Anthony Challinor IoA, KICC & DAMTP University of …...METRIC SCALAR PERTURBATIONS Need to go...

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Fundamentals of CMB anisotropy physics

Anthony Challinor

IoA, KICC & DAMTPUniversity of Cambridge

a.d.challinor@ast.cam.ac.uk

OUTLINE

• Overview

• Random fields on the sphere

• Reminder of cosmological perturbation theory

• Temperature anisotropies from scalar perturbations

• Cosmology from CMB temperature power spectrum

• Temperature anisotropies from gravitational waves

• Introduction to CMB polarization

USEFUL REFERENCES

• CMB basics

– Wayne Hu’s excellent website (http://background.uchicago.edu/~whu/)

– Hu & White’s “Polarization primer” (arXiv:astro-ph/9706147)

– AC’s summer school lecture notes (arXiv:0903.5158 andarXiv:astro-ph/0403344)

– Kosowsky’s “Introduction to Microwave Background Polarization”(arXiv:astro-ph/9904102)

• Textbooks covering most of the above

– The Cosmic Microwave Background by Ruth Durrer (CUP)

OVERVIEW

COSMIC HISTORY

• CMB and matter plausibly produced during reheating at end of inflation

• CMB decouples around recombination, 300 kyr later

• Universe starts to reionize somewhere in range z = 6–10 and around 6% of CMBre-scatters

10 Sec10 s-34 -6

3000 K10 K10 K10 K101328

THE COSMIC MICROWAVE BACKGROUND

• Almost perfect black-body spectrum with T = 2.7255 K (COBE-FIRAS)

• Fluctuations in photon density, bulk velocity and gravitational potential give rise tosmall temperature anisotropies (∼ 10−5)

• CMB cleanest probe of global geometry and composition of early universe andprimordial fluctuations

CMB TEMPERATURE FLUCTUATIONS: DENSITY PERTURBATIONS

GravityRadiation pressurePhoton diffusion

Θlm ∝∫

(2π)3/2Θl(k)R(k)Y ∗lm(k)

CMB CRITICAL IN DEFINING STANDARD COSMOLOGICAL MODEL

• Passive evolution of small amplitude primordial curvature perturbations (inflation)

– Nearly, but not exactly, scale-invariant primordial power spectrum

– Apparently acausal super-horizon correlations

– Gaussian

– Adiabatic

• Perturbations evolve under gravity (GR) and standard hydrodynamics/radiativetransfer on a spatially-flat FRW universe

• Composition of universe (now):

– 4.9% of energy density in baryons (and charged leptons)

– 26.4% in “non”-interacting cold dark matter

– 69% in dark energy consistent with cosmological constant (vacuum energy)

– Three families of neutrinos with temperature 1.95 K

– Blackbody photons with temperature TCMB = 2.7255 K

OPEN (BIG) QUESTIONS FOR ΛCDM

• Early universe:

– Did inflation happen, what energy scale, what action?

• Dark sector:

– Nature of dark energy?

– Nature of dark matter?

– Is General Relativity correct on cosmological scales?

– Neutrinos – masses, sterile neutrinos, extra relativistic degrees of freedom?

• Reionization:

– How and when did the universe reionize?

RANDOM FIELDS ON THE SPHERE

OBSERVABLE: TEMPERATURE FLUCTUATIONS

• In practice, have to deal with Galactic and extragalactic foregrounds

• Theory doesn’t predict this map, only its statistical properties

ANISOTROPIES AND THE POWER SPECTRUM

• Decompose temperature anisotropies in spherical harmonics

Θ ≡∆T (n)/T =∑lm

almYlm(n)

– Multipole l corresponds roughly to angular scale 180/l

– Under a rotation (R) of sky alm → Dlmm′(R)alm′

– Demanding statistical isotropy requires, for 2-point function

〈alma∗l′m′〉 = DlmMD

l′∗m′M ′〈alMa

∗l′M ′〉 ∀R

– Only possible (from unitarity Dl∗MmD

lMm′ = δmm′) if

〈alma∗l′m′〉 = Clδll′δmm′

– Symmetry restricts higher-order correlations also, but for Gaussian fluctuationsall information in power spectrum Cl

