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Daniel Baumann
University of Amsterdam &
National Taiwan University
AMPLITUDES MEET

COSMOLOGY
Simons Symposium
May 2022
Amplitudes Virtuous Circle
© JJ Carrasco (via Lance Dixon)
Compute
higher orders
Find new patterns
or principles
Invent better
algorithms
gluons
On-shell methods

Recursion relations

Generalized unitarity

…
Parke-Taylor
=
hiji4
h12ih23i · · · hn1i
i
j
Cosmology’s Virtuous Circle
Models
Observations
Principles
Equilateral Non-Gaussianity
[Babich, Creminelli and Zaldarriaga 2004]
DBI Inflation
[Silverstein and Tong 2004]
EFT of Inflation
[Cheung, Creminelli, Fitzpatrick, Kaplan and Senatore 2008]
Bootstrap
[Pajer 2020]
In-In Formalism
[Maldacena 2003]

[Weinberg 2005]
The Power of Locality
[DB and Green 2021]
Observations
Cosmological Correlations
By measuring cosmological correlations, we learn both about the evolution
of the universe and its initial conditions:
380 000 years
13.8 billion years
CMB Anisotropies
The correlations in the CMB temperature anisotropies have revealed a great
deal about the geometry and composition of the universe:
neutrinos
geometry, dark energy
dark matter
baryons
90◦
18◦
0
1000
2000
3000
4000
5000
6000
Power
[µK
2
]
2◦
0.2◦
0.1◦
0.07◦
Angular separation
Initial Conditions
Under relatively mild assumptions, the observed correlations can be traced
back to primordial correlations on the reheating surface:
We have learned a few interesting facts about these initial conditions.
2 10 100 1000 2000
Multipole
150
100
50
0
50
100
150
Power
[µK
2
]
TE
Superhorizon Fluctuations
The fluctuations span superhorizon scales at recombination:
The fluctuations were created before the hot Big Bang, during a period of
rapid expansion (= inflation) or slow contraction (= bouncing cosmology).
0.94 0.95 0.96 0.97 0.98 0.99 1
0
0.05
0.1
0.15
m2φ2
R2
1 − cos(φ/f)
ns
(r)
Scale Invariance
The primordial fluctuations were approximately scale-invariant:
This scale-invariance is built into the inflationary mechanism, but requires
some work in bouncing cosmologies.
ns < 1
P⇣(k)
k
Cl
l
Adiabaticity
The primordial fluctuations were adiabatic (= no fluctuations in composition):
Photons
Baryons
Dark matter
Adiabaticity arises naturally in single-field inflation and (in the absence of
conserved charges) is created by thermal equilibrium.
Curvature
perturbations
Gaussianity
The primordial fluctuations were highly Gaussian:
Pr[ ]
2
FNL 3/2
10 3
The Universe is more
Gaussian than flat.
Non-Gaussianity
A lot of the physics of inflation is encoded in non-Gaussian correlations:


This non-Gaussianity is a primary target of future LSS observations.
Local Equilateral Folded
flocal
NL = 0.9 ± 5.1 fequil
NL = 26 ± 47 ffolded
NL = 60 ± 54
Extra particles Derivative interactions Excited initial states
Tensor Modes
Tensor fluctuations were subdominant:
0.94 0.95 0.96 0.97 0.98 0.99 1
0
0.05
0.1
0.15
m2φ2
R2
1 − cos(φ/f)
ns
(r)
Ruled out
Favored
r < 0.03
The search for primordial tensor modes is a primary
target of future CMB observations.
CMB Experiments
EBEX
SPIDER
POLARBEAR
BICEP
SPTPol
ACTPol
ABS
CLASS
CMB Stage III
CMB Stage III.5
(2021-2028)
Atacama Desert
CMB Experiments
CMB Stage IV
(2028-2035)
Atacama Desert
South Pole
CMB Experiments
Selected by JAXA in 2019
Launch in 2028
LiteBIRD
(r) ⇠ 0.001
CMB Experiments
LSS Experiments
A number of galaxy surveys will also probe the primordial correlations:
The challenge is to separate primordial signals from late-time nonlinearities.
Euclid
SphereX
DESI
A Few Questions
• What observations would falsify inflation?

