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 CMB temperature fluctuations ⇄ fluctuations of expansion history ← NG modes for broken time trans. 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 δ T T ∼ 10 − 5 during inflation, φ ( x ) = ¯ φ ( t + π )

Non-Gaussianities from Baumann’s talk at Strings 2016 non-Gaussianities may probe interactions during inflation! # current bound on 3pt functions our window! 3pt and higher pt correlations non-Gaussian properties: [Chen-Wang ’09, Baumann-Green ’11, TN-Yamaguchi-Yokoyama ’12, ArkaniHamed-Maldacena ’15, …] cf. cosmological collider physics = new probe of high energy physics π π π F NL = h πππ i h ππ i 3 / 2 10 − 3 ∗ 10 − 7 1 F NL ruled out by gravitational ! non-perturbative Planck floor

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

EFT of inflation [Cheung et al ’08] → nonrelativistic matter theory on de Sitter space general action for NG boson π for broken time translation (diffs): nonlinear realization order parameter - generically nonrelativistic action - no significant interaction when - higher derivative terms: heavy field effects, loop effects, ... gravitational effects are negligible for the study of non-Gaussianities ∞ M 4 π + ( ∂ µ π ) 2 ⇤ n + . . . H ( ∂ µ π ) 2 + Pl ˙ X L π = M 2 ⇥ − 2 ˙ n n ! n =2 M n = 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( ∂ µ φ ) 2 + α Λ 4 ( ∂ µ φ ) 4 + . . . sign of α can be constrained by unitarity & causality: α ≥ 0

IR consistency on flat space [Adams et al ’06] 1. unitarity constraint # optical theorem → positivity of Im [forward scattering] Im consider a scalar field with a shift symmetry φ → φ + const L = − 1 2( ∂ µ φ ) 2 + α Λ 4 ( ∂ µ φ ) 4 + . . . sign of α can be constrained by unitarity & causality: α ≥ 0 2 X ≥ 0 n = n

IR consistency on flat space [Adams et al ’06] # optical theorem → positivity of Im [forward scattering] ※ s: Mandelstam variable. s 0 : mass of lightest intermediate particle 1. unitarity constraint # analyticity relates of scattering amplitudes imply Im Im ※ integral contour is around s = 0 consider a scalar field with a shift symmetry φ → φ + const L = − 1 2( ∂ µ φ ) 2 + α Λ 4 ( ∂ µ φ ) 4 + . . . sign of α can be constrained by unitarity & causality: α ≥ 0 2 X ≥ 0 n = n Z ∞ ds 2 ds I 1 ≥ 0 = s 3 s 3 2 π i π s 0

IR consistency on flat space [Adams et al ’06] ※ s: Mandelstam variable. s 0 : mass of lightest intermediate particle using low energy EFT 1. unitarity constraint Im # analyticity relates of scattering amplitudes imply Im ※ integral contour is around s = 0 # optical theorem → positivity of Im [forward scattering] consider a scalar field with a shift symmetry φ → φ + const L = − 1 2( ∂ µ φ ) 2 + α Λ 4 ( ∂ µ φ ) 4 + . . . sign of α can be constrained by unitarity & causality: α ≥ 0 2 = 4 α s 2 + O ( s 3 ) X ≥ 0 n = → α is constrained as α ≥ 0 n Z ∞ ds 2 ds I 1 ≥ 0 = s 3 s 3 2 π i π s 0

IR consistency on flat space [Adams et al ’06] 2. causality constraint (subluminality) → has a non-relativistic dispersion with consider a scalar field with a shift symmetry φ → φ + const L = − 1 2( ∂ µ φ ) 2 + α Λ 4 ( ∂ µ φ ) 4 + . . . sign of α can be constrained by unitarity & causality: α ≥ 0 consider fluctuations around time-dep. background φ ( x ) = ¯ φ ( t ) + ϕ ( x ) ϕ ˙ ¯ φ 2 � ˙ ϕ 2 − c 2 s ( ∂ i ϕ ) 2 ⇤ ⇥ ¯ c 2 φ 4 � ˙ L ϕ 2 ∝ s = 1 − 2 α Λ 4 + O

IR consistency on flat space [Adams et al ’06] with → has a non-relativistic dispersion 2. causality constraint (subluminality) consider a scalar field with a shift symmetry φ → φ + const L = − 1 2( ∂ µ φ ) 2 + α Λ 4 ( ∂ µ φ ) 4 + . . . sign of α can be constrained by unitarity & causality: α ≥ 0 consider fluctuations around time-dep. background φ ( x ) = ¯ φ ( t ) + ϕ ( x ) ϕ ˙ ¯ φ 2 � ˙ ϕ 2 − c 2 s ( ∂ i ϕ ) 2 ⇤ ⇥ ¯ c 2 φ 4 � ˙ L ϕ 2 ∝ s = 1 − 2 α Λ 4 + O ※ causality (subluminal propagations): c 2 s ≤ 1 ↔ α ≥ 0

