Lecture 3+1: Cosmic Microwave Background

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

Tie-ins to Morning Talk

Predicting the CMB: Nucleosynthesis Light element abundance depends on baryon/photon ratio Existence and temperature of CMB originally predicted (Gamow 1948) by light elements + visible baryons With the CMB photon number density fixed by the temperature light elements imply dark baryons Burles, Nollett, Turner (1999)

Spectrum FIRAS Spectrum Perfect Blackbody chemical potential is zero (re: morning talk) 12 GHz 200 400 600 10 B ν ( 10 5 ) 8 6 4 error 50 2 0 5 10 15 20 frequency (cm 1 )

Thermalization Compton upscattering: y distortion - seen in Galaxy clusters Redistribution: µ-distortion 0.1 T / Tinit 0 0.1 0.2 y distortion µ-distortion 10 3 10 2 10 1 1 10 1 10 2 p/t init

Outline Temperature Maps Thermal History Acoustic Oscillations Harmonic Extrema Dark Energy (First Peak) Testing Inflation (Cosmological Defects) Dark Baryons (Second Peak) Dark Matter (Third Peak) Damping Tail Polarization Secondary Anisotropies More info: http://background.uchicago.edu

Temperature Maps

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 imperfect reception

Understanding Maps Our best guess for the original map

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

Recent 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)

Physical Landscape

Thermal History

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

Acoustic Oscillations

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 /4ρ γ and R=3ρ b /4ρ 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

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...

Extrema=Peaks First peak = mode that just compresses Second peak = mode that compresses then rarefies: twice the wavenumber Recombination Recombination T/T Θ+Ψ First Peak T/T Θ+Ψ time time Ψ /3 k 1 =π/ sound horizon Ψ /3 k 2 =2k 1 Second Peak

Extrema=Peaks First peak = mode that just compresses Second peak = mode that compresses then rarefies: twice the wavenumber Harmonic peaks: 1:2:3 in wavenumber Recombination Recombination T/T Θ+Ψ First Peak T/T Θ+Ψ time time Ψ /3 k 1 =π/ sound horizon Ψ /3 k 2 =2k 1 Second Peak

Acoustic Landscape DT (mk) 100 80 60 40 20 FIRS COBE Ten Viper BAM QMAP SP BOOM IAB Sask 1 st extrema TOCO compression ARGO IAC MAX MSAM Pyth compression RING CAT BOOM 3 rd extrema 2WD nd extrema rarefaction Maxima OVRO ATCA SuZIE BIMA W. Hu 11/00 10 100 1000 l

Dark Energy

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

Score Card 100 IAB Sask 80 Viper BAM TOCO 1st critical density DT (mk) 60 40 20 FIRS COBE RING CAT QMAP SP MAX MSAM Ten IAC BOOM Pyth BOoM WD Maxima OVRO ATCA BIMA W. Hu 11/00 10 100 1000 l

1:2:3 Testing Inflation

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

Inflation and the Initial Conditions Inflation: (nearly) scale-invariant curvature (potential) perturbations Superluminal expansion superhorizon scales "initial conditions" Accompanying temperture perturbations due to cosmological redshift cold Time Newtonian Comoving hot Space Potential perturbation Ψ = time-time metric perturbation δt/t = Ψ δt/t = δa/a = 2/3δt/t = 2/3Ψ Sachs & Wolfe (1967); White & Hu (1997)

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.