• What observations would prove inflation?

• How is the microphysics of inflation encoded in observations?

• What are new observables signatures?

• How do we extract inflationary correlations in LSS observations?

• What are protected observables?
See discussion session on Friday.
How can inflation become part of the standard history of the
universe with the same level of confidence as BBN? Matias Zaldarriaga
The Challenge
See discussion session on Thursday.
We only measure a few parameters of a low-energy EFT:
ns r fNL
As H Ḣ cs · · ·
• How can we connect these IR parameters to the UV dynamics?

• What do their values teach us about the UV completion?
Models
Slow-Roll Inflation
Inflationary model-building is dominated by slow-roll inflation:
However, the physics of inflation is broader than just slow-roll inflation.
V (φ)
φ
Ultraviolet Completion
The UV completion of slow-roll inflation is an important challenge:
Landscape of EFTs
Islands of consistent
UV completions
Swampland
Observational constraints
• Does Starobinsky inflation have a consistent UV completion?

• Can amplitude methods sharpen constraints?
See talks on Thursday.
Beyond Slow-Roll
Slow-roll inflation is like the Higgs mechanism for electroweak symmetry
breaking:
However, we don’t have the analog of electroweak precision tests to
convince us that it must be true (even before the Higgs discovery).
It is therefore still useful to consider alternative inflationary mechanisms 

(the analog of technicolor in electroweak symmetry breaking).
Beyond Slow-Roll
To quantify the evidence for slow-roll inflation, we need to deform the theory:
Creminelli [2003]
perturbative non-perturbative
?
or
⇤2
. ˙
⇤2 ˙
We will analyze this in a framework that remains well-defined even for
nonperturbative corrections to the slow-roll dynamics.
EFT of Inflation
Inflation is a symmetry breaking phenomenon:
⇡(t, x)
ij(t, x)
H(t) (t)
The low-energy EFT is parameterized by two massless fields:

• Goldstone boson
of broken time translations
• Graviton
= (t + ⇡) ¯(t)
Creminelli et al. [2006]

Cheung et al. [2008]
EFT of Inflation
Inflation is a symmetry breaking phenomenon:

⇡(t, x)
ij(t, x)
H(t) (t)
Creminelli et al. [2006]

Cheung et al. [2008]
gij = a2
e2⇣
[e ]ij
In comoving gauge, the Goldstone boson gets eaten by the metric:
⇣ = H⇡
Goldstone Lagrangian
The Goldstone Lagrangian is
 Creminelli et al. [2006]

Cheung et al. [2008]
M2
plḢ( )2 +
n=2
M4
n
n!
2 ˙ + ( )2 n
M4
n
L = + · · ·
Slow-roll inflation Higher-derivative corrections
(@ )2n
1
2 (@ )2
V ( )
M2
plḢ(@⇡)2
Goldstone Lagrangian
The Goldstone Lagrangian is
 Creminelli et al. [2006]

