Fundamentals of CMB Anisotropy Physics

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Fundamentals of CMB anisotropy physics

Anthony Challinor

IoA, KICC & DAMTP University of Cambridge [email protected] 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 USEFULREFERENCES

• 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 and arXiv: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 COSMICHISTORY

• 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 CMB re-scatters

28 13 10 10 K 10 K 10 K 3000 K Recombination Nucleosynthesis Quarks -> Hadrons

-34 -6 10 s 10 Sec

4 THECOSMICMICROWAVEBACKGROUND

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

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

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

5 CMB TEMPERATURE FLUCTUATIONS: DENSITY PERTURBATIONS

Gravity Radiation pressure Photon diffusion

Z 3 d k ∗ ˆ Θlm ∝ Θl(k)R(k)Y (k) (2π)3/2 lm

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/radiative transfer 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) QUESTIONSFOR 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 ANISOTROPIESANDTHEPOWERSPECTRUM

• Decompose temperature anisotropies in spherical harmonics ¯ X Θ ≡ ∆T (ˆn)/T = almYlm(ˆn) lm – Multipole l corresponds roughly to angular scale 180◦/l l ( ) – Under a rotation (R) of sky alm → Dmm0 R alm0 – Demanding statistical isotropy requires, for 2-point function ∗ l l0∗ ∗ halmal0m0i = DmM Dm0M0halM al0M0i ∀R l∗ l = – Only possible (from unitarity DMmDMm0 δmm0) if ∗ halmal0m0i = Clδll0δmm0

– Symmetry restricts higher-order correlations also, but for Gaussian ﬂuctuations all information in power spectrum Cl ˆ P 2 • Estimator for power spectrum Cl = m |alm| /(2l + 1) has mean Cl and cosmic variance 2 var(Cˆ ) = C2 l 2l + 1 l 11 MEASUREDPOWERSPECTRUM

6000

5000

] 4000 2 K µ [ 3000 TT `

D 2000

1000

0 600 60 300 30 TT ` 0 0 D

∆ -300 -30 -600 -60 2 10 30 500 1000 1500 2000 2500 `

• Note: Dl ≡ l(l + 1)Cl/2π • Planck measurements cosmic-variance limited to l ∼ 2000 12 REMINDEROFRELATIVISTICCOSMOLOGICALPERTURBATION THEORY

13 METRICSCALARPERTURBATIONS

• 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: 2 2 2 i i j ds = a (η){(1 + 2ψ)dη − 2Bidx dη − [(1 − 2φ)δij + 2Eij]dx dx }

– Here, using conformal time dη ≡ dt/a and comoving coordinates xi i – ψ and φ are scalar functions of η and x ; Bi is a three-vector under i transformation of x ; Eij is a trace-free symmetric three-tensor (Eij = Eji and ij δ Eij = 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 scalar ﬁelds – Work in conformal Newtonian gauge: 2 2 2 i j ds = a (η)[(1 + 2ψ)dη − (1 − 2φ)δijdx dx ]

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 H(φ˙ + Hψ) R = −φ − 4πGa2(¯ρ + P¯)

•R (η, x) is conserved for adiabatic ﬂuctuations on super-Hubble scales irrespective of equation of state – Adiabatic ﬂuctuations like a spatially-dependent time shift of background:

δP − P¯˙ δρ/ρ¯˙ = 0

• Single-field inflation gives adiabatic initial fluctuations with primordial curvature perturbation R(x) and power spectrum 2π2 hR(k)R∗(k0)i = P (k)δ(3)(k − k0) k3 R 15 RADIATION-DOMINATED DYNAMICS AND INITIAL CONDITIONS

