The Cosmological Bootstrap
Daniel Baumann
University of Amsterdam

Web Seminar,April 2020

Based on work with

Nima Arkani-Hamed, Hayden Lee, Guilherme Pimentel,
Carlos Duaso Pueyo and Austin Joyce

The physics of the early universe is encoded in spatial correlations
between cosmological structures at late times:

A central challenge of modern cosmology is to construct a consistenthistory of the universe that explains these correlations.

The correlations can be traced back to primordial correlations at
the beginning of the hot big bang.

t=0To explain the observed fluctuations in the CMB, these fluctuations
must be created before the hot big bang!

What is the space of consistent histories?

t=0?
• What are the rules that consistent correlators have to satisfy?
• How are these rules encoded in the boundary observables?

Similar questions have been asked for scattering amplitudes:?
In that case, the rules of quantum mechanics and relativity are very
constraining.

Does a similar rigidity exist for cosmological correlators?

?
Goal: Develop an understanding of cosmological correlators thatparallels 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:particledecay

particlecreationtimespaceChen and Wang [2009]
DB and Green [2011]
Noumi, Yamaguchi and Yokoyama [2013]
Arkani-Hamed and Maldacena [2015]
Arkani-Hamed, DB, Lee and Pimentel [2018]

During inflation, the rapid expansion can produce very massive
particles (～1014 GeV) whose decays lead to nontrivial correlations.

At late times, these correlations will leave an imprint in the distributionof galaxies:

10 billion yrs<< 1 sec

Goal: Develop a systematic way to predict these signals.

Any Questions?

OutlineI.

The Cosmological
BootstrapII.

NewDevelopments

I.

The Cosmological
Bootstrap

Bootstrap Philosophy

Lagrangian
equations of motion

Physical Principles

spacetime evolution(Lorentz, locality, …)S=

Observables

Feynman diagramsZ1µd x L @µ @24

locality

Lorentz

1 2 2m+ ···2
parameters

Bootstrap Philosophy

LagrangianPhysical Principles(Lorentz, locality, …)

equations of motion
spacetime evolution

Observables

Feynman diagrams

Modern scattering
amplitudes programme
See Yu-tin’s book.

The S-Matrix Bootstrap
The structure of scattering amplitudes at tree level is fixed by Lorentz
invariance, locality and unitarity:A(s, t) =Xn m

anm s t+gs2

M2PS

✓

2t1+ 2M◆

M, S

contact
interactions

exchange
interactions• No Lagrangian or Feynman diagrams are needed to derive this.
• Basic principles allow only a small menu of possibilities.

The Challenge
Even tree-level processes are hard to compute in cosmology:

hi =⇠Zi(k1 +k2 )t i(k3 +k4 )t̃e

G |~k1 + ~k2 |, t, t̃

shown for conformallycoupled scalars

complicated functionof Hankel functions

dt dt̃ e

The Cosmological Bootstrap
In the cosmological bootstrap, the primordial correlators are determinedfrom consistency conditions alone:

Arkani-Hamed, DB, Lee and Pimentel [2018]
DB, Duaso Pueyo, Joyce, Lee and Pimentel [2019]DB, Duaso Pueyo, Joyce, Lee and Pimentel [2020]
Arkani-Hamed and Maldacena [2015]
Arkani-Hamed, Benincasa, and Postnikov [2017]
Sleight and Taronna [2019]
Sleight [2019]Inflation

De Sitter

If inflation is correct, then all primordial correlations live on the boundaryof an approximate de Sitter spacetime:• Isometries of dS become conformal symmetries on the boundary.
• This constrains the correlations of weakly interacting particles.

De Sitter

Inflation

Inflationary three-point functions are obtained from de Sitter four-point
functions by evaluating one of the external legs on the background:m4 !

p"Hk4 ! 0¯(t)We can therefore study de Sitter four-point functions as the fundamentalbuilding blocks of inflationary correlators.

Symmetries
If the couplings between particles are weak, then the primordialcorrelations inherit the symmetries of the quasi-de Sitter spacetime:

~k2
~k1

~k3
~k4

• Rotations• Translations}• Dilatation• Special Conformal}

Momentum conservation in Fourier space

Determine the allowed deformations (orshapes) of the correlators.