Cheung et al. [2008]
+
n=2
M4
n
n!
2 ˙ + ( )2 n
L = + · · ·
superluminal
ruled out by Planck allowed by current observations
L⇡ =
M2
pl|Ḣ|
c2
s
h
˙
⇡2
c2
s(@i⇡)2
(1 c2
s) ˙
⇡(@µ⇡)2
+ · · ·
i
The fluctuations can have a small sound speed and large interactions:
nonlinearly realized symmetry FNL / c 2
s
Energy Scales
superhorizon
freeze-out
subhorizon
energy scale of
the experiment
Mpl
H
Energy Scales
> 104
H
2⇡ =
H
Mpl
= ?
Mpl
H
Energy Scales
Mpl
H
f⇡
background
Goldstone
fluctuations
symmetry breaking
curvature
perturbations
⌘ (M2
pl|Ḣ|cs)1/4
= ˙1/2
Energy Scales
= 58H
2⇡ ⇣ =
✓
H
f⇡
◆2
⇠ 10 4
f⇡
Mpl
H
Energy Scales
f⇡
Mpl
H
strong coupling ⇤
strongly coupled
weakly coupled
⇤
Energy Scales
f⇡
Mpl
H
> 5H
FNL ⇠
✓
H
⇤
◆2
DB, Green, Lee and Porto [2015]
Grall and Mellville [2020]
Unitarity Bound
Goldstone scattering in flat space provides a precise unitarity bound for
theories with a small sound speed:
Can a similar result be derived in de Sitter space?
Perturbative unitarity holds if
|Re[a`]| <
1
2
!4
< 30⇡
f4
⇡c4
s
1 c2
s
⌘ ⇤4
u
⇤u
H
Mpl
slow-roll
non-slow-roll
Higgs
Technicolor
1
cs
0.02
DB, Green, Lee and Porto [2015]
Grall and Mellville [2020]
Unitarity Bound
Asking for the theory to be weakly coupled up to the symmetry
breaking scale implies a critical value for the sound speed:
Hard to measure!
⇤u
fequil
NL ⇠ 1
0.31
Ultraviolet Completion
Mpl
Ms
MKK
string scale
KK scale
The UV completion of inflation requires new scales below the Planck scale:
H
For high-scale inflation or models with large self-interactions, these scales
may not be far from the Hubble scale and could affect observables.
A Few Questions
• Was it a weakly or strongly coupled phenomenon?

• Was it an exotic state of matter?

• What is the space of consistent theories?

• Does Starobinsky inflation have a consistent UV completion?

• What are observable effects of the UV completion of inflation?

• Are there rigorous UV constraints on IR observables?

• Can amplitude methods give precise UV-IR relations?
See discussion sessions on Thursday and Friday.
What is inflation? Dan Green
Principles
Back to the Future
The pattern of correlations after inflation contains a memory of the physics
during inflation (evolution, symmetries, particle content, etc).
If inflation is correct, then all cosmological correlations can be traced back
to the future boundary of an approximate de Sitter spacetime:
Cosmological Bootstrap
Recently, there has been significant progress in bootstrapping correlators
using physical consistency conditions on the boundary:
symmetries, locality, 

causality, unitarity, …
See talks by Austin, Charlotte, Massimo and Enrico.
In-In Formalism
These boundary correlators are usually computed in the in-in formalism:
Weinberg [2005]
hO(⌘)i = h0| T̄ei
R ⌘
1
d⌘00
Hint(⌘00
)
O(⌘) Te i
R ⌘
1
d⌘0
Hint(⌘0
)
|0i
O(⌘)
U+
U
h0|
h0|
In-In Formalism
These boundary correlators are usually computed in the in-in formalism:
Weinberg [2005]
hO(⌘)i = h0| T̄ei
R ⌘
1
d⌘00
Hint(⌘00
)
O(⌘) Te i
R ⌘
1
d⌘0
Hint(⌘0
)
|0i
Diagrammatically, this can be written as
Giddings and Sloth [2010]
Wightman
Feynman
Wightman
I =
I++ =
I+ = I + =
h (x1) . . . (xN )i =
Z
D (x1) . . . (xN ) | [ ]|2
Wavefunction of the Universe
Alternatively, we can first define a wavefunction for the late-time fluctuations:
(t, x)
(x)
This wavefunction is typically given by
and the boundary correlators are
[ ] =
Z
D eiS[ ]
⇡ ieScl[ ]
[ ] ⌘ h |0i
Maldacena [2003]
[ ] = exp
X
N
Z
d3
x1 . . . d3
xN hO(x1) . . . O(xN )i (x1) . . . (xN )
!
Perturbation Theory
In perturbation theory, the wavefunction is
Bulk-to-boundary
Bulk-to-bulk propagator
(not quite Feynman!)
The wavefunction coefficients are usually easier to compute than in-in
correlators:
Wavefunction coefficients
hO1O2O3O4i =
Relation to In-In Correlators
In perturbation theory, there is a mapping from wavefunction coefficients to
boundary correlators:
I++ + I I+ + I +
This mapping reproduced the in-in computations.
h i =
1
2RehOOi
h i =
hOOOOi
hOOi4
+
hOOXi2
hXXihOOi4
h i =
2RehOOOi
Q3
i=1 2RehOiOii
Relation to AdS
The dS wavefunction is related by a double analytic continuation to the
partition functions in EAdS:
⌘ 7! iz
LdS 7! iLAdS
• This relates AdS Witten diagrams to dS wavefunction coefficients.
• AdS Witten diagram can also be mapped directly to dS in-in correlators.
Maldacena [2003]

