EFFECTIVE FIELD THEORY IN COSMOLOGY Daniel Baumann University of - PDF document

Cargese Lectures EFFECTIVE FIELD THEORY IN COSMOLOGY Daniel Baumann University of Amsterdam 1. INTRODUCTION TO EFFECTIVE FIELD THEORY - Motivation - Principles of EFT - Examples of EFTs - Outlook 2. EFT OF INFLATION - Motivation - Spontaneous Symmetry Breaking - EFT of Cosmological Perturbations - Cosmological Collider Physics - Outlook 3. EFT OF LARGE-SCALE STRUCTURE - Motivation - Standard Perturbation Theory - Effective Field Theory Approach - Outlook

Lecture 1. INTRODUCTION TO EFFECTIVE FIELD THEORY 1. MOTIVATION Nature comes with many scales: Planck Cells Observable scale LHC Nuclei Atoms Bacteria us Planets Galaxies universe Quantum Gravitation, Astrophysics, Cosmology Particle Biophysics gravity physics Subatomic Atomic physics, physics Nanoscience Condensed matter physics Science progresses because we can treat one scale at a time. Coarse-graining over short scales (high energies) leads to an effective field theories (EFTs) at long distances (low energies). Even if we don’t know the full microscopic theory, we can parameterize our ignorance as an EFT. Natural units : [see Nima Arkani-Hamed PIRSA/09080035] c = 3 × 10 8 m / s ≡ 1 [ L ] = [ T ] = [ E − 1 ] . Setting we have � = 10 − 34 J · s ≡ 1 A useful conversion is m p ∼ 1 GeV ∼ (10 − 16 m) − 1 ∼ (10 − 24 s) − 1 . 2

• Examples of EFTs : - Hydrogen atom H = p 2 E n = − m e α 2 − α Recall that ⇒ + Corrections n 2 2 m e r • Proton recoil: O ( m e /m p ) • Fine structure: O ( α 2 ) • Weak interactions: O ( m p /M W ) - Multipole expansion V ( r ) = 1 � a � l � c lm Y lm (Ω) r r l,m - High-energy physics Fermi Theory Standard Model String Theory QG + Gravity 3

- Hydrodynamics ∂ρ ∂t = − ∂ ∂x i ( ρv i ) ∂t ( ρv i ) = − ∂ ∂ ∂x j Π ij ↑ � � ∂ ( i v j ) − 2 Π ij = Pδ ij + ρv i v j + η + O ( ∂ 2 ) 3 δ ij ( ∂ k v k ) 2. PRINCIPLES OF EFT I will illustrate the basic principles of EFTs with the following toy model: − 1 2( ∂φ ) 2 − 1 2 m 2 φ 2 − 1 4! λφ 4 − 1 4 gφ 2 Ψ 2 L [ φ, Ψ] = − 1 2( ∂ Ψ) 2 − 1 2 M 2 Ψ 2 , where m ≪ M and g ≪ 1. • Integrating out We can integrate out the heavy fields to get an EFT for the light fields: � e iS eff [ φ ] = D Ψ e iS [ φ, Ψ] . • Matching In practice, the effective action is usually found by matching : Effective theory Full theory φ 0 = + · · · φ 2 = + + · · · φ 4 = + + · · · 4

φ 6 = + · · · • Renormalization Heavy fields renormalize the IR couplings � Λ 2 g � �� ∆ m 2 = Λ 2 − M 2 log = + 32 π 2 µ 2 = − 3 g 2 � Λ 2 � ∆ λ = 32 π 2 log µ 2 • Non-renormalizable interactions Heavy fields also add new non-renormalizable interactions : ∼ g 3 φ 6 M 2 • Decoupling These new higher-dimensional interactions decouple for M → ∞ . • Power counting EFTs are expansions in powers of δ ≡ E/M ≪ 1. Only a finite number of terms are relevant for observations with finite precision. • Effective actions “Everything that is allowed is compulsory.” For example, in the toy model we generate all terms that are consistent with the φ → − φ symmetry of the full theory: L eff [ φ ] = − 1 2( ∂φ ) 2 − 1 R − 1 2 m 2 4! λ R φ 4 � c i ∞ � M 2 i φ 4 + 2 i + d i M 2 i ( ∂φ ) 2 φ 2 i + · · · � − . i =1 5

Even if the full theory is not known, we can still parameterize the EFT: Wilson coefficient O i [ φ ] � L eff [ φ ] = L 0 [ φ ] + c i Λ δ i − 4 . cutoff operator i dimension • EFT approach - Identify the relevant degrees of freedom. - Determine the relevant symmetries. - Write all operators compatible with the symmetries. - Compute observables. - Measure parameters. 3. EXAMPLES OF EFTS • Photon-photon scattering Consider γγ scattering at energies E ≪ m e . The only dynamical degrees of freedom in the EFT are photons. Photons can interact via electron loops: Full theory Effective theory The EFT Lagrangian is 4 F µν F µν + α 2 L eff [ A µ ] = − 1 c 1 ( F µν F µν ) 2 + · · · , m 4 e where c 1 = 1 / 90. 6