• Consider radiation-dominated universe (a ∝ η, H = 1/η and P = ρ/3) and ignore neutrinos, baryons and CDM; Einstein equations then give φ = ψ and 3 1 1 Trace of G φ¨ + φ˙ − φ = δγ ij η η2 2η2 2 3 3 3 G00 k φ + φ˙ + φ = − δγ η η2 2η2 √ • Eliminate δγ ≡ δργ/ρ¯γ to get damped SHO with sound speed 1/ 3: 4 k2 φ¨ + φ˙ + φ = 0 η 3 √ – Solution that is constant outside sound horizon (kη 3): √ j (kη/ 3) 2 h i φ(η, k) = −2R(k) 1 √ = − R(k) 1 + O(kη)2 kη/ 3 3 √ √ √ – 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 in potential wells √ √ • For kη 3, δγ(η, k) = −4R(k) cos(kη/ 3) → constant amplitude oscillation with resonantly-enhanced amplitude 16 SUMMARYOFPERTURBATIONEVOLUTION

• Evolution of perturbations in ΛCDM model for adiabatic initial conditions: 3 3 δc = δb = 4δγ = 4δν

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

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

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

• After recombination baryons fall into CDM potential wells

17 TEMPERATUREANISOTROPIESFROMSCALAR PERTURBATIONS

18 CMB SPECTRUMANDDIPOLEANISOTROPY

• Microwave background almost perfect blackbody radiation – Temp. (COBE-FIRAS) 2.725 K

• Dipole anisotropy ∆T/T = β cos θ implies solar-system barycenter has velocity 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 with arcmin resolution

19 THERMALHISTORY — DETAILS

• Dominant element hydrogen recombines rapidly around z ≈ 1100 – Prior to recombination, Thomson scattering efﬁcient and mean free path 1/(neσT ) short cf. expansion time 1/H – Little chance of scattering after recombination → photons free stream keeping imprint of conditions on last scattering surface

• Optical depth back to (conformal) time η for Thomson scattering: Z η 0 0 τ(η) = aneσT dη η

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

• Visibility is probability density for last scattering at η:

visibility(η) = −τe˙ −τ

20 LINEARANISOTROPYGENERATION: REDSHIFTING

• Recall perturbed metric:

ds2 = a2(η)[(1 + 2ψ)dη2 − (1 − 2φ)dx2]

• Introduce orthonormal frame of vectors: µ −1 µ µ −1 µ (E0) = a (1 − ψ)δ0, (Ei) = a (1 + φ)δi

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

pµ = a−2[1 − ψ, (1 + φ)e]

µ ν • Geodesic equation p ∇µp = 0 becomes (overdot ≡ ∂/∂η)

d ln /dη = −dψ/dη + (ψ˙ + φ˙) de/dη = −(∇ − ee · ∇)(ψ + φ) dx/dη = (1 + φ + ψ)e

• Note, and e constant if no perturbations

21 BOLTZMANN EQUATION

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

d ln dΘ

(∂η + e · ∇) Θ − = | {z } dη dη ≡dΘ/dη scatt. 2 • Thomson scattering (kBT mec ) around recombination and reionization dominant scattering mechanism to affect CMB:

dΘ 3an σ Z e T 2 = −aneσT Θ + dmˆ Θ(, mˆ )[1 + (e · mˆ ) ] + aneσT e · vb dη | {z } 16π | {z } scatt. out-scattering | {z } Doppler in-scattering

• Doppler effect arises from electron bulk velocity vb

– Enhances ∆T/T for vb towards observer

• Neglecting anisotropic nature of Thomson scattering (1 + (e · mˆ )2 → 4/3): dΘ dψ + − φ˙ + ψ˙ ≈ −aneσ (Θ − Θ0 − e · v ) dη dη T b

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

22 TEMPERATUREANISOTROPIES

• Formal solution (with integrating factor e−τ ): Z η0 Z η0 [Θ(ˆn) + ψ] = − τe˙ −τ (Θ + ψ + e · v ) dη0 + e−τ φ˙ + ψ˙ dη0 (η0,x0) 0 b where integrals are along background line of sight and nˆ = −e −τ • On degree scales, can approximate −τe˙ ≈ δ(η − η∗) ignoring reionization: Z R ˙ ˙ [Θ(ˆn) + ψ](η ,x ) = Θ0|∗ + ψ|∗ + e · vb|∗ + (ψ + φ) dη 0 0 | {z } |{z} | {z } ∗ temp. gravity Doppler | {z } 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 late times (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 omissions

