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Consistency relations for non-standard models of inflation
Jorge NoreñaPUCV
Based mostly on:
P. Creminelli, J. Noreña, M. Simonović, arXiv: 1203.4595 [hep-th]L. Bordin, P. Creminelli, M. Mirbabayi, J. Noreña, arXiv: 1605.08424 [astro-ph.CO]L. Bordin, P. Creminelli, M. Mirbabayi, J. Noreña, arXiv: 1701.04382 [astro-ph.CO]
Outline
• Introduction, inflation and non-Gaussianity
• Adiabatic modes
• Scalar consistency relation
• Tensor consistency relation
• Solid consistency
k2
k11
1
k3
k1
Non-Gaussianityh⇣(~k1)⇣(~k2)⇣(~k3)i = (2⇡)3�(~k1 + ~k2 + ~k3)B(k1, k2, k3)
Squeezed limit: k1 ⌧ k2, k3
00.5
k2
k11
1
k3
k1
Non-Gaussianityh⇣(~k1)⇣(~k2)⇣(~k3)i = (2⇡)3�(~k1 + ~k2 + ~k3)B(k1, k2, k3)
Squeezed limit: k1 ⌧ k2, k3
Equilateral configurations: k1 = k2 = k3
00.5
k2
k11
1
k3
k1
Non-Gaussianityh⇣(~k1)⇣(~k2)⇣(~k3)i = (2⇡)3�(~k1 + ~k2 + ~k3)B(k1, k2, k3)
Squeezed limit: k1 ⌧ k2, k3
Equilateral configurations: k1 = k2 = k3
Enfolded configurations:
00.5
k1 = 2k2 = 2k3
k2
k11
1
k3
k1
Non-Gaussianityh⇣(~k1)⇣(~k2)⇣(~k3)i = (2⇡)3�(~k1 + ~k2 + ~k3)B(k1, k2, k3)
Squeezed limit: k1 ⌧ k2, k3
Equilateral configurations: k1 = k2 = k3
Enfolded configurations:
00.5
k1 = 2k2 = 2k3
f loc
NL
= 2.7± 5.8
fequiNL = �42± 75
forth
NL
= �25± 39
PLANCK, 2013
k2
k11
1
k3
k1
Non-Gaussianityh⇣(~k1)⇣(~k2)⇣(~k3)i = (2⇡)3�(~k1 + ~k2 + ~k3)B(k1, k2, k3)
Squeezed limit: k1 ⌧ k2, k3
Equilateral configurations: k1 = k2 = k3
Enfolded configurations:
00.5
k1 = 2k2 = 2k3
f loc
NL
= 2.7± 5.8
fequiNL = �42± 75
forth
NL
= �25± 39
PLANCK, 2013J. Maldacena, 2003
The information about the non-linearity of the evolution of the perturbations from inflation all the way to the LSS is contained in higher-order correlation functions
h�(q)�(k1)�(k2)i = (2⇡)3�(q+ k1 + k2)B(q, k1, k2)
Setup
The information about the non-linearity of the evolution of the perturbations from inflation all the way to the LSS is contained in higher-order correlation functions
We will be interested in the limit
h�(q)�(k1)�(k2)i = (2⇡)3�(q+ k1 + k2)B(q, k1, k2)
h�(q)�(k1) . . . �(kn)iq!0= h�(q)h�(k1) . . . �(kn)i�Li
�L
�S
q ⌧ k1, k2
Setup
The information about the non-linearity of the evolution of the perturbations from inflation all the way to the LSS is contained in higher-order correlation functions
We will be interested in the limit
h�(q)�(k1)�(k2)i = (2⇡)3�(q+ k1 + k2)B(q, k1, k2)
h�(q)�(k1) . . . �(kn)iq!0= h�(q)h�(k1) . . . �(kn)i�Li
�L
�S
q ⌧ k1, k2
Setup
Khouri, Hinterbickler, Hui, Joyce, 2012
Khouri, Hinterbickler, Hui, Joyce, 2013
Ghosh, Kundu, Raju, Trivedi, 2014
Goldberger, Hui, Nicolis, 2013
Adiabatic Modes
Let us begin by writing the unperturbed flat FLRW metric
ds
2 = �dt
2 + a
2(t)dx2
P. Creminelli, M. Zaldarriaga, 2004
J. Maldacena, 2003
Adiabatic Modes
Let us begin by writing the unperturbed flat FLRW metric
And we make the following coordinate transformation
x
i 7! x̃
i = (1 + �)xi
ds
2 = �dt
2 + a
2(t)dx2
P. Creminelli, M. Zaldarriaga, 2004
J. Maldacena, 2003
Adiabatic Modes
Let us begin by writing the unperturbed flat FLRW metric
And we make the following coordinate transformation
x
i 7! x̃
i = (1 + �)xi
ds
2 = �dt
2 + a
2(t)(1 + 2�)dx2
P. Creminelli, M. Zaldarriaga, 2004
J. Maldacena, 2003
Adiabatic Modes
Let us begin by writing the unperturbed flat FLRW metric
And we make the following coordinate transformation
⇣ = �
x
i 7! x̃
i = (1 + �)xi
ds
2 = �dt
2 + a
2(t)(1 + 2�)dx2
P. Creminelli, M. Zaldarriaga, 2004
J. Maldacena, 2003
⇣ = � = const.