• Rayleigh scattering Consider the scattering of photons off atoms at low energies. Let ψ ( x ) denote a field operator that creates an atom at the point x . The effective Lagrangian for the atom is i∂ t − ∂ 2 � � L eff [ ψ ] = ψ † ψ + L int . 2 M At low energies, the dominant interaction with photons is L int = a 3 0 ψ † ψF µν F µν ↑ size of the atom The corresponding cross section is 0 ω 4 . σ ∝ a 6 That is why the sky is blue! • Gravity Like Fermi’s theory, Einstein’s gravity requires a UV completion. However, at low energies, E ≪ M pl , gravity is described by an EFT: � � M 2 L eff [ g µν ] = √− g 2 R + c 1 R 2 + c 2 R µν R µν + 1 d 1 R 3 + · · · pl � � + · · · , Λ 2 where Λ � M pl . • Particle physics The most conservative way to described BSM physics is as an EFT: O i � L eff [ ψ, A µ , H ] = L SM + c i Λ δ i − 4 . i - Dim-0: CC problem. - Dim-2: Hierarchy problem. - Dim-5: Neutrino masses. → m ν = v 2 ∆ L ∼ 1 Λ ∼ 10 − 2 eV , for Λ ∼ 10 15 GeV . H = v Λ( LH )( LH ) − − − − - Dim-6: Proton decay. ∆ L ∼ 1 τ p > 10 33 yrs → Λ > 10 15 GeV . Λ 2 QQQL − − − − − − − 7

• Inflation The most conservative way to described the physics of inflation is as an EFT: � � M 2 L eff [ φ, Ψ , g ] = √− g 2 R − 1 O i [ φ, Ψ] 2( ∂φ ) 2 − V 0 ( φ ) + pl � c i . Λ δ i − 4 i - Dim-6: Eta problem. ≈ M 2 φ 2 V ′′ Λ <M pl pl → ∆ η ≡ M 2 ∆ V = V 0 − − − − − Λ 2 > 1. pl Λ 2 V - Dim-8: Non-Gaussianity. ˙ ∆ L = ( ∂φ ) 4 φ 2 Λ 2 < ˙ φ − − − − → f NL ∼ Λ 4 < 1 . Λ 4 - Dim- ∞ : Lyth bound. � r � 1 / 2 r> 0 . 01 ∆ φ ∼ M pl − − − − − → ∆ φ > M pl . 0 . 01 4. OUTLOOK In the rest of the lectures, I will describe two important EFTs in more detail: 1. EFT of Inflation 2. EFT of Large-Scale Structure References A. Manohar, Introduction to Effective Field Theories , [arXiv:1804.05863] D. Baumann and L. McAllister, Inflation and String Theory , [arXiv:1404.2601] 8

Lecture 2. EFFECTIVE FIELD THEORY OF INFLATION 1. MOTIVATION The origin of structure in the universe is one of the biggest open questions in cosmology: Although there is growing evidence that the primordial fluctuations originated from quantum fluctuations during inflation, the physics of inflation remains a mystery. In this lecture, I will describe inflation as a symmetry breaking phenomenon and derive an effective action for the inflationary perturbations. 2. SPONTANEOUS SYMMETRY BREAKING • Broken global symmetries Consider � µ 2 φ † φ − 1 � L = − ∂ µ φ † ∂ µ φ + 4 λ ( φ † φ ) 2 , which is invariant under the U (1) symmetry φ → e iβ φ . For µ 2 > 0, the symmetry is spontaneously broken: 9

Substituting φ = 1 v ≡ µ 2( v + ρ ( x )) e iπ ( x ) , √ √ with , λ we find √ L = − 1 λµρ 3 − λρ 4 − 1 2( ∂ µ ρ ) 2 − µ 2 ρ 2 − 2( v + ρ ) 2 ( ∂ µ π ) 2 . ↑ massless Goldstone boson Integrating out the massive field ρ , we get an effective Lagrangian for π ( ∂ µ π c ) 4 L π = − 1 2( ∂ µ π c ) 2 + c 1 + · · · , v 4 where π c ≡ vπ and c 1 = v 2 /µ 2 . From the bottom up, we can write the effective action of π as a derivative expansion of U ( x ) ≡ e iπ ( x ) : L π = − f 2 2 ∂ µ U † ∂ µ U + c 1 ( ∂ µ U † ∂ µ U ) 2 + c 2 ( ∂ µ U † ∂ ν U )( ∂ µ U † ∂ ν U ) + · · · , π where f π is the symmetry breaking scale. If a symmetry G is broken to a subgroup H , we obtain one massless Gold- stone boson π a for each broken generator T a . The effective Lagrangian of the Goldstone bosons is L π = − f 2 π 2 Tr[ ∂ µ U † ∂ µ U ] + c 1 Tr[( ∂ µ U † ∂ µ U ) 2 ] + · · · , where U ( x ) ≡ e iπ a ( x ) T a . • Broken gauge symmetries Consider scalar electrodynamics L = − D µ φ † D µ φ − V ( φ ) − 1 4 F 2 µν , with D µ = ∂ µ + igA µ . Let φ = 1 2( v + ρ ( x )) e iπ ( x ) . √ 10