =p�g

hR2 � 1

2@µ'@µ'� F

⇣tanh 'p

6

⌘i.

' ! 1

' ! 1,

tanh'p6! 1�.

Page 19: Conformal Description of Starobinsky Model and Beyond

T-model

• A typical (but not necessary) choice of F function is

!

• λ is a coupling constant and n is an (integer?) index.

• Inflation happens when φ goes to infinity.

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F = �n tanh2n 'p

6.

Page 20: Conformal Description of Starobinsky Model and Beyond

T-model

• There is a plateau around φ>>1

!

• It is similar to Starobinsky plateau when n=1/2.

20

V (�) =3M2

4

⇣1� e�

p2/3�

⌘2.

V (') ⇠ �n(1� 4ne�p

2/3' + · · · ).

Page 21: Conformal Description of Starobinsky Model and Beyond

T-model

• The slow-roll e.o.m. in the large field case is

!

!

• Integrate above we have

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d'

dN=

V 0

V= 4n

r2

3e�

p23' = 2n

r2

3⇠,

⇠ ⌘ 2e�p

23'.

⇠ =3

4nN.

Page 22: Conformal Description of Starobinsky Model and Beyond

T-model• The physical meaning of new variable

!

• Now the potential becomes

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⇠ = 1� �

�= 1� tanh

'p6⇡ 2e�

p2/3'.

V (⇠) = �(1� 2n⇠ +O(⇠2)).

Page 23: Conformal Description of Starobinsky Model and Beyond

T-model

23

✏1 ⌘ 1

2

✓V 0

V

◆2

=4

3n2⇠2 =

3

4N2,

✏2 =8

3n⇠ =

2

N.

r = 16✏1 =12

N2,

ns = 1� 2✏1 � ✏2 = 1� 2

N+O(N�2).

r is suppressed by 1/N independent the power of n, the same as Starobinsky model.

Page 24: Conformal Description of Starobinsky Model and Beyond

Beyond

• The above models are

!

• What if it is bounded but not smooth?

24

f 0

f=

1� z+ bounded function.

Page 25: Conformal Description of Starobinsky Model and Beyond

Outline• Introduction: Starobinsky model and predictions

• Starobinsky-like model 1: decimal index

• Starobinsky-like model 2: T-models

• Beyond Staobinsky: dynamical index

• Summary

25

Page 26: Conformal Description of Starobinsky Model and Beyond

A toy model• Suppose β=-2, but the other part is a function

which has a removable singularity at z=1.

!

• In this case the function f is

!

• Again, define ξ=1-z.

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f 0

f=

2

z � 1

� 2�(1� z) log(1� z).

f(z) = (1� z)2+�(1�z)e��(1�z).

Page 27: Conformal Description of Starobinsky Model and Beyond

Slow-roll parameters• The slow-roll parameter

!

• We can solve

!

• W is the Lambert function (lower branch).

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✏ =�2

3

⇠2 (log ⇠)2 ,

⇠ = �p3✏

�W�1

��p3✏/�

�= exp

"W�1

�p3✏

!#.

Page 28: Conformal Description of Starobinsky Model and Beyond

• tensor-to-scalar ratio is

!

• The e-folding dependence is

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ns = 1�r

r

3

"1 +

1

W�1

��p3r/�

�# "

1 +

p3r

8�W�1

��p3r/�

�#� r

8.

N =

3

Zd⇠

⇠2(2 + ⇠)| log ⇠| ⇡3

2�

Zd⇠

⇠2| log ⇠| ,

=

3

2�li

1

⇠.

⇠ =

✓li

(�1)�N

3

◆�1

.

Page 29: Conformal Description of Starobinsky Model and Beyond

λ=1/4

λ=1/2

λ=1λ=10000

λ->inf

Page 30: Conformal Description of Starobinsky Model and Beyond

Outline• Introduction: Starobinsky model and predictions

• Starobinsky-like model 1: decimal index

• Starobinsky-like model 2: T-models

• Beyond Staobinsky: dynamical index

• Summary

30

Page 31: Conformal Description of Starobinsky Model and Beyond

Can we recover power-law?

• Yes. Inflation happens elsewhere.

• We construct reversely from power-law chaotic inflation to get the form of f-function.

• Or add another parameter to control the form of effective potential. Kallosh et al 1311.0472, 1405.3646.

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Page 32: Conformal Description of Starobinsky Model and Beyond

Recover power-law• Take a new parameter in the gauge fixing condition, we

can have

!

• Here once α is of order 1, we go all back to the argument above.

• Once α is large and makes inflation happens elsewhere

32

� =

p6 cosh

'p6↵

, � =

p6 sinh

'p6↵

' ⌧p6↵

Page 33: Conformal Description of Starobinsky Model and Beyond

Recover power-law• And still take a typical T-model potential

!

!

• This is a standard quadratic potential which predicts

33

V = V0

⇣1� 2ne�

p2/3↵'

⌘2,

⇡ 1

2

4V0

3↵'2.

ns = 1� 2

N, r =

8

N.

Page 34: Conformal Description of Starobinsky Model and Beyond

↵ = O(1)

↵ = 10

↵ = 100

Page 35: Conformal Description of Starobinsky Model and Beyond

Summary• Conformal description is a good mechanism to

generate a class of Starobinsky-like and similar models.

• The “stretch” effect flatten the potential even if it is steep in the two-field case.

• They can also produce large tensor-to-scalor ratio as we introduce a new parameter.

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Page 36: Conformal Description of Starobinsky Model and Beyond

Thank you!