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Caltech Colloquium
May 2021
Daniel Baumann
University of Amsterdam &
National Taiwan University
The Cosmological Bootstrap
What was the Origin of
Structure in the Universe?
Cosmological structures are not distributed randomly, but display spatial
correlations over very large distances:
380 000 years
few billion years
13.8 billion years
These correlations can be traced back to the beginning of the hot Big Bang:
However, the Big Bang was not the beginning of time, but the end of an
earlier high-energy period:
?
t = 0
end of inflation
hot big bang
What exactly happened before the hot Big Bang?

How did it create the primordial correlations?
The challenge is to extract this information from the pattern of correlations in
the late universe.
?
end of inflation
hot big bang
?
end of inflation
hot big bang
• What correlations are consistent with basic physical principles?

• Can these correlations be bootstrapped directly?
locality, causality, 

unitarity, …
What is the space of consistent correlations?
?
In that case, the rules of quantum mechanics and relativity are very
constraining.
The bootstrap perspective has been very influential for scattering amplitudes:
?
Does a similar rigidity exist for cosmological correlators?
Goal: Develop an understanding of cosmological correlators that parallels
our understanding of flat-space scattering amplitudes.
The connection to scattering amplitudes is also relevant because the early
universe was like a giant cosmological collider:
Chen and Wang [2009]
DB and Green [2011]
Noumi, Yamaguchi and Yokoyama [2013]
Arkani-Hamed and Maldacena [2015]
Arkani-Hamed, DB, Lee and Pimentel [2018]
particle
creation
particle
decay
time
space
During inflation, the rapid expansion can produce very massive particles
(~1014 GeV) whose decays lead to nontrivial correlations.
10 billion yrs
<< 1 sec
Goal: Develop a systematic way to predict these signals.
At late times, these correlations leave an imprint in the distribution of galaxies:
Outline
Cosmological
Correlations II.
I. Cosmological
Bootstrap
Nima Arkani-Hamed, Wei Ming Chen, Carlos Duaso Pueyo, Hayden Lee,
Manuel Loparco, Guilherme Pimentel and Austin Joyce
Based on work with
I. Cosmological
Correlations
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
10−4
10−3
10−2
10−1
100
101
Separation [Mpc−1
]
100
101
102
103
104
Power
[Mpc
3
]
Galaxy Distribution
At late times, we observe correlations in the distribution of galaxies:
matter density
Cosmic Microwave Background
At early times, we observe correlations in the CMB temperature anisotropies:
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
90◦
18◦
0
1000
2000
3000
4000
5000
6000
Power
[µK
2
]
2◦
0.2◦
0.1◦
0.07◦
Angular separation
These correlations span superhorizon scales at photon decoupling:
Superhorizon
Superhorizon Correlations
Superhorizon Correlations
They must therefore have been created before the hot Big Bang:
Big Bang
Photon
decoupling
distance light travelled
since the Big Bang
?
Before the Hot Big Bang
bounce
inflation
This suggests two options: rapid expansion or slow contraction?
Scale Invariance
An important clue is that the primordial fluctuations are scale invariant:
This is a natural prediction of inflation.
Inflation and Quantum Fluctuations
During inflation, quantum fluctuations are stretched to macroscopic scales:
Mukhanov and Chibisov [1981]

Hawking [1982]

Starobinsky [1982]

Guth and Pi [1982]

Bardeen, Steinhardt and Turner [1983]
Inflation and Quantum Fluctuations
The predicted correlations are scale invariant because the inflationary
dynamics is approximately time-translation invariant:
Mukhanov and Chibisov [1981]

Hawking [1982]

Starobinsky [1982]

Guth and Pi [1982]

Bardeen, Steinhardt and Turner [1983]
What Caused Inflation?
So far, only two-point correlations are needed to describe the data. These are
fixed by symmetry, so they don’t reveal much about the physics of inflation.
Dynamical information is encoded in higher-point correlations:
?
?
Cosmological Collider Physics
Chen and Wang [2009]
DB and Green [2011]
Noumi, Yamaguchi and Yokoyama [2013]
Arkani-Hamed and Maldacena [2015]
Arkani-Hamed, DB, Lee and Pimentel [2018]
One way to obtain higher-point correlations is through the production and
decay of new massive particles during inflation:
particle
creation
particle
decay
time
space
Particles as Tracers
These particles are tracers of the inflationary dynamics:
The pattern of correlations after inflation contains a memory of the physics
during inflation (evolution, symmetries, particle content, etc.).
How do we extract this information?
Any Questions?
II. Cosmological
Bootstrap
Back to the Future
If inflation is correct, then all cosmological correlations can be traced back
to the future boundary of an approximate de Sitter spacetime:
?
end of inflation
Can the correlations on this boundary be bootstrapped directly?
• Conceptual advantage: focuses directly on observables.

