We present the application of the fast independent component analysis (fastica) technique for blind component separation to polarized astrophysical emission. We study how the cosmic microwave background (CMB) polarized signal, consisting of E and B modes, can be extracted from maps affected by substantial contamination from diffuse Galactic foreground emission and instrumental noise. We implement Monte Carlo chains varying the CMB and noise realizations in order to asses the average capabilities of the algorithm and their variance. We perform the analysis of all‐sky maps simulated according to the Planck satellite capabilities, modelling the sky signal as a superposition of the CMB and of the existing simulated polarization templates of Galactic synchrotron. Our results indicate that the angular power spectrum of CMB E mode can be recovered on all scales up to ℓ≃ 1000, corresponding to the fourth acoustic oscillation, while the B‐mode power spectrum can be detected, up to its turnover at ℓ≃ 100, if the ratio of tensor to scalar contributions to the temperature quadrupole exceeds 30 per cent. The power spectrum of the cross‐correlation between total intensity and polarization, TE, can be recovered up to ℓ≃ 1200, corresponding to the seventh TE acoustic oscillation.
We are right now in the epoch in which cosmological observations are revealing the finest structures in the cosmic microwave background (CMB) anisotropies. After the first discovery of CMB total intensity fluctuations as measured by the Cosmic Background Explorer (COBE) satellite (see Smoot 1999, and references therein), several balloon‐borne and ground‐based operating experiments were successful in detecting CMB anisotropies on degree and subdegree angular scales (Lee et al. 2001; Padin et al. 2001; De Bernardis et al. 2002; Halverson et al. 2002; see also Hu & Dodelson 2002, and references therein). The Wilkinson Microwave Anisotropy Probe (WMAP; see Bennett et al. 2003a) satellite released the first‐year, all‐sky CMB observations mapping anisotropies down to an angular scale of about 16 arcmin in total intensity and its correlation with polarization, on five frequency channels extending from 22 to 90 GHz. In the future, balloon‐borne and ground‐based observations will attempt to measure the CMB polarization on sky patches (see Kovac et al. 2002, for a first detection); the Planck satellite, scheduled for launch in 2007 (Mandolesi et al. 1998; Puget et al. 1998), will provide total intensity and polarization full‐sky maps of CMB anisotropy with resolution ≳5 arcmin and a sensitivity of a few μK, on nine frequencies in the range 30–857 GHz. A future satellite mission for polarization is currently under study.
Correspondingly, the data analysis science faces entirely new and challenging issues in order to handle the amount of incoming data, with the aim of extracting all the relevant physical information about the cosmological signal and the other astrophysical emissions, coming from extragalactic sources as well as from our own Galaxy. The sum of these foreground emissions, in total intensity, is minimum at about 70 GHz, according to the first‐year WMAP data (Bennett et al. 2003b). In the following we refer to low and high frequencies meaning the ranges below and above that of minimum foreground emission.
At low frequencies, the main Galactic foregrounds are synchrotron (see Haslam et al. 1982 for an all‐sky template at 408 MHz) and free–free (traced by Hα emission; see Haffner, Reynolds & Tufte 1999; Finkbeiner 2003, and references therein) emissions, as confirmed by the WMAP observations (Bennett et al. 2003b). At high frequencies, Galactic emission is expected to be dominated by thermal dust (Schlegel, Finkbeiner & Davies 1998; Finkbeiner, Davis & Schlegel 1999). Moreover, several populations of extragalactic sources, with different spectral behaviour, show up at all the frequencies, including radio sources and dusty galaxies (see Toffolatti et al. 1998), and the Sunyaev–Zel'dovich effect from clusters of galaxies (Moscardini et al. 2002). Because the various emission mechanisms have generally different frequency dependences, it is conceivable to combine multifrequency maps in order to separate them.
Much work has been recently dedicated to provide algorithms devoted to the component separation task, exploiting different ideas and tools from signal processing science. Such algorithms generally deal separately with point‐like objects such as extragalactic sources (Tenorio et al. 1999; Vielva et al. 2001), and diffuse emissions from our own Galaxy. In this work we focus on techniques developed to handle diffuse emissions; such techniques can be broadly classified into two main categories.
The ‘non‐blind’ approach consists of assuming priors on the signals to recover, on their spatial pattern and frequency scalings, in order to regularize the inverse filtering going from the noisy, multifrequency data to the separated components. Wiener filtering (WF; Tegmark & Efstathiou 1996; Bouchet, Prunet & Sethi 1999) and the maximum entropy method (MEM; Hobson et al. 1998) have been tested with good results, even if applied to the whole sky (Stolyarov et al. 2002). Part of the priors can be obtained from complementary observations, and the remaining ones have to be guessed. The WMAP group (Bennett et al. 2003b) exploited the available templates mentioned above as priors for a successful MEM‐based component separation.
The ‘blind’ approach consists instead of performing separation by only assuming the statistical independence of the signals to recover, without priors either for their frequency scalings or for their spatial statistics. This is possible by means of a novel technique in signal processing science, the independent component analysis (ica; see Amari & Chichocki 1998, and references therein). The first astrophysical application of this technique (Baccigalpi et al. 2000) exploited an adaptive (i.e. capable of self‐adjusting on time streams with varying signals) ica algorithm, working successfully on limited sky patches for ideal noiseless data. Maino et al. (2002) implemented a fast, non‐adaptive version of such algorithm (fastica; see Hyvärinen 1999) which was successful in reaching separation of CMB and foregrounds for several combinations of simulated all‐sky maps in conditions corresponding to the nominal performances of Planck, for total intensity measurements. Recently, Maino et al. (2003) were able to reproduce the main scientific results out of the COBE data exploiting the fastica technique. The blind techniques for component separation represent the most unbiased approach, because they only assume the statistical independence between cosmological and foreground emissions. Thus, they not only provide an independent check on the results of non‐blind separation procedures, but are likely to be the only viable way to go when the foreground contamination is poorly known.