23 SPATIAL-TO-ANGULARPROJECTION

• Consider angular projection at origin of potential ψ(η∗, x) over last-scattering surface; for a single Fourier component ik·nˆχ∗ ψ(ˆn) = ψ(η∗, χ∗nˆ) = ψ(η∗, k)e where χ∗ = ∆η = η0 − η∗ X l ∗ ˆ = ψ(η∗, k) 4πi jl(kχ∗)Ylm(ˆn)Ylm(k) lm l ∗ ˆ ψlm = 4πψ(η∗, k)i jl(kχ∗)Ylm(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∆η> l

k k∆η=l

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

24 ANGULARPOWERSPECTRUM

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

ˆ ik·nˆχ∗ e · vb|∗ = −nˆ · [ikvb(η∗, k)]e X l 0 ∗ ˆ = −vb(η∗, k) 4πi jl(kχ∗)Ylm(ˆn)Ylm(k) lm • Hence multipoles of Θ, ignoring ISW, are l h 0 i ∗ ˆ Θlm = 4πi (Θ0 + ψ)(η∗, k)jl(kχ∗) − vb(η∗, k)jl(kχ∗) Ylm(k)

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

• CMB anisotropies then statistically isotropic with power spectrum

Z " #2 (Θ0 + ψ)(η∗, k) vb(η∗, k) 0 C = 4π d ln k j (kχ∗) − j (kχ∗) P (k) l R(k) l R(k) l R

• Θ0 + ψ term dominates; slowly varying in k cf. jl(kχ∗) so " #2 l(l + 1) (Θ0 + ψ) (η∗, k) C ≈ P (k)(k = l/χ∗) 2π l R(k) R

25 ACOUSTICPHYSICS

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

• Large-scale approximation: ignore diffusion and slip between CMB and baryon bulk velocities (requires scattering rate k) – Photon–baryon plasma behaves like perfect ﬂuid responding to gravity (drives infall to wells), Hubble drag of baryons, gravitational redshifting and photon pressure (resists infall):

Θ¨ + HR Θ˙ + 1 2Θ = ¨ + HR ˙ 1 2 0 1+R 0 3(1+R)k 0 φ 1+Rφ − 3k ψ | {z } |{z} | {z } Hubble drag | pressure{z } redshift infall – R ≡ 3ρb/(4ργ) ∝ a is fraction of momentum density from baryons: 4 4 q = ρ¯γvγ +ρ ¯ v ≈ ρ¯γ(1 + R)v 3 b b 3 b – WKB solutions of homogeneous oscillator equation: −1/4 −1/4 (1 + R) cos krs , (1 + R) sin krs R η dη0 with sound horizon rs ≡ √ 0 3(1+R) 26 ACOUSTIC OSCILLATIONS

HR 1 2 HR 1 2 Θ¨ 0 + Θ˙ 0 + k Θ0 = φ¨ + φ˙ − k ψ 1 + R 3(1 + R) 1 + R 3

• Already solved this in radiation domination (R → 0 and taking φ = ψ) < – At end of radiation domination, ηeq,

1 < 3R(k) kηeq 1 Θ0(ηeq, k) = −R(k) cos krs kηeq 1 < −2 and ψ(ηeq, k) = −2R(k)/3 for kηeq 1 and O(kηeq) 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 on super-Hubble scales to conserve R (and Θ0 follows), have  1 −5R(k) [(1 + 3R) cos krs − 3(1 + R)] kηeq 1 Θ0(η, k) = −R(k) cos krs kηeq 1

27 CONDITIONS AT LAST SCATTERING

 1 −5R(k) [(1 + 3R) cos krs(η∗) − 3R] kηeq 1 (Θ0 + ψ)(η∗, k) = −R(k) cos krs(η∗) kηeq 1

• Modes with krs(η∗) = nπ are at extrema at last scattering ⇒ acoustic peaks in power spectrum • Baryon-dependent offset that disappears at high k modulates even/odd peaks • Resonant driving boosts amplitude by ∼ 5 on small scales (but have ignored diffusion damping!)