• Estimator for power spectrum Cl =∑m |alm|2/(2l + 1) has mean Cl and cosmic

variance

var(Cl) =2

2l + 1C2l

MEASURED POWER SPECTRUM

6000DTT

30 500 1000 1500 2000 2500`

-60-3003060

∆DTT

2 10-600-300

0300600

• Note: Dl ≡ l(l + 1)Cl/2π

• Planck measurements cosmic-variance limited to l ∼ 200012

REMINDER OF RELATIVISTIC COSMOLOGICAL PERTURBATIONTHEORY

METRIC SCALAR PERTURBATIONS

• Need to go beyond Newtonian theory for super-Hubble (coherence size > H−1)perturbations and relativistic matter (e.g., radiation)

• Most general perturbation to flat FRW background metric:

ds2 = a2(η)(1 + 2ψ)dη2 − 2Bidxidη − [(1− 2φ)δij + 2Eij]dx

– Here, using conformal time dη ≡ dt/a and comoving coordinates xi

– ψ and φ are scalar functions of η and xi; Bi is a three-vector undertransformation of xi; Eij is a trace-free symmetric three-tensor (Eij = Eji andδijEij = 0)

– 10 degrees of freedom in perturbed metric: 4 scalar (clumping), 4 vector(vortical motions) and 2 tensor (gravitational waves)

• For scalar perturbations all perturbed quantities are spatial gradients of scalarfields

– Work in conformal Newtonian gauge:

ds2 = a2(η)[(1 + 2ψ)dη2 − (1− 2φ)δijdxidxj]

COMOVING-GAUGE CURVATURE PERTURBATION R

• Perturbed 3D Ricci scalar of spatial surfaces orthogonal to comoving observers(see no energy flux):

a2(3)R(η,x) = −4∇2R(η,x)

• In terms of CNG potentials

R = −φ−H(φ+Hψ)

4πGa2(ρ+ P )

• R(η,x) is conserved for adiabatic fluctuations on super-Hubble scales irrespectiveof equation of state

– Adiabatic fluctuations like a spatially-dependent time shift of background:

δP − ˙Pδρ/ ˙ρ = 0

• Single-field inflation gives adiabatic initial fluctuations with primordial curvatureperturbation R(x) and power spectrum

〈R(k)R∗(k′)〉 =2π2

k3PR(k)δ(3)(k − k′)

RADIATION-DOMINATED DYNAMICS AND INITIAL CONDITIONS

• Consider radiation-dominated universe (a ∝ η, H = 1/η and P = ρ/3) and ignoreneutrinos, baryons and CDM; Einstein equations then give φ = ψ and

Trace of Gij φ+3

ηφ−

η2φ =

2η2δγ

G00 k2φ+3

η2φ = −

2η2δγ

• Eliminate δγ ≡ δργ/ργ to get damped SHO with sound speed 1/√

3φ = 0

– Solution that is constant outside sound horizon (kη √

φ(η,k) = −2R(k)j1(kη/

kη/√

3= −

[1 +O(kη)2

]– Asymptotic form for kη

√3: φ(η,k) = cos(kη/

√3)/(kη/

√3)2

• Outside sound horizon δγ(η,k) = 4R(k)/3 so constant and overdense inpotential wells

• For kη √

3, δγ(η,k) = −4R(k) cos(kη/√

3)→ constant amplitude oscillationwith resonantly-enhanced amplitude

SUMMARY OF PERTURBATION EVOLUTION

• Evolution of perturbations in ΛCDMmodel for adiabatic initial conditions:

δc = δb = 34δγ = 3

• Remain adiabatic and constant whensuper-Hubble (but δi gauge-dependent)

• Sub-Hubble, CDM growth slow untilmatter dominates, then power law

• Before recombination, tightly-coupledbaryons and photons oscillate acousti-cally inside Jeans’ scale

• After recombination baryons fall intoCDM potential wells

TEMPERATURE ANISOTROPIES FROM SCALARPERTURBATIONS

CMB SPECTRUM AND DIPOLE ANISOTROPY

• Microwave background almost perfectblackbody radiation

– Temp. (COBE-FIRAS) 2.725 K

• Dipole anisotropy ∆T/T = β cos θ im-plies solar-system barycenter has ve-locity v/c ≡ β = 0.00123 relative to‘rest-frame’ of CMB

• Variance of intrinsic fluctuations first de-tected by COBE-DMR: (∆T/T )rms =

16µK smoothed on 7 scale

• Now know (∆T/T )rms ≈ 115µK witharcmin resolution

THERMAL HISTORY — DETAILS

• Dominant element hydrogen recombines rapidly around z ≈ 1100

– Prior to recombination, Thomson scattering efficient and mean free path1/(neσT ) short cf. expansion time 1/H

– Little chance of scattering after recombination→ photons free stream keepingimprint of conditions on last scattering surface

• Optical depth back to (conformal) timeη for Thomson scattering:

τ(η) =∫ η0

ηaneσT dη

• e−τ is prob. of no scattering back to η

• Visibility is probability density for lastscattering at η:

visibility(η) = −τ e−τ

LINEAR ANISOTROPY GENERATION: REDSHIFTING

• Recall perturbed metric:

ds2 = a2(η)[(1 + 2ψ)dη2 − (1− 2φ)dx2]

• Introduce orthonormal frame of vectors:

(E0)µ = a−1(1− ψ)δµ0, (Ei)µ = a−1(1 + φ)δµi

• Parameterise photon 4-momentum with energy ε/a (seen by observer at rest incoordinates) and direction e (e2 = 1) on orthonormal spatial triad:

pµ = a−2ε[1− ψ, (1 + φ)e]

• Geodesic equation pµ∇µpν = 0 becomes (overdot ≡ ∂/∂η)

d ln ε/dη = −dψ/dη + (ψ + φ)

de/dη = −(∇− ee ·∇)(ψ + φ)

dx/dη = (1 + φ+ ψ)e

• Note, ε and e constant if no perturbations21

BOLTZMANN EQUATION

• Dimensionless temperature fluctuation Θ(η,x, e) evolves along photon path byBoltzmann equation:

(∂η + e ·∇) Θ︸︷︷︸≡dΘ/dη

−d ln ε

dη=dΘ

∣∣∣∣∣scatt.

• Thomson scattering (kBT mec2) around recombination and reionizationdominant scattering mechanism to affect CMB:

∣∣∣∣∣scatt.

= −aneσTΘ︸︷︷︸out-scattering

+3aneσT

∫dmΘ(ε, m)[1 + (e · m)2]︸︷︷︸

in-scattering

+ aneσTe · vb︸︷︷︸Doppler

• Doppler effect arises from electron bulk velocity vb

– Enhances ∆T/T for vb towards observer

• Neglecting anisotropic nature of Thomson scattering (1 + (e · m)2 → 4/3):

dη+dψ

dη−(φ+ ψ

)≈ −aneσT (Θ−Θ0 − e · vb)

so scattering tends to isotropise in rest-frame of electrons: Θ→ Θ0 + e · vb

TEMPERATURE ANISOTROPIES

• Formal solution (with integrating factor e−τ ):

[Θ(n) + ψ](η0,x0) = −∫ η0

τ e−τ (Θ0 + ψ + e · vb) dη′+∫ η0

e−τ(φ+ ψ

)dη′

where integrals are along background line of sight and n = −e

• On degree scales, can approximate −τ e−τ ≈ δ(η − η∗) ignoring reionization:

[Θ(n) + ψ](η0,x0) = Θ0|∗︸︷︷︸temp.

+ ψ|∗︸︷︷︸gravity

+ e · vb|∗︸︷︷︸Doppler

+∫ R∗

(ψ + φ) dη︸︷︷︸ISW

– Intrinsic temperature fluctuation at last-scattering (Θ0 = δγ/4)

– Gravitational redshift from difference in potentials

– Doppler effect from electron bulk velocity at last scattering

– Integrated Sachs–Wolfe effect from evolution of potentials important at latetimes (dark energy) and also around last-scattering

• Have ignored anisotropic scattering, finite width of visibility function (i.e.,last-scattering surface) and reionization

– Full calculations (e.g., in CAMB, CMBFAST, CLASS) fix these omissions23

SPATIAL-TO-ANGULAR PROJECTION

• Consider angular projection at origin of potential ψ(η∗,x) over last-scatteringsurface; for a single Fourier component

ψ(n) = ψ(η∗, χ∗n) = ψ(η∗,k)eik·nχ∗ where χ∗ = ∆η = η0 − η∗= ψ(η∗,k)

4πiljl(kχ∗)Ylm(n)Y ∗lm(k)

ψlm = 4πψ(η∗,k)iljl(kχ∗)Y∗lm(k)

• jl(kχ∗) peaks when kχ∗ ≈ l but for given l considerable power from k > l/χ∗ also(wavefronts perpendicular to line of sight)

k k∆η= l

k∆η> l

– CMB anisotropies at multipole l mostly sourced from fluctuations with linearwavenumber k ∼ l/χ∗ where distance to last scattering χ∗ ≈ 14 Gpc

ANGULAR POWER SPECTRUM

• For Doppler term, Θ ∼ e · vb|∗, have for a single Fourier mode

e · vb|∗ = −n · [ikvb(η∗,k)]eik·nχ∗

= −vb(η∗,k)∑lm

4πilj′l(kχ∗)Ylm(n)Y ∗lm(k)