In this paper we apply the fastica technique to astrophysical polarized emission. CMB polarization is expected to arise from Thomson scattering of photons and electrons at decoupling. Because of the tensor nature of polarization, physical information is coded in a entirely different way with respect to total intensity. Cosmological perturbations may be divided into scalars, such as density perturbations, vectors, for example vorticity, and tensors, i.e. gravitational waves (see Kodama & Sasaki 1984). Total intensity CMB anisotropies simply sum up contributions from all kinds of cosmological perturbations. For polarization, two non‐local combinations of the Stokes parameters Q and U can be built, commonly known as E and B modes (see Zaldarriaga & Seljak 1997 and Kamionkowski, Kosowsky & Stebbins 1997, featuring a different notation, namely gradient G for E and curl C for B). It can be shown that the E component sums up the contributions from all three kinds of cosmological perturbations mentioned above, while the B modes are excited via vectors and tensors only. Also, scalar modes of total intensity, which we label with T in the following, are expected to be strongly correlated with E modes. Indeed, the latter are merely excited by the quadrupole of density perturbations, coded in the total intensity of CMB photons, as seen from the rest frame of charged particles at last scattering (see Hu et al. 1999, and references therein). Therefore, for CMB, the correlation TE between T and E modes is expected to be the strongest signal from polarization. The latter expectation has been confirmed by WMAP (Kogut et al. 2003) with a spectacular detection on degree and superdegree angular scales; moreover, a first detection of CMB E modes has been obtained (Kovac et al. 2002).
This phenomenology is clearly much richer with respect to total intensity, and motivated a great interest toward CMB polarization, not only as a new data set in addition to total intensity, but as the best potential carrier of cosmological information via electromagnetic waves. Unfortunately, as we describe in the next section, foregrounds are even less known in polarization than in total intensity; see De Zotti (2002) and references therein for reviews. For this reason, it is likely that a blind technique will be required to clean CMB polarization from contaminating foregrounds. The first goal of this work is to present a first implementation of the ica techniques on polarized astrophysical maps. Secondly, we want to estimate the precision with which CMB polarized emission will be measured in the near future. We exploit the fastica technique on low frequencies where some foreground models have been carried out (Baccigalpi et al. 2001; Giardino et al. 2002).
The paper is organized as follows. In Section 2 we describe how the simulations of the synchrotron emission were obtained. In Section 3 we describe our approach to component separation for polarized radiation. In Section 4 we study the fastica performance on our simulated sky maps. In Section 5 we apply our technique to the Planck simulated data, studying its capabilities for polarization measurements in presence of foreground emission. Finally, Section 6 contains the concluding remarks.
2 SIMULATED POLARIZATION MAPS AT MICROWAVE FREQUENCIES
We adopt a background cosmology close to the model which best fits the WMAP data (Spergel et al. 2003). We assume a flat Friedman–Robertson–Walker (FRW) metric with an Hubble constant H0= 100 h km s−1 Mpc−1 with h= 0.7. The cosmological constant represents 70 per cent of the critical density today, ΩΛ= 0.7, while the energy density in baryons is given by Ωb h2= 0.022; the remaining fraction is in cold dark matter (CDM); we allow for a reionization with optical depth τ= 0.05 (Becker et al. 2001). Note that this is a factor of 2–3 lower than found in the first‐year WMAP data (see Bennett et al. 2003a, and references therein), because we built our reference CMB template before the release of the WMAP data. Cosmological perturbations are Gaussian, with spectral index for the scalar component leading to a not perfectly scale‐invariant spectrum, nS= 0.96, and including tensor perturbations giving rise to a B mode in the CMB power spectrum. We assume a ratio R= 30 per cent between tensor and scalar amplitudes, and the tensor spectral index is taken to be nT=−R/6.8 according to the simplest inflationary models of the very early Universe (see Liddle & Lyth 2000, and references therein). The cosmological parameters leading to our CMB template can be summarized as follows:Seljak & Zaldarriaga 1996), in the healpix environment (Górski et al. 1999). The maps are in antenna temperature, which is obtained at any frequency ν multiplying the thermodynamical fluctuations by a factor of x2 exp x/(exp x− 1)2, where x=hν/kTCMB, h and k are the Planck and the Boltzmann constants, respectively, while TCMB= 2.726 K is the CMB thermodynamical temperature.
The polarized emission from diffuse Galactic foregrounds in the frequency range which will be covered by the Planck satellite is very poorly known. On the high‐frequency side, the Galactic contribution to the polarized signal should be dominated by dust emission (Lazarian & Prunet 2002). The first detection of the diffuse polarized dust emission has been carried out recently (Benoit et al. 2003), and indicates a 3–5 per cent polarization on large angular scales and at low Galactic latitudes. On the low‐frequency side, the dominant diffuse polarized emission is Galactic synchrotron. Observations in the radio band cover about half of the sky at degree resolution (Brouw & Spoelstra 1976), and limited regions at low and medium Galactic latitudes with 10‐arcmin resolution (Duncan et al. 1997, 1999; Uyaniker et al. 1999). Analyses of the angular power spectrum of polarized synchrotron emission have been carried out by several authors (Tucci et al. 2000, 2002; Baccigalpi et al. 2001; Giardino et al. 2002; Bernardi et al. 2003).
Polarized foreground contamination is particularly challenging for CMB B‐mode measurements. In fact, the CMB B mode arises from tensor perturbations (see Liddle & Lyth 2000, for reviews), which are subdominant with respect to the scalar component (Spergel et al. 2003). In addition, tensor perturbations vanish on subdegree angular scales, corresponding to subhorizon scales at decoupling. On such scales, some B‐mode power could be introduced by weak lensing (see Hu 2002, and references therein). Anyway, the cosmological B‐mode power is always expected to be much lower than the E mode, while foregrounds are expected to have approximately the same power in the two modes (Zaldarriaga 2001).