28 ADIABATICANISOTROPYPOWERSPECTRUM

• Temperature power spectrum for scale-invariant curvature ﬂuctuations

29 COMPLICATION 1: PHOTON DIFFUSION

• Photons diffuse out of dense regions damping inhomogeneities in Θ0 (and creating higher moments of Θ) −1 – In time dη, when mean-free path `P = (aneσT ) = 1/|τ˙|, photon random walks mean square distance `P dη – Deﬁnes a diffusion length by last scattering: Z η −2 ∗ −1 2 −1/2 2 −1 k ∼ |τ˙| dη ∝ (Ωmh ) (Ωbh ) D 0

• Get exponential suppression of photons (and baryons) −k2/k2 Θ0 ∝ e D cos krs −1 on scales below kD ≈ 7 Mpc at last scattering

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

30 DAMPINGOFTHE CMB POWERSPECTRUM

−2l2/l2 • Diffusion damping implies e D 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 depth = R η0 τ η∗ aneσT dη – Produces a further low redshift peak in the visibility function (important for polarization – 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 −τ −τ Θ(e) + ψ → e [Θ(e) + ψ] + (1 − e )(Θ0 + ψ)

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

– Well inside horizon, Θ0 + ψ ≈ 0 and observed anisotropies −τ −2τ Θ(ˆn) → e Θ(ˆn) ⇒ Cl → e Cl

33 COSMOLOGYFROM CMB TEMPERATUREPOWERSPECTRUM

34 PARAMETERSFROM 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) 2 2 ∗ rs, keq and kD depend only on Ωbh and Ωmh 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 diameter distance to last-scattering dA, e.g., θs ∝ rs/dA 2 2 – Observable ratios, e.g., θs/θeq ∝ rskeq determine Ωmh once Ωbh ﬁxed from peak morphology

– Calibrated ruler rs then accurately determines dA

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

35 PARAMETERSFROM CMB: PRIMORDIALPOWERSPECTRUM

−2τ • Scalar power spectrum Cl essentially e PR(k) at k ≈ l/dA processed by acoustic physics

ns−1 – Inﬂation predicts almost scale-invariant power-law PR(k) = As(k/k0) – CMB probes scales 5 Mpc < k−1 < 5000 Mpc

36 ACOUSTICPEAKHEIGHTS: BARYON DENSITY

2 • Most distinctive effect of increasing Ωbh is boost of compressional (1, 3, etc.) low-order peaks cf. rarefaction peaks 2 • Planck: Ωbh = 0.02222 ± 0.00023 (i.e., to 1%)

2 Increasing Ωbh 6

37 ACOUSTICPEAKHEIGHTS: DARK MATTER DENSITY

2 • Unique effect of increase in Ωch is reduction in resonant driving of low-order peaks and early-ISW contribution to 1st peak 2 −1 2 2 – Since keq = aeqHeq ∝ (Ωmh ) (Ωmh ) , angular scale of equality relative 2 to peaks goes down if Ωmh increases: 2 0.75 θs/θeq = rskeq ∝ (Ωmh )

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

38 ACOUSTIC-SCALEDEGENERACYIN LCDM MODELS

• Peaks locations very well measured ⇒ θ∗ = rs/dA known to 0.05% precision! 3 • In LCDM, θ∗ is a function of mostly Ωmh – main (approximate) degeneracy in LCDM models – Along degeneracy direction, other parameters vary to try and maintain morphology of peaks

72 0.992

0.984 70 0.976

68 0.968 n 0 s H 0.960 66 0.952

64 0.944

0.936 0.26 0.30 0.34 0.38

Ωm

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 with different late-time parameters (ΩK, Ωde, w, . . .) in extended models if preserve dA – Disentangling these late-time parameters requires other datasets (e.g. Hubble, supernovae, shape of matter power spectrum or BAO) or CMB lensing (see later)

