Lecture II Polarization Anisotropy Spectrum Wayne Hu Crete, July 2008 Damping Tail Dissipation / Diffusion Damping • Imperfections in the coupled fluid → mean free path λC in the baryons • Random walk over diffusion scale: geometric mean of mfp & horizon λD ~ λC√N ~ √λCη >> λC • Overtake wavelength: λD ~ λ ; second order in λC/λ • Viscous damping for R<1; heat conduction damping for R>1 1.0 N=η / λC Power λD ~ λC√N 0.1 perfect fluid instant decoupling λ 500 1000 1500 Silk (1968); Hu & Sugiyama (1995); Hu & White (1996) l 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 (ΩK, ΩΛ) Recombination 1.0 Power 0.1 perfect fluid instant decoupling recombination 500 1000 1500 Silk (1968); Hu & Sugiyama (1995); Hu & White (1996) l 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 −1 λC = τ_ where τ_ = neσT a is the conformal opacity to Thomson scattering • Dissipation is related to the diffusion length: random walk approximation p p p λD = NλC = η=λC λC = ηλC the geometric mean between the horizon and mean free path • λD/η∗ ∼ few %, so expect the peaks :> 3 to be affected by dissipation Equations of Motion • Continuity k Θ_ = − v − Φ_ ; δ_ = −kv − 3Φ_ 3 γ b b where the photon equation remains unchanged and the baryons follow number conservation with ρb = mbnb • Euler k v_ = k(Θ + Ψ) − π − τ_(v − v ) γ 6 γ γ b a_ v_ = − v + kΨ +τ _(v − v )=R b a b γ b where the photons gain an anisotropic stress term πγ from radiation viscosity and a momentum exchange term with the baryons and are compensated by the opposite term in the baryon Euler equation Viscosity • Viscosity is generated from radiation streaming from hot to cold regions • Expect k π ∼ v γ γ τ_ generated by streaming, suppressed by scattering in a wavelength of the fluctuation. Radiative transfer says k πγ ≈ 2Avvγ τ_ where Av = 16=15 k k v_ = k(Θ + Ψ) − A v γ 3 v τ_ γ Viscosity & Heat Conduction • Both fluid imperfections are related to the gradient of the velocity kvγ by opacity τ˙: slippage of fluids vγ − vb. • Viscosity is an anisotropic stress or quadrupole moment formed by radiation streaming from hot to cold regions m=0 hot v cold hot v Damping & Viscosity • Quadrupole moments: damp acoustic oscillations from fluid viscosity generates polarization from scattering (next lecture) • Rise in polarization power coincides with fall in temperature power – ~ 1000 l Ψ damping πγ polarization driving Θ+Ψ 5 10 15 20 ks/π Oscillator: Penultimate Take • Adiabatic approximation ( ! a=a_ ) k Θ_ ≈ − v 3 γ • Oscillator equation contains a Θ_ damping term d k2c2 k2 d c2 (c−2Θ_ ) + s A Θ_ + k2c2Θ = − Ψ − c2 (c−2Φ_ ) s dη s τ_ v s 3 s dη s • Heat conduction term similar in that it is proportional to vγ and is suppressed by scattering k=τ_. Expansion of Euler equations to leading order in k=τ_ gives R2 A = h 1 + R since the effects are only significant if the baryons are dynamically important Oscillator: Final Take • Final oscillator equation d k2c2 k2 d c2 (c−2Θ_ ) + s [A + A ]Θ_ + k2c2Θ = − Ψ − c2 (c−2Φ_ ) s dη s τ_ v h s 3 s dη s • Solve in the adiabatic approximation Z Θ / exp(i !dη) k2c2 −!2 + s (A + A )i! + k2c2 = 0 τ_ v h s Dispersion Relation • Solve h ! i !2 = k2c2 1 + i (A + A ) s τ_ v h i ! ! = ±kc 1 + (A + A ) s 2 τ_ v h i kc = ±kc 1 ± s (A + A ) s 2 τ_ v h • Exponentiate Z Z 1 c2 exp(i !dη) = e±iks exp[−k2 dη s (A + A )] 2 τ_ v h ±iks 2 = e exp[−(k=kD) ] • Damping is exponential under the scale kD Diffusion Scale • Diffusion wavenumber Z 1 1 16 R2 k−2 = dη + D τ_ 6(1 + R) 15 (1 + R) • Limiting forms Z −2 1 16 1 lim kD = dη R!0 6 15 τ_ Z −2 1 1 lim kD = dη R!1 6 τ_ • Geometric mean between horizon and mean free path as expected from a random walk 2π 2π −1 1=2 λD = ∼ p (ητ_ ) kD 6 Damping Tail Measured 100 80 BOOM VSA K) µ 60 ( Maxima T ∆ 40 DASI CBI 20 W. Hu 9/02 10 100 1000 l Power Spectrum Present Standard Ruler • Damping length is a fixed physical scale given properties at recombination • Gemoetric mean of mean free path and horizon: depends on baryon-photon ratio and matter-radiation ratio Standard Rulers • Calibrating the Standard Rulers • 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 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Baryons Matter/Radiation Consistency Check on Recombinaton 100 CBI K) α: ± µ 10% ( T ∆ 10 10 100 1000 fixed lA, ρb/ργ, ρm/ρr l The Peaks IAB 100 Sask Viper TOCO 1st 80 BAM flat universe RING3rd K) µ 60 dark matter ( CAT QMAP T MAX Mchecksaxima ∆ SP MSAM Ten 40 FIRS OVROATCA ARGOIAC SuZIE BOOM WD BIMA BOOM 2nd 20 COBE Pyth baryonic dark matter W. Hu 11/00 10 100 1000 l The Peaks IAB 100 Sask Viper TOCO 1st 80 BAM flat universe RING3rd K) µ 60 dark matter ( CAT QMAP T MAX