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

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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

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Source: NASA/WMAP Science Team Observations Theory

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

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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

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

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

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

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

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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.

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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

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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

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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

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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

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

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

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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

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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

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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-

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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

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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)

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

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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)

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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

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

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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

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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

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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

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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

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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

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Complications E/B mixing Lensing of the CMB

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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

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

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

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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

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

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

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

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Comparison with lowest order harmonic and flat results

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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

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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!

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

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