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Effective Field Theory in Cosmology

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Effective Field Theory in Cosmology 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 Particle Biophysics Gravitation, Astrophysics, Cosmology 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] 8 c = 3 × 10 m=s ≡ 1 −1 Setting −34 we have [L] = [T ] = [E ]: ~ = 10 J · s ≡ 1 −16 −1 −24 −1 A useful conversion is mp ∼ 1 GeV ∼ (10 m) ∼ (10 s) . 2 • Examples of EFTs: - Hydrogen atom 2 2 p α meα Recall that H = − ) En = − 2 + Corrections 2me r n • Proton recoil: O(me=mp) • Fine structure: O(α2) • Weak interactions: O(mp=MW ) - Multipole expansion 1 X al V (r) = c Y (Ω) r lm r lm l;m - High-energy physics Fermi Theory Standard Model String Theory QG + Gravity 3 - Hydrodynamics @ρ @ = − (ρvi) @t @xi @ @ (ρvi)= − Πij @t @xj " 2 Πij = P δ + ρv v + η @ v − δ (@ v ) + O(@2) ij i j (i j) 3 ij k k 2. PRINCIPLES OF EFT I will illustrate the basic principles of EFTs with the following toy model: 1 2 1 2 2 1 4 − (@φ) − m φ − λφ 1 L[φ, Ψ] = 2 2 4! − gφ2Ψ2 1 1 4 − (@Ψ)2 − M 2Ψ2 ; 2 2 where m M and g 1. • Integrating out We can integrate out the heavy fields to get an EFT for the light fields: Z eiSeff [φ] = DΨ eiS[φ,Ψ] : • Matching In practice, the effective action is usually found by matching: Effective theory Full theory φ0 = + ··· φ2 = + + ··· φ4 = + + ··· 4 φ6 = + ··· • Renormalization Heavy fields renormalize the IR couplings g Λ2 ∆m2 = = + Λ2 − M 2 log 32π2 µ2 3g2 Λ2 ∆λ = = − log 32π2 µ2 • Non-renormalizable interactions Heavy fields also add new non-renormalizable interactions: φ6 ∼ g3 M 2 • Decoupling These new higher-dimensional interactions decouple for M ! 1. • 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: 1 1 1 L [φ] = − (@φ)2 − m2 − λ φ4 eff 2 2 R 4! R 1 X ci di − φ4 + 2i + (@φ)2φ2i + ··· : M 2i M 2i i=1 5 Even if the full theory is not known, we can still parameterize the EFT: Wilson coefficient X Oi[φ] Leff[φ] = L0[φ] + ci : cutoff Λδi−4 i operator 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 me. 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 2 1 µν α µν 2 Leff[Aµ] = − FµνF + 4 c1(FµνF ) + ··· ; 4 me where c1 = 1=90. 6 • Rayleigh scattering Consider the scattering of photons off atoms at low energies. Let (x) denote a field operator that creates an atom at the point x. The effective Lagrangian for the atom is @2 L [ ] = y i@ − + L : eff t 2M int At low energies, the dominant interaction with photons is 3 y µν Lint = a0 FµνF " size of the atom The corresponding cross section is 6 4 σ / a0 ! : That is why the sky is blue! • Gravity Like Fermi's theory, Einstein's gravity requires a UV completion. However, at low energies, E Mpl, gravity is described by an EFT: " # p M 2 1 L [g ] = −g pl R + c R2 + c R Rµν + d R3 + ··· + ··· ; eff µν 2 1 2 µν Λ2 1 where Λ . Mpl. • Particle physics The most conservative way to described BSM physics is as an EFT: X Oi Leff[ ; Aµ;H] = LSM + ci : Λδi−4 i - Dim-0: CC problem. - Dim-2: Hierarchy problem. - Dim-5: Neutrino masses. 1 v2 ∆L ∼ (LH)(LH) −−−−!H=v m = ∼ 10−2 eV ; for Λ ∼ 1015 GeV: Λ ν Λ - Dim-6: Proton decay. 