Di Pietro, Gorbenko and Komatsu [2021]

Sleight and Taronna [2021]
See talks by Charlotte and Massimo.
iV (⌘2)
0
Z
1
d⌘1d⌘2
The Challenge
Because energy is not conserved in cosmology, the computation of
cosmological correlators involves complicated time integrals:
K(⌘)
Since the bulk dynamics is not observable, there should be a more on-shell
way of deriving the correlators that does not include these integrals.
iV (⌘1)
G(⌘1, ⌘2)
Kinematics
• The total energy is not conserved in cosmology:
Correlators depend on the same external data as scattering amplitudes:
Two important differences:
• All energies must be positive:
Something interesting happens in the limit of would-be energy conservation:
En
=
AMPLITUDE
TOTAL ENERGY
0
Z
1
dt eiEt
f(t,~
kn) =
For amplitudes, we instead have an energy-conserving delta function:
Amplitudes live inside correlators.
+1
Z
1
dt eiEt
f(t, pn)
= / D(E)
Singularities
Raju [2012]

Maldacena and Pimentel [2011]
Additional singularities arise when the energy of a subgraph is conserved:
=
Arkani-Hamed, Benincasa and Postnikov [2017]

Arkani-Hamed, DB, Lee and Pimentel [2018]
DB, Duaso Pueyo, Joyce, Lee and Pimentel [2020]
Goodhew, Jazayeri and Pajer [2021]
PARTIAL ENERGY
CORRELATOR
AMPLITUDE
Singularities
Singularities
=
Additional singularities arise when the energy of a subgraph is conserved:
Amplitudes are the building blocks of correlators.
Arkani-Hamed, Benincasa and Postnikov [2017]

Arkani-Hamed, DB, Lee and Pimentel [2018]
DB, Duaso Pueyo, Joyce, Lee and Pimentel [2020]
Goodhew, Jazayeri and Pajer [2021]
Linking Singularities
These singularities occur only for unphysical kinematics:
En
Unphysical Physical
The challenge is to extend these singular limits to physical kinematics.
Bootstrap Tools
Unphysical Physical
En
See talks by Austin, Charlotte, Massimo and Enrico.
There has been significant progress in bootstrapping cosmological
correlators using symmetries, unitarity and locality.
A Few Questions
• Can we go beyond Feynman diagrams?

• What is the analytic structure beyond tree level?

• What are the constraints from causality?

• What are nonperturbative bootstrap conditions?

• Can we define rigorous dispersion relations?

• What is the space of UV-complete correlators?

• Does the bootstrap suggest new observational signatures?

• Is there a holographic theory of cosmology?
See discussion session by Enrico.
Where is the magic? Nima Arkani-Hamed
Open Questions
• Where is the magic?

• How to combine diagrams?

• Beyond tree level?

• Causality?

