The physics of the cosmic microwave background

Transcription

1 The physics of the cosmic microwave background V(φ) φ Hiranya Peiris STFC Advanced Fellow University of Cambridge

2 Before we start... Acknowledgment: Anthony Challinor (University of Cambridge) for invaluable help in preparing these lectures. Disclaimers: Time constraints necessitate brevity. Lectures intended to give only a flavour of the physics. Absolutely necessary to work through details (with the aid of a good textbook & the primary literature). Recommended: Scott Dodelson, Modern Cosmology (2003)

3 Roadmap Lecture 1: The physics of the cosmic microwave background. Lecture 2: What have we learnt about the early universe?

4 The Cosmic Microwave Background (CMB) 60 K Credit: NASA/WMAP Science Team

5 Space-time and CMB Physics (NOT to scale) time past light cone Universe transparent COBE resolution 379,000 yrs Last scattering Universe opaque WMAP resolution Big Bang space

6 Thermal History CMB and matter plausibly produced during reheating at end of inflation CMB decouples around recombination, 300 kyr later Universe starts to reionize once first stars (?) form (somewhere in range z = 10 20) and 10% of CMB re-scatters K K K 3000 K Quarks -> Hadrons Nucleosynthesis Recombination s 10 Sec

7 Mapping the CMB WMAP3 internal-linear combination map (left) and BOOMERanG03 (right) Aim to answer in these lectures: What is the physics behind these images? What have we learned about cosmology from them?

8 CMB spectrum and dipole anisotropy Microwave background almost perfect blackbody radiation Temp. (COBE-FIRAS) K Dipole anisotropy T/T = β cos θ implies solar-system barycenter has velocity v/c β = relative to rest-frame of CMB Variance of intrinsic fluctuations first detected by COBE-DMR: ( T/T ) rms = 16µK smoothed on 7 scale

9 Details of the thermal history Dominant element hydrogen recombines rapidly around z 1000 Prior to recombination, Thomson scattering efficient and mean free path 1/(n e σ T ) 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: τ(η) = ηr η an e σ T dη e τ is prob. of no scattering back to η Visibility is probability of last scattering at η per dη: visibility(η) = τe τ η z 1100

10 Anisotropies and the power spectrum Decompose temperature anisotropies in spherical harmonics Θ T (ˆn)/T = lm a lm Y lm (ˆn) Under a rotation (R) of sky a lm D l mm (R)a lm Demanding statistical isotropy requires, for 2-point function a lm a l m = D l mm Dl m M a lm a l M Only possible (from unitarity D l Mm Dl Mm = δ mm ) if R a lm a l m = C l δ ll δ mm Symmetry restricts higher-order correlations also, but for Gaussian fluctuations all information in power spectrum C l Estimator for power spectrum Ĉ l = m a lm 2 /(2l + 1) has mean C l and cosmic variance var(ĉ l )= 2 2l +1 C2 l 7

11 Measured power spectrum

12 Coupled Einstein-Boltzmann Equations Neutrinos Photons Metric Dark Matter Thomson scattering Electrons Coulomb scattering Protons

13 Einstein Equation Relates the geometry to the energy: G µν R µν 1 2 g µνr =8πGT µν g µν R µν metric tensor: machine to transform coordinates into physical invariants. Ricci tensor: depends on metric and its derivatives via Christoffel symbols. R Ricci scalar: contraction of Ricci tensor, R g µν R µν G µν T µν Einstein tensor: describes geometry. Energy-momentum tensor: describes energy content. Given a metric, LHS: (i) compute Christoffel symbols (ii) form Ricci tensor (iii) contact to form Ricci scalar (iv) form Einstein tensor. Finally equate to RHS.

14 Perturbed FRW metric in CNG The Conformal Newtonian Gauge describes the perturbed FRW metric. η ds 2 = a 2 (η)[(1 + 2Ψ)dη 2 δ ij (1 2Φ)dx i dx j ] where = conformal time. Metric perturbations described by scalar potentials (Newtonian potential) and (perturbation to spatial curvature). Ψ Φ Only describes scalar perturbations. By the Decomposition Theorem, scalars, vectors and tensors evolve independently. Only scalars couple to matter.

15 Boltzmann Equation Want to describe distributions of photons and matter inhomogeneities. Photons and neutrinos are only fully described by distribution functions (DFs) in phase space. Energy-momentum tensor is expressed through integrals over momenta of the DFs. df dt = C[f] RHS: all possible collision terms The DF gives number of particles in (invariant) phase space volume d 3 xd 3 p : dn = f( x, p, t)d 3 xd 3 p Zeroth order: Bose-Einstein (bosons) and Fermi-Dirac (fermions) DFs. Collision terms due to Thomson or Coulomb scattering need to be computed for all species except cold dark matter.