Kinematics
The kinematical data of correlators and amplitudes is similar:

k2
k2

k3

k3

k1
k4
AMPLITUDE

A+ ···nE
Raju [2012]
Maldacena and Pimentel [2011]

k1

k4

Ward Identities
Invariance under dilatations and SCTs imply the following Ward identities:"0= 90=4X

n=14 ⇣Xn=1"(nn~kn · @~kn3)@~kn

⌘

#F
#~kn
(~kn · @~kn )@~kn +(@~kn · @~kn ) F
2This is the analog of Lorentz invariance of the amplitude:

Ward Identities
These Ward identities dictate how the strength of the correlations
changes as we change the external momenta:

Singularities
The solutions to the Ward identities can be classified by their singularities:

lim

lim

EL !0E!0EnEFT
EXPANSION

Contact solutions only
have total-energy poles.

(EL )mSPINEXCHANGE

Exchange solutions haveadditional partial-energy poles.
Arkani-Hamed, DB, Lee and Pimentel [2018]

Exchange Solutions
There are distinct solutions for distinct microscopic processes during
inflation:

Arkani-Hamed, DB, Lee and Pimentel [2018]

Exchange Solutions
There are distinct solutions for distinct microscopic processes during
inflation:

m== DnWEIGHT-SHIFTINGOPERATORSp2H

M, S = 0

Remarkably, all solutions can be reduced to a unique building block.
Arkani-Hamed, DB, Lee and Pimentel [2018]

Seed Solution• The dilatation Ward identity for the seed is solved if
1
F = F̂ (u, v)
sk3

k2
k1sss
where we have introduced u ⌘and v ⌘.
k1 + k2
k3 + k4

k4

Seed Solution• The dilatation Ward identity for the seed is solved if
1
F = F̂ (u, v)
sk3

k2
k1sss
where we have introduced u ⌘and v ⌘.
k1 + k2
k3 + k4• The conformal Ward identity then becomes(whereuv )F̂ = 0.

k4

Seed Solution
For tree exchange, the conformal Ward identity reduces to:(u+M2)F̂ = F̂c

CONTACT SOLUTIONMASS OF THE EXCHANGE PARTICLE==

Seed Solution
For tree exchange, the conformal Ward identity reduces to:(u+M2)F̂ = F̂cNeed boundary conditions to solve this ODE:I.

Absence of singularity
in the folded limit:= regularII.

Correct normalizationin the collapsed limit:= 3pt x 3pt

Seed Solution
The explicit solution for the seed function is(uv)F =12 ±iMX

2 F1"

13±iM,
44 ± iM

1 ± iMcmn (M ) u2m+12|1|3F 2|0|1"12, 1
5+2iM 5 2iM, 44⇣ u ⌘nv

m,n1u2#2 F1"

13±iM,
44 ± iMv2

1 ± iM#⇡+g(u, v)
cosh(⇡M )5+2iM 5 2iM 1, 4 , 2 + iM
432 + iM

u2u , 2v
2#

Arkani-Hamed, DB, Lee and Pimentel [2018]

The Collapsed Limit
In the collapsed limit, the solution oscillates:lims!0= sin[M log(s/k12 )]Noumi, Yamaguchi and Yokoyama [2013]
Arkani-Hamed and Maldacena [2015]
Arkani-Hamed, DB, Lee and Pimentel [2018]

Particle Production
These oscillations are a key signature of particle production during
inflation:eiM t)

Oscillations in the superhorizon evolution become oscillations in theboundary correlations at late times.

Cosmological Collider Physics

Cross section (nb)Correlationfunction
strength
Four-point

This signal is the analog of resonances in collider physics:

Momentum ratio

Centre-of-mass energy (GeV)Any Questions?

II.

NewDevelopments

So far, we have studied the correlations of scalar fields.
Arkani-Hamed, DB, Lee and Pimentel [2018]
DB, Duaso Pueyo, Joyce, Lee and Pimentel [2019]

Now, we would like to extend the bootstrap to spinningcorrelators, especially to massless fields with spin.
DB, Duaso Pueyo, Joyce, Lee and Pimentel [2020]

Massless Particles in Flat Space• Massless bosons mediate long-range forces:

gravityelectromagnetism• The interactions of massless particles are highly constrained:
spin 2 = GR

spin 1 = YM

Beyond Feynman Diagrams• Computations using Feynman diagrams are complicated.
Parke and Taylor [1985]• Physical answers are simple.
De Witt [1967]• Bootstrap methods are a necessity, not a luxury:- Massless 3pt amplitudes are fixed by Poincare invariance:

+ =✓

3h12i
h13ih23i◆S- Higher-point amplitudes are constrained by locality:(p1 + p2 )2 ! 0A41A3A32(p1 + p2 )The Four-Particle Test
• Consistent factorisation is a nontrivial constraint:µ1 ...µShµ⌫• Only consistent for spins

S = { 0 , 21 , 1 , 32 , 2 }
GRYM

SUSY
Benincasa and Cachazo [2007]
McGady and Rodina [2010]

Massless Particles in de Sitter Space• Fluctuations of all massless fields are amplified during inflation.
• Every inflationary model has two massless modes:
ijscalartensor• Not much is known about tensor correlators beyond 3pt functions.
• Even less is known about the consistency of partially massless fields:⌃i1 ...iSpartially masslessBeyond Feynman Diagrams• Direct computations of spinning correlators are very complicated.