• Practical advantage: simplifies calculations.
hot big bang
Time Without Time
Time evolution is encoded in the spatial rescaling of the boundary correlations:
Although the bootstrap method describes static observables, we will see this
time-dependent physics emerging.
Bootstrap Philosophy
Lagrangian

equations of motion

spacetime evolution 

Feynman diagrams
Physical Principles Observables
see Cliff Cheung’s TASI lectures
locality, causality, 

unitarity, symmetries
S-matrix Bootstrap
fixed by Lorentz

and locality
Much of the physics of scattering amplitudes is controlled by SINGULARITIES:
constrained
by Lorentz
• Amplitudes have poles when intermediate particles go on-shell.

• On these poles, the amplitudes factorize.
The different singularities are connected by (Lorentz) SYMMETRY.
S-matrix Bootstrap
Consistent factorization is very constraining for massless particles:
µ1...µS
S = { 0 , 1
2 , 1 , 3
2 , 2 }
Only consistent for spins
YM
GR
SUSY
Benincasa and Cachazo [2007]

McGady and Rodina [2013]
A Success Story
The modern amplitudes program has been very successful:
1. New Computational Techniques 2. New Mathematical Structures
3. New Relations Between Theories
• Recursion relations

• Generalized unitarity

• Soft theorems
• Positive geometries

• Amplituhedrons

• Associahedrons
• Color-kinematics duality

• BCJ double copy
4. New Constraints on QFTs
• Positivity bounds

• EFThedron
5. New Applications
• Gravitational wave physics

• Cosmology
Cosmological Bootstrap
A similar logic constrains cosmological correlators:
• SINGULARITIES: 

Correlators have characteristic singularities as a function of the external
energies. Analog of resonance singularities for amplitudes.
• SYMMETRIES: 

These singularities are connected by causal time evolution, which is
constrained by the symmetries of the bulk spacetime.
Analog of Lorentz symmetry for amplitudes.
Arkani-Hamed, DB, Lee and Pimentel [2018]
DB, Duaso Pueyo, Joyce, Lee and Pimentel [2020]
Singularities
• 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:
Singularities
Something interesting happens in the limit of would-be energy conservation:
Raju [2012]

Maldacena and Pimentel [2011]
Amplitudes live inside correlators.
En
=
AMPLITUDE
TOTAL ENERGY POLE
Singularities
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]
PARTIAL ENERGY POLE
Singularities
=
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]
Amplitudes are the building blocks of correlators.
Singularities
At tree level, these are the only singularities of the correlator:
To determine the full correlator, we must connect these singular limits.
=
En
=
=
conformal symmetry
Symmetries
The isometries of the bulk de Sitter spacetime become rescaling symmetries
of the correlators on the boundary:
scaling symmetry
See Tom Hartman’s colloquium
Symmetries
These symmetries lead to a set of differential equations that control the
amplitude of the correlations as we deform the quadrilateral
The boundary conditions of this equation are the energy singularities.
Arkani-Hamed, DB, Lee and Pimentel [2018]
DB, Duaso Pueyo, Joyce, Lee and Pimentel [2020]
Symmetries
Symmetry relates the singularities in the unphysical region to the form of the
correlator in the physical region:
En
Analytic continuation Physical regime
Arkani-Hamed, DB, Lee and Pimentel [2018]
DB, Duaso Pueyo, Joyce, Lee and Pimentel [2020]
If the theory contains massive particles, then the correlator is forced to
have an oscillatory feature in the physical regime:
Particle Production
BRANCH CUT
OSCILLATIONS
COLLAPSED LIMIT
Time Without Time
These oscillations reflect the evolution of the massive particles during inflation:
Time-dependent effects emerge in the solution of the time-independent
bootstrap constraints.
Cosmological Collider Physics
Centre-of-mass energy (GeV)
Cross
section
(nb)
Four-point
function
Four-point
function
Momentum ratio
Correlation
strength
The oscillatory feature is the analog of a resonance in collider physics:
A =
g2
s M2
PS(cos ✓)
The angular dependence of the signal depends on the spin of the particles.
This is very similar to what
we do in collider physics:
Particle Spectroscopy
✓
kI / sin[M log kI ] PS(cos ✓)
The frequency of the oscillations depends on the mass of the particles:
Galaxy Distribution
Galaxies form in regions of high matter density:
0 2 4 6 8 10
0.0
0.5
1.0
1.5
2.0
2.5
x
δ
M
(x)
Correlations in the primordial matter density therefore become imprinted in the
distribution of galaxies on the sky.
Observational Prospects
SphereX
This signal will be an interesting target for future galaxy surveys:
Conclusions
• Much of this structure is controlled by singularities.