40 PARAMETERSFROMTHEDAMPINGTAIL: Neﬀ

• Effective number of relativistic (neutrino) degrees of freedom at recombination deﬁned by 4/3 ρν = Neﬀ(7/8)(4/11) ργ

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

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

θD 1 1 1/2 = ∝ = H∗ θ r k −1 1/2 ∗ s D H∗ H∗ – Increasing Neﬀ at ﬁxed θ∗ → damping kicks in at larger scales reducing power in damping tail

– Must also increase H0 to preserve θ∗

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

– Increased neutrino anisotropic stress with increased Neﬀ reduces initial amplitude 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 2 2 2 i j ds = a [dη − (δij + hij)dx dx ]

ij i – Trace-free δ hij = 0, transverse ∂ hij = 0

• Einstein equation (ij trace-free): ¨ ˙ 2 2 T hij + 2Hhij − ∇ hij = −16πGa Πij

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

– For k H, oscillator is over-damped and hij = const. or decays (as 1/a in radiation domination and a−3/2 in matter domination) – For k H use WKB solution to equation in form 2 2 ±ikη ∂η (ahij) + k − ¨a/a (ahij) = 0 ⇒ hij ∝ e /a

– Usual short wavelength description of gravitational waves with adiabatic decay of amplitude as 1/a −2 ij −4 – Note energy density ∼ a hh˙ ijh˙ i ∝ a like gas of massless gravitons

44 GRAVITATIONAL WAVES FROM INFLATION

• Quantum fluctuations in hij (note, semi-classical quantum gravity!) give primordial gravitational wave perturbations with power spectrum 2 Z 8 Hk ij Ph(k) = 2 where hhijh i = d ln k Ph(k) MPl 2π • Direct probe of H during inflation and hence energy scale of inflation where 2 4 2 H = Einf/(3MPl): 4 128 E ! E 4 ( ) √ inf = 1 93 10−11 inf Ph k ≈ . × 16 3 8πMPl 10 GeV

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

d ln Ph(k) 2 dH nt = ≈ = −2 d ln k H2 dt

• Conventional to use tensor-to-scalar ratio P (k ) r ≡ h 0 PR(k0)

45 GRAVITATIONAL WAVES AND THE CMB

• Evolution of comoving energy from shear of gravitational wave: 1 ˙ i j d ln /dη + 2hije e = 0

• Neglecting anisotropic scattering, Boltzmann equation gives 1 Z η0 Θ(η , x , e) = − e−τ h˙ eiej dη 0 0 2 ij −1 • Only signiﬁcant contribution on large scales since hij decays like a after entering horizon – Planck constraint r < 0.11 as good as will ever do with TT

46 INTRODUCTIONTO CMB POLARIZATION

47 POLARIZATION AND STOKES PARAMETERS

• For quasi-monochromatic plane wave along z, correlation tensor of electric ﬁeld E deﬁnes (transverse) polarization tensor: ∗ ∗ ! ! hExExi hExEyi 1 I + QU + iV ∗ ∗ ≡ hEyExi hEyEyi 2 U − iV I − Q • Thomson scattering of CMB anisotropies generates linear polarization described by symmetric trace-free part of correlation: ! ! 1h|E |2 − |E |2i h<(E E∗)i 1 QU P ≡ 2 x y x y ≡ ab ∗ 1 2 2 U −Q h<(ExEy)i −2h|Ex| − |Ey| i 2

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 hEθÊθˆ − EφÊφî ± ihEθÊφˆ EφÊθî = ( )( ∗ ∗) h Eθˆ ± iEφˆ Eθˆ ± iEφˆ i

– Q ± iU are components of P on null basis m± ≡ ˆθ ± iφˆ: ∗ a b Q ± iU = m± · hE ⊗ E i · m± = m±m±Pab

