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Conformal Description of Starobinsky Model and Beyond
Transcript of 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
Outline• Introduction: Starobinsky model and predictions
• Starobinsky-like model 1: decimal index
• Starobinsky-like model 2: T-models
• Beyond Staobinsky: dynamical index
• Summary
2
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.
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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)�
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.
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�2 � �2 = 6 .
� =
p6 cosh
✓'p6
◆, � =
p6 sinh
✓'p6
◆.
Starobinsky model• If we revise the potential a little bit, Starobinsky
model is recovered.
!
• After we take the gauge fixing, it becomes
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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)
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
1� ns =2
N, r =
12
N2.
1� ns =2
N, r =
8
N.
Outline• Introduction: Starobinsky model and predictions
• Starobinsky-like model 1: decimal index
• Starobinsky-like model 2: T-models
• Beyond Staobinsky: dynamical index
• Summary
8
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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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(�/�)
�.
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
Residue
• How to interpret the residue β?
• After integration we have
!
• When β=-2, it is equivalent to Starobinsky model.
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V (�) =3M2
4
⇣1� e�
p2/3�
⌘2.
f(z) = (1� z)��
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+�(�� �)��
�,
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
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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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.
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
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�.
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.
T-model
• There is a plateau around φ>>1
!
• It is similar to Starobinsky plateau when n=1/2.
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V (�) =3M2
4
⇣1� e�
p2/3�
⌘2.
V (') ⇠ �n(1� 4ne�p
2/3' + · · · ).
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.
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)).
T-model
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✏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.
Beyond
• The above models are
!
• What if it is bounded but not smooth?
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f 0
f=
�
1� z+ bounded function.
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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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).
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✏
�
!#.
• 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
.
λ=1/4
λ=1/2
λ=1λ=10000
λ->inf
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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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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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
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� =
p6 cosh
'p6↵
, � =
p6 sinh
'p6↵
' ⌧p6↵
Recover power-law• And still take a typical T-model potential
!
!
• This is a standard quadratic potential which predicts
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V = V0
⇣1� 2ne�
p2/3↵'
⌘2,
⇡ 1
2
4V0
3↵'2.
ns = 1� 2
N, r =
8
N.
↵ = O(1)
↵ = 10
↵ = 100
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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Thank you!