Basic concepts of CMB component separation 

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Galactic emissions

The most important foreground that needs to be taken into account during CMB cleaning is the galaxy. Its signal is several order of magnitudes higher than that of the CMB on some part of the sky. The galaxy itself is a mixture of different components and each of them can be simply described by a single physical process. The diffuse emissions are produced by the interstellar medium (ISM).
Figure 2.1 shows the extent of the problem. This figure was released with the Planck early results and shows a full sky map made up from a mixture of the Planck observation maps. It emphasize the angular extent of galactic dust emission. The galaxy is not only overwhelming all other components in the galactic plane but is also present over almost all the sky, mostly at high frequencies. The CMB fluctuations are barely distinguishable in some parts of the sky around the poles.

Galactic point sources

A compact source is a small source that spreads over a very few pixels. It is spatially resolved and is very bright compared to the sky around it. The main object causing the galactic point sources are stars and Ultra Compact Hii (UCHii) regions. UCHii regions are dense Hii regions whose size does not exceed the parsec.
Figure 2.11 shows a catalogue of galactic point source based on Wood and Churchwell (1989); Kurtz et al. (1994). There are many of them in the microwave domain. Diffuse components and point sources have very different generative models, close to Poisson process. Since a combined treatment of Poisson and Gaussian processes is difficult, point sources are treated separately from the diffuse emissions. There are several methods to remove the point sources from the observation maps prior to their use for component separation. A common way to remove them is to set to zero the pixels that contain a point source and then fill with fake data the holes thus made in the maps. These processes are respectively called masking and inpainting. Alternatively, the point source function (PSF) of the instrument can be fitted to the brightest point sources. The fit is then subtracted off the data.

CMB secondary anisotropies

CMB photons were emitted at a very early time and those that we receive have travelled through the late universe. During their journey, they crossed the structures of the universe and they interact with them and their content. The properties of the CMB radiation are found to be slightly altered as photons undergo physical processes in the structures. The difference between the CMB before the modifications due to late time effects and after the modifications are called CMB secondary anisotropies and are considered as components on their own.

Sunyaev-Zel’dovich effects

Galaxy clusters contain hot electron gas. Photons of the CMB scatter via inverse-Compton scattering on these electrons. The electrons that interacts with the CMB trans-fer part of their energy to the photons. Thus, after crossing a galaxy cluster, the CMB has a modified black body with slightly more energetic photons. This is known as the Sunyaev-Zel’dovich effect (SZ effect) (Sunyaev and Zeldovich (1972), see Carlstrom et al. (2002) for a review). The change in temperature is not the same for all lines of sight, since photons coming from different directions have crossed different structures. Thus, the SZ effect induces secondary CMB anisotropies. The SZ distortions due to the thermal energy of the electrons are called the thermal SZ (tSZ) effect. A second order effect, coined the kinetic SZ (kSZ), is due to the bulk velocity of the electrons with respect to the rest frame of the CMB. The interaction of the CMB photons with a moving gas of electrons results in a redshift or blueshift of the photons, depending on the direction of the cluster velocity with respect to the observer. Thus, it is a priori undistinguishable from primary anisotropies. However, in a typical galaxy cluster, the kSZ effect is small, inducing a change in temperature of the order of 10 5.
Figure 2.13 shows the 3K black body, i.e. the CMB electromagnetic spectrum as it has been emitted at the last scattering surface, the CMB electromagnetic spectrum of the anisotropies due to the tSZ effect and the kSZ effect. The tSZ spectrum is negative then positive because the whole CMB spectrum is blueshifted because of the kick of the hot electrons. It crosses zero at about 217GHz, one of the frequency observed by HFI. The kSZ spectrum is negative because of the broadening of the CMB spectrum, which is due to both blueshifting and redshifting of the CMB photons. Figure 2.12 shows a map of the CMB secondary anisotropies due to the SZ effect. Since the SZ effect is due to galaxy clusters and is very faint, this component looks like a point source map and therefore is highly non-Gaussian.

Extra-galactic point sources and the Cosmic Infrared Background

There are two kinds of extra-galactic point sources, one coming from the radio band, one from the infrared band. The radio sources correspond to galaxies hosting a radio loud active galactic nuclei (AGN). The infrared sources correspond to dusty galaxies in the process of forming stars.
As for the galactic point sources, the sources that are resolved need a special treat-ment, like fitting and removing or masking and inpainting. The unresolved infrared sources produce a diffuse anisotropic background known as the Cosmic Infrared Back-ground (CIB). This component is nearly a random Gaussian field and is handled as a diffuse component in the separation process.