Adiabatic Modes
For inflation, out of the horizon
Let us begin by writing the unperturbed flat FLRW metric
And we make the following coordinate transformation
⇣ = �
x
i 7! x̃
i = (1 + �)xi
ds
2 = �dt
2 + a
2(t)(1 + 2�)dx2
I never assumed dS!
P. Creminelli, M. Zaldarriaga, 2004
J. Maldacena, 2003
Adiabatic Modes
Let us begin by writing the unperturbed flat FLRW metric
And we make the following coordinate transformation
x̃
i = x
i + 2~x ·~b xi � x
2b
i
⇣ = ~
b · ~x
Missing a piece!
ds
2 = �dt
2 + a
2(t)(1 + 2~b · ~x)dx2
Creminelli, Noreña, Simonović, 2012
Creminelli, Noreña, Simonović, Vernizzi, 2013
Adiabatic Modes
Let us begin by writing the unperturbed flat FLRW metric
And we make the following coordinate transformation
⇣ = ~
b · ~x
ds2 = �dt2 � 2@i⇣
Hdxidt+ a2(t)(1 + 2~b · ~x)dx2
x̃
i = x
i + 2~x ·~b xi � x
2b
i
Missing a piece!Creminelli, Noreña, Simonović, 2012
Creminelli, Noreña, Simonović, Vernizzi, 2013
Adiabatic Modes
Let us begin by writing the unperturbed flat FLRW metric
And we make the following coordinate transformation
⇣ = ~
b · ~x
Valid in FLRW!
ds2 = �dt2 � 2@i⇣
Hdxidt+ a2(t)(1 + 2~b · ~x)dx2
x̃i = xi + 2~x ·~b xi � x2bi �Z
dt1
a2Hbi
Creminelli, Noreña, Simonović, 2012
Creminelli, Noreña, Simonović, Vernizzi, 2013
Squeezed limit informationThe squeezed limit contains model independent information aboutthe physics during inflation
H
B(q, k1, k2)q!0⇠
1
q
Single field
P. Creminelli, M. Zaldarriaga, 2004
J. Maldacena, 2003
P. Creminelli, G. D’Amico, M. Musso, JN, 2011
The squeezed limit contains model independent information aboutthe physics during inflation
H
B(q, k1, k2)q!0⇠
1
q
1
q3
Single field Multi field
Squeezed limit information
The squeezed limit contains model independent information aboutthe physics during inflation
H
B(q, k1, k2)q!0⇠
1
q
1
q↵1 < ↵ < 3
1
q3
Single field Multi field
X. Chen, J. Wang, 2009
Squeezed limit information
The squeezed limit contains model independent information aboutthe physics during inflation
H
B(q, k1, k2)q!0⇠
1
q
1
q↵1 < ↵ < 3
1
q31
q↵↵ > 3
?
Single field Multi field Not inflation
Assassi, Baumann, Green, 2012P. Creminelli, JN, M. Simonović, 2012P. Creminelli, G. D’Amico, M. Musso, JN, 2011
Squeezed limit information
The squeezed limit contains model independent information aboutthe physics during inflation
H
B(q, k1, k2)q!0⇠
1
q
1
q↵1 < ↵ < 3
1
q31
q↵↵ > 3
?
Single field Multi field Not inflation
Assassi, Baumann, Green, 2012P. Creminelli, JN, M. Simonović, 2012P. Creminelli, G. D’Amico, M. Musso, JN, 2011
Squeezed limit information
The squeezed limit contains model independent information aboutthe physics during inflation
H
B(q, k1, k2)q!0⇠
1
q
1
q↵1 < ↵ < 3
1
q31
q↵↵ > 3
?
Single field Multi field Not inflation?