• A quantity sη is spin s if isψ ±iψ sη → e sη for m± → e m±

– 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 b = ∇aVE + a∇bVB

• Consider spin ±1 components of V on null basis b a b – Since am± = ±im± have

a m± · V = m±∇a (VE ± iVB) = (∂θ ± icosecθ∂φ)(VE ± iVB)

• Deﬁne spin-weight derivatives via s −s ðsη = − sin θ(∂θ + icosecθ∂φ)(sin θsη) ¯ −s s ðsη = − sin θ(∂θ − icosecθ∂φ)(sin 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 c Pab(e) = ∇ha∇biPE + (a∇b)∇cPB

a b • Evaluating null components of covariant derivatives (recall m±m±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: r r ( ) = X (l−2)! ( ) ( ) = X (l−2)! ( ) PE e (l+2)!ElmYlm e ,PB e (l+2)!BlmYlm e lm lm – l-dependent factors “undo” ∼ l2 factors from double derivatives to give r ( ) = X( ) (l−2)! ðð Q ± iU Elm ± iBlm (l+2)! ¯¯ Ylm lm ðð X = (Elm ± iBlm)±2Ylm lm

51 E- AND B-MODE PATTERNS

• Consider axisymmetric potentials PE(µ) and PB(µ) with µ ≡ cos θ: d2 Q ± iU = (1 − µ2) (P ± iP ) dµ2 E B – Follows that d2 E−mode B−mode Q = (1 − µ2) P dµ2 E d2 U = (1 − µ2) P dµ2 B – Axisymmetric E modes produce pure-Q polarization • 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 ∗ E hElmEl0m0i = Cl δll0δmm0

• For Gaussian ﬂuctuations all information in power spectrum Cl

• Under parity transformations, PE is a scalar but PB is a pseudo-scalar so l l Elm → (−1) Elm and Blm → −(−1) Blm

– Cannot have E-B or T -B correlations if parity respected in mean TT EE TE BB – Expect non-zero spectra: Cl , Cl , Cl and Cl

53 GENERATION OF POLARIZATION:THOMSON SCATTERING

Hot • Photon diffusion around recombination → local temperature quadrupole Cold – Subsequent Thomson scattering generates (par- . tial) linear polarization with r.m.s. ∼ 5 µK from density perturbations Polarization

• Thomson scattering of radiation quadrupole produces linear polarization (dimensionless temperature units!)  s  3 X 1 d(Q ± iU)(e) = aneσ dη Y (e) E − Θ  5 T ±2 2m 2m 6 2m |m|≤2 – Purely electric quadrupole (l = 2)

• Ignoring reionization, observed polarization at (η0, x0) generated at x∗ = x0 − χ∗e on last-scattering surface: √ 6 X (Q ± iU)(η , x , e) ≈ − Θ (η∗, x∗) Y (e) 0 0 10 2m ±2 2m |m|≤2 54 LINEAR SCALAR PERTURBATIONS PRODUCE ONLY E-MODES

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

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

• Observed polarization from this Fourier mode is

−ikχ∗ˆz·e (Q ± iU)(η0, x0, e) ∝ Θ20(η∗, kˆz)e ±2Y20(e) | {z } ∝ sin2 θ • Generated as axisymmetric pure-Q, and this preserved by plane-wave modulation:

Scatter Modulate

- -

Plane-wave scalar quadrupole Electric quadrupole (m = 0) Pure E mode 55 QUADRUPOLESOURCETERM

• Consider scales large compared to diffusion-damping scale – Temperature ﬂuctuation seen by electron determined by conditions at previous scattering `p away:

Θ(e) + ψ ≈ (Θ0 + ψ)(−`pe) + e · v(b)(−`pe) i 1 2 i j ≈ (Θ + ψ) − `pe ∂ (Θ + ψ) + ` e e ∂ ∂ (Θ + ψ) 0 i 0 2 p i j 0 i j + e · v(b) − `pe e ∂jv(b)i + ···