• Behavior away from singularities captures local bulk dynamics.
Cosmological correlation functions have a rich structure:
The bootstrap approach has led to new conceptual insights:
• Complex correlators can be derived from simpler seed functions:
Flat space 

correlators
Transmutation De Sitter

correlators
Scalar

correlators
Spinning

correlators
Weight-shifting
It also has practical applications:
• Signatures of massive particles during inflation can be classified.
(ask me)
Thank you for your attention!
NTU, Taiwan
Consider the following slice through the shape of the four-point function:
Physical regime
Bootstrapping Correlators
Along this axis, the correlator has the following singularities:
Analytic continuation Physical regime
En
Bootstrapping Correlators
Along this axis, the correlator has the following singularities:
Bootstrapping Correlators
The singularities in the unphysical region determine the form of the correlator
in the physical region.
En
If the theory contains massive particles, then the correlator is forced to
have an oscillatory feature in the physical regime:
Particle Production
BRANCH CUT
COLLAPSED LIMIT
OSCILLATIONS
Observational
Cosmology
Inflation
Scattering
Amplitudes
Cosmological
Collider Physics
CFT/Holography
Cosmological Bootstrap
Much more remains to be discovered.
We have only scratched the surface of a fascinating subject, with relations to
many areas of physics:

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Caltech Physics Colloquium

  • 1. Caltech Colloquium May 2021 Daniel Baumann University of Amsterdam & National Taiwan University The Cosmological Bootstrap
  • 2. What was the Origin of Structure in the Universe?
  • 3. Cosmological structures are not distributed randomly, but display spatial correlations over very large distances: 380 000 years few billion years 13.8 billion years
  • 4. These correlations can be traced back to the beginning of the hot Big Bang:
  • 5. However, the Big Bang was not the beginning of time, but the end of an earlier high-energy period: ? t = 0 end of inflation hot big bang
  • 6. What exactly happened before the hot Big Bang? How did it create the primordial correlations? The challenge is to extract this information from the pattern of correlations in the late universe. ? end of inflation hot big bang
  • 7. ? end of inflation hot big bang • What correlations are consistent with basic physical principles? • Can these correlations be bootstrapped directly? locality, causality, 
 unitarity, … What is the space of consistent correlations?
  • 8. ? In that case, the rules of quantum mechanics and relativity are very constraining. The bootstrap perspective has been very influential for scattering amplitudes:
  • 9. ? Does a similar rigidity exist for cosmological correlators? Goal: Develop an understanding of cosmological correlators that parallels our understanding of flat-space scattering amplitudes.
  • 10. The connection to scattering amplitudes is also relevant because the early universe was like a giant cosmological collider: Chen and Wang [2009] DB and Green [2011] Noumi, Yamaguchi and Yokoyama [2013] Arkani-Hamed and Maldacena [2015] Arkani-Hamed, DB, Lee and Pimentel [2018] particle creation particle decay time space During inflation, the rapid expansion can produce very massive particles (~1014 GeV) whose decays lead to nontrivial correlations.
  • 11. 10 billion yrs << 1 sec Goal: Develop a systematic way to predict these signals. At late times, these correlations leave an imprint in the distribution of galaxies:
  • 12. Outline Cosmological Correlations II. I. Cosmological Bootstrap Nima Arkani-Hamed, Wei Ming Chen, Carlos Duaso Pueyo, Hayden Lee, Manuel Loparco, Guilherme Pimentel and Austin Joyce Based on work with
  • 14. 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
  • 15. 10−4 10−3 10−2 10−1 100 101 Separation [Mpc−1 ] 100 101 102 103 104 Power [Mpc 3 ] Galaxy Distribution At late times, we observe correlations in the distribution of galaxies: matter density
  • 16. Cosmic Microwave Background At early times, we observe correlations in the CMB temperature anisotropies: 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