IR consistency on flat space [Adams et al ’06] → has a non-relativistic dispersion with 2. causality constraint (subluminality) in this simplest setup, consider a scalar field with a shift symmetry φ → φ + const L = − 1 2( ∂ µ φ ) 2 + α Λ 4 ( ∂ µ φ ) 4 + . . . sign of α can be constrained by unitarity & causality: α ≥ 0 consider fluctuations around time-dep. background φ ( x ) = ¯ φ ( t ) + ϕ ( x ) ϕ ˙ ¯ φ 2 � ˙ ϕ 2 − c 2 s ( ∂ i ϕ ) 2 ⇤ ⇥ ¯ c 2 φ 4 � ˙ L ϕ 2 ∝ s = 1 − 2 α Λ 4 + O ※ causality (subluminal propagations): c 2 s ≤ 1 ↔ α ≥ 0 both the unitarity and causality require α ≥ 0

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 future boundary (end of inflation) time - conformal symmetry on future b.d. cf. AdS/CFT - inflation breaks dS symmetry special conf. is spontaneously broken symmetry of the problem: � h π k 1 ( τ ) π k 2 ( τ ) π k 3 ( τ ) π k 4 ( τ ) i � τ → 0 τ = 0

what is the analogue of optical theorem in cosmology?

unitarity constraints on dS let us consider 4pt functions # dS analogue of forward scattering h φ k 1 φ k 2 φ k 3 φ k 4 i k 1 k 4 k 1 k 4 k 2 k 3 k 1 + k 2 → 0 k 2 k 3 k 1 = k 4

unitarity constraints on dS let us consider 4pt functions let us assume that there exists a complete set of states # dS analogue of optical theorem # dS analogue of forward scattering for inflationary perturbations h φ k 1 φ k 2 φ k 3 φ k 4 i k 1 k 4 k 1 k 4 k 2 k 3 k 1 + k 2 → 0 k 2 k 3 k 1 = k 4 X | n ih n | 1 = n X h φ k 1 φ k 2 φ k 3 φ k 4 i = h φ k 1 φ k 2 | n k I ih n k I | φ k 3 φ k 4 i n � 2 � 0 X � � ! � h φ k 1 φ − k 1 | n 0 i n

extend this condition using dilatation & rotation symmetries

extend this condition using dilatation & rotation symmetries

for inflationary perturbations unitarity constraints on dS let us assume that there exists a complete set of states # dS analogue of optical theorem # dS analogue of forward scattering let us consider 4pt functions h φ k 1 φ k 2 φ k 3 φ k 4 i k 1 k 4 k 1 k 4 k 2 k 3 k 1 + k 2 → 0 k 2 k 3 k 1 = k 4 X | n ih n | 1 = n X h φ k 1 φ k 2 φ k 3 φ k 4 i = h φ k 1 φ k 2 | n k I ih n k I | φ k 3 φ k 4 i n � 2 � 0 X � � ! � h φ k 1 φ − k 1 | n 0 i n

unitarity constraints on dS # dS analogue of optical theorem for inflationary perturbations let us consider 4pt functions # dS analogue of forward scattering let us assume that there exists a complete set of states h φ k 1 φ k 2 φ k 3 φ k 4 i k 1 k 4 k 1 k 4 k 2 k 3 k 1 + k 2 → 0 k 2 k 3 k 1 ∝ k 4 X | n ih n | 1 = n X h φ k 1 φ k 2 φ k 3 φ k 4 i = h φ k 1 φ k 2 | n k I ih n k I | φ k 3 φ k 4 i n � 2 � 0 X � � ! � h φ k 1 φ − k 1 | n 0 i n

unitarity constraints on dS # dS analogue of optical theorem let us consider 4pt functions # dS analogue of forward scattering for inflationary perturbations let us assume that there exists a complete set of states h φ k 1 φ k 2 φ k 3 φ k 4 i k 1 k 4 k 1 k 4 k 2 k 3 k 1 + k 2 → 0 k 2 k 3 k 1 ∝ k 4 X | n ih n | 1 = n ⌘ ∗ ⇣ X h φ k 1 φ k 2 φ k 3 φ k 4 i ! h φ k 1 φ − k 1 | n 0 i h φ k 3 φ − k 3 | n 0 i n which is positive if h φ k 1 φ − k 1 | n 0 i = ( k 1 /k 3 ) real# h φ k 3 φ − k 3 | n 0 i