1 τ >1033 yrs ∆L ∼ QQQL −−−−−−−!p Λ > 1015 GeV . Λ2 7 • Inflation The most conservative way to described the physics of inflation is as an EFT: " 2 # p Mpl 1 2 X Oi[φ, Ψ] Leff[φ, Ψ; g] = −g R − (@φ) − V0(φ) + ci : 2 2 Λδi−4 i - Dim-6: Eta problem. 2 00 2 φ Λ<M V M ∆V = V −−−−−!pl ∆η ≡ M 2 ≈ pl > 1. 0 Λ2 pl V Λ2 - Dim-8: Non-Gaussianity. 4 2 (@φ) Λ2<φ_ φ_ ∆L = −−−−! f ∼ < 1 . Λ4 NL Λ4 - Dim-1: Lyth bound. r 1=2 ∆φ ∼ M −−−−−!r>0:01 ∆φ > M . 0:01 pl pl 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 1 L = −@ φy@µφ + µ2φyφ − λ(φyφ)2 ; µ 4 which is invariant under the U(1) symmetry φ ! eiβφ. For µ2 > 0, the symmetry is spontaneously broken: 9 Substituting 1 µ φ = p (v + ρ(x)) eiπ(x) ; with v ≡ p ; 2 λ we find 1 p 1 L = − (@ ρ)2 − µ2ρ2 − λµρ3 − λρ4 − (v + ρ)2(@ π)2 : 2 µ 2 µ " massless Goldstone boson Integrating out the massive field ρ, we get an effective Lagrangian for π 1 (@ π )4 L = − (@ π )2 + c µ c + ··· ; π 2 µ c 1 v4 2 2 where πc ≡ vπ and c1 = v /µ . From the bottom up, we can write the effective action of π as a derivative expansion of U(x) ≡ eiπ(x): f 2 L = − π @ U y@µU + c (@ U y@µU)2 + c (@ U y@ U)(@µU y@νU) + ··· ; π 2 µ 1 µ 2 µ ν where fπ is the symmetry breaking scale. If a symmetry G is broken to a subgroup H, we obtain one massless Gold- a stone boson πa for each broken generator T . The effective Lagrangian of the Goldstone bosons is f 2 L = − π Tr[@ U y@µU] + c Tr[(@ U y@µU)2] + ··· ; π 2 µ 1 µ a where U(x) ≡ eiπa(x)T . • Broken gauge symmetries Consider scalar electrodynamics 1 L = −D φyDµφ − V (φ) − F 2 ; µ 4 µν with Dµ = @µ + igAµ. Let 1 φ = p (v + ρ(x)) eiπ(x) : 2 10 and use the gauge symmetry to set π ≡ 0 (unitary gauge). After the SSB, the gauge field has become massive 1 1 L = − F 2 − m2A2 + ··· ; 4 µν 2 µ where m2 = g2v2. The Goldstone boson has become the longitudinal mode of the massive vector field (= Higgs mechanism). • St¨uckelberg trick To understand the behavior of the theory at high energies, it is useful to rein- troduce the Goldstone boson. This is achieved by imposing the following trans- formation on the vector field (i.e. by `undoing' the gauge fixing) @ π i A ! A + µ ≡ UD U y ; µ µ g g µ where U(x) ≡ eiπ(x). The Lagrangian then becomes 1 f 2 L = − F 2 − π D U yDµU; 4 µν 2 µ where fπ ≡ m=g. At quadratic order, this can be written as 1 1 1 L = − F 2 − (@ π )2 − m2A2 + m @ π Aµ ; 2 4 µν 2 µ c 2 µ µ c where πc ≡ fππ. • Decoupling limit µ 2 Because the mixing term @µπcA has one fewer derivative than (@µπc) , we expect it to become irrelevant at high energies. To see this, we take the so-called decoupling limit g ! 0 ; m ! 0 ; for fπ ≡ m=g = const: In this limit, there is no mixing between π and Aµ. 1 For E > Emix = m, the scattering of the longitudinal modes of the gauge fields is therefore described by the scattering of the Goldstone bosons, up to corrections of order m=E and g2 (= Goldstone boson equivalence theorem). 1 2 2 2 2 2 2 For non-Abelian gauge bosons, interactions of the form fππ (@µπ) = πc (@µπc) =fπ arise from expanding 2 y µ the universal kinetic term fπTr[DµU D U], while for Abelian gauge bosons they only arise from the non- universal higher-derivative terms. 11 3. EFT OF INFLATION • Broken time translations Time-dependent matter fields, m(t), break time translation invariance, i.e.
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