Baccigalpi et al. (2001) estimated the power spectrum of synchrotron as derived by two main data sets. As we already mentioned, on superdegree angular scales, corresponding to multipoles ℓ < 200, the foreground contamination is determined from the Brouw & Spoelstra (1976) data, covering roughly half of the sky with degree resolution. The Cℓ behaviour on smaller angular scales has been obtained by analysing more recent data reaching a resolution of about 10 arcmin (Duncan et al. 1997, 1999; Uyaniker et al. 1999). These data, reaching Galactic latitudes up to b≃ 20°, yield a flatter slope, Cℓ≃ℓ−(1.5–1.8) (see also Tucci et al. 2000; Giardino et al. 2002). Fosalba et al. (2002) interestingly provided evidence of a similar slope for the angular power spectrum of the polarization degree induced by the Galactic magnetic field as measured from starlight data. The synchrotron spectrum at higher frequencies was then inferred by scaling the one obtained in the radio band with a typical synchrotron spectral index of −2.9 (in antenna temperature).
Giardino et al. (2002) built a full‐sky map of synchrotron polarized emission based on the total intensity map by Haslam et al. (1982), assuming a synchrotron polarized component at the theoretical maximum level of 75 per cent, and a Gaussian distribution of polarization angles with a power spectrum estimated out of the high‐resolution radio band data (Duncan et al. 1997, 1999). The polarization map obtained, reaching a resolution of about 10 arcmin, was then scaled to higher frequencies by considering either a constant or a space‐varying spectral index as inferred by multifrequency radio observations.
In this paper we concentrate on low frequencies, modelling the diffuse polarized emission as a superposition of CMB and synchrotron. We used the synchrotron spatial template by Giardino et al. (2002), hereafter the SG model, as well as another synchrotron template, hereafter indicated as SB, obtained by scaling the spherical harmonics coefficients of the SG model to match the spectrum found by Baccigalpi et al. (2001). The Q Stokes parameters for the two spatial templates, in antenna temperature at 100 GHz, are shown in Fig. 1, plotted in a non‐linear scale to highlight the behaviour at high Galactic latitudes. Note how the contribution on smaller angular scales is larger in the SG model. This is evident in Fig. 2 where we compare the power spectra of the SG and SB models with the CMB model, for the cosmological parameters of equation (1). Both models imply a severe contamination of the CMB E mode on large angular scales, say ℓ≲ 200, which remains serious even if the Galactic plane is cut out. Cutting the region |b| ≤ 20° decreases the SG and SB signals by about a factor of 10 and 3, respectively; the difference comes from the fact that the SB power is concentrated more on large angular scales, which propagate well beyond the Galactic plane (see Fig. 2).
In the SG case, the contamination is severe also for the first CMB acoustic oscillation in polarization, as a result of the enhanced power on small angular scales with respect to the SB models (see also Fig. 1). On smaller scales, both models predict the dominance of CMB E modes. On the other hand, CMB B modes are dominated by foreground emission, even if the region around the Galactic plane is cut out as we commented above.
To obtain the synchrotron emission at different frequencies, we consider either a constant antenna temperature spectral index of −2.9, slightly shallower than indicated by the WMAP first‐year measurements (Bennett et al. 2003b), as well as a varying spectral index. In Fig. 3 we show the map of synchrotron spectral indices, in antenna temperature, which we adopt following Giardino et al. (2002). Note that this aspect is relevant especially for component separation, because all methods developed so far require a ‘rigid’ frequency scaling of all the components, which means that all components should have separable dependences on sky direction and frequency. Actually this requirement is hardly satisfied by real signals, and by synchrotron in particular. However, as we see in the next section, fastica results turn out to be quite stable as this assumption is relaxed, at least for the level of variation in Fig. 3. This makes this technique very promising for application to real data. A more quantitative study of how the fastica performance becomes degraded when realistic signals as well instrumental systematics are taken into account will be carried out in a future work.
3 COMPONENT SEPARATION FOR POLARIZED RADIATION
Component separation has been implemented so far for the total intensity signal (see Maino et al. 2002; Stolyarov et al. 2002, and references therein). In this section we expose how we extend the ica technique to treat polarization measurements.
3.1 E and B modes
As we stressed in the previous section, the relevant information for the CMB polarized signal can be conveniently read in a non‐local combination of Q and U Stokes parameters, represented by the E and B modes (see Zaldarriaga & Seljak 1997; Kamionkowski et al. 1997). There are conceptually two ways of performing component separation in polarization observations. Q and U can be treated separately, i.e. performing separation for each of them independently. However, in the hypothesis that Q and U have the same statistical properties, separation can be conveniently performed on a data set combining Q and U maps. This is surely the appropriate strategy if one is sure that the choice of polarization axes of the instrumental set‐up does not bias the signal distribution. In general, however, it may happen that accidentally the instrumental polarization axes are related to the preferred directions of the underlying signal, making Q and U statistically different, so that merging them in a single data set would not be appropriate. While for the Gaussian CMB statistics we do not expect such an occurrence, it may happen for foregrounds, especially if separation is performed on sky patches. For example, the Galactic polarized signal possesses indeed large‐scale structures with preferred directions, such as the Galactic plume, discovered by Duncan et al. (1998) in the radio band, extending up to 15° across the sky and reaching high Galactic latitudes. Therefore, in general, the most conservative approach to component separation in polarization is to perform it for Q and U separately. Note that in our Galactic model no coherence in the polarization detection is present. This allow us to verify, in the next section, that the results obtained by merging Q and U in a single data set are quite equivalent or more accurate than those obtained by treating them separately. In the following we report the relevant fastica formalism for the latter case. Otherwise, when Q and U form a single template, the same formulae developed in Maino et al. (2002) do apply.