• Bootstrap methods are a necessity, not a luxury.

Two Approaches
In our new paper, we derive a large class of spinning correlators in deSitter space. We use two diﬀerent approaches:1) Spin-raising operators=XnSn

spinning correlator

scalar seed2) Singularities
In the following, I will describe the second approach.
DB, Duaso Pueyo, Joyce, Lee and Pimentel [2020]

Singularities of Cosmological Correlators
The four-point function is controlled by three singularities:

A3 ⇥ F3EL

F3 ⇥ A3
ERA4
ESingularities occur when energies add up to zero.

Raju [2012]
Maldacena and Pimentel [2011]
Arkani-Hamed, Benincasa, and Postnikov [2017]

A Simple Example
Consider Compton scattering in de Sitter space.

• The factorisation limits of the s-channel arek3k2lim

EL !0lim

ER !0⇠~1 · ~k2⇠~3 · ~k4=
EL ER (k34 s)⇠~3 · ~k4⇠~1 · ~k2=
ER EL (k12 s)

k1s

k4

EL ⌘ k12 + s

ER ⌘ k34 + s
E ⌘ k12 + k34

A Simple Example
Consider Compton scattering in de Sitter space.

• The factorisation limits of the s-channel are
k3k2
lim

EL !0lim

ER !0⇠~1 · ~k2⇠~3 · ~k4=
EL ER (k34 s)⇠~3 · ~k4⇠~1 · ~k2=
ER EL (k12 s)

k1s

k4

EL ⌘ k12 + s

ER ⌘ k34 + s
E ⌘ k12 + k34• The unique solution that is consistent with these limits is(⇠~1 · ~k2 )(⇠~3 · ~k4 )hJ J is =EL E R E• The total energy singularity has the correct residue.

EL E R

E!0!S

A More Complicated Example
Consider Compton scattering of gravitons.

• The solution in the s-channel isfixed by factorisation✓◆

fixed by totalenergy singularity

fixed by conformal
symmetry= (⇠~1 · ~k2 )2 (⇠~3 · ~k4 )2

1
2k1 k3 + EL k3 + ER k1
2sk1 k3+22
2
EL E REE◆✓
12k1 k3
k13
1+ 2 +3EL E REEEA More Complicated Example
Consider Compton scattering of gravitons.

• The solution in the s-channel is= (⇠~1 · ~k2 )2 (⇠~3 · ~k4 )2

✓

12k1 k3 + EL k3 + ER k12sk1 k3+22
2EL E REE◆✓
12k1 k3
k13
1+ 2 +
3EL E REEE• The solution in the u-channel is✓fixed by factorisation◆

1EL
2k1 k3~1 , ⇠~3 , ~k2 , ~k4 )= 2 2+N(⇠EL ERE2E◆✓
12k1 k3
k13
1~1 , ⇠~3 , ~k2 , ~k4 )M(⇠+++EL ER
E3E2E
fixed by totalenergy singularity

fixed by conformalsymmetry◆

One Channel Is Not EnoughConsider the correlator of one photon and three scalars:Aie2

e4s-channel

e3t-channel

u-channel✓◆• Can we have e3 = e4 = 0 ?
Aie2!
e4 EE!0e2

h12ih2̄4̄ih41i
STflat-space
amplitude

h14ih4̄1i 1
2k1 e3T

not Lorentz-invariant

One Channel Is Not EnoughConsider the correlator of one photon and three scalars:Aie2e4s-channel

e3t-channel

u-channel• Let e4 6= 0 :+1!
eE3E!0e4

Lorentz-violation disappears when✓h12ih2̄4̄ih41i
e2ST

e2 + e4 = 0h14ih4̄1i 1(e2 + e4 )2k1 Tchargeconservation◆Discovering Yang-Mills
Consider two gluons and two scalars:
GiAaj
GBb

BTcb

ATac+ATbc

s-channelBTcat-channel• The sum of all channels is only consistent if
the couplings satisfy the Lie algebra:
A+ f ABC

B[T , T ]ab = fABCCTabCTabu-channel+(A B)
Tac Tcb

contact• Consistency also fixes the contact term required by gauge invariance.

