Amplitudes Meet CosmologyDaniel BaumannUniversity of Amsterdam

Amplitudes 2020 Zoom @ Brown

Pimentel Lee Joyce

Arkani-HamedDuaso Pueyo

Based on work with

Arkani-Hamed, DB, Lee and Pimentel [1811.00024]DB, Duaso Pueyo, Joyce, Lee and Pimentel [1910.14051]DB, Duaso Pueyo, Joyce, Lee and Pimentel [2005.04234]

See also talk by C. Sleight

Cosmological Correlations

Spatial correlations are the fundamental observables in cosmology:

These correlations encode the history of the early universe.

= h�⇢(~x1)�⇢(~x2) · · · �⇢(~xN )i

Back to the Future

If inflation is correct, then all correlations can be traced back to the future boundary of an approximate de Sitter spacetime:

Can we bootstrap these boundary correlations from consistency conditions alone, without reference to the bulk time evolution?

• Conceptual advantage: focus directly on observables.• Practical advantage: simplify calculations.

?

END OF INFLATION

Cosmological Collider Physics

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

Chen and Wang [2009]DB and Green [2011]

Noumi, Yamaguchi and Yokoyama [2013] Arkani-Hamed and Maldacena [2015]

Lee, DB and Pimentel [2016]Arkani-Hamed, DB, Lee and Pimentel [2018]

PARTICLECREATION

• These correlations are tracers of the inflationary dynamics.• Time dependence is encoded in the analytic properties of the correlators.

END OF INFLATION

PARTICLE DECAY

Amplitudes Meet Cosmology

The bootstrap approach also makes manifest a beautiful connection between scattering amplitudes and cosmological correlators:

Amplitudes live inside correlators

limE!0

=1

En

p1

p2

k1 k2

Raju [2012]Maldacena and Pimentel [2011]

This suggests that insights from the modern amplitudes program should be relevant for cosmology.

The Cosmological Bootstrap

Outline

I.

Future DirectionsIII.II. Spinning

Correlators

I. The Cosmological Bootstrap

Key Ideas

If the couplings between particles are weak, then the primordial correlations inherit the symmetries of the quasi-de Sitter spacetime.

1) Symmetries

2) SingularitiesPhysical processes can be classified by their singularities.

3) Scalar SeedsAll correlations arise from a unique seed solution.

4) Inflation from dSInflationary correlators are deformed de Sitter correlators.

Symmetries I

Pi� = @i�

Jij� = (xi@j � xj@i)�

• TRANSLATIONS and ROTATIONS

• DILATATIONS

D� = �(�+ x

i@i)�

• SPECIAL CONFORMAL TRANSFORMATIONS

~k1

~k2

~k3

~k4

F ⌘

The de Sitter isometries become conformal symmetries on the boundary.This symmetry constrains the boundary correlators:

Ki� =�2xi�+ 2xix

j@j � x

2@i

��

Symmetries II

0 =

"9�

4X

n=1

⇣�n � ~kn · @~kn

⌘#F

0 =4X

n=1

"(�n � 3)@~kn

� (~kn · @~kn)@~kn

+~kn2(@~kn

· @~kn)

#F

Bzowski, McFadden and Skenderis [2014]Arkani-Hamed and Maldacena [2015]

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

• In general, these equations are hard to solve.• Physical solutions can be classified by their singularities.

The conformal Ward identities (in momentum space) are

Singularities I

Every correlator has a singularity when the sum of all energies vanishes:

• The residue of the singularity is the flat-space amplitude.• For contact interactions this is the only type of singularity.

limE!0 En

EFT EXPANSION

=

Z 0

�1dt eiEt f(t,~kn) =

Raju [2012]Maldacena and Pimentel [2011]

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

AMPLITUDE

Singularities II

Exchange interactions give rise to additional partial energy singularities:

• This factorisation limit is an important constraint on physical correlators.• It encodes the signature of new particles.

limEL!0

=(EL)

m

SPIN EXCHANGE

Arkani-Hamed, Benincasa and Postnikov [2017]Arkani-Hamed, DB, Lee and Pimentel [2018]

DB, Duaso Pueyo, Joyce, Lee and Pimentel [2020]

limk12!kI

Singularities III

Vacuum initial conditions demand regularity in the folded limit:

~k1

~k2

=

= REGULAR

Flauger, Green and Porto [2013]Arkani-Hamed and Maldacena [2015]

Green and Porto [2020]

• The absence of folded singularities is a nontrivial constraint on all physical solutions of the Ward identities.

kI

Seed Solution

There are many distinct solutions for all distinct microscopic processes during inflation:

Seed Solution

There are many distinct solutions for all distinct microscopic processes during inflation:

Remarkably, all solutions can be reduced to a unique building block.

M,S = 0

= Dn

WEIGHT-SHIFTING OPERATORS

CONFORMALLY COUPLED SCALARS

m =p2H

MASSIVE SCALAR

DB, Duaso Pueyo, Joyce, Lee and Pimentel [2019]Costa, Penedones, Poland and Rychkov [2011]Karateev, Kravchuk and Simmons-Duffin [2017]

F =X

m,n

cmn(M)u2m⇣uv

⌘2n

+ e�⇡M�eiM log ug(u, v) + c.c.

�

Seed Solution

ANALYTIC NON-ANALYTIC

The explicit solution for the seed function is

sk1

k2k3

k4u ⌘ s/(k1 + k2)where and v ⌘ s/(k3 + k4) .

• The analytic part corresponds to the EFT expansion.• The non-analytic part corresponds to particle production.

The particle production piece is forced on us by the factorization limit.

We have succeeded in obtaining a purely boundary description of time-dependent physics in the bulk = time without time.

