Thermal History of the Universe and the Cosmic Microwave Background. II. Structures in the Microwave Background Matthias Bartelmann Max Planck Institut für Astrophysik IMPRS Lecture, March 2003
Part 2: Structures in the Microwave Background 1. expectations 2. physical ingredients 3. primary physical mechanisms 4. description of fluctuations 5. secondary physical mechanisms 6. measurements 7. cosmological consistency 8. satellite experiments
1. Expectations 1. today s structures 2. growth factor 3. expected temperature fluctuations 4. necessity of dark matter
we see pronounced structures in the Universe today: galaxies, galaxy clusters, filaments, voids voids have typical diameters of 50Mpc density contrast of structures: δ ρ ρ ρ > 1 1.1. Today s Structures structures (should have) formed via gravitational collapse from seed fluctuations
gravitational instability picture: density contrast evolves according to δ + 2ȧ a δ = 4πGρδ two solutions; one growing, one decaying; growth factor: δ(a) = δ 0 D + (a) 1.2. Growth Factor simple case (Einstein-de Sitter universe): D + (a) = a = (1 + z) 1
1.3. Expected Temperature Fluctuations today δ 0 > 1; redshift of last-scattering surface z ls 1100; growth factor: δ ls 1 1 + z ls 10 3 relative density fluctuations of this order expected at last scattering suppose similar fluctuations in radiation energy density u: u T 4 δu u 4δT T relative temperature fluctuations of order milli-kelvin expected should have been observed in the 1970 s and 1980 s, but could not be found: What s wrong?
1.4. Necessity of Dark Matter suppose matter consists mainly of weakly interacting particles if so, most of the cosmic matter decoupled from radiation well before last scattering photon-baryon interaction prevented baryons from forming structures before last scattering, but weakly interacting matter could already start forming structures after recombination, baryons fell into existing potential wells much smaller temperature fluctuations compatible with δ > 1 today: δt T 10 5 expected, and found by COBE in 1992 level of CMB fluctuations strong argument for Dark Matter
2. Physical Ingredients 1. Boltzmann equation 2. curved spacetime 3. collision terms
2.1. Boltzmann Equation physics of CMB fluctuations: cosmic fluid consisting of photons, baryons, dark matter tight coupling between photons and (charged) baryons through Thomson scattering gravitational interaction between all components, dominated by dark matter description: Boltzmann equation for photon phase-space distribution f : f t + v f x Φ f v = C[ f ] integration over momentum space: eq. for temperature fluctuations
2.2. Curved Spacetime photons propagate along geodesic curves: d 2 x µ dλ + dx α dx β 2 Γµ αβ dλ dλ = 0 (Γ µ αβ : Christoffel symbols; λ: affine parameter) photon momentum can be written: p µ = dxµ dλ weakly perturbed metric (c 2 Φ: Newtonian potential): ds 2 = (1 + 2Φ)dt 2 a 2 (1 2Φ)d x 2 change of photon momentum: 1 dp p dt = with γ i = p i / p ( 1 da a dt Φ t + c ) a γi Φ x i
2.3. Collision Terms collisions between photons and electrons: Compton scattering photon energy at recombination: kt 0.3eV m e c 2 ; Compton scattering can be approximated by Thomson scattering: dσ dω = 3 16π σ T ( 1 + cos 2 θ ) collision term: C[ f ] = 3 16π σ Tcn e ( f f )(1 + cos 2 θ)d 2 Ω additional equations describing matter flow: hydrodynamic equations (continuity, Euler equation)
3. Primary Physical Mechanisms 1. Sachs-Wolfe effect 2. acoustic oscillations 3. Silk damping 4. polarisation
3.1. Sachs-Wolfe Effect dark matter was already forming structures at recombination: gravitational potential wells were forming at recombination, photons were released in a hilly landscape : some had to climb out of potential wells, other could run down from potential hills potential wells: overdensities, energy loss, lower temperature; potential hills: higher temperature Sachs & Wolfe: δt T = 1 3 δφ c 2
3.2. Acoustic Oscillations gravity exerted by matter overdensities wants to compress cosmic fluids photons and baryons build up counter-acting pressure contraction until pressure wins, expansion until gravity takes over: acoustic oscillations set in sound speed very high because photon-baryon ratio is so high: c s = c 3 0.6c sound horizon defines maximum size of oscillating structures: r s (t) = c s t phase correlation: structures of size r s (t) start oscillating together at time t