Equivalence Principle (without falling elevators)
Consider one graviton and three scalars:hij2

4s-channel

3t-channel• The individual channels are not consistent.
• The sum of all channels is consistent if and only if
2 = 3 = 4u-channel

Equivalence Principle (without falling elevators)
Consider two gravitons and two scalars:+s-channel+
t-channel• The sum of all channels is only consistent if
all gravitational couplings are universal:

abu-channel+
2ca = b = ccontact

Ruling Out Theories
The bootstrap approach will also allow us to rule out theories:• Couplings of massless gravitinos will not be supersymmetric.
• Couplings of higher-spin particles will not be local.
• Multiple gravitons must be decoupled.
• Interactions of partially massless particles will be highly constrained.
•…

We hope to report on such no-go results in the future.

Any Questions?

III.

FutureChallenges

We have only scratched the surface of a fascinating subject:

Observational
Cosmology

Inflation

Cosmological
Collider Physics

Cosmological Bootstrap

CFT/Holography

Much more remains to be discovered.

ScatteringAmplitudes

Open Problems• Beyond Feynman Diagrams• What is the on-shell formulation of cosmological correlators?
• What are the fundamental building blocks?
• How are these building blocks connected?
• Is there a cosmological analog of Parke-Taylor?
• Where is the hidden simplicity?

• Ultraviolet Completion• What are the rules?
• How is unitarity encoded in the boundary correlators?
• Are there interesting positivity constraints?
• How does this constrain the space of consistent correlators?
• Does this motivate new observational targets?

Thank you for your attention!

Dramatic progress in the study of scattering amplitudes:• on-shell recursion relations• color-kinematics duality• soft theorems• hidden positivities• spinor helicity formalism• generalised unitarity• momentum twistors•…
Do these insights translate to cosmological correlators?

Amplitudes live inside correlators.

lim

E!0=1En

Raju [2012]
Maldacena and Pimentel [2011]

Insights from the modern scattering amplitudes programme must
therefore be relevant for cosmology.

What is the goal?
Relate low-energy predictions to high-energy physics:

UVIR
Define observational targets.

What are the rules?
The closer we get to the real world, the less we understand:de Sitter space?

flat spacetree-level(some loops)

anti-de Sitter space

understanding
non-perturbative

Causality
Consistent time evolution is encoded in the analytic structure (polesand branch cuts) of amplitudes and correlators:

one-loop

tree-level

Scattering amplitudes

tree-level

Cosmological correlatorsLocalityLocality is encoded in factorization:lim

EL !0(EL )mScattering amplitudes

Cosmological correlators

Unitarity
Unitarity is encoded in positivity:?
Scattering amplitudes

Cosmological correlators

Landscape vs Swampland
The ultraviolet completion of scattering amplitudes is highly
constrained by these basic physical principles:Exchange of Spinning Particles

Strategy
We wish to find diﬀerential operators that relate scalar exchange to spin
exchange:spin-raising
operator

It turns out that the spin raising is best implemented in embedding space
and then Fourier transformed.

CFTs in Embedding SpaceConsider the following embedding of d-dimensional Euclidean space into(d+2)-dimensional Minkowski space:

Dirac [1936]
Costa, Penedones, Poland and Rychkov [2011]

CFTs in Embedding Space
Lorentz transformations in embedding space become conformaltransformations on the Euclidean section:

Dirac [1936]
Costa, Penedones, Poland and Rychkov [2011]

CFTs in Embedding Space
Conformal correlators in embedding space are simply the most general
Lorentz-invariant expressions with the correct scaling behavior:

where.
Dirac [1936]
Costa, Penedones, Poland and Rychkov [2011]

Spin-Raising Operator
Correlators of spinning fields can be written in terms of scalar seeds.
For example:,where.

In Fourier space, this becomes.

Karateev, Kravchuk and Simmons-Duﬃn [2018]
Costa, Penedones, Poland and Rychkov [2011]

Bootstrapping Spin Exchange
Using this spin-raising operator, we have,spin-exchangesolutionpolarization
tensorspin-raising
operatorscalar-exchangesolution

which can be written ase.g.
Arkani-Hamed, DB, Lee and Pimentel [2018]