Inflationary Dynamics Any quantum field during inflation acquires quantum fluctuations Inflaton is only special in that it carries the energy density Gravitational waves also acquire quantum fluctuations Power in quantum fluctuations H 2 V E 4 Power in gravity waves measures energy scale of inflation Current upper limits: E < 2 x 10 16 GeV Joint constraint with scalar slope distinguishes classes of inflationary models Tensor/Scalar power vs tensor slope tests the idea of slow-roll inflation itself Meaningful test by Planck only possible if tensors close to current upper limit

Score Card 100 IAB Sask 80 Viper BAM TOCO 1st critical density DT (mk) 60 40 20 FIRS COBE RING CAT QMAP SP MAX MSAM Ten IAC BOOM Pyth BOoM 1:2:3 WD inflation Maxima OVRO ATCA BIMA W. Hu 11/00 10 100 1000 l

Dark Baryons

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)

Baryon & Inertia Baryons add inertia to the fluid Equivalent to adding mass on a spring Same initial conditions Same null in fluctuations Unequal amplitudes of extrema

A Baryon-meter Low baryons: symmetric compressions and rarefactions T time Low Baryons

A Baryon-meter Load the fluid adding to gravitational force Enhance compressional peaks (odd) over rarefaction peaks (even) T time Baryon Loading

A Baryon-meter Enhance compressional peaks (odd) over rarefaction peaks (even) e.g. relative suppression of second peak T time

Baryons in the Power Spectrum

Second Peak Detected 80 DT (mk) 60 40 l 2 ~550 20 Boom98 CBI Maxima-1 DASI 500 1000 1500 l (multipole)

Score Card 100 IAB Sask 80 Viper BAM TOCO 1st critical density DT (mk) 60 40 20 FIRS COBE RING CAT QMAP SP MAX MSAM Ten ARGO IAC BOOM Pyth 2nd BOOM dark WD baryons Maxima OVRO ATCA SuZIE BIMA W. Hu 11/00 10 100 1000 l

Dark Matter

Oscillator: Take Three and a Half The not-quite-so simple harmonic oscillator equation is a forced harmonic oscillator 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) changes in the gravitational potentials alter the form of the acoustic oscillations If the forcing term has a temporal structure that is related to the frequency of the oscillation, this becomes a driven harmonic oscillator

Relativistic Term in Continuity Continuity equation contains relativistic term from changes in the spatial curvature perturbation to the scale factor For w=0 (matter), simply density dilution; for w=1/3 (radiation) density dilution plus (cosmological) redshift a.k.a. ISW effect photon redshift from change in grav. potential

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) )

Radiation and Dark Matter Radiation domination: potential wells created by CMB itself Pressure support potential decay driving Heights measures when dark matter dominates

Driving Effects and Matter/Radiation Potential perturbation: Radiation Potential: k 2 Ψ = 4πGa 2 δρ generated by radiation inside sound horizon δρ/ρ pressure supported δρ hence Ψ decays with expansion Ψ T/T η Θ+Ψ Hu & Sugiyama (1995)

Driving Effects and Matter/Radiation Potential perturbation: Radiation Potential: Potential Radiation: Feedback stops at matter domination k 2 Ψ = 4πGa 2 δρ generated by radiation inside sound horizon δρ/ρ pressure supported δρ hence Ψ decays with expansion Ψ decay timed to drive oscillation 2Ψ + (1/3)Ψ = (5/3)Ψ 5x boost Ψ T/T Θ+Ψ η Hu & Sugiyama (1995)

Dark Matter in the Power Spectrum

Third Peak Constrained 80 DT (mk) 60 40 l 3 20 Boom98 CBI Maxima-1 DASI 500 1000 1500 l (multipole)

Score Card 100 IAB Sask 80 Viper BAM TOCO 1st critical density DT (mk) 60 40 20 FIRS COBE QMAP SP Ten ARGO IAC BOOM Pyth MAX MSAM RING 3rd CAT dark matter 2nd BOOM dark WD baryons Maxima OVRO ATCA SuZIE BIMA W. Hu 11/00 10 100 1000 l

Damping Tail

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 %, soexpectthe peaks >3to be affected by dissipation

Dissipation / Diffusion Damping Imperfections in the coupled fluid mean free path λ C in the baryons Random walk over diffusion scale: geometric mean of mfp & horizon λ D ~ λ C N ~ λ C η >> λ C Overtake wavelength: λ D ~ λ ; second order in λ C /λ Viscous damping for R<1; heat conduction damping for R>1 N=η / λ C 1.0 λ D ~ λ C N λ Power 0.1 perfect fluid instant decoupling Silk (1968); Hu & Sugiyama (1995); Hu & White (1996) 500 1000 1500 l