• Hence multipoles of Θ, ignoring ISW, are

Θlm = 4πil[(Θ0 + ψ)(η∗,k)jl(kχ∗)− vb(η∗,k)j′l(kχ∗)

]Y ∗lm(k)

• Θ0(η∗,k) etc. linearly related to primordial R(k)

• CMB anisotropies then statistically isotropic with power spectrum

Cl = 4π∫d ln k

[(Θ0 + ψ)(η∗,k)

R(k)jl(kχ∗)−

vb(η∗,k)

R(k)j′l(kχ∗)

• Θ0 + ψ term dominates; slowly varying in k cf. jl(kχ∗) so

l(l + 1)

2πCl ≈

[(Θ0 + ψ) (η∗,k)

PR(k) (k = l/χ∗)

ACOUSTIC PHYSICS

• Photon isotropic temperature Θ0 and electron velocity vb at last scattering dependon acoustic physics of pre-recombination plasma

• Large-scale approximation: ignore diffusion and slip between CMB and baryonbulk velocities (requires scattering rate k)

– Photon–baryon plasma behaves like perfect fluid responding to gravity (drivesinfall to wells), Hubble drag of baryons, gravitational redshifting and photonpressure (resists infall):

Θ0 + HR1+RΘ0︸︷︷︸

Hubble drag

+ 13(1+R)k

2Θ0︸︷︷︸pressure

= φ︸︷︷︸redshift

+ HR1+Rφ−

2ψ︸︷︷︸infall

– R ≡ 3ρb/(4ργ) ∝ a is fraction of momentum density from baryons:

3ργvγ + ρbvb ≈

3ργ(1 +R)vb

– WKB solutions of homogeneous oscillator equation:

(1 +R)−1/4 cos krs , (1 +R)−1/4 sin krs

with sound horizon rs ≡∫ η0

dη′√3(1+R)

ACOUSTIC OSCILLATIONS

Θ0 +HR

1 +RΘ0 +

3(1 +R)k2Θ0 = φ+

HR1 +R

φ−1

• Already solved this in radiation domination (R→ 0 and taking φ = ψ)

– At end of radiation domination, η<eq,

Θ0(η<eq,k) =

13R(k) kηeq 1

−R(k) cos krs kηeq 1

and ψ(η<eq,k) = −2R(k)/3 for kηeq 1 and O(kηeq)−2 for kηeq 1

• In matter domination, have oscillations in constant potential with midpoint−(1 +R)ψ where pressure balances infall

– Matching to above, noting that φ decays by 10% across transition onsuper-Hubble scales to conserve R (and Θ0 follows), have

Θ0(η,k) =

−15R(k) [(1 + 3R) cos krs − 3(1 +R)] kηeq 1

−R(k) cos krs kηeq 1

CONDITIONS AT LAST SCATTERING

(Θ0 + ψ)(η∗,k) =

−15R(k) [(1 + 3R) cos krs(η∗)− 3R] kηeq 1

−R(k) cos krs(η∗) kηeq 1

• Modes with krs(η∗) = nπ are at extrema at last scattering⇒ acoustic peaks inpower spectrum

• Baryon-dependent offset that disappears at high k modulates even/odd peaks

• Resonant driving boosts amplitude by ∼ 5 on small scales (but have ignoreddiffusion damping!)

ADIABATIC ANISOTROPY POWER SPECTRUM

• Temperature power spectrum for scale-invariant curvature fluctuations

COMPLICATION 1: PHOTON DIFFUSION

• Photons diffuse out of dense regions damping inhomogeneities in Θ0 (andcreating higher moments of Θ)

– In time dη, when mean-free path `P = (aneσT )−1 = 1/|τ |, photon randomwalks mean square distance `Pdη

– Defines a diffusion length by last scattering:

k−2D ∼

∫ η∗0|τ |−1dη ∝ (Ωmh

2)−1/2(Ωbh2)−1

• Get exponential suppression ofphotons (and baryons)

Θ0 ∝ e−k2/k2

D cos krs

on scales below k−1D ≈ 7 Mpc at

last scattering

• Mixing blackbodies → generatesspectral distortions at higher order(see Chluba)

DAMPING OF THE CMB POWER SPECTRUM

• Diffusion damping implies e−2l2/l2D damping tail in angular power spectrum

– Softened by finite width of visibility function

COMPLICATION 2: REIONIZATION

• IGM reionized between z = 6–10

• CMB Thomson scatters off all (re-)ionized gas back to z∗ with optical depthτ =

∫ η0η∗ aneσT dη

– Produces a further low redshift peak in the visibility function (important forpolarization – see later)

