Issue 
A&A
Volume 557, September 2013



Article Number  A53  
Number of page(s)  20  
Section  Astrophysical processes  
DOI  https://doi.org/10.1051/00046361/201321160  
Published online  28 August 2013 
Online material
Appendix A: Harmonicdomain CCA
The sky radiation, , from direction r at frequency ν results from the superposition of signals coming from N_{c} different physical processes : (A.1)The signal is observed through a telescope, the beam pattern of which can be modelled at each frequency as a spatially invariant point spread function B(r,ν). For each value of ν, the telescope convolves the physical radiation map with B. The frequencydependent convolved signal is input to an N_{d}channel measuring instrument, which integrates the signal over frequency for each of its channels and adds noise to its outputs. The output of the measurement channel at a generic frequency ν is (A.2)where t_{ν}(ν′) is the frequency response of the channel and n_{ν}(r) is the noise map. The data model in Eq. (A.2) can be simplified by virtue of the following assumptions:

Each source signal is a separable function of direction and frequency, i.e.,(A.3)

B(r,ν) = B_{ν}(r)is constant within the bandpass of the measurement channel.
These two assumptions lead us to a new data model: (A.4)where ∗ denotes convolution, and (A.5)For each location, r, we define

the N_{c}vector s (sources vector), whose elements are s_{j}(r);

the N_{d}vector x (data vector), whose elements are x_{ν}(r);

the N_{d}vector n (noise vector), whose elements are n_{ν}(r);

the diagonal N_{d}matrix B, whose elements are B_{ν}(r);

the N_{d} × N_{c} matrix H, containing all h_{νj} elements.
Then, we can rewrite Eq. (A.4) in vector form: (A.6)The matrix H is called the mixing matrix and contains the frequency scaling of the components for all the data maps involved.
Under the assumption that B does not depend on the frequency, when working in the pixel domain, we can simplify Eq. (A.6) to (A.7)where the components in the source vector s are now convolved with the instrumental beam.
Equation (A.6) can be translated into the harmonic domain, where for each transformed mode it becomes (A.8)where X, S, and N are the transforms of x, s, and n, respectively, and is the transform of matrix B. Relying on this data model, we can derive the following relation between the crossspectra of the data , sources and noise, , all depending on the multipole ℓ: (A.9)where the dagger superscript denotes the adjoint matrix. To reduce the number of unknowns, the mixing matrix is parametrized through a parameter vector p (such that H = H(p)), using the fact that its elements are proportional to the spectra of astrophysical sources (see Sect. 3.2).
Since the foreground properties are expected to be spatially variable, we work on relatively small square patches of data. This allows us to use the 2D Fourier transform to approximate the harmonic spectra (see, e.g., Bond & Efstathiou 1987).
The HEALPix (Górski et al. 2005) data on the sphere are projected on the plane tangential to the centre of the patch and are regridded with a suitable number of bins to correctly sample the original resolution. Each pixel in the projected image is associated with a specific vector normal to the tangential plane and assumes the value of the HEALPix pixel nearest to the corresponding position on the sphere. Clearly, the projection and regridding process will create some distortion in the image at small scales and will modify the noise properties. However, we verified that this has only a negligible impact on the spectra in Eq. (A.9) for the scales considered in this work and, therefore, on the spectral parameters. If x(i,j) contains the data projected on the planar grid and X(i,j) is its twodimensional discrete Fourier transform, the energy of the signal at a certain scale, which corresponds to the power spectrum, can be obtained as the average of X(i,j)X^{†}(i,j) over annular bins , (Bedini & Salerno 2007): (A.10)where is the number of pairs (i,j) contained in the spectral bin denoted by . Every spectral bin is related to a specific ℓ in the spherical harmonic domain by (A.11)where p is the thickness of the annular bin, Δ_{ℓ} = 180/L_{deg}(N_{pix} − 1), and L_{deg}, N_{pix} are the size in degrees and the number of pixels on the side of the square patch, respectively.
If we reorder the matrices and into vectors and , respectively, we can rewrite Eq. (A.9) as (A.12)where , and the symbol ⊗ denotes the Kronecker product. The vector is now computed using the approximated data crossspectrum matrix in Eq. (A.10) and represents the error on the noise power spectrum.
The parameter vector p and the source crossspectra are finally obtained by minimizing the functional (A.13)The vectors d_{V} and c_{V} contain the elements and , respectively, and the diagonal matrices H_{kB} and N_{ϵ} the elements and the covariance of error for all relevant spectral bins. The term is a quadratic stabilizer for the source power crossspectra: the matrix C is in our case the identity matrix, and the parameter λ must be tuned to balance the effects of data fit and regularization in the final solution. The functional in Eq. (A.13) can be considered as a negative joint logposterior for p and c_{V}, where the first quadratic form represents the loglikelihood, and the regularization term can be viewed as a logprior density for the source power crossspectra.
Appendix B: Spectral model for AME
Theoretical spinningdust models predict a variety of spectra that can be substantially different in shape, depending on a large number of parameters describing the physics of the medium. The number of these physical parameters is too large to be constrained by the data in the available frequency range. For the purpose estimating the spectral behaviour of the AME we adopted a simple formula depending on only a few parameters. The CCA componentseparation method used in this work implements the parametric relation proposed by Bonaldi et al. (2007) (Eq. (6)), depending on the peak frequency, ν_{p}, and slope at 60 GHz, m_{60}. To verify the adequacy of this parametrization we produced spinningdust spectra for different input physical parameters with the SpDust code and fitted each of them with the proposed relation by minimizing the χ^{2} for the set of frequencies used in this work. The input models we considered are weak neutral medium (WNM), cold neutral medium (CNM), weak ionized medium (WIM), and molecular cloud (MC). Both the input SpDust parameters and the bestfit m_{60}, ν_{p} parameters for each model are reported in Table B.1. For comparison, we also considered alternative parametric relations and in particular

the model implemented in the Commander componentseparation method (Pietrobonet al. 2012; PlanckCollaboration 2013d) which is a Gaussian in the T_{CMB} − ln(ν)plane, parametrized in terms of central frequency and width;

the Tegmark et al. (2000) model, which is a modified blackbody relation (Eq. (1)) with a temperature of around 0.25K and an emissivity index of about 2.4;
Because this test neither accounts for the presence of the other components and includes any data simulation, it verifies the intrinsic ability of the parametric model to reproduce the actual spectra. Realistic estimation errors for the CCA model are derived through simulations in Appendix C.
Figure B.1 compares the input spectra with the bestfit models for the different parametrizations. In general, the fits are accurate at least up to ν = 50–60 GHz, while at higher frequencies the parametric relations may not be able to reproduce the input spectra in detail. This is a consequence of fitting complex spectra with only a few parameters. The fit tends to fail where the AME signal is weaker.
Over the frequency range considered, CCA and Commander models fit the input spectrum generally better than the Tegmark et al. (2000) model (which decreases too rapidly at high frequencies). When adding lower frequency data, however, CCA and Commander models will be increasingly inaccurate, because they are symmetric with respect to the peak of the emission. The models implemented by CCA and Commander perform quite similarly, despite the different formulation. As a result, these methods are able to give consistent answers, which ensures consistency between different analyses within Planck (e.g. Planck Collaboration 2013d).
The CCA model used in this work provides a reasonable fit to theoretical spinningdust models for a variety of physical conditions. The bestfit parameters that we obtain, reported in Table B.1, vary significantly from one input model to another and have a straightforward interpretation in terms of the spectrum.
Fig.B.1
Theoretical spinning dust models produced with SpDust (solid lines) and fitted with CCA (triangles), Commander (diamonds), and Tegmark et al. (2000) (asterisks) models. Input SpDust parameters and bestfit parameters for the CCA model are provided in Table B.1. 

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Appendix C: Description of the simulations
Fig.C.1
Freefree templates at 23 GHz used for the analysis. The reference template FF_{REF} is in the upper left corner; the other columns (left to right) are FF_{1}, FF_{2} and FF_{3}, respectively. The differences in the lower panels (FF_{i} − FF_{REF})/FF_{REF} are on average of the order of 10%, but reach 50% in regions of strong dust emission. 

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We simulated Planck and WMAP 7yr data by assuming monochromatic bandpasses positioned at the central frequency of the bands, Gaussian beams at the nominal values indicated in Tables 1 and 2, and Gaussian noise generated according to realistic, spatially varying noise rms. Our model of the sky consists of the following components:

CMB emission given by the bestfit power spectrum model fromWMAP 7yr analyses;

synchrotron emission given by the Haslam et al. (1982) template scaled in frequency with a powerlaw model with a spatially varying synchrotron spectral index β_{s}, as modelled by Giardino et al. (2002);

freefree emission given by the Dickinson et al. (2003) Hα corrected for dust absorption with the E(B − V) map from Schlegel et al. (1998) with a dust absorption fraction f_{d} = 0.33, and scaled in frequency according to Eq. (2) with T_{e} = 7000K;

thermal dust emission modelled with the 100 map from Schlegel et al. (1998), scaled in frequency according to Eq. (1) with T_{d} = 18K and a spatially varying β_{d} with an average value of 1.7;

AME modelled by the E(B − V) map from Schlegel et al. (1998) with the intensity at 23 GHz calibrated using the results of Ghosh et al. (2012) for the same region of the sky.
We adopted more than one spectral model for the AME. We first considered two convex spectra, generated with the SpDust code: one peaking around 26 GHz and the other peaking around 19 GHz. We also tested a spatially varying powerlaw model (with spectral index of −3.6 ± 0.6, Ghosh et al. 2012), which could result from the superposition of multiple convex components along the line of sight.
It is worth noting that the simulated sky is more complex than the model assumed in the component separation. This has been done intentionally, to reflect a more realistic situation. Another realistic feature we included are errors in the synchrotron and freefree templates. The spatial variability of the synchrotron spectral index modifies the morphology of the component with respect to that traced by the 408MHz map from Haslam et al. (1982). The use of Hα as a tracer of freefree emission is affected by even larger uncertainties. Our uncertainties on the dust absorption fraction f_{d} (estimated to be at intermediate latitudes by Dickinson et al. 2003) and on the scattering of Hα photons from dust grains can create dustcorrelated biases in the template. This is illustrated in Fig. C.1, where we compare different versions of the freefree template. FF_{REF} is our reference template, adopted for the analysis of real data and for simulating the component, which is corrected for f_{d} = 0.33 as described in Dickinson et al. (2003). Two more templates (FF_{1} and FF_{2}) were obtained by correcting Hα for f_{d} = 0.33−0.15 and f_{d} = 0.33 + 0.1 (± 1σ according to Dickinson et al. 2003). A final template (FF_{3}) was obtained by correcting FF_{REF} for scattered light at the 15% level by subtracting from the freefree map the 1 map of Schlegel et al. (1998) multiplied by a suitable constant factor (Witt et al. 2010). Difference maps (FF_{i} − FF_{REF})/FF_{REF}, presented in the lower panels of Fig. C.1, are on average of order of 10%, but can be much higher (up to 50–60%) in regions of strong dust emission.
When analysing the simulated data, we used both FF_{1} and FF_{2} as freefree templates in place of FF_{REF}, which corresponds to the simulated component. For synchrotron emission the morphological mismatch between the simulated component and the template was achieved by scaling the component from 23 GHz to 408MHz with a spatially varying spectral index. The comparison between component and template is presented in Fig. C.2; the differences are of the order of 10%.
Fig.C.2
Upper panel: simulated synchrotron component (left) and synchrotron template (right) at 23 GHz. Lower panel: difference map divided by the simulated component. 

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The simulated datasets described above were analysed with the CCA method using the same procedure as applied to the real data; the results of this assessment are presented in Sect. 4.1. As a separate test, we verified the impact of the CMB component on the results for ν_{p} and m_{60}. We generated 100 sets of mock data with the same foreground emission and different realizations of CMB and instrumental noise, and repeated the estimation of the AME frequency scaling. For this test we used the simulation with a spatially constant AME spectrum peaking at 26 GHz. As this analysis is computationally demanding, the CCA estimation was performed only on the ten independent patches covering the Gould Belt region (centred on latitudes −20° and −40° and longitudes of 140°, 160°, 180°, 200°, and 220°). In Fig. C.3 we show the average (diamonds) and rms (error bars) ν_{p} and m_{60} over the 100 realizations for each
patch for different patches on the xaxis. The scatter between the results obtained for different patches (indicated by the grey area in the plots) is typically larger than the error bars, measuring the scatter due to different CMB realizations. This means that the foreground emission generally dominates over the CMB as a source of error. Larger error bars associated with the CMB are obtained for three patches with faint foreground emission. For these patches the estimated errors on ν_{p} and m_{60} are consistently larger. The CMB variation results on average in Δν_{p} = 0.1 GHz and Δm_{60} = 0.3, which reach 0.3 GHz and 0.8, respectively, for the poorest sky patch. These values are below the error bars resulting from the analysis of the data, which amount to 1–1.5 GHz for ν_{p} and 1.5–2 for m_{60}. The CMB has limited impact on the results because this component is modelled in the mixing matrix. Having a known frequency scaling, the statistical constraint used by CCA is able to trace the pattern of the CMB through the frequencies with good precision and hence identify it correctly.
Fig.C.3
Average and rms of ν_{p} and m_{60} estimated over simulations with different CMB and noise realizations for different patches on the xaxis. The grey area is the average and rms over different patches, which is typically larger than that due to noise and CMB. 

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© ESO, 2013
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