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

Dissipation / Diffusion Damping Rapid increase at recombination as mfp Independent of (robust to changes in) perturbation spectrum Robust physical scale for angular diameter distance test (Ω K, Ω Λ ) Recombination 1.0 Power 0.1 perfect fluid instant decoupling recombination Silk (1968); Hu & Sugiyama (1995); Hu & White (1996) 500 1000 1500 l

Damping Length as Ruler Damping scale: physical scale for curvature test Independent of phase of oscillation

Spotless Laundry: Standard Rulers Calibrated the Standard Ruler Sound Horizon 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Baryons Matter/Radiation Damping Scale Cross-Check 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Baryons Matter/Radiation

The Peaks 100 IAB Sask 80 Viper TOCO 1st BAM critical density DT (mk) 60 40 20 FIRS COBE Ten QMAP SP BOOM ARGO IAC MAX MSAM Pyth RING dark matter CAT BOOM 3rd WD 2nd baryonic dark matter Maxima checks OVRO ATCA SuZIE BIMA W. Hu 11/00 10 100 1000 l

Further Implications of Damping CMB anisotropies 10% polarized dissipation viscosity quadrupole quadrupole linear polarization Secondary anisotropies are observable at arcminute scales dissipation exponential suppression of primary anisotropy uncovery of secondary anisotropy from CMB photons traversing the large scale structure of the universe

Polarization

Polarization from Thomson Scattering Differential cross section depends on polarization and angle

Polarization from Thomson Scattering Isotropic radiation scatters into unpolarized radiation

Polarization from Thomson Scattering Quadrupole anisotropies scatter into linear polarization aligned with cold lobe

Isotropization by Scattering Catch-22: polarization generated by Thomson scattering Thomson scattering destroys quadrupole source

Polarization Power Spectrum Quadrupole generation on diffusion scale (optically thin regime) Primary peak coincides with beginning of damping tail l~1000 Secondary peak coincides with horizon scale at reionization l<10 6 0.4 T (µk) 4 0.3 0.2 0.1 < 10% 10 2 10 l 100 1000

Electric & Magnetic Polarization (a.k.a. gradient & curl) Alignment of principal vs polarization axes (curvature matrix vs polarization direction) E B Kamionkowski, Kosowsky, Stebbins (1997) Zaldarriaga & Seljak (1997)

Inflationary B-Modes Inflationary B-modes appear at intermediate scales and test the energy scale of inflation (Seljak & Zaldarriaga 1997; Kamionkowski, Kosowsky, Stebbins 1997) 100 10 reionization QE T (mk) 1 EE BB 0.1 0.01 gravitational waves 10 100 1000 l (multipole) gravitational lensing

Secondary Anisotropies

Gravitational Secondaries 100 DT (mk) 10 1 0.1 ISW lensing moving halo 10 100 1000 l un lensed

Scattering Secondaries 100 DT (mk) 10 1 Doppler suppression density mod linear ion-mod SZ 0.1 10 100 1000 l

Clusters in Power Spectrum? Excess in arcminute scale CMB anisotropy from CBI 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

Sensitivity of SZE Power Amplitude of fluctuations

Dark energy Sensitivity of SZE Power

Summary Precision (accurate!) cosmology has arrived Sound physics seen (pun intended) Consistent with inflationary initial conditions

Summary Precision (accurate!) cosmology has arrived Sound physics seen (pun intended) Consistent with inflationary initial conditions First peak nailed: nearly flat universe Second detected: baryonic dark matter (consistent with Big Bang Nucleosynthesis) Third constrained: cold dark matter required Mysteries at two ends of time dark energy: secondary anisotropies gravitational waves: polarization structure formation and dark matter: secondaries