16 Temperature anisotropies On degree scales, scattering time short c.f. wavelength of fluctuations and (local!) temperature is uniform plus dipole: Θ 0 + e v b Observed temperature anisotropy is snapshot of this at last scattering but modified by gravity: R [Θ(ˆn)+ψ] R = Θ 0 }{{ + ψ }}{{} + e v b }{{ + } ( ψ + φ) dη } {{} temp. gravity Doppler ISW with line of sight ˆn = e, and Θ 0 isotropic part of Θ Ignores anisotropic scattering, finite width of visibility function (i.e. last-scattering surface) and reionization Will fix these omissions shortly

17 Spatial to angular projection Consider angular projection at origin of potential ψ(x, η ) over last-scattering surface; for a single Fourier component ψ(ˆn) =ψ(ˆn η, η ) η η R η = ψ(k, η ) 4πi l j l (k η)y lm (ˆn)Y lm (ˆk) lm ψ lm 4πψ(k, η )i l j l (k η)ylm (ˆk) j l (k η) peaks when k η l but for given l considerable power from k > l/ η 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/ η where conformal distance to last scattering 14 Gpc 13

18 CMB as a sound wave Last scattering surface : snapshot of the photon-baryon fluid (graphic by W. Hu) On large scales : primordial ripples, purely GR Effects On smaller scales: Photons radiation pressure { } Gravity compression Smaller than photon mean free path: Exponentially damped by photon diffusion Forced damped harmonic oscillator Sound waves Stop oscillating at recombination Horizon size at last scattering surface is fundamental mode Peebles & Yu (1970), Sunyaev & Zel dovich (1970), Sachs and Wolfe (1968), Silk (1968)

19 SHO analogy I Consider simple harmonic oscillator with mass m, force constant k driven by external force F0. ẍ + k m x = F 0 m Assuming oscillator is initially at rest, x = A cos(ωt)+ F 0 mω 2 Peaks at F 0 t = nπ ω. ω = k m unforced: peak heights equal forced: odd (even) peaks higher (lower) forcing disparity greater for lower Even peaks correspond to negative x Figure from Dodelson, Modern Cosmology

20 SHO analogy II Cartoon gravitational instability: δ + [Pressure Gravity]δ = 0 Oscillator analogy: Θ 0 + k 2 c 2 sθ 0 = F F 0 c 2 s F is force due to gravity, is sound speed of entire photonbaryon fluid. Add more baryons; sound speed (frequency) goes down. Modes enter horizon and start to oscillate. See their phases frozen at recombination. Peaks are maximum amplitudes. Figure from Dodelson, Modern Cosmology

21 Acoustic physics Photon isotropic temperature Θ 0 and electron velocity v b 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 baryon pressure (resists infall): Θ 0 + HR 1+R Θ 0 }{{} Hubble drag where R 3ρ b /(4ρ γ ) a + 1 3(1+R) k2 Θ 0 }{{} pressure WKB solutions of homogeneous equation: with sound horizon r s η 0 = }{{} φ + 1+R HR φ 1 3 k2 ψ }{{} redshift (1 + R) 1/4 cos kr s, (1 + R) 1/4 sin kr s dη 3(1+R) sound speed c s = infall 1 3(1 + R) 14

22 Gravitational potential and acoustic driving For adiabatic initial conditions (e.g. simple inflation models), no relative perturbations between number densities of species Density perturbations of all species vanish on same hypersurface its curvature equals comoving curvature R on super-hubble scales Adiabatic driving term mimics cos kη Oscillator is resonantly driven inside sound horizon whilst CDM sub-dominant Potentials constant in matter domination then decay as DE dominates dominates 15

23 Acoustic oscillations: adiabatic models δ γ /4 Θ 0 starts out constant at ψ(0)/2 cosine oscillation cos kr s about equilibrium point (1 + R)ψ Modes with k η 0 c s dη = nπ are at extrema at last scattering acoustic peaks in power spectrum v b v γ follows from continuity equation (π/2 out of phase with Θ 0 so Doppler effect fills in zeroes of Θ 0 + ψ) 16

24 Adiabatic anisotropy power spectrum Temperature power spectrum for scale-invariant curvature fluctuations

25 Tightly coupled solution Elegant solution first due to Sugiyama & Hu (1995): Θ 0 (η)+φ(η) = [Θ 0 (0) + Φ(0)] cos(kr s ) + 1 η dη [Θ 0 (η )+Φ(η )] sin[k(r s (η) r s (η ))]. 3 0 Gets peak locations right and obtains odd/even height disparities fairly well. Neatly divides problem into (i) calculation of external gravitational potentials generated by CDM. (ii) effect of these potentials on the anisotropies. Clearly illustrates that cosine mode is the one excited by inflationary models.

