1. Primordial perturbations and precision cosmology from the Cosmic Microwave Background Antony Lewis CITA, University of Toronto http://cosmologist.info

2. Outline Introduction and current data Parameter estimation General primordial perturbations Current constraints CMB Polarization: E and B modes Future constraints Complications: E/B mixing; CMB lensing

3. Source: NASA/WMAP Science Team Observations Theory

4. Hu & White, Sci. Am., 290 44 (2004) Evolution of the universe Opaque Transparent

5. Perturbation evolution CMB monopole source till 380 000 yrs (last scattering), linear in conformal time scale invariant primordial adiabatic scalar spectrum photon/baryon plasma + dark matter, neutrinos Characteristic scales: sound wave travel distance; diffusion damping length

6. Hu & White, Sci. Am., 290 44 (2004) CMB temperature power spectrum Primordial perturbations + later physics diffusion damping acoustic oscillations primordial power spectrum

7. Source: NASA/WMAP Science Team O(10 -5 ) perturbations (+galaxy) Dipole (local motion) (almost) uniform 2.726K blackbody Observations: the microwave sky today

8. CMB observation history Source: NASA/WMAP Science Team + numerous balloon and ground based observations

9. WMAP + other CMB data Redhead et al: astro-ph/0402359 + Galaxy surveys, galaxy weak lensing, Hubble Space Telescope, supernovae, etc...

10. What can we learn from the CMB? Initial conditions What types of perturbations, power spectra, distribution function (Gaussian?); => learn about inflation or alternatives. What and how much stuff Matter densities (Ω b, Ω cdm ); ; neutrino mass Geometry and topology global curvature Ω K of universe; topology Evolution Expansion rate as function of time; reionization - Hubble constant H 0 ; dark energy evolution w = pressure/density Astrophysics S-Z effect (clusters), foregrounds, etc.

11. CMB C l and statistics Theory: Linear physics + Gaussian primordial fluctuations Theory prediction - variance (average over all possible sky realizations) ClCl CAMB: http://camb.info Initial conditions + cosmological parameters linearized GR + Boltzmann equations

12. Observations: only one sky Assume a lm gaussian: Cosmic Variance Use estimator for variance: - inverse gamma distribution (+ noise, sky cut, etc). WMAP low l l

13. Parameter Estimation Can compute P( {ө} | data) = P( C l ({ө}) | c l obs ) Often want marginalized constraints. e.g. BUT: Large n integrals very hard to compute! If we instead sample from P( {ө} | data) then it is easy: Can easily learn everything we need from set of samples

14. Markov Chain Monte Carlo sampling Metropolis-Hastings algorithm Number density of samples proportional to probability density At its best scales linearly with number of parameters (as opposed to exponentially for brute integration) CosmoMC code at http://cosmologist.info/cosmomc Lewis, Bridle: astro-ph/0205436http://cosmologist.info/cosmomc

15. CMB data alone color = optical depth Samples in 6D parameter space

16. Contaldi, Hoekstra, Lewis: astro-ph/0302435 e.g. CMB+galaxy lensing +BBN prior Plot number density of samples as function of parameters

17. Primordial Perturbations fluid at redshift < 10 9 Photons Nearly massless neutrinos Free-streaming (no scattering) after neutrino decoupling at z ~ 10 9 Baryons + electrons tightly coupled to photons by Thomson scattering Dark Matter Assume cold. Coupled only via gravity. Dark energy probably negligible early on

18. Perturbations O(10 -5 ) Linear evolution Fourier k mode evolves independently Scalar, vector, tensor modes evolve independently Various linearly independent solutions Scalar modes: Density perturbations, potential flows Vector modes: Vortical perturbations Tensor modes: Anisotropic space distortions – gravitational waves http://www.astro.cf.ac.uk/schools/6thFC2002/GravWaves/sld009.htm

19. General regular perturbation Scalar Adiabatic (observed) Matter density Cancelling matter density (unobservable) Neutrino density (contrived) Neutrino velocity (very contrived) Vector Neutrino vorticity (very contrived) Tensor Gravitational waves + irregular modes, neutrino n-pole modes, n-Tensor modes Rebhan and Schwarz: gr-qc/9403032 + other possible components, e.g. defects, magnetic fields, exotic stuff… General regular linear primordial perturbation -isocurvature-

20. Bridle, Lewis, Weller, Efstathiou: astro-ph/0302306 Adiabatic modes What is the primordial power spectrum? Parameters are primordial power spectrum bins P(k i ) + cosmological parameters On most scales P(k) ~ 2.3 x 10 -9 Close to scale invariant

21. Matter isocurvature modes Possible in two-field inflation models, e.g. curvaton scenario Curvaton model gives adiabatic + correlated CDM or baryon isocurvature, no tensors CDM, baryon isocurvature indistinguishable – differ only by cancelling matter mode Constrain B = ratio of matter isocurvature to adiabatic; n s = power law spectrum tilt No evidence, though still allowed. Not very well constrained. Gordon, Lewis: astro-ph/0212248 CDM = baryon + (CDM-baryon)

