The ILC method falls in the category of those methods aimed at obtaining a map of the CMB free from foreground contamination, without studying the nature and properties of the single foreground components. Proposed by Tegmark & Efstathiou (1996), it has been used in a simplified way by the W MAP science team as a quick blind method to derive a clean map of CMB emission.
The approach to the problem is indeed simple. Let us assume to have k observed CMB maps at different frequencies, but with identical beams. Each of them can be written in thermodynamic temperature as:
T (νk)=TCMB+Tresidual(νk), (2.1)
where TCMBand Tresidual(νk) are statistically independent. It is possible to use them to generate a CMB map as cleaned as possible from any contribution but the cosmological emission, by simply linearly combining them as follow:
T =
k
X
i=1
wiT (νi). (2.2)
Therefore, it is not necessary to have any information about the foreground components, since anything we need to know is already contained in the observations.
The frequency independence of the CMB spectrum requires that: k
X
i=1
wi =1, (2.3)
Therefore, the resulting map may be written as
T =TCMB+
k
X
i=1
wiTresidual(νi). (2.4)
and the k−1 free weights can be chosen to minimize the contribution from the residuals. This means minimizing the variance of T , assuming that the CMB component is statistically independent of the foregrounds and the noise: indeed, in these conditions, the variance of the error is minimum when the
variance of the ILC map itself is minimum:
Var(T )= Var(TCMB)+Var
Xk
i=1
wiTresidual(νi)
. (2.5)
Following the prescription given by Eriksen et al. (2004), the variance minimization can be achieved by means of Lagrange multipliers. Indeed, it turns out that the solution is obtained by solving the following linear system of equations:
2C −1 1T 0 w λ = 0 1 , (2.6)
whereλis an arbitrary constant, and w = (w1, . . . ,wk)T are the ILC weights. Their definition is given by a straightforward linear algebra:
wi = Pk j=1C−i j1 P jkC−jk1 . (2.7) Finally, Ci j ≡ h∆Ti∆Tji= 1 Npix Npix X p=1 (Ti(p)−T¯i)(Tj(p)−T¯j) (2.8) is the map-to-map covariance matrix. This is the procedure which has been adopted by the W MAP science team since the data of three years of observations have been released.
Away from the galactic plane and on small scales, the best linear combination for cleaning the CMB of foregrounds and noise may be very different from what it is close to the galactic plane and on large scales. In fact, the foreground properties vary strongly over the sky as a result of spatially dependent spectral indexes. A very natural idea to improve the ILC is to decompose the sky maps in several regions. They can be independently used to derive ILC maps which can consequently be added together to obtain a full-sky final map.
It is important to note that the ILC method does not work properly (biasing the result) if some correlation actually affects the components of interest. In particular, small data sets are always empirically correlated at some level, although they are realisations of uncorrelated random processes. This is why it is inadvisable to apply the ILC method on too small subsets of the original data (very small regions). Besides that, however, there is a level of correlation which can not be overcome. It has been generally referred to as Cosmic Covariance (Chiang et al., 2009) and it is due to the fact that the foregrounds and the CMB are correlated amongst themselves. Over an ensemble of universes the true CMB and foreground are indeed expected to be uncorrelated, but any particular sky pattern (such as the one we happen to observe) will generate nonzero measured correlations simply by chance.
Different partitions of the sky have been proposed in the literature as well as different implementations of the ILC method itself. Some of them have been considered as a reference for the results of the work described later on in Chapter 4.
Figure 2.1: Full-sky map color-coded to show the 12 regions that were used to generate the five-year ILC map.
as shown in Figure 2.1, 11 regions lie in the inner Galactic plane, within the Kp2 cut, while the rest of the sky is treated as a single area. Thus, a full sky map is obtained by co-adding the maps derived with the individual regions: in order to suppress boundary effects, they also smoothed the edges of the regions with an effective Gaussian beam of amplitude 1.5◦. Although the ILC method has been adopted
by the W MAP science team as a possible solution for cleaning the data, they have also recognized the limitation of the results in terms of a possible usage for cosmological purposes. It is essentially due to the not perfect cleaning of foregrounds, which leaves a significant residual (a bias), mostly concentrated along the Galactic plane, but also quite significant at higher latitudes. The W MAP science team itself suggested to use this maps only up to a multipoleℓ =32, unless a mask is applied to exclude the region of the Plane. Moreover, the level and properties of the noise, as well as the limited angular resolution are other significant limits which have to be taken into account. They have estimated the bias due to the CMB-foreground correlation by Monte-Carlo simulations, and subtracted it from the composite map. Even though a bias correction has been done, the resultant map has still some drawbacks: it is assumed that the foreground emissions have constant properties in the regions, which is probably not realistic. Moreover, it is difficult to access the impact on the final map of the arbitrariness of the choice of the regions driven by a priori knowledge of foreground emissions. Finally, it is assumed that the foreground model used to determine the bias is accurate, but there might be a mismatch with the real data which is difficult to quantify.
The work proposed by Park et al. (2007) goes in the direction of improving all these aspects: in fact, a new set of regions is defined as the result of a more detailed study of the spatial variations of the spectral properties of the signals. 400 regions are defined from 20×20 spectral index bins: 20 of them describe the spectral properties of the sky between the K and V band, and 20 the spectral index between V and W. Although the work is certainly a good attempt of improving the ILC performance, the method proposed is weakened by the fact that the spectral variations are defined on the basis of the MEM maps derived by W MAP (Bennett et al., 2003), which is not the optimal estimation of the foreground contaminants of the CMB emission. Moreover, the MEM solution uses the result of the W MAP ILC as a prior, which is subtracted from the data before using the MEM method to separate galactic foregrounds, hence automatically including all the drawbacks of the ILC map.
Another attempt in terms of taking into account spectral and spatial variations of the foregrounds, are the methods suggested by Kim et al. (2008) and Delabrouille & Cardoso (2007). In the first case the ILC method is implemented deriving pixel dependent weights, which are computed as maps via harmonic decomposition. Moreover, the method has been improved with an estimation of the bias: an iterative foreground reduction method has been developed, where perturbative corrections are made for the cross term which appears not to be negligible. In the second case, Delabrouille & Cardoso (2007) combined the ILC method with a Wiener filtering, however everything is now implemented on the frame of spherical wavelets called needlets. Needlets were introduced by Narkowich et al. (2006) as a particular construction of a wavelet frame on the sphere. They have been studied in a statistical context (e.g. Baldi et al. (2009, 2007)). The most distinctive property of the needlets is the fact that they can be simultaneously localised in the spherical harmonic domain and in the spatial domain (as is generally the case with wavelets). In fact, they are quasi-exponentially (i.e., faster than any polynomial) concentrated in pixel space and exactly localized on a finite number of multipoles.