REIONIZATION: EFFECT OF RE-SCATTERING

• CMB re-scatters off re-ionized gas; ignoring anisotropic (Doppler and quadrupole)scattering terms, locally at reionization have

Θ(e) + ψ → e−τ [Θ(e) + ψ] + (1− e−τ)(Θ0 + ψ)

– Outside horizon at reionization, Θ(e) ≈ Θ0 and scattering has no effect

– Well inside horizon, Θ0 + ψ ≈ 0 and observed anisotropies

Θ(n)→ e−τΘ(n) ⇒ Cl → e−2τCl

COSMOLOGY FROM CMB TEMPERATURE POWER SPECTRUM

PARAMETERS FROM CMB: MATTER AND GEOMETRY

• Acoustic physics (dark energy and curvature negligible):

– Peak locations in k depend on sound horizon rs at last scattering

– Potential envelope (resonant driving) is f(k/keq) where keq = 2/ηeq

– Damping scale 1/kD(roughly geometric mean of horizon and mean free path)

∗ rs, keq and kD depend only on Ωbh2 and Ωmh2 for fixed TCMB in standard

models

– Baryons further affect peak heights through R (baryon offset)

• Scales rs, 1/keq and 1/kD are seen in projection generally with angular diameterdistance to last-scattering dA, e.g., θs ∝ rs/dA– Observable ratios, e.g., θs/θeq ∝ rskeq determine Ωmh2 once Ωbh

2 fixed frompeak morphology

– Calibrated ruler rs then accurately determines dA

∗ Fixes H0 in LCDM, but degenerate with e.g., geometry, dark energy andsub-eV massive neutrinos in extended models

PARAMETERS FROM CMB: PRIMORDIAL POWER SPECTRUM

• Scalar power spectrum Cl essentially e−2τPR(k) at k ≈ l/dA processed byacoustic physics

– Inflation predicts almost scale-invariant power-law PR(k) = As(k/k0)ns−1

– CMB probes scales 5 Mpc < k−1 < 5000 Mpc

ACOUSTIC PEAK HEIGHTS: BARYON DENSITY

• Most distinctive effect of increasing Ωbh2 is boost of compressional (1, 3, etc.)

low-order peaks cf. rarefaction peaks

• Planck: Ωbh2 = 0.02222± 0.00023 (i.e., to 1%)

Increasing Ωbh2

ACOUSTIC PEAK HEIGHTS: DARK MATTER DENSITY

• Unique effect of increase in Ωch2 is reduction in resonant driving of low-orderpeaks and early-ISW contribution to 1st peak

– Since keq = aeqHeq ∝ (Ωmh2)−1(Ωmh2)2, angular scale of equality relativeto peaks goes down if Ωmh2 increases:

θs/θeq = rskeq ∝ (Ωmh2)0.75

• Planck (only): Ωch2 = 0.1197± 0.0022 (i.e., to 2%)

ACOUSTIC-SCALE DEGENERACY IN LCDM MODELS

• Peaks locations very well measured⇒ θ∗ = rs/dA known to 0.05% precision!

• In LCDM, θ∗ is a function of mostly Ωmh3 – main (approximate) degeneracy inLCDM models

– Along degeneracy direction, other parameters vary to try and maintainmorphology of peaks

0.26 0.30 0.34 0.38

GEOMETRIC DEGENERACY

• Some parameters not determined by linear T anisotropies alone

– For same primordial perturbation spectrum and physical densities of CDM,baryons, photons and neutrinos, can get (almost) identical primary Cls withdifferent late-time parameters (ΩK,Ωde, w, . . .) in extended models if preservedA

– Disentangling these late-time parameters requires other datasets (e.g. Hubble,supernovae, shape of matter power spectrum or BAO) or CMB lensing (seelater)

PARAMETERS FROM THE DAMPING TAIL: Neff

• Effective number of relativistic (neutrino) degrees of freedom at recombinationdefined by

ρν = Neff(7/8)(4/11)4/3ργ

– Equals 3.04 in standard scenario since neutrinos not fully decoupled at e+e−

annihilation

• Consider increasing Neff , hence H∗, while preserving peak locations:

θDθ∗

rskD∝

H−1∗ H

1/2∗

= H1/2∗

– Increasing Neff at fixed θ∗→ damping kicks in at larger scales reducing powerin damping tail

