Lecture 3+1: Cosmic Microwave Background

Lecture 3+1: Cosmic Microwave Background Structure Formation and the Dark Sector Wayne Hu Trieste, June 2002

Large Angle Anisotropies Actual Temperature Data Really Isotropic!

Large Angle Anisotropies dipole anisotropy 1 part in 1000

Large Angle Anisotropies 10º 90º anisotropy 1 part in 100000

Understanding Maps COBE's fuzzy vision

Understanding Maps COBE's fuzzy vision

Understanding Maps COBE's imperfect reception

Understanding Maps Our best guess for the original map

Small Angle Anisotropies? <1º anisotropy seeing inside the horizon

New DASI Data 3.5 100µK 100µK

MAP Satellite has Launched!

What MAP Should See Simulated Data http://map.gsfc.nasa.gov

Original Power Spectra of Maps 64º Band Filtered

Ringing in the New Cosmology

Power Spectrum Today 100 IAB Sask T (µk) 80 60 40 20 FIRS COBE Viper BAM QMAP SP Ten ARGO IAC BOOM Pyth TOCO BOOM RING CAT MAX MSAM DASI BOOM WD VSA Maxima OVRO CBI ATCA SuZIE BIMA W. Hu 5/02 10 100 1000 l

Projected MAP Errors 10 1 θ (degrees) 100 80 T (µk) 60 40 20 W. Hu Feb. 1998 10 100 l (multipole)

A Brief Thermal History CMB photons hotter at high redhift z At z~1000, T~3000K: photons ionize hydrogen

Very Brief History CMB g 3 min 3x10 5 yrs 5x10 9 yrs e p n He Nucleo- Synthesis Last Scattering Galaxy Formation

A Brief Thermal History Rapid scattering couples photons and baryons Plasma behaves as perfect fluid

From Lecture 2: Photon-Baryon System Photons and baryons described as a single system. Conservation law for the photon-baryon system [ ] d dη + 3ȧ δρ γb + 3ȧ a a δp γb = (ρ γb + p γb )(kv γb + 3 Φ), [ ] d dη + 4ȧ [(ρ γb + p γb )v γb /k] = δp γb + (ρ + p)ψ, a or with Θ = δρ γ /ρ γ and R = 3ρ b /ρ γ Θ = k 3 v γb Φ v γb = Ṙ 1 + R v γb + 1 kθ + kψ 1 + R Forced oscillator equation see Zaldarriaga s (now my) talk(s) Anisotropic stresses and entropy generation through non-adiabatic stress Silk damps fluctuations Silk 1968

Basics Take a simple photon dominated system, neglecting gravity Continuity Equation: Θ = 1 3 kv γ where Θ = δρ γ /4ρ γ is the temperature fluctuation in the Newtonian gauge expresses number conservation n γ T 3 Euler Equation: v γ = kθ expresses momentum conservation q = F where the force is provided by pressure gradients δp γ = δρ γ /3 = 4Θ/3.

Oscillator: Take One Combine these to form the simple harmonic oscillator equation Θ + c 2 sk 2 Θ = 0 where the adiabatic sound speed is defined through c 2 s ṗγ ρ γ here c 2 s = 1/3 since we are photon-dominated Given an initial temperature perturbation and no velocity perturbation, solution is Θ(η) = Θ(0) cos(ks) where the sound horizon is defined as s c s dη

A Brief Thermal History Redshift-ionization Photon-baryon fluid with pressure

Harmonic Extrema All modes are frozen in at recombination (denoted with a subscript ) yielding temperature perturbations of different amplitude for different modes Θ(η ) = Θ(0) cos(ks ) Modes caught in the extrema of their oscillation will have enhanced fluctuations k n s = nπ yielding a fundamental scale or frequency, related to the inverse sound horizon k A = π/s and a harmonic relationship to the other extrema as 1 : 2 : 3...

Peak Location The fundmental physical scale is translated into a fundamental angular scale by simple projection according to the angular diameter distance D θ A = λ A /D l A = k A D In a flat universe, the distance is simply D = d η 0 η η 0, the horizon distance, and k A = π/s = 3π/η so θ A η η 0 In a matter-dominated universe η a 1/2 so θ A 1/30 2 or l A 200

Peak Locations: Dark Energy? 80 l 1 ~200 DT (mk) 60 40 20 Boom98 CBI Maxima-1 DASI 500 1000 1500 l (multipole)

Curvature In a curved universe, the apparent or angular diameter distance is no longer the conformal distance D = R sin(d/r) d α D d=rθ λ Objects in a closed universe are further than they appear! gravitational lensing of the background...

Curvature in the Power Spectrum Features scale with angular diameter distance Angular location of the first peak

Restoring Gravity Take a simple photon dominated system with gravity Continuity Equation: Θ = 1 3 kv γ Φ includes a redshift effect from the distortion of the spatial metric Recall that in the absence of stress fluctuations Φ =constant and Φ = 0. Euler Equation: v γ = k(θ + Ψ) includes graviational infall generating momentum.