22. General isocurvature models General mixtures currently poorly constrained Bucher et al: astro-ph/0401417

23. Primordial Gravitational Waves (tensor modes) Well motivated by some inflationary models - Amplitude measures inflaton potential at horizon crossing - distinguish models of inflation Observation would rule out other models - ekpyrotic scenario predicts exponentially small amplitude - small also in many models of inflation, esp. two field e.g. curvaton Weakly constrained from CMB temperature anisotropy Look at CMB polarization: B-mode smoking gun - cosmic variance limited to 10% - degenerate with other parameters (tilt, reionization, etc)

24. CMB Polarization - - Q U Rank 2 trace free symmetric tensor Observe Stokes parameters Generated during last scattering (and reionization) by Thomson scattering of anisotropic photon distribution Hu astro-ph/9706147

25. E and B polarization gradient modes E polarization curl modes B polarization e.g.

26. Why polarization? E polarization from scalar, vector and tensor modes (constrain parameters, break degeneracies, reionization ) B polarization only from vector and tensor modes (curl grad = 0) + non-linear scalars smoking gun for primordial vector and tensor modes

27. CMB polarization from primordial gravitational waves (tensors) Adiabatic E-mode Tensor B-mode Tensor E-mode Planck noise (optimistic) Weak lensing Amplitude of tensors unknown Clear signal from B modes – there are none from scalar modes Tensor B is always small compared to adiabatic E Seljak, Zaldarriaga: astro-ph/9609169

28. Regular vector mode: neutrino vorticity mode logical possibility but unmotivated (contrived). Spectrum unknown. Lewis: astro-ph/0403583 Similar to gravitational wave spectrum on large scales: distinctive small scale B-modes

29. Pogosian, Tye, Wasserman, Wyman: hep-th/0304188 Topological defects Seljak, Pen, Turok: astro-ph/9704231 10% local strings from brane inflation: lensing r=0.1 global defects: Other B-modes? Non-Gaussian signals

30. Primordial inhomogeneous magnetic fields - Lorentz force on Baryons - Anisotropic stress sources vector and tensor metric perturbations e.g. Inhomogeneous field B = 3x10 -9 G, spectral index n = -2.9 Lewis, astro-ph/0406096. Subramanian, Seshadri, Barrow, astro-ph/0303014 Tensor amplitude uncertain. Non-Gaussian signal. vector tensor Banerjee and Jedamzik: astro-ph/0410032 Observable amplitudes probably already ruled out by cluster field observations

31. Complications E/B mixing Lensing of the CMB

32. Partial sky E/B separation problem Pure E: Pure B: Inversion non-trivial with boundaries Likely important as reionization signal same scale as galactic cut Use set of E/B/mixed harmonics that are orthogonal and complete over the observed section of the sphere. Project onto the `pure B modes to extract B. (Nearly) pure B modes do exist Lewis, Challinor, Turok astro-ph/0106536

33. Underlying B-modesPart-sky mix with scalar E Recovered B modes map of gravity waves Separation method Observation Lewis: astro-ph/0305545

34. Weak lensing of the CMB Last scattering surface Inhomogeneous universe - photons deflected Observer

35. Lensing potential and deflection angles LensPix sky simulation code: http://cosmologist.info/lenspix Lensing effect can be largely subtracted if only scalar modes + lensing present, but approximate and complicated (especially posterior statistics). Hirata, Seljak : astro-ph/0306354, Okamoto, Hu: astro-ph/0301031

36. Lensed CMB power spectra Few % on temperature 10% on TE/EE polarization New lensed BB signal How to calculate it accurately?

37. Series expansion method? Doesnt converge (though works surprisingly well given this plot!)

38. Accurate lensed C l calculation: correlation function method Need non-perturbative term; account for sky curvature Challinor and Lewis 2005

39. Comparison with lowest order harmonic and flat results

40. Planck (2007+) parameter constraint simulation (neglect non-Gaussianity of lensed field; BB noise dominated so no effect on parameters) Important effect, but using lensed CMB power spectrum gets right answer Lewis 2005LensPix lensed sky simulation code:http://cosmologist.info/lenspix

41. Conclusions CMB contains lots of useful information! - primordial perturbations + well understood physics (cosmological parameters) Precision cosmology - sampling methods used to constrain many parameters with full posterior distribution Currently no evidence for any deviations from standard near scale-invariant purely adiabatic primordial spectrum Large scale B-mode polarization from primordial gravitational waves: - energy scale of inflation - rule out most ekpyrotic and pure curvaton/ inhomogeneous reheating models and others Small scale B-modes - Strong signal from any vector vorticity modes, strong magnetic fields, topological defects Weak lensing of CMB : - B-modes potentially confuse primordial signals - Using lensed CMB power spectra good enough for precision parameter estimation with Planck Foregrounds, systematics, etc, may make things much more complicated!

42. http://CosmoCoffee.info arXiv paper discussion and comments