– Must also increase H0 to preserve θ∗

– Increasing Neff reduces zeq (hence increases early-ISW) but can compensatewith increased ωm

– Increased neutrino anisotropic stress with increased Neff reduces initialamplitude of δγ but can compensate by increasing primordial power

CONSTRAINING Neff

• CMB damping tail consistent with standard Neff

– Planck: Neff = 3.13± 0.3

TEMPERATURE ANISOTROPIES FROM GRAVITATIONAL WAVES

COSMOLOGICAL GRAVITATIONAL WAVES

• Tensor metric perturbations

ds2 = a2[dη2 − (δij + hij)dxidxj]

– Trace-free δijhij = 0, transverse ∂ihij = 0

• Einstein equation (ij trace-free):

hij + 2Hhij −∇2hij = −16πGa2ΠTij

– Consider ΠTij = 0 case→ wave equation (speed c) damped by expansion

– For k H, oscillator is over-damped and hij = const. or decays (as 1/a inradiation domination and a−3/2 in matter domination)

– For k H use WKB solution to equation in form

∂2η (ahij) +

(k2 − a/a

)(ahij) = 0 ⇒ hij ∝ e±ikη/a

– Usual short wavelength description of gravitational waves with adiabatic decayof amplitude as 1/a

– Note energy density ∼ a−2〈hijhij〉 ∝ a−4 like gas of massless gravitons

GRAVITATIONAL WAVES FROM INFLATION

• Quantum fluctuations in hij (note, semi-classical quantum gravity!) give primordialgravitational wave perturbations with power spectrum

Ph(k) =8

(Hk2π

)2where 〈hijhij〉 =

∫d ln kPh(k)

• Direct probe of H during inflation and hence energy scale of inflation whereH2 = E4

inf/(3M2Pl):

Ph(k) ≈128

(Einf√8πMPl

= 1.93× 10−11(

1016 GeV

• In slow-roll inflation, Ph(k) almost power-law with spectral index

nt =d lnPh(k)

d ln k≈

dt= −2ε

• Conventional to use tensor-to-scalar ratio

r ≡Ph(k0)

PR(k0)

GRAVITATIONAL WAVES AND THE CMB

• Evolution of comoving energy from shear of gravitational wave:

d ln ε/dη + 12hije

iej = 0

• Neglecting anisotropic scattering, Boltzmann equation gives

Θ(η0,x0, e) = −1

∫ η0e−τ hije

iej dη

• Only significant contribution on large scales since hij decays like a−1 afterentering horizon

– Planck constraint r < 0.11 as good as will ever do with TT

INTRODUCTION TO CMB POLARIZATION

POLARIZATION AND STOKES PARAMETERS

• For quasi-monochromatic plane wave along z, correlation tensor of electric field Edefines (transverse) polarization tensor:(

〈ExE∗x〉〈ExE∗y〉〈EyE∗x〉〈EyE∗y〉

(I +Q U + iVU − iV I −Q

• Thomson scattering of CMB anisotropies generates linear polarization describedby symmetric trace-free part of correlation:

Pab ≡( 1

2〈|Ex|2 − |Ey|2〉〈<(ExE∗y)〉

〈<(ExE∗y)〉 −12〈|Ex|

2 − |Ey|2〉

(Q UU −Q

Q > 0 Q < 0 U > 0 U < 0

• Under right-handed rotation of x and y through ψ about propagation direction (z)

Q± iU → (Q± iU)e∓2iψ

SPIN-s FIELDS ON THE SPHERE

• For propagation along e, Stokes parameters defined on θ, φ unit basis⇒

Q± iU = 〈EθE∗θ− EφE

∗φ〉 ± i〈EθE

+ EφE∗θ〉

= 〈(Eθ ± iEφ)(E∗θ± iE∗

φ)〉

– Q± iU are components of P on null basis m± ≡ θ ± iφ:

Q± iU = m± · 〈E ⊗E∗〉 ·m± = ma±m

b±Pab

• A quantity sη is spin s if

sη → eisψsη for m± → e±iψm±

– Q± iU is spin ±2

E AND B MODES: WARM-UP WITH VECTOR FIELDS

• As a warm-up, can always write vector field in 2D as

Va = gradient + divergence-free vector

= ∇aVE + εba∇bVB

• Consider spin ±1 components of V on null basis

– Since εbama± = ±imb

± have

m± · V = ma±∇a (VE ± iVB)

= (∂θ ± icosecθ∂φ)(VE ± iVB)

• Define spin-weight derivatives via

ðsη = − sins θ(∂θ + icosecθ∂φ)(sin−s θsη)