  • 17. 90◦ 18◦ 0 1000 2000 3000 4000 5000 6000 Power [µK 2 ] 2◦ 0.2◦ 0.1◦ 0.07◦ Angular separation These correlations span superhorizon scales at photon decoupling: Superhorizon Superhorizon Correlations
  • 18. Superhorizon Correlations They must therefore have been created before the hot Big Bang: Big Bang Photon decoupling distance light travelled since the Big Bang ?
  • 19. Before the Hot Big Bang bounce inflation This suggests two options: rapid expansion or slow contraction?
  • 20. Scale Invariance An important clue is that the primordial fluctuations are scale invariant: This is a natural prediction of inflation.
  • 21. Inflation and Quantum Fluctuations During inflation, quantum fluctuations are stretched to macroscopic scales: Mukhanov and Chibisov [1981] Hawking [1982] Starobinsky [1982] Guth and Pi [1982] Bardeen, Steinhardt and Turner [1983]
  • 22. Inflation and Quantum Fluctuations The predicted correlations are scale invariant because the inflationary dynamics is approximately time-translation invariant: Mukhanov and Chibisov [1981] Hawking [1982] Starobinsky [1982] Guth and Pi [1982] Bardeen, Steinhardt and Turner [1983]
  • 23. What Caused Inflation? So far, only two-point correlations are needed to describe the data. These are fixed by symmetry, so they don’t reveal much about the physics of inflation. Dynamical information is encoded in higher-point correlations: ? ?
  • 24. Cosmological Collider Physics Chen and Wang [2009] DB and Green [2011] Noumi, Yamaguchi and Yokoyama [2013] Arkani-Hamed and Maldacena [2015] Arkani-Hamed, DB, Lee and Pimentel [2018] One way to obtain higher-point correlations is through the production and decay of new massive particles during inflation: particle creation particle decay time space
  • 25. Particles as Tracers These particles are tracers of the inflationary dynamics: The pattern of correlations after inflation contains a memory of the physics during inflation (evolution, symmetries, particle content, etc.). How do we extract this information?
  • 28. Back to the Future If inflation is correct, then all cosmological correlations can be traced back to the future boundary of an approximate de Sitter spacetime: ? end of inflation Can the correlations on this boundary be bootstrapped directly? • Conceptual advantage: focuses directly on observables. • Practical advantage: simplifies calculations. hot big bang
  • 29. Time Without Time Time evolution is encoded in the spatial rescaling of the boundary correlations: Although the bootstrap method describes static observables, we will see this time-dependent physics emerging.
  • 30. Bootstrap Philosophy Lagrangian equations of motion spacetime evolution Feynman diagrams Physical Principles Observables see Cliff Cheung’s TASI lectures locality, causality, 
 unitarity, symmetries
  • 31. S-matrix Bootstrap fixed by Lorentz and locality Much of the physics of scattering amplitudes is controlled by SINGULARITIES: constrained by Lorentz • Amplitudes have poles when intermediate particles go on-shell. • On these poles, the amplitudes factorize. The different singularities are connected by (Lorentz) SYMMETRY.
  • 32. S-matrix Bootstrap Consistent factorization is very constraining for massless particles: µ1...µS S = { 0 , 1 2 , 1 , 3 2 , 2 } Only consistent for spins YM GR SUSY Benincasa and Cachazo [2007] McGady and Rodina [2013]
  • 33. A Success Story The modern amplitudes program has been very successful: 1. New Computational Techniques 2. New Mathematical Structures 3. New Relations Between Theories • Recursion relations • Generalized unitarity • Soft theorems • Positive geometries • Amplituhedrons • Associahedrons • Color-kinematics duality • BCJ double copy 4. New Constraints on QFTs • Positivity bounds • EFThedron 5. New Applications • Gravitational wave physics • Cosmology
  • 34. Cosmological Bootstrap A similar logic constrains cosmological correlators: • SINGULARITIES: 
 Correlators have characteristic singularities as a function of the external energies. Analog of resonance singularities for amplitudes. • SYMMETRIES: 
 These singularities are connected by causal time evolution, which is constrained by the symmetries of the bulk spacetime. Analog of Lorentz symmetry for amplitudes. Arkani-Hamed, DB, Lee and Pimentel [2018] DB, Duaso Pueyo, Joyce, Lee and Pimentel [2020]