Let these multifrequency maps be represented by xQ and xU respectively, where x is made of two indexes, labelling frequencies on rows and pixel on columns. If the unknown components to be recovered from the input data scale rigidly in frequency, which means that each of them can be represented by a product of two functions depending on frequency and space separately, we can define a spatial pattern for them, which we indicate with sQ and sU. Then we can express the inputs xQ,U asMaino et al. (2002). yQ and yU can be combined together to obtain the E and B modes for the independent components present in the input data (see Kamionkowski et al. 1997; Zaldarriaga & Seljak 1997). Note that failures in separation for even one of Q and U affect in general both E and B, because each of them receive contributions from both Q and U. It is possible, even in the noisy case, to check the quality of the resulting separation by looking at the product W A, which should be the identity in the best case. This means that the frequency scalings of the recovered components can be estimated. Following Maino et al. (2002), by denoting as xQ,Uν j the jth component in the data x at frequency ν, it can be easily seen that the frequency scalings are simply the ratios of the column elements of the matrices W−1: equations (2) and (3) imply that noise is transmitted to the fastica outputs, even if it can be estimated and, to some extent, taken into account during the separation process.
3.2 Instrumental noise
Our method to deal with instrumental noise in a fastica‐based separation approach is described in Maino et al. (2002), for total intensity maps. Before starting the separation process, the noise correlation matrix, which for a Gaussian, uniformly distributed noise is null except for the noise variances at each frequency on the diagonal, is subtracted from the total signal correlation matrix; the ‘denoised’ signal correlation matrix enters then as an input in the algorithm performing separation. The same is done also here, for Q and U separately. Moreover, in Maino et al. (2002) we described how to estimate the noise of the fastica outputs. In a similar way, let us indicate the input noise patterns as nQ,U. Then from equations (2) and (3) it can be easily seen that the noise on fastica outputs is given byMaino et al. (2002), if the noise correlation matrix is known, it is possible to subtract it from the signal correlation matrix, greatly reducing the influence of the noise on the estimation of WQ,U; however, sample variance and, in general, any systematics will make the separation matrix noisy in a way which is not accounted by equation (6).
On whole‐sky signals, the contamination to the angular power spectrum coming from a uniformly distributed, Gaussian noise characterized by σ is Cℓ= 4πσ2/N, where N is the pixel number on the sphere. The noise contamination to Cℓ on Q and U can therefore be estimated easily on fastica outputs once in equation (6) are known. Gaussianity and uniformity also make it very easy to calculate the noise level on E and B modes, because they contribute at the same level. Thus, we can estimate the noise contamination in the E and B channels asKamionkowski et al. (1997), whereas the other common version (Zaldarriaga & Seljak 1997) would yield a factor of 2. The quantities defined in equation (7) represent the average noise power, which can be simply subtracted from the output power spectra by virtue of the uncorrelation between noise and signal; the noise contamination is then represented by the power of noise fluctuations around the mean (equation 7): equations (7) and (8).
4 PERFORMANCE STUDY
In this section we apply our approach to simulated skies to assess: (i) the ultimate capability of fastica to clean the CMB maps from synchrotron in ideal noiseless conditions; (ii) how the results are degraded by noise.
4.1 Noiseless separation
We work with angular resolution of 3.5 arcmin, corresponding to nside= 1024 in a healpix environment (Górski et al. 1999); this is enough to test the performance of the CMB polarization reconstruction, in particular for the undamped subdegree acoustic oscillations, extending up to ℓ≃ 2000 in Fig. 2. In all the cases, we show that the computing time to achieve separation was of the order of a few minutes on a Pentium IV 1.8‐GHz processor with 512 Mb RAM memory. We perform separation by considering the CMB model defined by equation (1) and both the SG (Giardino et al. 2002) and SB (Baccigalpi et al. 2001) models for synchrotron emission. We have considered Q and U maps both separately and combined.
As we have already mentioned, the fastica performance turns out to be stable against relaxation of rigid frequency scalings, at least for the spectral index variations shown in Fig. 3. In order to illustrate quantitatively this point, we compare the quality of the CMB reconstruction assuming either constant and varying synchrotron spectral indices, β. In Fig. 4 we plot the original (dotted) and reconstructed (solid) CE,Bℓ for CMB, in the case of the SB foreground model with constant (left‐hand panels) or spatially varying β. The upper (lower) panels refer to the two frequency combinations 70, 100 (30, 44) GHz channels. Fig. 5 shows the results of the same analysis, but using the SG model.
As can be seen, the CMB signal is well reconstructed on all relevant scales, down to the pixel size. The same is true for the synchrotron emission, not shown. Per cent precision in frequency scaling recovery for CMB and synchrotron is achieved (see Table 1). As we stressed in the previous section, the precision on frequency scaling recovery corresponds to the precision on the estimation of the elements in the inverse of the matrices WQ and WU. Remarkably, fastica is able to recover the CMB B modes on all the relevant angular scales, even if they are largely subdominant with respect to the foreground emission, as can be seen in Fig. 2. This is due to two main reasons: the difference of the underlying statistics describing the distribution of CMB and foreground emission, and the high angular resolution of templates (3.5 arcmin). Such resolution allows the algorithm to converge close to the right solution by exploiting the wealth of statistical information contained in the maps. Note also that in the noiseless case with constant synchrotron spectral index, the CMB power spectrum is reconstructed at the same good level both for the 100 and 70 GHz and for the 30 and 44 GHz channel combinations, although in this frequency range the synchrotron emission changes amplitude by a factor of about 10. Indeed, by comparing the top‐left and bottom‐left‐hand panels of Figs 4 and 5, we can note that there is only a minimum difference between the B spectra at 44 and 100 GHz, arising at high ℓ, while the E spectra exhibit no appreciable difference at all.
In the case of spatially varying spectral index (right‐hand panels of Figs 4 and 5) a rigorous component separation is virtually impossible, because the basic assumption of rigid frequency scaling is badly violated. However, fastica is able to approach convergence by estimating a sort of ‘mean’ foreground emission, scaling roughly with the mean value of the spectral index distribution. Some residual synchrotron contamination of the CMB reconstructed maps cannot be avoided, however. This residual is proportional to the difference between the ‘true’ synchrotron emission and that corresponding to the ‘mean’ spectral index and is thus less relevant at the higher frequencies, where synchrotron emission is weaker. As shown by the upper right‐hand panels of Figs 4 and 5, when the 70–100 GHz combination is used, the power spectrum of the CMB E mode is still well reconstructed on all scales, and even that of the CMB B mode is recovered at least up to ℓ≃ 100. On smaller scales, synchrotron contamination of the B mode is strong in the SG case, but not in the SB case (at least up to ℓ≃ 1000). As expected, the separation quality degrades substantially if the 30–44 GHz combination is used (right‐hand bottom panels of Figs 4 and 5; see also Table 1 where the quoted error on frequency scaling for varying spectral index is the percentage difference between the average values 〈(ν1/ν2)β〉 computed on input and reconstructed synchrotron maps).