Features
Non-trivialvacua
Squeezed limit information
GravitonsThe ones above are not the only adiabatic modes
Hinterbichler, Hui, Khoury, 2012
Hinterbichler, Hui, Khoury, 2013
Make the following coordinate transformation
Sii = 0x̃ = Sijxj
ds
2 = �dt
2 + a
2(t)dx2
GravitonsThe ones above are not the only adiabatic modes
Hinterbichler, Hui, Khoury, 2012
Hinterbichler, Hui, Khoury, 2013
Make the following coordinate transformation
Sii = 0x̃ = Sijxj
ds
2 = �dt
2 + a
2(t)(�ij + 2Sij)dxidx
j
GravitonsThe ones above are not the only adiabatic modes
Hinterbichler, Hui, Khoury, 2012
Hinterbichler, Hui, Khoury, 2013
Make the following coordinate transformation
Sii = 0x̃ = Sijxj
ds
2 = �dt
2 + a
2(t)(�ij + 2Sij)dxidx
j
�ij = 2Sij
GravitonsThe ones above are not the only adiabatic modes
Hinterbichler, Hui, Khoury, 2012
Hinterbichler, Hui, Khoury, 2013
Make the following coordinate transformation
Sii = 0x̃ = Sijxj
ds
2 = �dt
2 + a
2(t)(�ij + 2Sij)dxidx
j
�ij = 2Sij
Implies that the correlation is trivial for a long wavelengthgraviton.
h�⇣⇣i
Tensor CR is very generalThe scalar consistency relation can be broken by additional scalarfields sourcing the curvature perturbation. But scalar fields can’tsource tensors.
J. Maldacena, N. Arkani-Hamed, 2015
Tensor CR is very generalThe scalar consistency relation can be broken by additional scalarfields sourcing the curvature perturbation. But scalar fields can’tsource tensors.
What about fields with s > 0 ?
J. Maldacena, N. Arkani-Hamed, 2015
Tensor CR is very generalThe scalar consistency relation can be broken by additional scalarfields sourcing the curvature perturbation. But scalar fields can’tsource tensors.
What about fields with s > 0 ?
If the background is approximately de Sitter
At late times ⌘ ! 0 �± =3
2±r
9
4� m2
H2t / ⌘�±
h✏2.S~k ✏̃2.S�~ki
0 / e2i + 43��
�ei + 6
(3��)(2��)
(��1)�+ 4
3��
�e�i + e�2i ,
J. Maldacena, N. Arkani-Hamed, 2015
Tensor CR is very generalThe scalar consistency relation can be broken by additional scalarfields sourcing the curvature perturbation. But scalar fields can’tsource tensors.
What about fields with s > 0 ?
If the background is approximately de Sitter
At late times ⌘ ! 0 �± =3
2±r
9
4� m2
H2t / ⌘�±
h✏2.S~k ✏̃2.S�~ki
0 / e2i + 43��
�ei + 6
(3��)(2��)
(��1)�+ 4
3��
�e�i + e�2i ,
� > 1 Higuchi bound
J. Maldacena, N. Arkani-Hamed, 2015
e.g.
How to break it ?
A. Riotto, A. Kehagias, 2017
We need to break the de Sitter isometries:
1. No breaking of isotropy
2. Breaking isotropy
⇤hµ⌫ = 0+ r↵f(t)r↵hµ⌫
You can arrange it to get h / ⌘0
H. Lee, D. Baumann, G. Pimentel, 2016
Several theories have a preferred spatial frame, e.g. solid inflation,gauge-flation.