  • 35. Singularities • 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:
  • 36. Singularities Something interesting happens in the limit of would-be energy conservation: Raju [2012] Maldacena and Pimentel [2011] Amplitudes live inside correlators. En = AMPLITUDE TOTAL ENERGY POLE
  • 37. Singularities 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] PARTIAL ENERGY POLE
  • 38. Singularities = 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] Amplitudes are the building blocks of correlators.
  • 39. Singularities At tree level, these are the only singularities of the correlator: To determine the full correlator, we must connect these singular limits. = En = =
  • 40. conformal symmetry Symmetries The isometries of the bulk de Sitter spacetime become rescaling symmetries of the correlators on the boundary: scaling symmetry See Tom Hartman’s colloquium
  • 41. Symmetries These symmetries lead to a set of differential equations that control the amplitude of the correlations as we deform the quadrilateral The boundary conditions of this equation are the energy singularities. Arkani-Hamed, DB, Lee and Pimentel [2018] DB, Duaso Pueyo, Joyce, Lee and Pimentel [2020]
  • 42. Symmetries Symmetry relates the singularities in the unphysical region to the form of the correlator in the physical region: En Analytic continuation Physical regime Arkani-Hamed, DB, Lee and Pimentel [2018] DB, Duaso Pueyo, Joyce, Lee and Pimentel [2020]
  • 43. If the theory contains massive particles, then the correlator is forced to have an oscillatory feature in the physical regime: Particle Production BRANCH CUT OSCILLATIONS COLLAPSED LIMIT
  • 44. Time Without Time These oscillations reflect the evolution of the massive particles during inflation: Time-dependent effects emerge in the solution of the time-independent bootstrap constraints.
  • 45. Cosmological Collider Physics Centre-of-mass energy (GeV) Cross section (nb) Four-point function Four-point function Momentum ratio Correlation strength The oscillatory feature is the analog of a resonance in collider physics:
  • 46. A = g2 s M2 PS(cos ✓) The angular dependence of the signal depends on the spin of the particles. This is very similar to what we do in collider physics: Particle Spectroscopy ✓ kI / sin[M log kI ] PS(cos ✓) The frequency of the oscillations depends on the mass of the particles:
  • 47. Galaxy Distribution Galaxies form in regions of high matter density: 0 2 4 6 8 10 0.0 0.5 1.0 1.5 2.0 2.5 x δ M (x) Correlations in the primordial matter density therefore become imprinted in the distribution of galaxies on the sky.
  • 48. Observational Prospects SphereX This signal will be an interesting target for future galaxy surveys:
  • 50. • Much of this structure is controlled by singularities. • Behavior away from singularities captures local bulk dynamics. Cosmological correlation functions have a rich structure: The bootstrap approach has led to new conceptual insights: • Complex correlators can be derived from simpler seed functions: Flat space correlators Transmutation De Sitter correlators Scalar correlators Spinning correlators Weight-shifting It also has practical applications: • Signatures of massive particles during inflation can be classified. (ask me)
  • 51. Thank you for your attention! NTU, Taiwan
  • 52. Consider the following slice through the shape of the four-point function: Physical regime Bootstrapping Correlators
  • 53. Along this axis, the correlator has the following singularities: Analytic continuation Physical regime En Bootstrapping Correlators
  • 54. Along this axis, the correlator has the following singularities: Bootstrapping Correlators The singularities in the unphysical region determine the form of the correlator in the physical region. En
  • 55. If the theory contains massive particles, then the correlator is forced to have an oscillatory feature in the physical regime: Particle Production BRANCH CUT COLLAPSED LIMIT OSCILLATIONS
  • 56. Observational Cosmology Inflation Scattering Amplitudes Cosmological Collider Physics CFT/Holography Cosmological Bootstrap Much more remains to be discovered. We have only scratched the surface of a fascinating subject, with relations to many areas of physics: