POLARIZATION OF THE WMAP POINT SOURCES

We have carried out a systematic investigation of the polarized flux of the 516 sources detected at ⩾5σ by Massardi et al. (2009) in the five-year WMAP temperature maps, and listed in the NEWPS_5yr_5s catalogue.⁷ 484 of these sources have a clear identification in low-frequency catalogs (including 27 Galactic objects); the five objects detected in polarization by Wright et al. (2009) belong to this group. The remaining 32 candidate sources do not have plausible counterparts in all-sky low radio frequency surveys and may therefore be just high peaks in the highly non-Gaussian distribution of the other components present in the maps. If they are all spurious, the reliability of the sample is 94% and its completeness is of 91% above 1 Jy.

Since one of our goals is to define a sample as large as possible of potential calibrators for CMB polarization experiments, we have added three extended sources (Cygnus A, Taurus A, Cas A), not included in the catalog because they lie in very noisy regions, close to the Galactic equator, but known to be very bright and significantly polarized. The full sample (Input Catalog) is thus made of 519 sources.

Methods to extract astrophysical foregrounds from multi-frequency CMB maps frequently exploit the prior knowledge of their frequency dependence. This approach however does not work well for radio sources because of the broad variety of their spectral properties in the relevant frequency range (Massardi et al. 2008; Sadler et al. 2008). On the other hand, with few exceptions, extragalactic radio sources look point-like when observed with the beams used by CMB experiments, and therefore have, in the maps, the shape of the effective angular response function of the instrument. In the literature one can find several methods exploiting this property and using linear filters to detect point sources. The standard matched filter approach has been used for years (Nailong 1992; Vikhlinin et al. 1995; Malik & Subramanian 1997; Tegmark & de Oliveira-Costa 1998; Sanz, Herranz & Martínez-Gónzalez 2001; Herranz et al. 2002; Stewart 2006). More recently, a multi-frequency approach based on matched filters has been elaborated (Herranz et al. 2009; Herranz & Sanz 2008). An approach based on optimal wavelets (Vielva et al. 2001, 2003; Barnard et al. 2004; Sanz et al. 2006; González-Nuevo et al. 2006) was successfully applied to WMAP maps (López-Caniego et al. 2007; Massardi et al. 2009) as well as to realistic simulations of Planck maps (López-Caniego et al. 2006; Leach et al. 2008). Moreover, filters based on the Neyman–Pearson approach, using the distribution of maxima, have been proposed (López-Caniego et al. 2005a, 2005b) and a Bayesian approach has been developed (Hobson & McLachlan 2003; Feroz & Hobson 2008; Carvalho et al. 2009).

In this work, following Argüeso et al. (2009), we use the same matched filter over Q and U images. The matched filter is a circularly symmetric filter, Ψ(x; R, b), such that the filtered map, w(R, b) satisfies the following two conditions: (1) 〈w(R₀, 0)〉 = s(0) ≡ A, i.e., w(R, 0) is an unbiased estimator of the flux density of the source; (2) the variance of w(R, b) has a minimum on the scale R₀, i.e. it is an efficient estimator. In Fourier space, the matched filter writes as

$$\begin{equation} \psi _{\rm MF} = \frac{1}{a}\frac{\tau (q)}{\Delta (q)}, \qquad a = 2 \pi \int dq q \frac{\tau ^2 (q)}{\Delta (q)}, \end{equation} \tag{ 1 }$$

where Δ(q) is the power spectrum of the background and τ(q) is the Fourier transform of the source profile (equal to the beam profile for point sources).

Since, in this application, each patch is centered on the position of a source detected in total intensity, we describe the source as

$$\begin{equation} s(\vec{x})=A\tau (\vec{x}), \end{equation} \tag{ 2 }$$

where A is its unknown polarized flux density and $\tau (\vec{x})$ is its profile. We assume circular symmetry, so that $\tau (\vec{x})=\tau (x)$, $x=|\vec{x}|$. For point sources the profile is equal to the beam response function of the detector. The WMAP beams are not Gaussian and we use the real symmetrized radial beam profiles for the different WMAP channels to construct our filters.

The matched filter gives directly the maximum amplification of the source and yields the best linear estimation of the flux. As extensively discussed in the literature, it is a very powerful tool to detect point sources, but it has to be used with care because its performance may degrade rapidly if it is not properly implemented. In particular, the power spectrum Δ(q) of the image has to be obtained directly from the data.

The WMAP team has used a matched filter that operates on the sphere and takes into account the non-Gaussian profile of the beam. It is described by Equation (1), replacing the flat limit quantities τ(q) and Δ(q) with their harmonic equivalents b_ℓ and C_ℓ. As pointed out by López-Caniego et al. (2006), the use of the C_ℓ's computed from all-sky maps to construct a global-matched filter that operates on the sphere is a reasonable first approach, but we think that it can be improved by operating locally.

We have followed the scheme named filtered fusion by its authors (Argüeso et al. 2009). For each object in the input catalog, and for each WMAP frequency between 23 and 61 GHz, we have projected two patches (one for Q and one for U), each of 14.65 × 14.65 deg², centered on the source position. We have left aside the 94 GHz channel because of the normalization problems discussed by López-Caniego et al. (2007) and González-Nuevo et al. (2008). Each patch is made of 128 × 128 6.87 arcmin pixels (HealPix Nside = 512; Górski et al. 2005). The projection has been done using the CPACK library.⁸ Then, each pair of patches has been filtered using a matched filter exploiting the power spectrum determined within each patch. After filtering, the Q² and U² are added together and the square root of the resulting image has been calculated. The noise bias can be removed by subtracting from Q² and U² the corresponding noise contribution $\sigma ^2_{\hat{Q}}$ and $\sigma ^2_{\hat{U}}$. This correction turns out to be negligible. In this way we have obtained a map of the polarized intensity P within each patch. This approach differs from the usual one, where a $\hat{P}$ map is constructed adding together the unfiltered $\hat{Q^2}$ and $\hat{U^2}$ maps, and taking the square root of the resulting map (see Figure 1).

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We have then looked for the brightest maximum inside a circle centered on the center of each patch and covering the area of the WMAP beam. Next, we estimated the noise level of the patch and the significance of the possible detection. Finally, we constructed catalogs containing all sources whose polarized flux P was detected above a chosen significance level.

The reason why we need to use the significance of the detection instead of the usual signal-to-noise ratio is that the noise does not obey a Gaussian distribution but a Rayleigh distribution, since we are dealing with maps that have been squared. The significance was derived from the distribution of the values of P for the pixels within the patch. An example of such distribution is in Figure 2.

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The polarized flux densities and their errors were estimated in a way similar to that applied for total intensity (Massardi et al. 2009). Point sources appear in the image with a profile identical to the beam profile. For example, if the beams were Gaussian, the source flux could be obtained multiplying the flux in the brightest pixel by the ratio between the beam and the pixel area, 2π(R_s)²/L²_p, where $R_s=\mathrm{FWHM}/(2\sqrt{2\log {2}})$ and L_p is the pixel side.

In our case the beams are not Gaussian and we need to calculate this relationship integrating over the real symmetrized beam profile for each channel. In doing that we have to take into account that we work with HEALPix pixelization (Górski et al. 2005) at Nside=512. Although the image is centered at the position of the source, after the projection to the flat patch the object does not always lie in the central pixel, but may end up in an adjacent one. Thus, to estimate its flux we make reference not to the intensity in the central pixel but to that of the brightest pixel close to the center of the filtered image within an area equal to that of the beam. As discussed in Section 3.3 of López-Caniego et al. (2007), this method for flux estimation through linear filtering is almost optimal in many circumstances.

Note that the method adopted here to estimate the flux differs from the one used by the WMAP team. Assuming that they have followed similar procedures in intensity and polarization, they used a matched filter taking into account the non-Gaussian profile of the beam to detect point sources in the filtered full-sky maps, but their fluxes have been derived fitting the pixel intensities around the point source to a Gaussian profile plus a plane baseline (in the unfiltered image).

We calculated the rms noise for each patch containing an input source. This value can be easily overestimated if border effects and strong fluctuations due to other point sources or small-scale structure of the diffuse emissions in the patch are not removed or filtered out. In order to avoid this, first we find the 5% brightest pixels in the patch and flag them. Second, we flag pixels within a distance equal to 15 pixels from the border (see Figure 3). Finally, we select a shell around the source with inner radius equal to the FWHM of the beam (since polarized fluxes are never very high, this is enough for them to have decreased well below the noise level), and an outer radius of 3 × FWHM. The rms noise, σ, is then obtained as the square root of the variance of the pixels included in this shell, excluding flagged pixels (if any). From the distribution of the values of P in the unflagged pixels we calculate the levels exceeding those of 95.00%, 99.00%, 99.90%, and 99.99% of pixels. These are the probabilities that signals at those levels are real rather than noise fluctuations. An example is shown in Figure 2.

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Unfortunately, the number of objects with polarized flux above 400 mJy and ground-based polarization measurement at frequencies close to WMAP ones is small, so that our comparison has a poor statistics. To better assess the reliability of our flux estimates it is thus necessary to resort to simulations. We have selected a sample of 738 positions with |b|>5° and far from each other more than 4R_s (see the first paragraph of Section 2.3) and more than 2R_s away from each object in the NEWPS_5yr_3s catalogue. These "blank" positions (free of sources brighter than 3σ in total intensity) constitute our control fields. We have then chosen 10 values of P, ranging from 0.2 to 2 Jy, with a constant step in log P, i.e. P = [0.2, 0.26, 0.33, 0.43, 0.56, 0.719, 0.928, 1.20, 1.55, 2.0]. For each value of P we have injected a source with that polarized flux density in the projected Q and U patches centered on 100 randomly chosen control field positions. This was done randomly selecting Q between −P and P and setting $U=\pm \sqrt{P^2-Q^2}$, the sign of U being again chosen at random. The inserted source was convolved with the WMAP symmetrized beam at 23 GHz. We avoided using the same patch more than once, except when all the control fields were already used. In any case the same patch was never used twice for the same value of P.

Figure 5 shows the percentage error in the values of P of the simulated sources recovered with our filtering and flux estimation process with respect to the input values. This comparison confirms the reliability of recovered fluxes for P_input ≳ 400 mJy, while the Eddington bias becomes increasingly important below this value. For P_input = 400 mJy, the recovered fluxes are, on average, overestimated by less than 10%.

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The median polarization degree at 23 GHz of the 11 sources detected with a confidence level ⩾99.99%, with low radio frequency counterparts, and with P ⩾ 400 mJy is ≃7.5%. This value, however, is not representative of the mean polarization level of sources in a complete sample for two reasons. First, the sample is obviously biased toward sources with the highest polarization degrees. Second, as mentioned above, our photometry is not optimal for extended sources which make up a substantial fraction of the sample, so that their estimated polarization degrees are highly uncertain.

An unbiased estimate of the mean polarization degree of sources at WMAP frequencies can be obtained from a comparison of the distribution of P values for a suitably chosen complete subsample with that of control fields. The completeness limit, in total flux, cannot be too faint, otherwise the mean polarization level is too low to be detectable. On the other hand, if the flux limit is too high, the number of sources is too small for a meaningful statistical inference. The optimal flux limits turn out to be of 5, 4, 3, and 4 Jy at 23, 33, 41, and 61 GHz, respectively. We find a highly significant detection of polarized flux density only at 23 GHz (see Figure 6): the probability that the distribution of P values of sources is drawn from the same parent distribution as control fields is 7.3 × 10⁻⁵. The median polarization degree, Π, can be estimated as

$$\begin{equation} \Pi = {P_{\rm med, sources}- P_{\rm med, control fields}\over S_{\rm med, sources}}, \end{equation} \tag{ 3 }$$

where S_med,sources is the median flux of sources and we have neglected the median flux of control fields, which is close to zero. The result at 23 GHz is $\Pi _{23\;\rm GHz}= 1.7\pm 1.1$%, consistent with the median polarization degree for the AT20G BSS, $\Pi _{20\;\rm GHz}\simeq 2.5$% (Massardi et al. 2008). Note that the lower resolution of WMAP, compared to AT20G, observations make them more liable to beam depolarization (due to chaotic components of the magnetic field within the unresolved region) in the case of extended sources.

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The probabilities that the distributions of P values of sources are drawn from the same parent distribution as control fields are of 0.02, 0.012, and 0.065 at 33, 41, and 61 GHz, respectively. The derived values of Π are $\Pi _{33\;\rm GHz}= 0.91\pm 0.83$%, $\Pi _{41\;\rm GHz}= 0.68\pm 1.0$%, and $\Pi _{61\;\rm GHz}= 1.3\pm 1.8$%.

The Planck satellite is expected to measure the polarization of the sources with greater sensitivity than WMAP, and therefore to detect more sources, down to fainter polarized flux limits. Building on the results of our analysis of WMAP maps, we have carried out simulations to estimate the minimum polarized flux detectable and reliably measurable by Planck. We have adopted the nominal mean instrumental noise levels expected after two complete sky surveys (one year).¹⁰

Assuming that the Q and U images are dominated by instrumental white noise, and adopting a pixel size of 6.87 arcmin and idealized matched filters for the nominal beam sizes, we have computed the ratios between the σ_P levels for the 30–100 GHz Planck frequency channels and those for the closest WMAP frequency channels. We find $\sigma _{P,\rm WMAP\,5yr}/\sigma _{P,\rm Planck\,1yr}= 2.2$, 1.6, 2.0, and 6.8 at about 30, 44, 70, and 100 GHz, respectively. Note that the higher Planck sensitivity is partly compensated by the longer WMAP exposure time. An extension of the Planck mission for one more year would decrease $\sigma _{P,\rm Planck}$ by a factor of $\sqrt{2}$.

The above calculations take into account only instrumental noise. To investigate the effect also of polarization fluctuations due to diffuse foregrounds and to the CMB we have performed simulations analogous to those we did for WMAP at 23 GHz for the Planck 30 GHz channel, using the Planck Sky Model simulation software (J. Delabrouille et al. 2009, in preparation). In simulated Q and U maps containing polarized diffuse foregrounds and the CMB we have injected sources with P in the range 0.05–0.6 Jy. Our algorithm recovered sources with P down to 300 mJy with systematic offsets due to the Eddington bias of few percent or less.

This result already highlights the difficulty of finding suitable polarization calibrators for Planck. Not many point-like sources have P substantially larger than 300 mJy at 30 GHz. Combining the results by Ricci et al. (2004), of the AT20G BSS (Massardi et al. 2008) and the VLA calibrator observations, we have found 11 extragalactic sources above this limit. The count may be incomplete, particularly in the northern hemisphere, but it is unlikely that many bright sources have been missed. Our analysis of 33 GHz WMAP maps, including bright Galactic sources, has detected 12 sources at a confidence level >99.99%; polarized flux densities for most of them, however, are overestimated because of the Eddington bias. Also, some of the brightest sources are extended (compared to the Planck beam) and others (3C273 and 3C279) are highly variable.