ðsη = − sin−s θ(∂θ − icosecθ∂φ)(sins θsη)

– Then spin components of V are spin-weight derivatives of complex potential:

m+ · V = −ð(VE + iVB), m− · V = −ð(VE − iVB)

E AND B MODES FOR POLARIZATION

• Generalisation of E-B decomposition to 2nd-rank STF tensors

Pab(e) = ∇〈a∇b〉PE + εc(a∇b)∇cPB

• Evaluating null components of covariant derivatives (recall ma±m

b±Pab = Q± iU )

Q+ iU = ðð(PE + iPB), Q− iU = ðð(PE − iPB)

• PE and PB are scalar fields⇒ can expand in usual spherical harmonics:

PE(e) =∑lm

√(l−2)!(l+2)!ElmYlm(e), PB(e) =

√(l−2)!(l+2)!BlmYlm(e)

– l-dependent factors “undo” ∼ l2 factors from double derivatives to give

Q± iU =∑lm

(Elm ± iBlm)

√(l−2)!(l+2)!

ðððð

=∑lm

(Elm ± iBlm)±2Ylm

E- AND B-MODE PATTERNS

• Consider axisymmetric potentials PE(µ) andPB(µ) with µ ≡ cos θ:

Q± iU = (1− µ2)d2

dµ2(PE ± iPB)

– Follows that

Q = (1− µ2)d2

dµ2PE

U = (1− µ2)d2

dµ2PB

– Axisymmetric E modes produce pure-Q po-larization

E−mode B−mode

• On flat patch of sky, plane-wave E and B generate following polarization:

TWO-POINT STATISTICS

• Statistical isotropy demands 2-point correlations of form

〈ElmE∗l′m′〉 = CEl δll′δmm′

• For Gaussian fluctuations all information in power spectrum Cl

• Under parity transformations, PE is a scalar but PB is a pseudo-scalar so

Elm → (−1)lElm and Blm → −(−1)lBlm

– Cannot have E-B or T -B correlations if parity respected in mean

– Expect non-zero spectra: CTTl , CEEl , CTEl and CBBl

GENERATION OF POLARIZATION: THOMSON SCATTERING

• Photon diffusion around recombination→ local tem-perature quadrupole

– Subsequent Thomson scattering generates (par-tial) linear polarization with r.m.s. ∼ 5µK fromdensity perturbations

Polarization

Cold.• Thomson scattering of radiation quadrupole produces linear polarization

(dimensionless temperature units!)

d(Q± iU)(e) =3

5aneσTdη

∑|m|≤2

±2Y2m(e)

E2m −√

– Purely electric quadrupole (l = 2)

• Ignoring reionization, observed polarization at (η0,x0) generated atx∗ = x0 − χ∗e on last-scattering surface:

(Q± iU)(η0,x0, e) ≈ −√

∑|m|≤2

Θ2m(η∗,x∗)±2Y2m(e)

LINEAR SCALAR PERTURBATIONS PRODUCE ONLY E-MODES

• For scalar perturbations, anisotropies are azimuthally-symmetric about wavevector

• For single Fourier component, with k along z, have Θ2m ∝ δm0

• Observed polarization from this Fourier mode is

(Q± iU)(η0,x0, e) ∝ Θ20(η∗, kz)e−ikχ∗z·e±2Y20(e)︸︷︷︸∝ sin2 θ

• Generated as axisymmetric pure-Q, and this preserved by plane-wave modulation:

Plane-wave scalar quadrupole Electric quadrupole (m = 0) Pure E mode

Scatter Modulate

QUADRUPOLE SOURCE TERM

• Consider scales large compared to diffusion-damping scale

– Temperature fluctuation seen by electron determined by conditions at previousscattering `p away:

Θ(e) + ψ ≈ (Θ0 + ψ)(−`pe) + e · v(b)(−`pe)

≈ (Θ0 + ψ)− `pei∂i(Θ0 + ψ) +1

iej∂i∂j(Θ0 + ψ)

+ e · v(b) − `peiej∂jv(b)i + · · ·

– Dominant temperature quadrupole from trace-free part of eiej components:∑m

Θ2mY2m(e) ∼(eiej − 1

3δij) [

2p∂i∂j(Θ0 + ψ)− `p∂jv(b)i

]– Intrinsic temperature contribution suppressed by factor ∼ k`p cf. Doppler

• Polarization traces baryon velocity at recombination⇒ peaks at troughs of ∆T

• Large-angle polarization from recombination small since quadrupole sourcegenerated causally

