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 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 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 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 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 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 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 ∗ ^ 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) 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 ¯ X Θ ≡ ∆T (^n)=T = almYlm(^n) 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 8R 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 ^ P 2 • Estimator for power spectrum Cl = m jalmj =(2l + 1) has mean Cl and cosmic variance 2 var(C^ ) = C2 l 2l + 1 l 11 MEASURED POWER SPECTRUM 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 REMINDER OF RELATIVISTIC COSMOLOGICAL PERTURBATION THEORY 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: 2 2 2 i i j ds = a (η)f(1 + 2 )dη − 2Bidx dη − [(1 − 2φ)δij + 2Eij]dx dx g – Here, using conformal time dη ≡ dt=a and comoving coordinates xi i – 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 + 2 )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) = −4r2R(η; x) • In terms of CNG potentials H(φ_ + H ) 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 = 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 p • Eliminate δγ ≡ δργ=ρ¯γ to get damped SHO with sound speed 1= 3: 4 k2 φ¨ + φ_ + φ = 0 η 3 p – Solution that is constant outside sound horizon (kη 3): p j (kη= 3) 2 h i φ(η; k) = −2R(k) 1 p = − R(k) 1 + O(kη)2 kη= 3 3 p p p – 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 p p • For kη 3, δγ(η; k) = −4R(k) cos(kη= 3) ! constant amplitude oscillation with resonantly-enhanced amplitude 16 SUMMARY OF PERTURBATION EVOLUTION • 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 TEMPERATURE ANISOTROPIES FROM SCALAR PERTURBATIONS 18 CMB SPECTRUM AND DIPOLE ANISOTROPY • Microwave background almost perfect blackbody 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 with arcmin resolution 19 THERMAL HISTORY — 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 LINEAR ANISOTROPY GENERATION: REDSHIFTING • Recall perturbed metric: ds2 = a2(η)[(1 + 2 )dη2 − (1 − 2φ)dx2] • Introduce orthonormal frame of vectors: µ −1 µ µ −1 µ (E0) = a (1 − )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 − ; (1 + φ)e] µ ν • Geodesic equation p rµp = 0 becomes (overdot ≡ @=@η) d ln /dη = −d /dη + ( _ + φ_) de/dη = −(r − ee · r)( + φ) dx/dη = (1 + φ + )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 · r) 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 + dm^ Θ(, m^ )[1 + (e · m^ ) ] + 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 · m^ )2 ! 4=3): dΘ d + − φ_ + _ ≈ −aneσ (Θ − Θ0 − e · v ) dη dη T b so scattering tends to isotropise in rest-frame of electrons: Θ ! Θ0 + e · vb 22 TEMPERATURE ANISOTROPIES • Formal solution (with integrating factor e−τ ): Z η0 Z η0 [Θ(^n) + ] = − τe_ −τ (Θ + + e · v ) dη0 + e−τ φ_ + _ dη0 (η0;x0) 0 b where integrals are along background line of sight and n^ = −e −τ • On degree scales, can approximate −τe_ ≈ δ(η − η∗) ignoring reionization: Z R _ _ [Θ(^n) + ](η ;x ) = Θ0j∗ + j∗ + e · vbj∗ + ( + 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