3.3. Silk Damping photon free streaming counter-acts structure formation in baryon gas baryon structures which are too small are damped by photon diffusion: Silk damping diffusion scale: mean free path: λ D Nλ λ (n e σ T ) 1 number of collisions: dn = n e σ T cdt = dτ dt dt hence the diffusion scale is: λ 2 D = Silk damping: λ 2dτ dt dt = ) exp ( k2 kd 2 cdt n e σ T, k D = 2π λ D
3.4. Polarisation Compton scattering is anisotropic electron irradiated by unpolarised light with anisotropic intensity emits linearly polarised light; quadrupole anisotropy needed CMB expected to be linearly polarised; polarisation fluctuations 10% of temperature fluctuations
4. Description of Fluctuations 1. expansion into spherical harmonics 2. power spectrum 3. examples 4. sensitivity to cosmological parameters
4.1. Expansion into Spherical Harmonics description of structures in the CMB: harmonic expansion T (θ,φ) = l,m a lm Y m l (θ, φ) ( Fourier transform on the sphere ) rotational invariance: expansion coefficients statistically independent of m, average: C l = a lm a lm m C l is the variance of structures on scale l, or angular scale φ π l 180 l if temperature fluctuations are Gaussian, C l contains complete information cosmic variance: on scale l, there are l(l + 1) independent modes
4.2. Power Spectrum similar to Fourier space, C l is called power spectrum ; characteristic shape expected: Sachs-Wolfe part on large scales (low l) onset of acoustic fluctuations where l corresponds to sound horizon at recombination Silk damping towards smaller structures (larger l)
4.3. Examples flat universe open universe
4.4. Sensitivity to Cosmological Parameters amplitude of Sachs-Wolfe tail, location of peaks, absolute peak heights, relative peak heights, amplitude of Silk-damping tail all depend on cosmological parameters main parameters: Ω 0, Ω B, Ω Λ, H 0, spatial curvature, amplitude and shape of dark-matter power spectrum, reionisation optical depth,... polarisation power spectrum contains partially independent information explains interest in accurate CMB measurements; information down to angular scales of 5 10 CMB should contain
5. Secondary Physical Mechanisms 1. Sunyaev-Zel dovich effect 2. emission by the Galaxy
5.1. Sunyaev-Zel dovich Effect hot gas in galaxy clusters Comptonscatters incoming CMB photons; quantified by Compton-y parameter photons gain energy, their number is unchanged; intensity of scattered radiation reduced at low, increased at high frequencies zero point at 217GHz independent of cluster temperature and redshift clusters cast shadows below, shine above 217 GHz
5.2. Emission by the Galaxy Galaxy contains (dipolar) magnetic field; hot electrons emit synchrotron radiation, power law falling from radio into microwave regime Galaxy contains warm dust, T 20 K; warmer than CMB, becomes important at frequencies 300GHz Galaxy contains clouds of ionised hydrogen ( HII regions ); their freefree emission drops from radio into microwave bands
6. Measurements 1. balloon experiments 2. ground-based experiments 3. current parameter estimates 4. example: curvature
Boomerang, Maxima and others: balloons carry telescopes into stratosphere; observe between hours and weeks detectors: bolometers, cooled to the milli-kelvin level observing frequencies: 90, 150, 400 GHz cover small fraction of sky (typically 1%) 6.1. Balloon Experiments
6.2. Ground-Based Experiments DASI (Degree Angular Scale Interferometer): 13 interferometer elements, operating between 26 36 GHz, bandwidth 1 GHz, angular resolution 20 ; located at South Pole CBI (Cosmic Background Imager): similar to DASI, but higher resolution (4.5 10 ); located at 5000m altitude in Chilean Andes VSA (Very Small Array): 14 interferometer elements, operating between 26 36 GHz, bandwidth 1.5GHz, angular resolution 12 30 ; located on Tenerife
6.3. Current Parameter Estimates