To compare the fastica performances when Q and U maps are dealt with separately or together (case QU), we have carried out a Monte Carlo chain on the CMB realizations, referring to the 70‐ and 100‐GHz channels, and we have computed the rms error on the CMB frequency scaling reconstruction, σQ, σU and σQU. The results are shown in Table 2. Again, the reconstruction is better when the weaker synchrotron model SB is considered. The slight difference between σQ and σU is probably due to the particular realization of the synchrotron model we have used (not changed through this Monte Carlo chain). The fact that the difference is present for both the SB and SG models is not surprising because the two models have different power as a function of angular scales but have the same distribution of polarization angles. The reason why we have not varied the foregrounds templates in our chain is the present poor knowledge of the underlying signal statistics.
The separation precision when Q and U are considered together is equivalent or better than if they are treated separately, as expected because the statistical information in the maps which are processed by fastica is greater. On the other hand, the fact that σQU is so close to σQ and σU clearly indicates that for a pixel size of about 3.5 arcmin the statistical information in the maps is such that the results are not greatly improved if the pixel number is doubled. In the following we just consider the most general case in which Q and U maps are considered separately.
4.2 Effect of noise
To study the effect of the noise on fastica component separation we use a map resolution of about 7 acrmin, corresponding to nside= 512 in a healpix scheme (Górski et al. 1999). At this resolution, the all‐sky separation runs take a few seconds. Moreover, we consider only the combination of 70‐ and 100‐GHz channels, and a space‐varying synchrotron spectral index. We give the results for one particular noise realization and then we show that the quoted results are representative of the typical fastica performance within the present assumptions. Moreover, we investigate how the foreground emission affects the recovered CMB map. The noise is assumed to be Gaussian and uniformly distributed, with rms parametrized with the signal‐to‐noise (S/N) ratio, where the signal stays for CMB. As we have already stressed, noise is subtracted both during the separation process and on the reconstructed Cℓ, according to the estimate in equation (7). The results are affected by the residual noise fluctuations, with power given by equation (8).
As expected, the noise primarily affects the reconstruction of the CMB B mode. In Fig. 6 we plot the reconstructed and original CMB E‐ and B‐mode power spectra, for SG (left) and SB (right) foreground emission, in the case S/N = 2. With this level of noise, separation is still successful: the E‐mode power spectrum comes out very well, while that of the B mode is well reconstructed up to the characteristic peak at ℓ≃ 100. Table 3 shows that the error on the frequency scaling recovery, for CMB, remains, in the noisy case, at the per cent level both for Q and U.
If the S/N ratio is decreased, B modes become quickly lost, while the algorithm is still successful in recovering the E power spectrum for S/N ≳ 0.2 (see Fig. 7). The algorithm starts failing at low multipoles, say ℓ≲ 100, where synchrotron dominates over the CMB both in the SG and SB cases. The results in Fig. 7 for the SG case are averages over eight multipoles, to avoid excessive oscillations of the recovered spectrum. For the S/N ratios in this figure, the B modes are lost on all scales. Because the SG model has a higher amplitude, fastica is able to catch up the statistics more efficiently than for SB, thus being able to work with a lower S/N. Table 3 shows the degradation of the separation matrix for the S/N ratios of Fig. 7, compared to the case with S/N = 2.
We stress that the noise levels quoted here are not the maximum that the algorithm can support. The quality of the separation depends on the noise level as well as on the number of channels considered; adding more channels, while keeping constant the number of components to recover, generally improves the statistical sample with which fastica deals and so the quality of the reconstruction as well as the amount of noise supported. In the next section we show an example where a satisfactory separation can be obtained with higher noise by considering a combination of three frequency channels.
We now investigate to what extent the results quoted here are representative of the typical fastica performance and we study how the foreground emission biases the CMB maps recovered by fastica. To this end, we have performed a Monte Carlo chain of separation runs, building for each of them a map of residuals by subtracting the input CMB template from the recovered one and studying the ensemble of those residual maps.
The residuals in the noiseless case are just a copy of the foreground emission. Their amplitude is greatly reduced with respect to the true foreground amplitude, in proportion to the accuracy of the recovered separation matrix. Because the latter accuracy is at a level of per cent of better, as can be seen in Table 1, the residual foreground emission in the CMB recovered map is roughly the true one divided by 100. In terms of the angular power spectrum (see, for example, Fig. 2), the residual foreground contamination to the CMB recovered power spectrum is roughly a factor of 104 less than the true one.
In the noisy separation, a key feature is that at the present level of architecture, the fastica outputs are just a linear combination of the input channels. Thus, even if the separation goes perfectly, the noise is present in the output just as the same linear combination of the input noise templates. Note, however, that this does not mean that the noise is transmitted linearly to the outputs. The way the separation matrix is found depends non‐linearly on the input data including the noise. In other words, the noise affects directly the estimation of the separation matrix, as we explained in Section 3.2. Equation (6) describes only the amount of noise which affects the outputs after the separation matrix is found. As we shall see in a moment, at least in the case S/N = 2 the main effect of the noise is that given by equation (6), dominant over the error induced by the noise on the separation matrix estimation. Moreover, the noise in the outputs reflects the input noise statistics, which is Gaussian and uniformly distributed in the sky. As we shall see now, this is verified if the foreground contamination is the stronger SG one.
