Unitarity constraints on EFT of inflation Jockey Club Institute for - - PowerPoint PPT Presentation

8th Aug 2016 @ YITP workshop

Unitarity constraints on EFT of inflation

Jockey Club Institute for Advanced Study Hong Kong U. of Science & Technology ( → Kobe Univ. from October)

Toshifumi Noumi

based on work in progress with Gary Shiu

• 1. Introduction

inflation: accelerated expansion of early universe

• generates primordial curvature fluctuations

→ seeds of the structure in the universe

Primordial perturbations

CMB as seen by Planck

δT T ∼ 10−5

CMB temperature fluctuations ⇄ fluctuations of expansion history ← NG modes for broken time trans. during inflation, φ(x) = ¯

φ(t + π)

we are looking for small deviation from that such as tensor modes (graviton) and non-Gaussian features π has an approximately scale invariant & Gaussian distribution

Non-Gaussianities

non-Gaussianities may probe interactions during inflation!

• cf. cosmological collider physics = new probe of high energy physics

[Chen-Wang ’09, Baumann-Green ’11, TN-Yamaguchi-Yokoyama ’12, ArkaniHamed-Maldacena ’15, …] non-Gaussian properties: 3pt and higher pt correlations

π π π

from Baumann’s talk at Strings 2016

# current bound on 3pt functions

non-perturbative gravitational ! floor

FNL

1 10−7

ruled out by Planck

10−3∗

FNL = hπππi hππi3/2

• ur window!

EFT of inflation is a model-insensitive framework particularly useful for the study of non-Gaussianities

EFT of inflation [Cheung et al ’08]

Lπ = M 2

Pl ˙

H(∂µπ)2 +

∞

X

n=2

M 4

n

n! ⇥ −2 ˙ π + (∂µπ)2⇤n + . . .

• rder parameter

nonlinear realization general action for NG boson π for broken time translation (diffs): gravitational effects are negligible for the study of non-Gaussianities → nonrelativistic matter theory on de Sitter space

• generically nonrelativistic action
• no significant interaction when
• higher derivative terms: heavy field effects, loop effects, ...

Mn = 0

we know that theoretical consistency such as causality, unitarity, renormalizability etc is useful to connect UV theory and IR EFT IR EFT UV theory

constraints how to UV complete

(W bosons, higher spin fields, strings, ... )

• cf. talk by Huang

(subluminality, positive energy, weak gravity, ... )

in this talk, I discuss theoretical constraints on EFT of inflation based on unitarity of the inflationary perturbations IR EFT UV theory

constraints unitary EFT of inflation

Plan of my talk:

• 1. Introduction
• 2. IR consistency (review of flat space & dS extension)
• 3. constraints on EFT of inflation
• 4. Summary and discussion

✔

• 2. IR consistency
• review of unitarity & causality on flat space

[Adams-Arkani Hamed-Dubovsky-Nicolis-Rattazzi ’06]

• positivity constraints on de Sitter space

IR consistency on flat space [Adams et al ’06]

consider a scalar field with a shift symmetry φ → φ + const

L = −1 2(∂µφ)2 + α Λ4 (∂µφ)4 + . . .

sign of α can be constrained by unitarity & causality:α ≥ 0

IR consistency on flat space [Adams et al ’06]

• 1. unitarity constraint

consider a scalar field with a shift symmetry φ → φ + const

L = −1 2(∂µφ)2 + α Λ4 (∂µφ)4 + . . .

sign of α can be constrained by unitarity & causality:α ≥ 0

# optical theorem → positivity of Im [forward scattering]

X

=

Im

≥ 0

2

n n

IR consistency on flat space [Adams et al ’06]

• 1. unitarity constraint

consider a scalar field with a shift symmetry φ → φ + const

L = −1 2(∂µφ)2 + α Λ4 (∂µφ)4 + . . .

sign of α can be constrained by unitarity & causality:α ≥ 0

# optical theorem → positivity of Im [forward scattering]

X

=

Im

≥ 0

2

n n

# analyticity relates of scattering amplitudes imply

Im

=

I ds 2πi 1 s3 2 π Z ∞

s0

ds s3

≥ 0

※ s: Mandelstam variable. s0: mass of lightest intermediate particle ※ integral contour is around s = 0

IR consistency on flat space [Adams et al ’06]

• 1. unitarity constraint

consider a scalar field with a shift symmetry φ → φ + const

L = −1 2(∂µφ)2 + α Λ4 (∂µφ)4 + . . .

sign of α can be constrained by unitarity & causality:α ≥ 0

# optical theorem → positivity of Im [forward scattering]

X

=

Im

≥ 0

2

n n

# analyticity relates of scattering amplitudes imply

Im

=

I ds 2πi 1 s3 2 π Z ∞

s0

ds s3

≥ 0

※ s: Mandelstam variable. s0: mass of lightest intermediate particle ※ integral contour is around s = 0

using low energy EFT

= 4αs2 + O(s3)

→ α is constrained as α ≥ 0

IR consistency on flat space [Adams et al ’06]

• 2. causality constraint (subluminality)

consider a scalar field with a shift symmetry φ → φ + const

L = −1 2(∂µφ)2 + α Λ4 (∂µφ)4 + . . .

sign of α can be constrained by unitarity & causality:α ≥ 0

consider fluctuations around time-dep. background φ(x) = ¯

φ(t) + ϕ(x)

→ has a non-relativistic dispersion

ϕ

Lϕ2 ∝ ⇥ ˙ ϕ2 − c2

s(∂iϕ)2⇤

with

c2

s = 1 − 2α

˙ ¯ φ2 Λ4 + O ˙ ¯ φ4

IR consistency on flat space [Adams et al ’06]

• 2. causality constraint (subluminality)

consider a scalar field with a shift symmetry φ → φ + const

L = −1 2(∂µφ)2 + α Λ4 (∂µφ)4 + . . .

sign of α can be constrained by unitarity & causality:α ≥ 0

consider fluctuations around time-dep. background φ(x) = ¯

φ(t) + ϕ(x)

→ has a non-relativistic dispersion

ϕ

Lϕ2 ∝ ⇥ ˙ ϕ2 − c2

s(∂iϕ)2⇤

with

c2

s = 1 − 2α

˙ ¯ φ2 Λ4 + O ˙ ¯ φ4

※ causality (subluminal propagations): c2

s ≤ 1 ↔ α ≥ 0

IR consistency on flat space [Adams et al ’06]

• 2. causality constraint (subluminality)

consider a scalar field with a shift symmetry φ → φ + const

L = −1 2(∂µφ)2 + α Λ4 (∂µφ)4 + . . .

sign of α can be constrained by unitarity & causality:α ≥ 0

consider fluctuations around time-dep. background φ(x) = ¯

φ(t) + ϕ(x)

→ has a non-relativistic dispersion

ϕ

Lϕ2 ∝ ⇥ ˙ ϕ2 − c2

s(∂iϕ)2⇤

with

c2

s = 1 − 2α

˙ ¯ φ2 Λ4 + O ˙ ¯ φ4

※ causality (subluminal propagations): c2

s ≤ 1 ↔ α ≥ 0

both the unitarity and causality require α ≥ 0 in this simplest setup,

would like to argue similar constraints on EFT of inflation

• unitarity constraints vs causality condition?
• constraints on primordial non-Gaussianities?

a class of positivity conditions on dS late time correlators based on unitarity and de Sitter space symmetry

• cf. constraints from analyticity of non-relativistic scattering

in the flat space limit [Baumann et al 15’]

de Sitter late time correlators

late time correlators = initial conditions of standard cosmology

hπk1(τ)πk2(τ)πk3(τ)πk4(τ)i

• τ→0

future boundary (end of inflation) time

τ = 0

• conformal symmetry on future b.d.
• cf. AdS/CFT
• inflation breaks dS symmetry

special conf. is spontaneously broken

symmetry of the problem:

what is the analogue of optical theorem in cosmology?

unitarity constraints on dS

hφk1φk2φk3φk4i

let us consider 4pt functions

# dS analogue of forward scattering

k4 k3

k2

k1 k1 k4

k3

k2

k1 = k4 k1 + k2 → 0

unitarity constraints on dS

hφk1φk2φk3φk4i = X

n

hφk1φk2|nkIihnkI|φk3φk4i ! X

n

• hφk1φ−k1|n0i
• 2 0

hφk1φk2φk3φk4i

let us consider 4pt functions

# dS analogue of forward scattering

k4 k3

k2

k1 k1 k4

k3

k2

k1 = k4 k1 + k2 → 0

1 = X

n

|nihn|

# dS analogue of optical theorem let us assume that there exists a complete set of states for inflationary perturbations

extend this condition using dilatation & rotation symmetries

extend this condition using dilatation & rotation symmetries

unitarity constraints on dS

hφk1φk2φk3φk4i = X

n

hφk1φk2|nkIihnkI|φk3φk4i ! X

n

• hφk1φ−k1|n0i
• 2 0

hφk1φk2φk3φk4i

let us consider 4pt functions

# dS analogue of forward scattering

1 = X

n

|nihn|

# dS analogue of optical theorem let us assume that there exists a complete set of states for inflationary perturbations

k4

k3

k2 k1

k1 + k2 → 0 k1 = k4

k1 k4

k3

k2

unitarity constraints on dS

hφk1φk2φk3φk4i = X

n

hφk1φk2|nkIihnkI|φk3φk4i ! X

n

• hφk1φ−k1|n0i
• 2 0

hφk1φk2φk3φk4i

let us consider 4pt functions

# dS analogue of forward scattering

1 = X

n

|nihn|

# dS analogue of optical theorem let us assume that there exists a complete set of states for inflationary perturbations

k4

k3

k2 k1 k1 k4 k3 k2

k1 + k2 → 0 k1 ∝ k4

unitarity constraints on dS

hφk1φk2φk3φk4i

let us consider 4pt functions

# dS analogue of forward scattering

1 = X

n

|nihn|

# dS analogue of optical theorem let us assume that there exists a complete set of states for inflationary perturbations

k4

k3

k2 k1 k1 k4 k3 k2

k1 + k2 → 0 k1 ∝ k4

hφk1φk2φk3φk4i ! X

n

hφk1φ−k1|n0i ⇣ hφk3φ−k3|n0i ⌘∗

which is positive if hφk1φ−k1|n0i = (k1/k3)real#hφk3φ−k3|n0i

unitarity constraints on dS

hφk1φk2φk3φk4i

let us consider 4pt functions

# dS analogue of forward scattering

1 = X

n

|nihn|

# dS analogue of optical theorem let us assume that there exists a complete set of states for inflationary perturbations

k4

k3

k2 k1 k1 k4 k3 k2

k1 + k2 → 0 k1 ∝ k4

hφk1φk2φk3φk4i ! X

n

hφk1φ−k1|n0i ⇣ hφk3φ−k3|n0i ⌘∗

which is positive if hφk1φ−k1|n0i = (k1/k3)real#hφk3φ−k3|n0i generically holds for inflation thanks to dilatation symmetry

extend this condition using dilatation & rotation symmetries

recall partial wave unitarity of scattering amplitudes non-forward scatterings → angular dependence

partial wave expansions (flat space)

X

× =

Im

θ

(s, m) (s, m)

X

s

αsPs(cos θ) βseimψP m

s (cos γ)

β⇤

seimψ0P m s (cos γ0)

rotational symmetry → Legendre polynomial expansion

: relative angles between ingoing momenta and z-axis

(γ, ψ)

: spin and helicty (angular momentum around z-axis)

(s, m)

partial wave expansions (flat space)

X

× =

Im

θ

(s, m) (s, m)

X

s

αsPs(cos θ) βseimψP m

s (cos γ)

β⇤

seimψ0P m s (cos γ0)

# summation over the helicity m in r.h.s. gives

s

X

m=s

eim(ψψ0)P m

s (cos γ)P m s (cos γ0) = Ps(cos θ)

spin s states

→ each Legendre coefficient of l.h.s. is positive αs =

X |βs|2 ≥ 0

rotational symmetry → Legendre polynomial expansion

: relative angles between ingoing momenta and z-axis

(γ, ψ)

: spin and helicty (angular momentum around z-axis)

(s, m)

partial wave decomposition (de Sitter space)

hφk1φk2φk3φk4i

# dS analogue of non-forward scattering → collapsed limit momentum configuration (with general angle) ※ characterized by the ratio and the angle

k1/k3 θ

k4

k3

k2 k1 k1 k4 k3 k2

θ

partial wave decomposition (de Sitter space)

hφk1φk2φk3φk4i

# dS analogue of non-forward scattering → collapsed limit momentum configuration (with general angle) as long as the s-channel expansion is valid [in progress] similar argument to the flat space case + dilatation symmetry → Legendre coefficient of collapsed limit 4pt func. is positive with αs ≥ 0

hφk1φk2φk3φk4i = X

s

αs(k1/k3)Ps(cos θ)

※ characterized by the ratio and the angle

k1/k3 θ

k4

k3

k2 k1 k1 k4 k3 k2

θ

Plan of my talk:

• 1. Introduction
• 2. IR consistency (review of flat space & dS extension)
• 3. constraints on EFT of inflation
• 4. Summary and discussion

✔ ✔

• 3. Constraints on EFT of inflation

EFT of inflation [Cheung et al ’08]

# 3pt interactions from and

M2 M3

# 4pt interactions from , and M4

M2 M3

# non-relativistic dispersion relation when M2 6= 0 sound speed (propagation speed): c2

s =

−M 2

Pl ˙

H −M 2

Pl ˙

H + 2M 4

2

※ subluminal condition is 0 ≤ c2

s ≤ 1

general action for NG boson π for broken time translation (diffs):

Lπ = M 2

Pl ˙

H(∂µπ)2 +

∞

X

n=2

M 4

n

n! ⇥ −2 ˙ π + (∂µπ)2⇤n + . . .

• rder parameter

nonlinear realization etc

∂2π

M4 M2 M3

positivity of 4pt functions → constraints on , and we use dimensionless parameters , and defined by

cs c3 c4

c2

s =

−M 2

Pl ˙

H −M 2

Pl ˙

H + 2M 4

2

cn = c2(n−1)

s

M 4

n

−2M 2

Pl ˙

H

, ※ corresponds to strong coupling

c3, c4 & 1

constraints from positivity

hφk1φk2φk3φk4i =

+

c3, c2

3, c4

c3, c4 dependence:

generically, the positivity condition takes the form

c4 ≥ a function of cs and c3

• ex. for dS analogue of forward scattering,

k1 k4 k3 k2

qualitatively the same as what [Baumann et al 15’] obtained using analyticity of non-relativistic scatterings c4 ≥ 17 3 c2

3 +

✓22 3 − 187 3 c2 + 102c2

2

◆ c3 + ✓ −86 3 c2 + 1105 6 c2

2 − 314c3 2 + 306c4 2

◆

c3, c2

3

more interesting constraints appear in the limit k1 ⌧ k3

constraints from positivity

→ (positive factor) × 3 8

• 1 − c4

s

• + 5

18

• 1 − c2

s

• − c3 + 5

6

• 1 − c2

s

• P2(cos θ)
• hφk1φk2φk3φk4i

k1 k4 k3 k2

θ

k1 ⌧ k3

with

• 1. contribution is higher oder in this limit
• 2. no angular dependence from c3

c4

constraints from positivity

hφk1φk2φk3φk4i

k1 k4 k3 k2

θ

k1 ⌧ k3

with

• 1. contribution is higher oder in this limit
• 2. no angular dependence from

c4

5 6

• 1 − c2

s

• P2(cos θ)
• c3

→ (positive factor) × 3 8

• 1 − c4

s

• + 5

18

• 1 − c2

s

• − c3 +

θ)

• positivity of → subluminality

P2(cos θ) c2

s ≤ 1

extension of [Adams-Arkani Hamed-Dubovsky-Nicolis-Rattazzi ’06] to

• finite sound speed deviation ∆cs = 1 − cs
• de Sitter space (inflation)

constraints from positivity

→ (positive factor) × 3 8

• 1 − c4

s

• + 5

18

• 1 − c2

s

• − c3 + 5

6

• 1 − c2

s

• P2(cos θ)
• hφk1φk2φk3φk4i

k1 k4 k3 k2

θ

k1 ⌧ k3

with

• 1. contribution is higher oder in this limit
• 2. no angular dependence from c3

c4

can be used as a constraint on 3pt functions!

※ experimentally, 3pt functions are more accessible

c3 ≤ 3 8

• 1 − c4

s

• + 5

18

• 1 − c2

s

• positivity of →

P0(cos θ) = 1

comparison to current bound by Planck 2015

DBI Planck

• ur theoretical upper bound on c3

from forward type from Legendre coef.

Planck: based on bound on 3pt func.

※ 3pt functions are not detected yet

further constraints may be obtained for concrete models

• DBI inflation is consistent with positivity constraints
• positivity implies when , ...

c3 = 0 cs = 1

Summary and prospects

summary and prospects

# summary

• introduced a class of positivity conditions on dS correlators

based on unitarity of inflationary perturbations & dS symmetry ※ collapsed limit of 4pt functions → analogue of optical theorem

• application to EFT of inflation

→ subluminal condition, constraints on 3pt functions from 4pt, ... ※ the limit and angular dependence are useful

k1 ⌧ k3

• in particular, constraints on 3pt functions are relevant

to the target of ongoing and coming observational experiments

summary and prospects

# prospects

cs = 1

• no higher derivative corrections when ?

conjectured by [Baumann et al ’15]

• connections to & implication from developments in AdS/CFT

(crossing symmetry, bootstrap, Witten diagram, ...)

• effective interactions from (massive) higher spin fields
• cf. string theory as a causal UV completion [Camanho et al ’14]

※ any observables more accessible than graviton 3pt functions?

[Maldacena-Pimentel ’11]

Thank you!