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

Cosmological Collider Physics

Centre-of-mass energy (GeV)

Cros

s sec

tion

(nb)

Four

-poi

nt fu

nctio

n

Momentum ratioCentre-of-mass energy (GeV)

Cros

s sec

tion

(nb)

Four

-poi

nt fu

nctio

n

Momentum ratio

Cor

rela

tion

stre

ngth

The solution oscillates for small exchange momentum. This feature is the analog of resonances in collider physics:

II. Spinning Correlators

Only massless fields survive until the end of inflation and can therefore be observed on the boundary.

Massless Fields in Inflation

scalar tensor

�

Every inflationary model has two massless modes:

• Not much is known about tensor correlators beyond 3pt functions.• Direct computations of spinning correlators are very complicated.• Bootstrap methods are a necessity, not a luxury.

hij

densityfluctuations

gravitationalwaves

=X

n

Sn

Two Approaches

I. Spin-raising operatorsGenerate spinning correlators by acting with spin-raising operator on scalar seeds:

DB, Duaso Pueyo, Joyce, Lee and Pimentel [2020]Costa, Penedones, Poland and Rychkov [2011]Karateev, Kravchuk and Simmons-Duffin [2017]

• This is a very systematic and algorithmic procedure.• But: it becomes cumbersome for multiple fields of high spin.

In our new paper, we derived a large class of spinning correlators in de Sitter space. We use two different approaches:

Two Approaches

DB, Duaso Pueyo, Joyce, Lee and Pimentel [2020]Arkani-Hamed, Benincasa, and Postnikov [2017]

(see extra slides for examples)

II. SingularitiesFix correlators by their singularities alone (without having to solve the four-point Ward identities):

1) Total energy singularity

2) Partial energy singularities

3) Absence of folded singularities

• In many cases, these singularities fix the correlator completely.• Sometimes, subleading poles are required by conformal invariance.

In our new paper, we derived a large class of spinning correlators in de Sitter space. We use two different approaches:

Consistency of Multiple Channels

1) Gauge invariance

2) Lorentz symmetry

current conservation = Ward-Takahashi identity

qihAi~q �~k2

�~k3�~k4

i =4X

a=2

eah�~ka+~q �~k3�~k4

i

Conformal invariance of the correlator implies Lorentz invariance of the total energy singularity. Neither is automatic!

This allows us to construct the correlators for the s, t and u-channels.But, these channels are not physical (like Feynman diagrams).

The sum of all channels is constrained by

DB, Duaso Pueyo, Joyce, Lee and Pimentel [2020]

Consistency of Multiple Channels

• For spin 1, consistency requires the structure of Yang-Mills theory:

GiA �a Gj

B �b

TAac TB

cb TAbc TB

ca fABC TCab

[TA, TB ]ab = fABCTCab

• For spin 2, it requires the equivalence principle.

++

Benincasa and Cachazo [2007]Schuster and Toro [2008]

McGady and Rodina [2013]Arkani-Hamed, Huang and Huang [2017]

Applications to Inflation

��

�� ��

�̄(t)

Our results for the scalar and tensor four-point functions in de Sitter space are related in a simple way to three-point functions in slow-roll inflation:

Creminelli [2003]Ghosh, Kundu, Raju and Trivedi [2014]Arkani-Hamed and Maldacena [2015]

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

Applications to Inflation

• The bootstrap perspective unifies many classic results in the literature:

• It also provides a systematic classification of the effects of new massive particles during inflation.

Maldacena [2002]

Creminelli [2003]Seery, Sloth, Vernizzi [2008]

Arroja and Koyama [2008]

hµ⌫

hµ⌫hµ⌫

Maldacena [2002]

(@�)4

III. Future Directions

Amplitudes Meet Cosmology

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:

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 are hidden simplicity of cosmological correlators?

• Towards UV 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?

• Can we relax our symmetry assumptions?• What are the remaining bootstrap constraints?

• Beyond Conformal Symmetry

Thank you for your attention!

A Simple Example

• The factorisation limits of the s-channel are

• The unique solution that is consistent with these limits is

hJ�J�is =(~⇠1 · ~k2)(~⇠3 · ~k4)

ELERE

• The total energy singularity has the correct residue. ELERE!0����! S

=~⇠1 · ~k2EL

~⇠3 · ~k4ER(k34 � s)

=~⇠3 · ~k4ER

~⇠1 · ~k2EL(k12 � s)

limEL!0

limER!0

EL ⌘ k12 + s

ER ⌘ k34 + s

E ⌘ k12 + k34

sk1

k2k3

k4

Consider Compton scattering in de Sitter space.

A More Complicated Example

• The solution in the s-channel isfixed by factorisation

fixed by total energy singularity

fixed by conformal symmetry

= (~⇠1 · ~k2)2(~⇠3 · ~k4)2

1

E2LE

2R

✓2sk1k3E2

+2k1k3 + ELk3 + ERk1

E

◆

1

ELER

✓2k1k3E3

+k13E2

+1

E

◆�

Consider Compton scattering of gravitons.

=1

E2LE

2R

✓2k1k3E2

+EL

E

◆N (~⇠1, ~⇠3,~k2,~k4)

+1

ELER

✓2k1k3E3

+k13E2

+1

E

◆M(~⇠1, ~⇠3,~k2,~k4)

= (~⇠1 · ~k2)2(~⇠3 · ~k4)2

1

E2LE

2R

✓2sk1k3E2

+2k1k3 + ELk3 + ERk1

E

◆

1

ELER

✓2k1k3E3

+k13E2

+1

E

◆�

A More Complicated Example

• The solution in the s-channel is

• The solution in the u-channel is

fixed by total energy singularity

fixed by conformal symmetry

fixed by factorisation

Consider Compton scattering of gravitons.