Key ideas to solve the problem

As a general rule, to separate different components mixed together, we have to make use of what makes them different from one another. In CMB experiments the feature that differentiates the most the components is their different frequency behaviour. The electromagnetic spectrum of the CMB is accurately known, since it has been measured with high precision by previous missions (Mather et al., 1994). All component separation methods assume that the CMB is a perfect black body since primordial CMB spectral distortions are too small to be detected (Chluba, 2014).
Although our knowledge on spectral emissions of most of the individual components in the data is rather poor, we know that their electromagnetic spectra are distinct. Fur-thermore, the data can be interpreted as a superposition of the components, i.e. each frequency contain a specific amount of each component. Thus a multi-frequency data set is the starting point of many component separation methods.
As an example, figure 3.2 shows the frequency spectra of the main galactic components in the frequency domain covered by the WMAP satellite. As the components mix linearly, each band of observation (K, Ka, Q, V, W) is a linear combination of the component maps, the linear coefficients depending on the emission laws of the components. The total contribution of the galaxy is also shown. But choosing just one map that scales through frequencies for the whole galaxy is not enough because the spatial distribution of the entire galaxy vary from one frequency to another, unlike the individual components that are approximately coherent through frequency.
Performing blind source separation requires at least one statistical assumption about the source, either independence (Cardoso, 1998; Cardoso et al., 2008) or sparsity (Zibulevsky and Pearlmutter, 2000; Bobin et al., 2007). Both are a measure of diversity of the sources. Independence force the separation of sources that have been independently emitted whereas sparsity use clear morphological diversity to disentangle between several signals.
The goal of component separation is to recover the individual components by combin-ing a linear mixture of them. In other words, it is an inverse problem, from the frequency map space to the component map space. The next section discusses methods to solve this problem.

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Review of component separation methods

The optimal component separation method for CMB data does not exist yet. Source separation specifically dedicated to the analysis of the CMB signal has been an active field for a couple of decades now and several fundamental ideas have been introduced.
This section gives an overview of the principal approaches to component separation to- gether with their respective advantages and disadvantages. I will particularly stress the differences between the methods that are blind source separation methods and those that makes use of physical parametric modelling.

Data model

I will first describe the usual description of the data in the space of the spherical harmon- ics, without loss of generality. The available data is a collection of maps, each of them is an observation of the full sky at a given frequency and each of them is a specific mixture of all the components presented in chapter 2. To good approximation, the flux of the physical emissions has no influence on each others. This assumption allows us to model the mixing as a linear combination. Hence the following decomposition for the piece of data di‘m contained in the spherical harmonic coefficient (‘; m) of the observation maps of frequency i (out of Nf frequency bands) Nc Xk (3.1) di‘m = Aiksk‘m + ni‘m ; =1 where the sum runs over the assumed number of components Nc, sk = fsk‘m; ‘ = [[‘min; ‘max]]; m 2 [[ ‘; ‘]]g is the spatial distribution, or map, of the kth component, Aik is the amount of component k in frequency band i and ni‘m is the instrumental noise present in di‘m. Equation 3.1 reads in matrix form d‘m = As‘m + n‘m : (3.2).

Internal Linear Combination

The idea of the Internal Linear Combination (ILC) method (Bennett et al., 1992; Tegmark, 1997; Delabrouille et al., 2009) is encapsulated in its name: taking a linear combination of the multi-frequency data maps in order to cancel all components but one. When all components are random Gaussian fields, the optimal choice of coefficients is the one that minimises the variance of the linear combination.
The method can be applied in any basis that can describe the data. The resulting map is then expressed in the same basis. Historically it has been developed to work in pixel space and spherical harmonic space. A version of ILC using a needlet decomposition of the data, named NILC, was one of the four component separation methods that were retained by the Planck collaboration for the 2013 results and data release. The needlet basis is a basis for functions defined on the sphere. Its basis vectors look like localised waves on the sphere, as in figure 3.3. Needlets are then well defined to describe the complex structures of the galaxy while still having some of the properties of spherical harmonic basis.
For a recovering of non-CMB component, the method is parametric since the emission law of the component to be recovered is needed to perform the separation. ILC is therefore a powerful method for CMB cleaning because its electromagnetic spectrum is accurately known but is not suitable for blind separation of any other component whose frequency behaviour is uncertain.

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

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