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.
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)�
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
◆.
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)
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.
9
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.
10
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:
11
����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.
12
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).
13
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
15
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.
16
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.
17
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
18
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.
19
F = �n tanh2n 'p
6.
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' + · · · ).
T-model
• The slow-roll e.o.m. in the large field case is
!
!
• Integrate above we have
21
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
22
⇠ = 1� �
�= 1� tanh
'p6⇡ 2e�
p2/3'.
V (⇠) = �(1� 2n⇠ +O(⇠2)).
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.
Beyond
• The above models are
!
• What if it is bounded but not smooth?
24
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
25
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.
26
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).
27
✏ =�2
3
⇠2 (log ⇠)2 ,
⇠ = �p3✏
�W�1
��p3✏/�
�= exp
"W�1
�p3✏
�
!#.
• tensor-to-scalar ratio is
!
• The e-folding dependence is
28
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
30
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.
31
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↵
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.
↵ = 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.
35
Thank you!
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