sound horizon at recombination sets maximum wavelength for acoustic oscillations; time until recombination t rec 400000yr; hence λ max = c 3 t rec 240000Lj 6.4. Example: Curvature apparent angular size of that physical size is determined by spatial curvature curvature can be read off location of first acoustic peak
7. Cosmological Consistency 1. microwave background results 2. supernovae of type Ia 3. gravitational lensing 4. galaxy clusters 5. baryon density
7.1. Microwave Background Results temperature T = 2.726 K, compatible with light-element abundances excellent black-body spectrum despite finite width of last-scattering shell dipole of milli-kelvin order, compatible with typical peculiar motions first acoustic peak clearly seen, second at 3-σ significance; curvature very accurately zero slope of dark-matter power spectrum compatible with inflationary models polarisation recently discovered by DASI; fully compatible with temperature fluctuations amplitude of primordial fluctuations compatible with structure formation from gravitational instability
binary systems: white dwarf plus giant star; matter flows from giant star to white dwarf; collapses once mass exceeds 1.4M (Chandrasekhar mass limit) detonation of fixed amount of explosives: fixed luminosity, standard candles ; distance can be inferred from apparent brightness result: expansion of Universe accelerates! 7.2. Supernovae of Type Ia
light from distant sources propagates through inhomogeneous matter distribution differential light deflection causes distortion; imprints distortion pattern on images of distant galaxies distant galaxies are very dense, 30arcmin 2 autocorrelation function of cosmic shear contains cosmological information 7.3. Gravitational Lensing
largest gravitationally bound objects in the Universe; form late in cosmic history formation time depends on cosmic matter density: if low, expansion is fast, and clusters have to form early if at all clusters at high redshift indicate early formation, thus low matter density cluster evolution argues for Ω 0 0.3 7.4. Galaxy Clusters
nucleosynthesis of light elements depends on expansion rate when the Universe was 3min old expansion rate determined by radiation density; critical parameter is baryon-photon-ratio η η can be determined from measured abundances of light elements; baryon density can be determined: Ω B h 2 0.024 ± 0.002 7.5. Baryon Density
8. Satellite Experiments 1. WMAP mission 2. WMAP first-year map and spectra 3. WMAP first-year parameters 4. Planck 5. expectations
8.1. WMAP Mission NASA satellite measuring full-sky CMB temperature maps; operating at outer Lagrangian point (L2) angular resolution 15 frequency coverage 23 94GHz operating very well; released in Feb. 2003 first results third sky coverage to be completed in Apr. 2003 (WMAP home page)
8.2. WMAP First-Year Map and Spectra Internal Linear Combination map: linearly combined, foregroundsubtracted signal from five channels (for visualisation only)
8.3. WMAP First-Year Parameters CMB temperature T CMB 2.275 ± 0.002K total density Ω tot 1.02 ± 0.02 matter density Ω m 0.27 ± 0.04 baryon density Ω b 0.044 ± 0.004 Hubble constant h 0.71 +0.04 0.03 baryon-to-photon ratio η 6.1 +0.3 0.2 10 10 fluctuation amplitude σ 8 0.84 ± 0.04 scalar spectral index n s 0.93 ± 0.03 decoupling redshift z dec 1089 ± 1 age of the Universe t 0 13.7 ± 0.2Gyr age at decoupling t dec 379 +8 7 kyr reionisation redshift (95% c.l.) z r 20 +10 9 reionisation optical depth τ 0.17 ± 0.04
8.4. Planck ESA satellite, launch scheduled for early 2007; to measure full-sky temperature and polarisation maps; also operating at L2 angular resolution 5 frequency coverage 30 857GHz wide frequency coverage very important for accurate foreground subtraction
8.5. Expectations verification of high-order acoustic peaks, Silk damping, polarisation power spectrum determination of all relevant cosmological parameters with relative accuracies 1% detection and quantification of astrophysical foregrounds: 30000 galaxy clusters through thermal Sunyaev-Zel dovich effect; 10 4 high-redshift galaxies; microwave emission of the Galaxy and of solar-system bodies consistency achieved so far is remarkable; cosmological framework model will be rigorously determined or confusion will set in