ORDER OF MAGNITUDE POLARIZATION

(Q± iU)(η0,x0, e) ≈ −√

∑|m|≤2

Θ2m(η∗,x∗)±2Y2m(e)

• Follows that mean-square of observed polarization

〈Q2 + U2〉 = 〈(Q+ iU)(Q+ iU)∗〉 =3

40πCTT2 (η∗)

• Tight-coupled quadrupole around last scattering:∑m

Θ2mY2m(e) ∼ −`p(eiej − 1

3δij)∂iv(b)j

⇒ 54πC

TT2 (η∗) ∼ 4

45`2p〈(∇ · vb)2

∣∣∣η∗〉

• Continuity equation, δγ + 4∇ · vγ/3 = 0, and tight-coupling vb ≈ vγ gives

∇ · vb ≈ −34δγ ≈ −3R(k)kcs sin krs

⇒ 〈(∇ · vb)2〉 ∼ 9c2s

∫ kD0

dk k sin2 krsPR(k) ∼ 94c

• Noting that (kD`p)2 ∼ `p/η∗ ∼ 10−2 and PR ≈ 2× 10−9 from TT , have

〈Q2 + U2〉 ∼ 1250(kD`p)

2PR ∼ 1250(`p/η∗)PR ∼ (fewµK)2

SCALAR POLARIZATION POWER SPECTRA

• Polarization mostly probing vb at lastscattering

– CEl peaks at minima of CTl

• Correlations between T and E

• Additional large-angle polarization fromscattering around reionization

• B-modes are generated at second or-der, e.g., by lensing (see Benabed)

LARGE-ANGLE POLARIZATION FROM REIONIZATION

• Temperature quadrupole at reionization peaks around k(ηre − η∗) ∼ 2

– Re-scattering generates polarization on this linear scale→ projects tol ∼ 2(η0 − ηre)/(ηre − η∗)

– Amplitude of polarization ∝ optical depth through reionization→ best way tomeasure τ with CMB

CURRENT STATUS OF POLARIZATION MEASUREMENTS

• Low S/N maps over full sky from WMAP and Planck

– Foreground-cleaned Planck LFI 70 GHz:

!2 !1 0 1 2

• High S/N maps over small fractions ofsky (SPTpol, ACTPol, POLARBEAR,BICEP/Keck)

PLANCK POWER SPECTRUM MEASUREMENTS: EE

0−5µ

30 500 1000 1500 2000`

∆CEE

2 10-5000

• Optical depth through reionization: τ = 0.074± 0.013 from Planck+WMAP9

PLANCK POWER SPECTRUM MEASUREMENTS: TE

30 500 1000 1500 2000`

∆DTE

2 10-16

CMB POLARIZATION SCIENCE

• Reionization

• Large-angle anomalies

• B-modes and gravitational waves

• Lensing reconstruction and delensing

• High-l E-modes

– Parameters from the damping tail

– Primordial non-Gaussianity

• Cluster science

– Transverse velocities and remote quadrupole

– Lensing-calibrated masses

E- AND B-MODES FROM GRAVITATIONAL WAVES

• Free streaming through `p generates quadrupole∑mΘ2mY2m(e) ∼ `phijeiej

• For p-polarized wave (p = ±2 helicity states) with k = kz, have Θ2m ∝ δmp

• Observed polarization from such a Fourier mode is

(Q± iU)(η0,x0, e) ∝ Θ2p(η∗, kz)e−ikχ∗z·e±2Y2p(e)

• This for + polarization (h(+2) = h(−2)):

HHHHHH

HHHHHj

Plane-wave tensor quadrupole Electric quadrupole (|m| = 2)

Scatter

Modulate E mode

B mode

SCALAR AND TENSOR POWER SPECTRA (r = 0.2)

GRAVITATIONAL WAVES AND B-MODES

• Tiny signal – r.m.s. < 200 nK – but not confused by dominant densityperturbations like TT

CURRENT LIMITS ON B-MODE POWER SPECTRUM

10−4

10−3

10−2

10−1

lensingr=0.05

r=0.01

BK14CMB component

DASICBICAPMAPBoomerangWMAP

QUADBICEP1QUIET−QQUIET−W

PolarbearSPTpolACTpol

Multipole

Keck Array and BICEP2 Collaborations

• Direct BB constraints on gravitational waves have now surpassed TT

– r < 0.09 (95% CL) from BICEP2/Keck (+Planck+WMAP for foregrounds)67

Anthony Challinor IoA, KICC & DAMTP University of …...METRIC SCALAR PERTURBATIONS Need to go...

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