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Table of contents
General Introduction
1 Introduction to Cosmic Microwave Background
1.1 Physical context
1.2 Our understanding of the CMB
1.2.1 Acoustic oscillations
1.2.2 Gravitational effects
1.2.3 Baryonic effects
1.2.4 Damping
1.2.5 The transfer function and the growth factor
1.2.6 Projection on the sphere
1.3 Our understanding from the CMB
1.3.1 Parameters of the standard model of cosmology
1.3.2 Beyond the power spectrum
1.4 Polarisation of the CMB
2 The microwave sky
2.1 The Planck Sky Model
2.2 Galactic emissions
2.2.1 Dust emission
2.2.2 Synchrotron emission
2.2.3 Free-free emission
2.2.4 Molecular lines
2.2.5 Galactic point sources
2.3 Extra-galactic components
2.3.1 CMB secondary anisotropies
2.3.2 Extra-galactic point sources and the Cosmic Infrared Background
3 Basic concepts of CMB component separation
3.1 The component separation challenge
3.2 Key ideas to solve the problem
3.3 Review of component separation methods
3.3.1 Data model
3.3.2 Internal Linear Combination
3.3.3 Independent Component Analysis
3.3.4 Sparse blind source separation
3.3.5 Template fitting
3.3.6 Physical parametrisation
4 BICA: a semi-blind Bayesian approach to component separation
4.1 Constructing the component separation PDF
4.1.1 The blind Bayesian formulation of the problem
4.1.2 Likelihood distribution
4.1.3 Prior distributions
4.1.4 Hierarchical model and power spectrum inference
4.1.5 Posterior distribution
4.2 Deriving the sampling equations
4.2.1 First attempt
4.2.2 Marginalisation
4.2.3 Sampling scheme
4.3 Comparison to previous methods
4.3.1 Relevance of the method
4.3.2 Comparison to SMICA
4.3.3 Comparison to Commander
4.3.4 Comparison to ILC
4.3.5 Comparison to SEVEM
5 Application to simulations
5.1 Description of the simulations
5.1.1 The components
5.1.2 The mixing matrix
5.1.3 The noise
5.1.4 The data
5.2 Model approximations to the simulations
5.2.1 Isotropic noise
5.2.2 Lack of correlation between components
5.2.3 Gaussianity
5.3 Results
5.3.1 Full Gibbs sampling treatment
5.3.2 Self consistent treatments
5.3.3 Products of the method
5.3.4 CMB power spectrum inference
5.3.5 CMB map inference
5.3.6 Inference of non-CMB components
5.4 Model checking
5.4.1 Construction of the mismatch
5.4.2 Consistency of the results on simulations
5.4.3 Consistency of the results with modified priors
5.5 Discussion
6 Application to Planck data
6.1 The Planck data
6.2 Additional modelling
6.2.1 Noise
6.2.2 Cross-spectra
6.2.3 Point sources
6.2.4 Beaming
6.2.5 Masking, apodising, inpainting
6.3 Possible configurations of the data
6.3.1 Mask
6.3.2 Frequency range
6.3.3 Multipole range
6.3.4 Number of components
6.3.5 Point source model
6.3.6 Choice of prior
6.3.7 Test cases
6.4 Results
6.4.1 CMB map and power spectrum inference
6.4.2 Inference of the non-CMB components
6.4.3 Consistency of the results
6.4.4 Comparison with SMICA
General conclusion
Appendices
A Statistical basics
A.1 Random variables, distributions and probability density functions
A.2 Gaussian distribution and related distributions
A.3 Kullback-Leibler divergence
A.4 Shannon entropy
B Bayesian inference
B.1 Bayes’ theorem
B.2 Jeffreys priors
C PDF evaluation techniques
C.1 Simple approaches
C.2 Metropolis-Hastings sampling
C.3 Gibbs sampling
C.4 Collapsed sampling
D Isotropic Gaussian random field on the sphere
D.1 Spherical harmonics
D.2 Power spectrum
E HEALPix
F Link between ILC and BICA
F.1 Data and notations
F.2 ILC and « BICA derived » formulas
F.3 Expanding the ILC formula
F.4 Relation between the two formulas