The results presented in Table 4 show the ensemble average of the mean of the residuals 〈r̄〉 together with its Gaussian expectation 〈r̄2〉G1/2, and the mean rms error on the CMB frequency scaling recovery, σ. The most important feature is that a non‐zero mean value, at almost 10σ with respect to its Gaussian expectation, is detected. This is the only foreground contamination we find in the residuals. Note that the separation matrix precision recovery is at the per cent level. This means that the present amount of noise does not affect significantly the accuracy of the separation process. Of course, if the noise is increased, the separation matrix estimation starts to be affected and eventually the foreground residual in the CMB reconstructed map will be relevant.
We made a further check by verifying that the residuals obey a Gaussian statistics with rms given by equation (6) on all Galactic latitudes. We constructed a map having in each pixel the variance built out of the 50 residual maps in our Monte Carlo chain. In Fig. 8 we show the rms of such a map, plus/minus the standard deviation, calculated on rings with constant latitude with width equal to 1°. Together with the curves built out of our Monte Carlo chain, we report the theoretical values according to a Gaussian statistics, i.e. the average given by equation (6) equal to (3.90 × 10−6 K)2, and the standard deviation over N= 50 samples, given by . The agreement demonstrates that the Gaussian expectation is satisfied at all latitudes, especially at the lowest, where the foreground contamination is expected to be maximum. Note that the fluctuations around the Gaussian theoretical levels are larger near the poles because of the enhanced sample variance.
We conclude that, within the present assumptions, for a successful separation the residual foreground contamination in the recovered CMB map is subdominant with respect to the noise. On the other hand, further tests are needed to check this result against a more realistic noise model, featuring the most important systematic effects such as a non‐uniform sky distribution, the presence of non‐Gaussian features, etc.
5 AN APPLICATION TO PLANCK
In this section, we study how fastica behaves in conditions corresponding to the instrumental capabilities of Planck. While this work was being completed, the polarization capabilities at 100 GHz were lost because of a funding problem of the Low Frequency Instrument (LFI), but that capability could be restored if the 100‐GHz channel of the High Frequency Instrument (HFI) is upgraded, as is presently under discussion. The Planck polarization sensitivity in all its channels has a crucial importance, and it is our intention to support this issue. Thus, we work assuming Planck polarization sensitivity at 100 GHz, highlighting the fact that our results have been obtained under this assumption. At 30, 44, 70 and 100 GHz, the Planck beams have full width at half‐maximum (FWHM) of 33, 23, 14 and 10 arcmin, respectively. We study the fastica effectiveness in recovering E, B and TE modes, separately. We adopted, for polarization, the nominal noise level for total intensity measurements increased by a factor of (note also that due to the 1.10 and lower healpix version convention to normalize Q and U following the prescription by Kamionkowski et al. (1997), a further has to be taken into account when generating Q and U maps out of a given power in E and B). We neglect all instrumental systematics in this work. The Planck instrumental features assumed here, with noise rms in antenna temperature calculated for a pixel size of about 3.52 arcmin corresponding to nside= 1024 in the healpix scheme, are summarized in Table 5. By looking at the numbers, it can be immediately realized that the level of noise is sensibly higher than that considered in the previous section, so that the same method would not work in this case and an improved analysis, involving more channels as described below, is necessary.
5.1 E mode
Because of the high noise level, we found it convenient to include in the analysis the lower‐frequency channels together with those at 70 and 100 GHz. Because the fastica algorithm is unable to deal with channels having different FWHM, as in Maino et al. (2002), we had to degrade the maps, containing both signal and noise, to the worst resolution in the channels considered. However, a satisfactory recovery of the CMB E modes, extending on all scales up to the instrument best resolution, is still possible by making use of the different angular scale properties of both synchrotron and CMB. Indeed, as can be seen in Fig. 2, the Galaxy is likely to be a substantial contaminant on low multipoles, say ℓ≲ 200.
For the present application, we found it convenient to use a combination of three Planck channels, 44, 70 and 100 GHz for the SG model and 30, 70 and 100 GHz for the SB model. The reason for the difference is that the SB contamination is weaker, and the 30‐GHz channel is necessary for fastica to catch synchrotron with enough accuracy. Including a fourth channel does not imply relevant improvements. The maps, including signals properly smoothed and noise according to the Table 5, were simulated at 3.52‐arcmin resolution, corresponding to nside= 1024. Higher‐frequency maps were then smoothed to the FWHM of the lowest‐frequency channel and then re‐gridded to nside= 128, corresponding to a pixel size of about 28 arcmin and to a maximum multipole ℓ≃ 400. In all the cases shown, the spectral index for synchrotron has been considered variable. Fig. 9 shows the resulting CMB E‐mode power spectrum after separation, for the SG (left) and SB (right) synchrotron models. An average every four (left) and three (right) coefficients was applied to eliminate fluctuations becoming negative on the lowest signal part at ℓ≃ 10. The agreement between the original spectrum and the reconstructed one is good on all the scales probed at the present resolution, up to ℓ≃ 400. The reionization bump is clearly visible as well as the first polarization acoustic oscillation at ℓ≃ 100. Moreover, there is no evident difference in the quality of the reconstruction between the two synchrotron models adopted.
Let us turn now to the degree and subdegree angular scales, ℓ≳ 200. As we have already stressed, the Galaxy is expected to yield approximately equal power on E and B modes (see Fig. 2). On the other hand, CMB E and B modes are dramatically different on subdegree angular scales. Summarizing, on ℓ≳ 200, we expect to haveFig. 10 shows the results of this technique applied to the Planck HFI 100‐GHz channel, assumed to have polarization capabilities, for both the SG and SB synchrotron models. Residual fluctuations are higher in the SG case because the synchrotron contamination is stronger. In both cases, CMB E modes are successfully recovered in the whole interval 100 ≲ℓ≲ 1000. It has also to be noted that the same subtraction technique would not help on the lower multipoles considered before, because in that case the foreground contamination is so strong that the tiny fluctuations making Galactic E and B modes different are likely to hide the CMB signal anyway.