26 Complications: photon diffusion Photons diffuse out of dense regions damping inhomogeneities in Θ 0 (and creating higher moments of Θ) In time dη, when mean-free path l =(an e σ T ) 1 =1/ τ, photon random walks mean square distance ldη Defines a diffusion length by last scattering: k 2 D η 0 τ 1 dη 0.2(Ω m h 2 ) 1/2 (Ω b h 2 ) 1 (a/a ) 5/2 Mpc 2 Get exponential suppression of photons (and baryons) Θ 0 e k2 /k 2 D cos kr s on scales below 30 Mpc at last scattering Implies e 2l2 /l 2 D damping tail in power spectrum 18

27 19 Integrated Sachs-Wolfe Effect Linear Θ ISW ( φ + ψ) dη from late-time dark-energy domination and residual radiation at η ; non-linear small-scale effect from collapsing structures In adiabatic models early ISW adds coherently with SW at first peak since Θ 0 + ψ ψ/2 same sign as ψ Late-time effect is large scale (integrated effect peak trough cancellation suppresses small scales) Late-time effect in dark-energy models produces positive correlation between large-scale CMB and LSS tracers for z<2

28 Reionization: effect of rescattering Lyman-α optical depth, as measured by quasar absorption spectra, rises rapidly around z 6 probing the end of the epoch of reionization CMB Thomson scatters off all (re-)ionized gas back to z with optical depth τ = η R η an e σ T dη Produces a further low redshift peak in the visibility function (important for polarization see later) 20

29 Reionization: effect of rescattering 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 )(Θ 0 + ψ) Outside horizon at reionization, Θ(e) Θ 0 and scattering has no effect Well inside horizon, Θ 0 + ψ 0 and observed anisotropies Θ(ˆn) e τ Θ(ˆn) C l e 2τ C l 21

30 Gravitational waves Tensor metric perturbations ds 2 = a 2 [dη 2 (δ ij + h ij )dx i dx j ] with δ ij h ij =0 Shear ḣ ij gives anisotropic redshifting Θ(ˆn) 1 2 dη ḣ ijˆn iˆn j Only contributes on large scales since h ij decays like a 1 after entering horizon 22

31 Isocurvature modes CDM isocurvature most physically motivated (perturb CDM relative to everything else) Starts off with δ γ (0) = φ(0) = 0 so matches onto sin kr s modes Temperature power spectrum for n iso =2entropy fluctuations (CDM isocurvature mode) 23

32 CMB polarization: Stokes parameters For plane wave along z, symmetric trace-free correlation tensor of electric field E defines (transverse) linear polarization tensor: ( 12 E 2 P ab x E2 y E ) xe y E x E y 1 2 E2 x E2 y 1 ( ) Q U 2 U 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 Q + iu is spin -2

33 E and B modes Decomposition into E and B modes (use θ, φ basis to define Q and U) P ab (ˆn) = a b P E + ɛ c (a b) cp B Q + iu = ð ð(p E ip B ) Spin-lowering operator: ð s η = sin s θ( θ icosec φ )(sin s θ s η) Expand P E and P B in spherical harmonics, e.g. P E (ˆn) = (l 2)! (l+2)! E lmy lm (ˆn) (Q ± iu)(ˆn) = (E lm ib lm ) 2 Y lm (ˆn) lm lm Spin-weight harmonics s Y lm provide orthonormal basis for spin-s functions Only three power spectra if parity respected in mean: C E l, CB l and C TE l

34 CMB polarization: Thomson scattering Photon diffusion around recombination local temperature quadrupole Subsequent Thomson scattering generates (partial) linear polarization with r.m.s. 5 µk from density perturbations Hot. Cold Polarization Thomson scattering of radiation quadrupole produces linear polarization (dimensionless temperature units!) d(q ± iu)(e) = 3 5 an eσ T dη m ±2 Y 2m (e) E 2m Purely electric quadrupole (l =2) 1 6 Θ 2m In linear theory, generated Q + iu then conserved for free-streaming radiation Suppressed by e τ if further scattering at reionization

35 CMB polarization: scalar perturbations Single plane wave of scalar perturbation has Θ 2m Y 2m (ˆk) with ˆk along z, dq sin 2 θ and du =0 Scatter Modulate Plane-wave scalar quadrupole Electric quadrupole (m =0) Pure E mode Linear scalar perturbations produce only E-mode polarization Mainly traces baryon velocity at recombination peaks at troughs of T 27

36 CMB polarization: tensor perturbations For single +-polarized gravity wave with ˆk along z, Θ 2m δ m2 + δ m 2 so dq (1 + cos 2 θ) cos 2φ and du cos θ sin 2φ Modulate E mode Scatter B mode Plane-wave tensor quadrupole Electric quadrupole ( m =2) Gravity waves produce both E- and B-mode polarization (with roughly equal power) 28