Oscillator: Take Two Combine these to form the simple harmonic oscillator equation Θ + c 2 sk 2 Θ = k2 3 Ψ Φ In a CDM dominated expansion Φ = Ψ = 0. Also for photon domination c 2 s = 1/3 so the oscillator equation becomes Θ + Ψ + c 2 sk 2 (Θ + Ψ) = 0 Solution is just an offset version of the original [Θ + Ψ](η) = [Θ + Ψ](0) cos(ks) Θ + Ψ is also the observed temperature fluctuation since photons lose energy climbing out of gravitational potentials at recombination

Gravitational Ringing Potential wells = inflationary seeds of structure Fluid falls into wells, pressure resists: acoustic oscillations

Sachs-Wolfe Effect and the Magic 1/3 From comoving gauge, where the CMB temperature perturbation is initially negligible, to Newtonian gauge involves a temporal shift δt t = Ψ Temporal shift implies that during matter domination δa a = 2 δt 3 t CMB temperature is cooling as T a so Θ δt T = δa a = 2 3 Ψ Observed or effective temperature Θ + Ψ = 1 3 Ψ

Causality and the CMB The harmonic series of peaks of comparable amplitude, now observed, depend on the existence of an initial spatial curvature fluctuation that is nearly scale-invariant. The Bardeen curvature ζ, from Lecture 2 is constant above the horizon ζ = 0 so that if ζ(0) = 0, ζ(η) = 0. A more severe form of the horizon problem: the mechanism which ensures the homogeneity of the universe must also imprint fluctuation to the spatial curvature that are scale invariant and correlated above the apparent horizon. Inflation to the rescue... Given the observations, restrictions on the yellwould-be alternates to inflation are severe, e.g. cosmological defects cannot be the main source of structure

Baryon Loading Baryons add extra mass to the photon-baryon fluid Controlling parameter is the momentum density ratio: R p b + ρ b p γ + ρ γ 30Ω b h 2 of order unity at recombination ( a 10 3 ) Θ = k 3 v γb Φ v γb = Ṙ 1 + R v γb + 1 kθ + kψ 1 + R Baryons add to the inertial and gravitational mass but not to the radiation pressure

Oscillator: Take Three Combine these to form the not-quite-so simple harmonic oscillator equation c 2 s d dη (c 2 s Θ) + c 2 sk 2 Θ = k2 3 Ψ c2 s d dη (c 2 s Φ) where c 2 s ṗ γb / ρ γb c 2 s = 1 3 1 1 + R In a CDM dominated expansion Φ = Ψ = 0 and the adiabatic approximation Ṙ/R ω = kc s [Θ + (1 + R)Ψ](η) = [Θ + (1 + R)Ψ](0) cos(ks)

Matter-to-radiation ratio Radiation Driving ρ m ρ r 24Ω m h 2 ( a 10 3 of order unity at recombination in a low Ω m universe Radiation is not stress free and so impedes the growth of structure k 2 Φ = 4πGa 2 ρ 4Θ oscillates around a constant value, ρ a 4 so the Netwonian curvature decays. Decay is timed precisely to drive the oscillator 5 the amplitude of the Sachs-Wolfe effect! N.B. analytic solutions to E.O.M. exist but derivation too lengthy to present here see Hu & Sugiyama (1996) )

Damping Tight coupling equations assume a perfect fluid: no viscosity, no heat conduction Fluid imperfections are related to the mean free path of the photons in the baryons λ C = τ 1 where τ = n e σ T a is the conformal opacity to Thompson scattering Dissipation is related to the diffusion length: random walk approximation λ D = Nλ C = η/λ C λ C = ηλ C the geometric mean between the horizon and mean free path λ D /η few %, so expect the peaks 3 to be affected by dissipation

Equations of Motion Continuity Θ = k 3 v γ Φ, δb = kv b 3 Φ where the photon equation remains unchanged and the baryons follow number conservation with ρ b = m b n b Euler v γ = k(θ + Ψ) k 6 π γ τ(v γ v b ) v b = ȧ a v b + kψ + τ(v γ v b )/R where the photons gain an anisotropic stress term π γ from radiation viscosity and a momentum exchange term with the baryons and are compensated by the opposite term in the baryon Euler equation

Oscillator: Final Take Both viscosity and heat conduction through v γ v b are generated by gradients in the velocity field and add terms proportional to Θ to the oscillator equation c 2 s d dη (c 2 s Θ) + k2 c 2 s τ [A v + A h ] Θ + c 2 sk 2 Θ = k2 3 Ψ d c2 s dη (c 2 s where a leading order expansion in k/ τ in kinetic theory gives A v = 16 15, A h = R2 1 + R solution leads to a damping factor to the amplitude of order Φ) or k D exp( k 2 η/ τ) τ/η as expected from the random walk description