Our results here can be summarized as follows. In the case of Planck capabilities, the fastica technique makes it possible to remove substantially the foreground contamination in the regions in which that is expected to be relevant. Planck is likely to measure the CMB E modes over all multipoles up to ℓ≃ 1000.
5.2 B mode
In Fig. 11 the B‐mode power spectrum after fastica separation described in the previous section is shown for the SG (left) and SB (right) synchrotron models. An average over 13 multipoles has been applied to both cases in order to avoid fluctuations becoming negative. The reconstructed signal approaches the original one at very low multipoles, say ℓ≲ 5. At higher multipoles, where the B signal is generated by gravitational waves, the overall amplitude appears to be recovered, even if with major contaminations especially in the region where the signal is low, i.e. right between the reionization bump and the rise toward the peak at ℓ≃ 100. Needless to say, such contaminations are due to a residual foreground emission.
In the insets of Fig. 11 we show (data points) the recovered B‐mode power spectrum in the range between 30 ≤ℓ≤ 120, averaged over 20 multipoles, with error bars given by equation (8). Even if the contamination is substantial, especially for ℓ≥ 100 and for the SG case, the results show a sign of the characteristic rise of the spectrum due to cosmological gravitational waves.
Concluding, our results indicate that the fastica technique is able to remove substantially the foreground contamination of the B mode, up to the peak at ℓ≲ 100 if the tensor to scalar perturbation ratio is at least 30 per cent.
5.3 TE mode
While the cosmological TE power spectrum is substantially stronger than that of any other polarized CMB mode, the opposite should happen in the case of foregrounds. On degree and subdegree angular scales, a measure of the synchrotron TE power spectrum can be achieved in the radio band. In the Parkes data at 1.4 GHz, Uyaniker et al. (1999) were able to isolate a region exhibiting low rotation measures, called the ‘fan region’, which is therefore expected to be only weakly affected by Faraday depolarization. This and other regions from the existing surveys in the radio band were used to predict the synchrotron power for the SB scenario (Baccigalpi et al. 2001).
In Fig. 12 we show the T, E, B and TE power spectra for the fan region. Total intensity anisotropies are represented by the upper curve (solid). E and B modes (light lines) have very similar behaviour. The TE mode (heavy solid line) is the weakest and, as can be easily seen by scaling the TE amplitude in Fig. 12 with the typical spectra index for synchrotron, it is markedly below the expected cosmological TE signal at CMB frequencies. Both synchrotron models SG and SB are consistent with this result, as illustrated in Fig. 13. It is straightforward to check that our models for the synchrotron emission have a TE power spectrum not far from the one in Fig. 12, when scaled to the appropriate frequency.
From the point of view of CMB observations, this means that, if the synchrotron contamination at microwave frequencies is well represented by its signal in the radio band, at least on degree and subdegree angular scales the contamination from synchrotron is almost absent due to the change in the magnetic field orientation along the line of sight. On the other hand, on larger scales, as can be seen in Fig. 13, the contamination could be relevant both in the SG and SB cases and we perform component separation as described in Section 5.1 for the Planck case. In Fig. 14 we show the recovery of the CMB TE mode, obtained by combining the templates of Q and U maps obtained after fastica application as in Section 5.1, with the CMB T template obtained, still with fastica‐based component separation strategy, in Maino et al. (2002). Oscillations due to residual noise are visible in the recovered CTEℓ. However, as in the case of the E mode, the procedure was successful in substantially removing the contamination.
Both in the SG and SB cases, the synchrotron contamination is almost absent in the acoustic oscillation region of the spectrum, as is evident again from Fig. 13; neglecting it, we obtain the results shown in Fig. 15. The combination of Planck angular resolution and sensitivity allows the recovery of the TE power spectrum up to ℓ≃ 1200, corresponding roughly to the seventh CMB acoustic oscillation.
6 CONCLUDING REMARKS
Forthcoming experiments are expected to measure CMB polarization. The first detections have been obtained on pure polarization (Kovac et al. 2002), as well on its correlation with total intensity CMB anisotropies, by the WMAP satellite.
The foreground contamination is mildly known for total intensity measurements, and poorly known for polarization (see De Zotti 2002, and references therein). It is therefore crucial to develop data analysis tools able to clean the polarized CMB signal from foreground emission by exploiting the minimum number of a priori assumptions. In this work, we implemented the fastica technique in an astrophysical context; see Amari & Chichocki (1998), Hyvärinen (1999), Baccigalpi et al. (2000) and Maino et al. (2002) for blind component separation to deal with astrophysical polarized radiation.
In our scheme, component separation is performed both on the Stokes parameters Q and U maps independently and by joining them in a single data set. E and B modes, coding CMB physical content in the most suitable way (see Kamionkowski et al. 1997; Zaldarriaga & Seljak 1997), are then built out of the separation outputs. We have described how to estimate the noise on fastica outputs, on Q and U as well as on E and B.
We tested this strategy on simulated polarization microwave all‐sky maps containing a mixture of CMB and Galactic synchrotron. CMB is modelled close to the current best fit (Spergel et al. 2003), with a component of cosmological gravitational waves at the 30 per cent level with respect to density perturbations. We also included reionization, although with an optical depth lower than indicated by the WMAP results (Bennett et al. 2003a) because they came while this work was being completed, but consistent with the Gunn–Peterson measurements by Becker et al. (2001). Galactic synchrotron was modelled with the two existing templates by Giardino et al. (2002) and Baccigalpi et al. (2001). These models yield approximately equal power on angular scales above the degree, dominating over the expected CMB power. On subdegree angular scales, the Giardino et al. (2002) model predicts a higher power, but still subdominant compared to the CMB E‐mode acoustic oscillations. Note that, at microwave frequencies, the fluctuations at high multipoles (ℓ≳ 1000), corresponding to a few arcmin angular scales, are likely dominated by compact or flat spectrum radio sources (Baccigalpi et al. 2002b; Mesa et al. 2002). Their signal is included in the maps used to estimate the synchrotron power spectrum.