• Non-perturbative?

• Dispersion relations?

• Observational signatures?

• Holography?
• What is inflation?

• Weakly or strong coupled?

• Space of consistent theories?

• Starobinsky inflation?

• Signatures of UV completion?

• UV constraints on IR observables?

• New insights from amplitudes methods?
Models
Observations
Principles
• How to falsify inflation?

• How to prove inflation?

• Microphysics of inflation?

• New observational signatures?

• Protected LSS observables?

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Cosmology

  • 1. Daniel Baumann University of Amsterdam & National Taiwan University AMPLITUDES MEET
 COSMOLOGY Simons Symposium May 2022
  • 2. Amplitudes Virtuous Circle © JJ Carrasco (via Lance Dixon) Compute higher orders Find new patterns or principles Invent better algorithms gluons On-shell methods
 Recursion relations
 Generalized unitarity
 … Parke-Taylor = hiji4 h12ih23i · · · hn1i i j
  • 3. Cosmology’s Virtuous Circle Models Observations Principles Equilateral Non-Gaussianity [Babich, Creminelli and Zaldarriaga 2004] DBI Inflation [Silverstein and Tong 2004] EFT of Inflation [Cheung, Creminelli, Fitzpatrick, Kaplan and Senatore 2008] Bootstrap [Pajer 2020] In-In Formalism [Maldacena 2003]
 [Weinberg 2005] The Power of Locality [DB and Green 2021]
  • 5. Cosmological Correlations By measuring cosmological correlations, we learn both about the evolution of the universe and its initial conditions: 380 000 years 13.8 billion years
  • 6. CMB Anisotropies The correlations in the CMB temperature anisotropies have revealed a great deal about the geometry and composition of the universe: neutrinos geometry, dark energy dark matter baryons 90◦ 18◦ 0 1000 2000 3000 4000 5000 6000 Power [µK 2 ] 2◦ 0.2◦ 0.1◦ 0.07◦ Angular separation
  • 7. Initial Conditions Under relatively mild assumptions, the observed correlations can be traced back to primordial correlations on the reheating surface: We have learned a few interesting facts about these initial conditions.
  • 8. 2 10 100 1000 2000 Multipole 150 100 50 0 50 100 150 Power [µK 2 ] TE Superhorizon Fluctuations The fluctuations span superhorizon scales at recombination: The fluctuations were created before the hot Big Bang, during a period of rapid expansion (= inflation) or slow contraction (= bouncing cosmology).
  • 9. 0.94 0.95 0.96 0.97 0.98 0.99 1 0 0.05 0.1 0.15 m2φ2 R2 1 − cos(φ/f) ns (r) Scale Invariance The primordial fluctuations were approximately scale-invariant: This scale-invariance is built into the inflationary mechanism, but requires some work in bouncing cosmologies. ns < 1 P⇣(k) k Cl l
  • 10. Adiabaticity The primordial fluctuations were adiabatic (= no fluctuations in composition): Photons Baryons Dark matter Adiabaticity arises naturally in single-field inflation and (in the absence of conserved charges) is created by thermal equilibrium. Curvature perturbations
  • 11. Gaussianity The primordial fluctuations were highly Gaussian: Pr[ ] 2 FNL 3/2 10 3 The Universe is more Gaussian than flat.
  • 12. Non-Gaussianity A lot of the physics of inflation is encoded in non-Gaussian correlations: 
 This non-Gaussianity is a primary target of future LSS observations. Local Equilateral Folded flocal NL = 0.9 ± 5.1 fequil NL = 26 ± 47 ffolded NL = 60 ± 54 Extra particles Derivative interactions Excited initial states
  • 13. Tensor Modes Tensor fluctuations were subdominant: 0.94 0.95 0.96 0.97 0.98 0.99 1 0 0.05 0.1 0.15 m2φ2 R2 1 − cos(φ/f) ns (r) Ruled out Favored r < 0.03 The search for primordial tensor modes is a primary target of future CMB observations.