Mchecksaxima ∆ SP MSAM Ten 40 FIRS OVROATCA ARGOIAC SuZIE BOOM WD BIMA BOOM 2nd 20 COBE Pyth baryonic dark matter W. Hu 11/00 10 100 1000 l Polarized Landscape 100 10 reionization ΘE K) µ 1 EE ( ∆ BB 0.1 gravitational lensing gravitational waves 0.01 10 100 1000 Hu & Dodelson (2002) l (multipole) Recent Data Ade et al (QUAD, 2007) Power Spectrum Present QUAD: Pryke et al (2008) 100 50 TE 0 −50 −100 30 20 EE 10 0 −10 10 5 BB 0 −5 −10 0 500 1000 1500 2000 multipole Instantaneous Reionization • WMAP data constrains optical depth for instantaneous models of τ=0.087±0.017 • Upper limit on gravitational waves weaker than from temperature Why is the CMB polarized? Polarization from Thomson Scattering • Differential cross section depends on polarization and angle dσ 3 0 2 = |εˆ · εˆ| σT dΩ 8π dσ 3 0 2 = |εˆ · εˆ| σT dΩ 8π Polarization from Thomson Scattering • Isotropic radiation scatters into unpolarized radiation Polarization from Thomson Scattering • Quadrupole anisotropies scatter into linear polarization aligned with cold lobe Whence Quadrupoles? • Temperature inhomogeneities in a medium • Photons arrive from different regions producing an anisotropy hot cold hot (Scalar) Temperature Inhomogeneity Hu & White (1997) Whence Polarization Anisotropy? • Observed photons scatter into the line of sight • Polarization arises from the projection of the quadrupole on the transverse plane Polarization Multipoles • Mathematically pattern is described by the tensor (spin-2) spherical harmonics [eigenfunctions of Laplacian on trace-free 2 tensor] • Correspondence with scalar spherical harmonics established via Clebsch-Gordan coefficients (spin x orbital) • Amplitude of the coefficients in the spherical harmonic expansion are the multipole moments; averaged square is the power E-tensor harmonic l=2, m=0 Modulation by Plane Wave • Amplitude modulated by plane wave → higher multipole moments • Direction detemined by perturbation type → E-modes Polarization Pattern Multipole Power 1.0 B/E=0 π/2 0.5 Scalars φ l A Catch-22 • Polarization is generated by scattering of anisotropic radiation • Scattering isotropizes radiation • Polarization only arises in optically thin conditions: reionization and end of recombination • Polarization fraction is at best a small fraction of the 10-5 anisotropy: ~10-6 or µK in amplitude Polarization Peaks Fluid Imperfections • Perfect fluid: no anisotropic stresses due to scattering isotropization; baryons and photons move as single fluid • Fluid imperfections are related to the mean free path of the photons in the baryons −1 λC = τ˙ where τ˙ = neσT a is the conformal opacity to Thomson scattering • Dissipation is related to the diffusion length: random walk approximation √ p p λD = NλC = η/λC λC = ηλC the geometric mean between the horizon and mean free path • λD/η∗ ∼ few %, so expect the peaks >3 to be affected by dissipation Viscosity & Heat Conduction • Both fluid imperfections are related to the gradient of the velocity kvγ by opacity τ˙: slippage of fluids vγ − vb. • Viscosity is an anisotropic stress or quadrupole moment formed by radiation streaming from hot to cold regions m=0 hot v cold hot v Dimensional Analysis • Viscosity= quadrupole anisotropy that follows the fluid velocity k π ≈ v γ τ˙ γ • Mean free path related to the damping scale via the random walk 1/2 2 kD = (τ/η ˙ ∗) → τ˙ = kDη∗ • Damping scale at ` ∼ 1000 vs horizon scale at ` ∼ 100 so kDη∗ ≈ 10 • Polarization amplitude rises to the damping scale to be ∼ 10% of anisotropy k 1 ` 1 πγ ≈ vγ ∆P ≈ ∆T kD 10 `D 10 • Polarization phase follows fluid velocity Damping & Polarization • Quadrupole moments: damp acoustic oscillations from fluid viscosity generates polarization from scattering • Rise in polarization power coincides with fall in temperature power – ~ 1000 l Ψ damping πγ polarization driving Θ+Ψ 5 10 15 20 ks/π Acoustic Polarization • Gradient of velocity is along direction of wavevector, so polarization is pure E-mode • Velocity is 90◦ out of phase with temperature – turning points of oscillator are zero points of velocity: Θ + Ψ / cos(ks); vγ / sin(ks) • Polarization peaks are at troughs of temperature power Cross Correlation • Cross correlation of temperature and polarization (Θ + Ψ)(vγ) / cos(ks) sin(ks) / sin(2ks) • Oscillation at twice the frequency • Correlation: radial or tangential around hot spots • Partial correlation: easier to measure if polarization data is noisy, harder to measure if polarization data is high S=N or if bands do not resolve oscillations • Good check for systematics and foregrounds • Comparison of temperature and polarization is proof against features