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

1

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)

2

OVERVIEW

3

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

Quar

ks ->

Had

rons

Nucl

eosy

nthe

sis

Reco

mbi

natio

n

10 Sec10 s-34 -6

3000 K10 K10 K10 K101328

4

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

5

CMB TEMPERATURE FLUCTUATIONS: DENSITY PERTURBATIONS

GravityRadiation pressurePhoton diffusion

Θlm ∝∫

d3k

(2π)3/2Θl(k)R(k)Y ∗lm(k)

6

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

7

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?

8

RANDOM FIELDS ON THE SPHERE

9

OBSERVABLE: TEMPERATURE FLUCTUATIONS

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

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

10

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

11

MEASURED POWER SPECTRUM

0

1000

2000

3000

4000

5000

6000DTT

`[µ

K2]

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

13

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

idxj

– 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]

14

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′)

15

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−

1

η2φ =

1

2η2δγ

G00 k2φ+3

ηφ+

3

η2φ = −

3

2η2δγ

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

3:

φ+4

ηφ+

k2

3φ = 0

– Solution that is constant outside sound horizon (kη √

3):

φ(η,k) = −2R(k)j1(kη/

√3)

kη/√

3= −

2

3R(k)

[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

16

SUMMARY OF PERTURBATION EVOLUTION

• Evolution of perturbations in ΛCDMmodel for adiabatic initial conditions:

δc = δb = 34δγ = 3

4δν

• 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

17

TEMPERATURE ANISOTROPIES FROM SCALARPERTURBATIONS

18

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

19

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

20

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

dη

∣∣∣∣∣scatt.

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

dΘ

dη

∣∣∣∣∣scatt.

= −aneσTΘ︸ ︷︷ ︸out-scattering

+3aneσT

16π

∫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):

dΘ

dη+dψ

dη−(φ+ ψ

)≈ −aneσT (Θ−Θ0 − e · vb)

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

22

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)

∑lm

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

24

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∗)

]2

PR(k)

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

l(l + 1)

2πCl ≈

[(Θ0 + ψ) (η∗,k)

R(k)

]2

PR(k) (k = l/χ∗)

25

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−

13k

2ψ︸ ︷︷ ︸infall

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

q =4

3ργvγ + ρbvb ≈

4

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)

26

ACOUSTIC OSCILLATIONS

Θ0 +HR

1 +RΘ0 +

1

3(1 +R)k2Θ0 = φ+

HR1 +R

φ−1

3k2ψ

• 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

27

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!)

28

ADIABATIC ANISOTROPY POWER SPECTRUM

• Temperature power spectrum for scale-invariant curvature fluctuations

29

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)

30

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

31

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)

32

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

33

COSMOLOGY FROM CMB TEMPERATURE POWER SPECTRUM

34

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

35

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

36

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%)

6

6

Increasing Ωbh2

37

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%)

38

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

Ωm

64

66

68

70

72

H0

0.936

0.944

0.952

0.960

0.968

0.976

0.984

0.992

ns

39

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)

40

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∗

=1

rskD∝

1

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

41

CONSTRAINING Neff

• CMB damping tail consistent with standard Neff

– Planck: Neff = 3.13± 0.3

42

TEMPERATURE ANISOTROPIES FROM GRAVITATIONAL WAVES

43

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

44

GRAVITATIONAL WAVES FROM INFLATION

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

Ph(k) =8

M2Pl

(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

3

(Einf√8πMPl

)4

= 1.93× 10−11(

Einf

1016 GeV

)4

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

nt =d lnPh(k)

d ln k≈

2

H2

dH

dt= −2ε

• Conventional to use tensor-to-scalar ratio

r ≡Ph(k0)

PR(k0)

45

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

2

∫ 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

46

INTRODUCTION TO CMB POLARIZATION

47

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〉

)≡

1

2

(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〉

)≡

1

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

48

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

∗φ

+ 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

49

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)

50

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) =

∑lm

√(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

Ylm

=∑lm

(Elm ± iBlm)±2Ylm

51

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:

52

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

53

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

Hot

Cold.• Thomson scattering of radiation quadrupole produces linear polarization

(dimensionless temperature units!)

d(Q± iU)(e) =3

5aneσTdη

∑|m|≤2

±2Y2m(e)

E2m −√

1

6Θ2m

– 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) ≈ −√

6

10

∑|m|≤2

Θ2m(η∗,x∗)±2Y2m(e)

54

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

55

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

2`2pe

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) [

12`

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

56

ORDER OF MAGNITUDE POLARIZATION

(Q± iU)(η0,x0, e) ≈ −√

6

10

∑|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

2sk

2DPR

• 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

57

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)

58

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

59

CURRENT STATUS OF POLARIZATION MEASUREMENTS

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

– Foreground-cleaned Planck LFI 70 GHz:

Q U

!2 !1 0 1 2

µK

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

60

PLANCK POWER SPECTRUM MEASUREMENTS: EE

0

20

40

60

80

100

CEE

`[1

0−5µ

K2]

0

5000

10000

15000

30 500 1000 1500 2000`

-4

0

4

∆CEE

`

2 10-5000

0

5000

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

61

PLANCK POWER SPECTRUM MEASUREMENTS: TE

-140

-70

0

70

140

DTE

`[µ

K2]

30 500 1000 1500 2000`

-10

0

10

∆DTE

`

2 10-16

-808

16

62

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

63

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

64

SCALAR AND TENSOR POWER SPECTRA (r = 0.2)

65

GRAVITATIONAL WAVES AND B-MODES

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

66

CURRENT LIMITS ON B-MODE POWER SPECTRUM

101

102

103

10−4

10−3

10−2

10−1

100

101

102

lensingr=0.05

r=0.01

BK14CMB component

DASICBICAPMAPBoomerangWMAP

QUADBICEP1QUIET−QQUIET−W

PolarbearSPTpolACTpol

Multipole

l(l+

1)C

lBB/2

π [

µK

2]

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