  • 15. CMB Stage III.5 (2021-2028) Atacama Desert CMB Experiments
  • 16. CMB Stage IV (2028-2035) Atacama Desert South Pole CMB Experiments
  • 17. Selected by JAXA in 2019 Launch in 2028 LiteBIRD (r) ⇠ 0.001 CMB Experiments
  • 18. LSS Experiments A number of galaxy surveys will also probe the primordial correlations: The challenge is to separate primordial signals from late-time nonlinearities. Euclid SphereX DESI
  • 19. A Few Questions • What observations would falsify inflation? • What observations would prove inflation? • How is the microphysics of inflation encoded in observations? • What are new observables signatures? • How do we extract inflationary correlations in LSS observations? • What are protected observables? See discussion session on Friday. How can inflation become part of the standard history of the universe with the same level of confidence as BBN? Matias Zaldarriaga
  • 20. The Challenge See discussion session on Thursday. We only measure a few parameters of a low-energy EFT: ns r fNL As H Ḣ cs · · · • How can we connect these IR parameters to the UV dynamics?
 • What do their values teach us about the UV completion?
  • 22. Slow-Roll Inflation Inflationary model-building is dominated by slow-roll inflation: However, the physics of inflation is broader than just slow-roll inflation. V (φ) φ
  • 23. Ultraviolet Completion The UV completion of slow-roll inflation is an important challenge: Landscape of EFTs Islands of consistent UV completions Swampland Observational constraints • Does Starobinsky inflation have a consistent UV completion? • Can amplitude methods sharpen constraints? See talks on Thursday.
  • 24. Beyond Slow-Roll Slow-roll inflation is like the Higgs mechanism for electroweak symmetry breaking: However, we don’t have the analog of electroweak precision tests to convince us that it must be true (even before the Higgs discovery). It is therefore still useful to consider alternative inflationary mechanisms 
 (the analog of technicolor in electroweak symmetry breaking).
  • 25. Beyond Slow-Roll To quantify the evidence for slow-roll inflation, we need to deform the theory: Creminelli [2003] perturbative non-perturbative ? or ⇤2 . ˙ ⇤2 ˙ We will analyze this in a framework that remains well-defined even for nonperturbative corrections to the slow-roll dynamics.
  • 26. EFT of Inflation Inflation is a symmetry breaking phenomenon: ⇡(t, x) ij(t, x) H(t) (t) The low-energy EFT is parameterized by two massless fields: • Goldstone boson of broken time translations • Graviton = (t + ⇡) ¯(t) Creminelli et al. [2006] Cheung et al. [2008]
  • 27. EFT of Inflation Inflation is a symmetry breaking phenomenon: ⇡(t, x) ij(t, x) H(t) (t) Creminelli et al. [2006] Cheung et al. [2008] gij = a2 e2⇣ [e ]ij In comoving gauge, the Goldstone boson gets eaten by the metric: ⇣ = H⇡
  • 28. Goldstone Lagrangian The Goldstone Lagrangian is Creminelli et al. [2006] Cheung et al. [2008] M2 plḢ( )2 + n=2 M4 n n! 2 ˙ + ( )2 n M4 n L = + · · · Slow-roll inflation Higher-derivative corrections (@ )2n 1 2 (@ )2 V ( )
  • 29. M2 plḢ(@⇡)2 Goldstone Lagrangian The Goldstone Lagrangian is Creminelli et al. [2006] Cheung et al. [2008] + n=2 M4 n n! 2 ˙ + ( )2 n L = + · · · superluminal ruled out by Planck allowed by current observations L⇡ = M2 pl|Ḣ| c2 s h ˙ ⇡2 c2 s(@i⇡)2 (1 c2 s) ˙ ⇡(@µ⇡)2 + · · · i The fluctuations can have a small sound speed and large interactions: nonlinearly realized symmetry FNL / c 2 s