A. Maleknejad, M. Sheikh-Jabbari, 2011
S. Endlich, A. Nicolis, J. Wang, 2013
Observations ?Observing the squeezed limit of h�⇣⇣i
1. Look for it directly in the CMB hBTT i
(S/N)
2 ' f�
NL2A
s
r
✓`T,max
`T,min
◆2
log
✓`B,max
`B,min
◆
Meerburg et. al., 2016
Observations ?Observing the squeezed limit of h�⇣⇣i
1. Look for it directly in the CMB hBTT i
2. Local modulation of the matter power spectrum
(S/N)
2 ' f�
NL2A
s
r
✓`T,max
`T,min
◆2
log
✓`B,max
`B,min
◆
Meerburg et. al., 2016
P⇣(~k) = P⇣(k)h1 + f�
NL✏sij k̂ik̂j�
s(q)i
(S/N)2 ' 2fs
45⇡f�
NL2rA
s
✓kmax
kmin
◆3
�N
Bartolo et. al., 2014
Observations ?Observing the squeezed limit of h�⇣⇣i
1. Look for it directly in the CMB hBTT i
2. Local modulation of the matter power spectrum
(S/N)
2 ' f�
NL2A
s
r
✓`T,max
`T,min
◆2
log
✓`B,max
`B,min
◆
Meerburg et. al., 2016
P⇣(~k) = P⇣(k)h1 + f�
NL✏sij k̂ik̂j�
s(q)i
(S/N)2 ' 2fs
45⇡f�
NL2rA
s
✓kmax
kmin
◆3
�N
3. Matter 4-point function
Bartolo et. al., 2014
(S/N)2 ' ⇡
6075f�
NL4r2A2
s
✓kmax
kmin
◆6
Jeong, Kamionkowski, 2012
Solid inflation: a non-trivial caseSeveral scalar fields with an isotropy-breaking vev h�Ii / x
I
But the background is isotropic if are a vector under SO(3)�I
Perturbations: �I = �̄I + ⇡I
Action written in terms of SO(3) invariants, e.g. @µ�I@µ�I
@µ�I@⌫�I@
µ�J@⌫�J
S. Endlich, A. Nicolis, J. Wang, 2013
Inflation happens if the energy densitydepends very weakly on the volume.
(approximate )�I 7! ��I
Solid inflation breaks isotropy and violates the standard consistencyrelations.
Solid inflation: a non-trivial case
Solid inflation breaks isotropy and violates the standard consistencyrelations.
Solid inflation: a non-trivial case
However, it does have an adiabatic mode:
This cancels the perturbations.
�I = �̄I + ⇡Ih�Ii / x
I
x
i 7! x
i � ⇡
i
Solid inflation breaks isotropy and violates the standard consistencyrelations.
Solid inflation: a non-trivial case
However, it does have an adiabatic mode:
This cancels the perturbations.
But it’s not physical! It doesn’t leave the metric untouched.
�I = �̄I + ⇡Ih�Ii / x
I
x
i 7! x
i � ⇡
i
Solid inflation breaks isotropy and violates the standard consistencyrelations.
Solid inflation: a non-trivial case
However, it does have an adiabatic mode:
This cancels the perturbations.
But it’s not physical! It doesn’t leave the metric untouched.
However, the changes in the metric go to zero after an angular average.
This implies a consistency relation for angular-averaged correlationfunctions.
�I = �̄I + ⇡Ih�Ii / x
I
x
i 7! x
i � ⇡
i
Conclusions• Non-Gaussianity allows us to measure the interactions
and non-linearity of the very early universe.
• The effect of a very long wavelength perturbation can be
written as a change of frame, if the evolution of the
Universe is adiabatic.
• Tensor consistency relations hold for a very large class of
models of inflation.
• A non-trivial case study of this logic is solid inflation.
A homogeneus gravitational potential has no physical meaning
� ! 0
An “average equivalence principle”
A homogeneus gravitational potential has no physical meaning
A homogeneus gravitational force can be set to zero by going to a freely falling frame
� ! 0
An “average equivalence principle”
A homogeneus gravitational potential has no physical meaning
A homogeneus gravitational force can be set to zero by going to a freely falling frame
� ! 0
r� ! 0
~V ! ~V � tr�
An “average equivalence principle”
Electromagnetic adiabatic modes
Mirbababyi, Simonović, 2016
Electromagnetism also has adiabatic modes:
Aµ 7! Aµ + @µ↵ � 7! ei�Q� A0 = 0
But the gauge is not yet completely fixed @0↵ = 0
These solutions have zero frequency, but they can be continued tofinite-frequency solutions.
The constraint equation is @0@iAi = 0 =) r2↵ = 0
(�@20 +r2)Ai � @i@jA
j = 0The dynamic equation is
One such solution is for example ↵ = ✏ixi
Observations
Dark Energy Task Force stage IV: ⇥�3⇤ � 1/q3/2+�
kmax 0.1 h/Mpc
JN, L. Verde, G. Barenboim, C. Bocsh, 2012
Sefusatti, et. al., 2012
Gank, Komatsu, 2012
Observations
Dark Energy Task Force stage IV: ⇥�3⇤ � 1/q3/2+�
kmax 0.1 h/Mpc
JN, L. Verde, G. Barenboim, C. Bocsh, 2012
Sefusatti, et. al., 2012
Gank, Komatsu, 2012
Observations
Dark Energy Task Force stage IV: ⇥�3⇤ � 1/q3/2+�
kmax 0.1 h/Mpc
JN, L. Verde, G. Barenboim, C. Bocsh, 2012
Sefusatti, et. al., 2012
Gank, Komatsu, 2012