– Dominant temperature quadrupole from trace-free part of eiej components: X i j 1 ij h1 2 i Θ2mY2m(e) ∼ e e − 3δ 2`p∂i∂j(Θ0 + ψ) − `p∂jv(b)i m – 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 source generated causally

56 ORDEROFMAGNITUDEPOLARIZATION √ 6 X (Q ± iU)(η , x , e) ≈ − Θ (η∗, x∗) Y (e) 0 0 10 2m ±2 2m |m|≤2

• Follows that mean-square of observed polarization

2 2 ∗ 3 TT hQ + U i = h(Q + iU)(Q + iU) i = C (η∗) 40π 2 • Tight-coupled quadrupole around last scattering: X i j 1 ij Θ2mY2m(e) ∼ −`p e e − 3δ ∂iv(b)j m 5 TT 4 2 2 ⇒ C2 (η∗) ∼ `ph(∇ · vb) i 4π 45 η∗ ˙ • Continuity equation, δγ + 4∇ · vγ/3 = 0, and tight-coupling vb ≈ vγ gives 3 ˙ ∇ · vb ≈ −4δγ ≈ −3R(k)kcs sin krs k 2 2 Z D 2 9 2 2 ⇒ h(∇ · vb) i ∼ 9cs dk k sin krsPR(k) ∼ cs kDPR 0 4 2 −2 −9 • Noting that (kD`p) ∼ `p/η∗ ∼ 10 and PR ≈ 2 × 10 from TT , have 2 2 1 2 1 2 hQ + U i ∼ 250(kD`p) PR ∼ 250(`p/η∗)PR ∼ (few µK) 57 SCALARPOLARIZATIONPOWERSPECTRA

• Polarization mostly probing vb at last scattering E T – Cl peaks at minima of Cl • Correlations between T and E

• Additional large-angle polarization from scattering around reionization

• B-modes are generated at second order, 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 to l ∼ 2(η0 − ηre)/(ηre − η∗) – Amplitude of polarization ∝ optical depth through reionization → best way to measure 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 of sky (SPTpol, ACTPol, POLARBEAR, BICEP/Keck)

60 PLANCKPOWERSPECTRUMMEASUREMENTS: EE

100

15000 80 ] 2 K

µ 10000 60 5 − [10 5000 40 EE ` C 20 0

0 5000 4 EE ` 0 0 C ∆ -4 -5000 2 10 30 500 1000 1500 2000 `

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

61 PLANCKPOWERSPECTRUMMEASUREMENTS: TE

140

70 ] 2 K µ [ 0 TE ` D -70

-140

16 10 8 TE ` 0 0 D

∆ -8 -16 -10 2 10 30 500 1000 1500 2000 `

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 P ˙ i j • Free streaming through `p generates quadrupole m Θ2mY2m(e) ∼ `phije e

• For p-polarized wave (p = ±2 helicity states) with k = kˆz, have Θ2m ∝ δmp • Observed polarization from such a Fourier mode is −ikχ∗ˆz·e (Q ± iU)(η0, x0, e) ∝ Θ2p(η∗, kˆz)e ±2Y2p(e)

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

Modulate E mode

¡ ¡ ¡ ¡ Scatter ¡ ¡ ¡ ¡ ¡ -¡ H HH H HH H HH HHj B mode

Plane-wave tensor quadrupole Electric quadrupole (|m| = 2) 64 SCALARANDTENSORPOWERSPECTRA (r = 0.2)

65 GRAVITATIONAL WAVES AND B-MODES

• Tiny signal – r.m.s. < 200 nK – but not confused by dominant density perturbations like TT 66 CURRENTLIMITSON B-MODEPOWERSPECTRUM

2 10

DASI QUAD 1 CBI BICEP1 10 CAPMAP QUIET−Q Boomerang QUIET−W WMAP

0 10

]

[

π −1 /2 10

BB l

Polarbear

l(l+1)C −2 SPTpol 10 BK14 ACTpol CMB component

−3 10 r=0.05

lensing −4 r=0.01 10 1 2 3 10 10 10 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)