  • 31. Energy Scales > 104 H 2⇡ = H Mpl = ? Mpl H
  • 33. Energy Scales = 58H 2⇡ ⇣ = ✓ H f⇡ ◆2 ⇠ 10 4 f⇡ Mpl H
  • 34. Energy Scales f⇡ Mpl H strong coupling ⇤ strongly coupled weakly coupled
  • 36. DB, Green, Lee and Porto [2015] Grall and Mellville [2020] Unitarity Bound Goldstone scattering in flat space provides a precise unitarity bound for theories with a small sound speed: Can a similar result be derived in de Sitter space? Perturbative unitarity holds if |Re[a`]| < 1 2 !4 < 30⇡ f4 ⇡c4 s 1 c2 s ⌘ ⇤4 u
  • 37. ⇤u H Mpl slow-roll non-slow-roll Higgs Technicolor 1 cs 0.02 DB, Green, Lee and Porto [2015] Grall and Mellville [2020] Unitarity Bound Asking for the theory to be weakly coupled up to the symmetry breaking scale implies a critical value for the sound speed: Hard to measure! ⇤u fequil NL ⇠ 1 0.31
  • 38. Ultraviolet Completion Mpl Ms MKK string scale KK scale The UV completion of inflation requires new scales below the Planck scale: H For high-scale inflation or models with large self-interactions, these scales may not be far from the Hubble scale and could affect observables.
  • 39. A Few Questions • Was it a weakly or strongly coupled phenomenon? • Was it an exotic state of matter? • What is the space of consistent theories? • Does Starobinsky inflation have a consistent UV completion? • What are observable effects of the UV completion of inflation? • Are there rigorous UV constraints on IR observables? • Can amplitude methods give precise UV-IR relations? See discussion sessions on Thursday and Friday. What is inflation? Dan Green
  • 41. Back to the Future The pattern of correlations after inflation contains a memory of the physics during inflation (evolution, symmetries, particle content, etc). If inflation is correct, then all cosmological correlations can be traced back to the future boundary of an approximate de Sitter spacetime:
  • 42. Cosmological Bootstrap Recently, there has been significant progress in bootstrapping correlators using physical consistency conditions on the boundary: symmetries, locality, 
 causality, unitarity, … See talks by Austin, Charlotte, Massimo and Enrico.
  • 43. In-In Formalism These boundary correlators are usually computed in the in-in formalism: Weinberg [2005] hO(⌘)i = h0| T̄ei R ⌘ 1 d⌘00 Hint(⌘00 ) O(⌘) Te i R ⌘ 1 d⌘0 Hint(⌘0 ) |0i O(⌘) U+ U h0| h0|
  • 44. In-In Formalism These boundary correlators are usually computed in the in-in formalism: Weinberg [2005] hO(⌘)i = h0| T̄ei R ⌘ 1 d⌘00 Hint(⌘00 ) O(⌘) Te i R ⌘ 1 d⌘0 Hint(⌘0 ) |0i Diagrammatically, this can be written as Giddings and Sloth [2010] Wightman Feynman Wightman I = I++ = I+ = I + =
  • 45. h (x1) . . . (xN )i = Z D (x1) . . . (xN ) | [ ]|2 Wavefunction of the Universe Alternatively, we can first define a wavefunction for the late-time fluctuations: (t, x) (x) This wavefunction is typically given by and the boundary correlators are [ ] = Z D eiS[ ] ⇡ ieScl[ ] [ ] ⌘ h |0i Maldacena [2003]
  • 46. [ ] = exp X N Z d3 x1 . . . d3 xN hO(x1) . . . O(xN )i (x1) . . . (xN ) ! Perturbation Theory In perturbation theory, the wavefunction is Bulk-to-boundary Bulk-to-bulk propagator (not quite Feynman!) The wavefunction coefficients are usually easier to compute than in-in correlators: Wavefunction coefficients hO1O2O3O4i =
