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Statistics of the CMB From Boltzmann equation of photons to power spectra 2-point statistics on the sphere: ML, quadratic estimators, polarisation-specifics Outline of the lectures

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Boltzmann equation of CMB Homogeneous solution Perturbed metric reads FLAT SCLICING GAUGE NEWTONIAN GAUGE Perturbed photon energy-momentum

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Boltzmann equation of photons Geodesic parametrization Geodesic equation of particles (interacting gravitationally only) Homogeneous evolution Unperturbed ======== background

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Perturbed Boltzmann equation Geodesic equation for the energy, in perturbed metric Boltzmann equation for perturbed distribution, in perturbed background Collisional cross-section is frequency independent: can integrate over frequency: STF TENSORS

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SVT, STF, Spherical harmonics… Vector field: potential plus solenoidal: STF tensor of rank 2: NORMAL MODES Generalization: Fourier space, with k=e 3

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Thomson scattering term (temperature) COLLISION TERM THOMSON PHASE FUNCTION Energy as seen by observer comoving with baryons/photons fluid

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Gauge-invariant phase-space density perturbation Gauge-invariant Boltzmann equation reads (Newtonian gauge)

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USING THE FOLLOWING Monopole is unaffected by scatteringForward photons are scattered awayBaryon-photon drag Anisotropic pressure Temperature hierarchy, scalar modes TRANSPORT GRAVITYTHOMSON SCATTERING

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Boltzmann hierarchy, tensor modes

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Einstein and conservation equations Scalar modes, Einstein equations Constraint equations (Poisson) Evolution equations Scalar modes, conservation equations Energy Momentum (Euler) Tensor modes, Einstein equation

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DO YOU (REALLY) THINK THIS IS IT ?? NOT YET…. POLARISATION OK, I’ll make it soft !

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Polarisation Due to quadrupolar anisotropy in the electron rest frame Linked to velocity field gradients at recombination

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E and B modes of polarisation Scalar quantity Pseudo-scalar quantity Scalar perturbations cannot produce B modes B modes are model-independent tracers of tensor perturbations

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Normal modes As for temperature, we have normal modes for polarisation Temperature and polarisation get decomposed on these modes are gauge-invariant (Stewart-Walker lemma)

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Boltzmann equation for Stokes Q,U Stokes parameters are absent in unperturbed background Their evolution does not couple to metric perturbations at linear order Redefining SIMPLE, ISN’T IT ?

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Polarized scattering term SCATTERING GEOMETRY

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Boltzmann polarization hierarchy As for the temperature case, express gradient term in terms of spherical harmonics Using the following recurrence formula: SCALARS DO NOT PRODUCE B MODES ONLY E-MODES COUPLE TO TEMPERATURE QUADRUPOLE

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Interpretation Normal modes and integral solutions Develop the plane wave into radial modes Using recurrence relations of spherical Bessels: State of definite total angular momentum results in a weighted sum of PLANE WAVE MODULATION SOURCE DEPENDANCE

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Normal modes and integral solutions These normal modes are the solutions of the equations of free-streaming !! (Boltzmann equation without gravity and collisions) Line-of-sight integration codes Sources depend on monopole, dipole and quadrupole only Linear dependance in the primordial perturbations amplitudes

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CMB power spectra Wayne Hu

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CMB imaging: scanning experiments Time-response of the instrument (detector + electronics) EM filters band-passAngular response: beam and scanning strategy Detector noise Simplified linear model (pixelized sky) Archeops, Kiruna BICEP focal planeSpider web bolometer

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Imagers: map-making BAYES theorem Linear data model Sufficient statistics Covariance matrix of the map Uniform signal prior Huge linear system to solve: use iterative methods (PCG) + FFTs

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Imagers: power spectrum Signal covariance matrix BAYES again… Marginalize over the map TO BE MAXIMIZED WITH RESPECT TO POWER SPECTRUM

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Imagers: power spectrum (cont.) Second order Taylor expansion For each iteration and each band, N pix 3 operation scaling !! PSEUDO-NEWTON (FISHER)

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Imagers: too many pixels ! New (fast) analysis methods needed Fast harmonic transforms Heuristically weighted maps Quite ugly at first sight !!

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Imagers (cont.) Power spectrum expectation value…simplifies, after summation over angles (m):

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Imagers: “Master” method Finite sky coverage loss of spectral resolution need to regularize inversion MC estimation of covariance matrix of PS estimates Spectral binning of the kernel Unbiased estimator Works also for polarization (easier regularization on correlation function)

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Imagers: polarised map-making One polarised detector (i) Let us consider n measurements of the same pixel, indexed by their angle ML solution

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Polarisation: optimal configurations Assume uncorrelated and equal variance measurements, look for optimal configuration of angles : General expression of the covariance matrix Stokes parameters errors are uncorrelated Covariance determinant is minimized

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Imagers: polarised spectrum estimation Stokes parameter in the great circle basis

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Polarisation: correlation functions Polynomials in cos( ): integrate exactly with Gauss-Legendre quadrature

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Polarisation: (fast) CF estimators Heuristic weighting (w P,w T ): Normalization: correlation function of the weights Using for m=n=2 involves with Weighted polarization field Using We get

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Polarisation: (fast) CF and PS estimators Define the pseudo-Cls estimates: These can be computed using fast SPH transforms in O(n pix 3/2 ) (compare to o(n pix 3 ) scaling of ML…) If CF measured at all angles: integrate with GL quadrature Assuming parity invariance

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Polarisation: CF estimators on finite surveys Incomplete measurement of correlation function: apodizing function f( ): Normalization of the window functions Results in E/B modes leakage

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Polarisation: E/B coupling of cut-sky Leakage window functions (not normalized) Recovered BB spectra (dots) No correlation function information over max =20 ±

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Polarisation: E/B coupling of cut-sky Leakage window functions (not normalized) Recovered BB spectra (dots) No correlation function over Gaussian apodization

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Polarisation: E/B leakage correction Define: Then: As a function of + We have obtained pure E and B spectra (in the mean)

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Quadratic estimators: covariances RAPPELS Edge-corrected estimators covariances in terms of pseudo-Cls covariances As long as M ll’ is invertible, same information content in edge- corrected Cls and pseudo-Cls

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Pseudo-Cls estimators: cosmic variance Forget noise for the moment, consider signal only: Case of high ells and/or almost full sky If simple weighting (zeros and ones)

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The case of interferometers CBI – Atacama desert

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Interferometers: data model Visibilities: sample the convolved UV space: Idem for Q and U Stokes parameters RL and LR baselines give (Q § iU) Relationship between (Q,U) and (E,B) in UV (flat) space Visibilities correlation matrix UV coverage of a single pointing of CBI (10 freq. bands) ( Pearson et al. 2003)

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Pixelisation in UV/pixel space Redundant measurements in UV-space Possibility to compress the data ~w/o loss Hobson and Maisinger 2002 Myers et al. 2003 Park et al. 2003 Least squares solution For an NGP pointing matrix: Resultant noise matrix Use in conjonction with an ML estimator Newton-like iterative maximisation Fisher matrix Covariance derivatives for one visibility

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