We studied in detail the limiting performance in the noiseless case, as well as the degradation induced by a Gaussian, uniformly distributed noise, by considering two frequency combinations: 30, 44 GHz and 70, 100 GHz. In the noiseless case, the algorithm is able to recover CMB E and B modes on all the relevant scales. In particular, this result is stable against the space variations of the synchrotron spectral index indicated by the existing data. In this case, fastica is able to converge to an average synchrotron component, characterized by a ‘mean’ spectral index across the sky, and to remove it efficiently from the map. The output CMB map, also containing residual synchrotron due to its space‐varying spectral index, is mostly good as far as the frequencies considered are those where the synchrotron contamination is weaker.
By switching on the noise we found that separation, at least for what concerns the CMB E mode, is still satisfactory for noise exceeding the CMB but not the foreground emission. The reason is that in these conditions the algorithm is still able to catch and remove the synchrotron component efficiently. We implemented a Monte Carlo chain varying the CMB and the noise realizations in order to show that the performance quoted above is typical and does not depend on the particular case studied. Moreover, we studied how the foreground emission biases the recovered CMB map, by computing maps of residuals, i.e. subtracting the true CMB map out of the recovered one. In the noiseless case, the residual is just a copy of the foreground emission, with amplitude decreased proportionally to the accuracy of the separation matrix. In the noisy case, for interesting noise amplitudes the residual maps are dominated by the noise in the input data, linearly mixed with the separation matrix. The situation is obviously worse for the weaker CMB B mode.
We applied these tools making reference to the Planck polarization capabilities, in terms of frequencies, angular resolution and noise, to provide a first example of how the fastica technique could be relevant for high precision large polarization data sets. We addressed separately the analysis of the CMB E, B and TE modes. While this work was being completed, the LFI lost its 100‐GHz channel, having polarization sensitivity. However, polarimetry at this frequency could be restored if the 100‐GHz channel of the HFI is upgraded, as is presently under discussion. Because of the scientific content of the CMB polarization signal, the Planck polarization sensitivity deserves great attention. Within our context here, it is our intention to support the importance of having polarization capabilities in all the cosmological channels of Planck, and in particular at 100 GHz. Our results have been obtained under this assumption.
To improve the signal statistics, we found it convenient to consider at least three frequency channels in the separation procedure, including those where the CMB is strongest, 70 and 100 GHz, plus one out of the two lower frequency channels, at 30 and 44 GHz. Because the latter have lower resolution we had to degrade the higher‐frequency maps because the present fastica architecture cannot deal with maps having different resolutions. CMB E and TE modes were accurately recovered for both the synchrotron models considered. The B‐mode power spectrum is recovered on very large angular scales in the presence of a conspicuous reionization bump. On smaller scales, where the B‐mode power mainly comes from cosmological gravitational waves, the recovery is only marginal for a 30 per cent tensor to scalar perturbation ratio.
On the subdegree angular scales, the contamination from synchrotron is almost irrelevant according to both models (Baccigalpi et al. 2001; Giardino et al. 2002). Moreover, it is expected that Galactic E and B modes have approximately the same power (Zaldarriaga 2001), while for CMB the latter are severely damped down because they are associated with vector and tensor perturbations, vanishing on subhorizon scales at decoupling corresponding to a degree or less in the sky (see Hu et al. 1999). This argument holds also if the B‐mode power is enhanced by weak lensing effects from matter structures along the line of sight (see Hu 2002, and references therein). Therefore, on these scales, we expect the E power spectrum to be a sum of Galactic and CMB contributions, while the B power comes essentially from foregrounds only. In these conditions, the CMB E power spectrum is recovered by simply subtracting the B power spectrum.
We also estimated the TE contamination from synchrotron to be irrelevant for CMB, because of the strength of the CMB TE component due to the intrinsic correlation between scalar and quadrupole modes exciting E polarization. By applying these considerations on subdegree angular scales, as well as the results of the fastica procedure described above on larger scales, we show how the Planck instrument is capable of recovering the CMB E and TE spectra on all scales down to the instrumental resolution, corresponding to a few arcmin scales. In terms of multipoles, the E and TE angular power spectra are recovered up to 1000 and 1200, respectively.
Summarizing, we found that the fastica algorithm, when applied to a Planck‐like experiment, could be able to substantially clean the foreground contamination on the relevant multipoles, corresponding to degree angular scales and above. Because the foreground contamination on subdegree angular scales is expected to be subdominant, the CMB TE and E modes are recovered on all scales extending from the whole sky to a few arcmin. In particular, the fastica algorithm can clean the B‐mode power spectrum up to the peak due to primordial gravitational waves if the cosmological tensor amplitude is at least 30 per cent of the scalar one. In particular, we find that on large angular scales, of a degree and more, foreground contamination is expected to be severe and the known blind component separation techniques are able to efficiently clean the map from such contamination, as is presently known or predicted.
Still, despite these good results, the main limitation of the present approach is the neglect of any instrumental systematics. While it is important to assess the performance of a given data analysis tool in the presence of the nominal instrumental features, as we do here, a crucial test is checking the stability of such tool with respect to relaxation of the assumptions regarding the most common sources of systematic errors, such as beam asymmetry, non‐uniform and/or non‐Gaussian noise distribution, etc., as well as the idealized behaviour of the signals to recover. In this work we have had a good hint about the second aspect, because we have shown that fastica is stable against relaxation of the assumption, common to all component separation algorithms developed so far, about the separability between space and frequency dependence for all the signal to recover. In a forthcoming work we will investigate how ica‐based algorithms for blind component separation deal with maps affected by the most important systematics errors.
C. Baccigalpi warmly thanks R. Stompor for several useful discussions. We are also grateful to G. Giardino for providing all‐sky maps of simulated synchrotron emission (Giardino et al. 2002), which has been called the SG model in this paper. The healpix sphere pixelization scheme, available at , by A. J. Banday, M. Bartelmann, K. M. Gorski, F. K. Hansen, E. F. Hivon and B. D. Wandelt, has been extensively used.