  • 47. Relation to In-In Correlators In perturbation theory, there is a mapping from wavefunction coefficients to boundary correlators: I++ + I I+ + I + This mapping reproduced the in-in computations. h i = 1 2RehOOi h i = hOOOOi hOOi4 + hOOXi2 hXXihOOi4 h i = 2RehOOOi Q3 i=1 2RehOiOii
  • 48. Relation to AdS The dS wavefunction is related by a double analytic continuation to the partition functions in EAdS: ⌘ 7! iz LdS 7! iLAdS • This relates AdS Witten diagrams to dS wavefunction coefficients. • AdS Witten diagram can also be mapped directly to dS in-in correlators. Maldacena [2003]
 Di Pietro, Gorbenko and Komatsu [2021] Sleight and Taronna [2021] See talks by Charlotte and Massimo.
  • 49. iV (⌘2) 0 Z 1 d⌘1d⌘2 The Challenge Because energy is not conserved in cosmology, the computation of cosmological correlators involves complicated time integrals: K(⌘) Since the bulk dynamics is not observable, there should be a more on-shell way of deriving the correlators that does not include these integrals. iV (⌘1) G(⌘1, ⌘2)
  • 50. Kinematics • The total energy is not conserved in cosmology: Correlators depend on the same external data as scattering amplitudes: Two important differences: • All energies must be positive:
  • 51. Something interesting happens in the limit of would-be energy conservation: En = AMPLITUDE TOTAL ENERGY 0 Z 1 dt eiEt f(t,~ kn) = For amplitudes, we instead have an energy-conserving delta function: Amplitudes live inside correlators. +1 Z 1 dt eiEt f(t, pn) = / D(E) Singularities Raju [2012] Maldacena and Pimentel [2011]
  • 52. Additional singularities arise when the energy of a subgraph is conserved: = Arkani-Hamed, Benincasa and Postnikov [2017] Arkani-Hamed, DB, Lee and Pimentel [2018] DB, Duaso Pueyo, Joyce, Lee and Pimentel [2020] Goodhew, Jazayeri and Pajer [2021] PARTIAL ENERGY CORRELATOR AMPLITUDE Singularities
  • 53. Singularities = Additional singularities arise when the energy of a subgraph is conserved: Amplitudes are the building blocks of correlators. Arkani-Hamed, Benincasa and Postnikov [2017] Arkani-Hamed, DB, Lee and Pimentel [2018] DB, Duaso Pueyo, Joyce, Lee and Pimentel [2020] Goodhew, Jazayeri and Pajer [2021]
  • 54. Linking Singularities These singularities occur only for unphysical kinematics: En Unphysical Physical The challenge is to extend these singular limits to physical kinematics.
  • 55. Bootstrap Tools Unphysical Physical En See talks by Austin, Charlotte, Massimo and Enrico. There has been significant progress in bootstrapping cosmological correlators using symmetries, unitarity and locality.
  • 56. A Few Questions • Can we go beyond Feynman diagrams? • What is the analytic structure beyond tree level? • What are the constraints from causality? • What are nonperturbative bootstrap conditions? • Can we define rigorous dispersion relations? • What is the space of UV-complete correlators? • Does the bootstrap suggest new observational signatures? • Is there a holographic theory of cosmology? See discussion session by Enrico. Where is the magic? Nima Arkani-Hamed
  • 58. • Where is the magic? • How to combine diagrams? • Beyond tree level? • Causality? • Non-perturbative? • Dispersion relations? • Observational signatures? • Holography? • What is inflation? • Weakly or strong coupled? • Space of consistent theories? • Starobinsky inflation? • Signatures of UV completion? • UV constraints on IR observables? • New insights from amplitudes methods? Models Observations Principles • How to falsify inflation? • How to prove inflation? • Microphysics of inflation? • New observational signatures? • Protected LSS observables?