37 Correlated polarization in real space On largest scales, infall into potential wells at last scattering generates e.g. tangential polarization around large-scale hot spots Sign of correlation scale-dependent inside horizon 29

38 Polarization from reionization (Q ± iu)(ˆn) = 6 10 dd dτ dd e τ(d) 2 m= 2 T 2m (Dˆn) ±2 Y 2m (ˆn) x!dn x Dn observer D rec reionization recombination

39 30 Large angle polarization from reionization Temperature quadrupole at reionization peaks around k(η re η ) 2 Re-scattering generates polarization on this linear scale projects to l 2(η 0 η re )/(η re η ) Amplitude of polarization optical depth through reionization best way to measure τ with CMB

40 Scalar & tensor power spectra For scalar perturbations (left), δ γ oscillates π/2 out of phase with v γ C E l peaks at minima of C T l scalar tensor 31

41 Parameters from CMB: matter & geometry Acoustic physics (dark energy and curvature negligible): Peak locations depend on sound horizon r s at last scattering Damping scale 1/k D (roughly geometric mean of horizon and mean free path) Both depend only on Ω b H 2 0 and Ω mh 2 0 for fixed T CMB Peak heights depend on baryon loading (Ω b H0 2 ) and gravitational driving (Ω m H0 2; see shortly) r s and k D then calibrated standard rulers Main influence of geometry, dark energy and sub-ev massive neutrinos then through angular diameter distance to last scattering d A accurately determined from angular size of standard rulers r s and k D Weak influence on large scales (where cosmic variance bad) through ISW

42 Parameters from CMB: primordial power spectrum Scalar power spectrum C l essentially e 2τ P R (k) at k l/d A processed by acoustic physics CMB probes scales 5 Mpc <k 1 < 5000 Mpc Tensor power spectra sensitive to e 2τ P h (k)

43 Degeneracies Some parameters not determined by linear T anisotropies alone: Angular diameter test gives only d A = d A (Ω K, Ω de, w,...) once matter densities determined from peak morphology Disentangling dark energy and K relies on large-scale anisotropies, where cosmic variance large, or other datasets (e.g. Hubble, supernovae, shape of matter power spectrum or baryon oscillations) Addition of gravity waves and renormalisation mimics reionization but can break with polarization 35

44 Current temperature data Sachs-Wolfe Plateau and late-time ISW effect Acoustic peaks at adiabatic locations Damping tail/photon diffusion Weak gravitational lensing (see later)

45 Current polarization data Acoustic peaks at adiabatic locations E-mode polarization and cross-correlation with T Large-angle polarization from reionization 37

46 Acoustic peak heights: baryon density Peak spacing fixed by r s (Ω m h 2, Ω b h 2 ) and angular diameter distance d A Θ 0 oscillates with midpoint (1 + R)ψ so Θ 0 + ψ (S-W source) oscillates around Rψ Increasing Ω b h 2 (hence R) boosts compressional peaks (1, 3 etc. for adiabatic) and reduces r s Current constraints from CMB alone (weak priors): Ω b h 2 = ± (i.e. to 3%; Dunkley et al. 2008) Should improve to sub-percent level with Planck data 38

47 Acoustic peak heights: CDM density Increasing Ω c h 2 reduces d A and shifts matter-radiation equality to earlier times Reduces resonant driving φ for low-order peaks and reduces early-isw contribution to 1st peak Current constraints from CMB alone (weak priors): Ω c h 2 = ± (i.e. to 6%; Dunkley et al. 2008) Should improve to sub-percent level with Planck data 39

48 Peak locations: curvature & dark energy Mainly affect CMB through d A ; small effects from ISW and mode quantisation for K>0 CMB alone only well constrains d A = 14.1 ± 0.2 Gpc Λ =0, closed models fit CMB alone but have very low h, high Ω m h cf. LSS, and don t fit ISW-LSS correlation (see later) WMAP5 plus BAO gives Ω K = ± (w = 1) and Ω Λ =0.734 ±

49 Reionization WMAP5 EE large-angle correlation τ = ± (Dunkley et al. 2008) Requires aggressive cleaning of polarized Galactic foregrounds (synchrotron and thermal dust emission)

50 CMB spectrum: parameter dependences 100 (a) Curvature (b) Dark Energy (c) Baryons (d) Matter Image from W. Hu

51 Timeline Scale a(t) reheating BBN recombination dark ages reionization galaxy formation 21 cm dark energy Ia LSS QSO BAO Inflation CMB Lyα lensing gravitational waves B-mode polarization? s GeV 3 min 1 MeV 4 10 neutrinos 370, billion 1, ev 1 mev Time [years] Redshift Energy Image from D. Baumann