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Appendix A: Derivation of the CC coefficients

This appendix details how we compute the dust SED using the CC analysis in Sect. 4.2. For simplicity, we present the simplest case for intensity from the fit with one template.

We minimize the between the data and the 353 GHz template maps, as expressed in Eq. (7). The CC coefficient is then given as (A.1)where and Î₃₅₃ are the data and 353 GHz template with mean values (computed over the N_pix) subtracted. The observed Planck map at a given frequency is written as the sum of the CMB signal, the Galactic signals (synchrotron, free-free, dust and AME) and noise as (A.2)where the superscripts c, d, s, f, e, and n represent the CMB, dust, synchrotron, free-free, AME, and noise, respectively. Combining Eqs. (A.1) and (A.2) we find (A.3)where is the normalization factor. At HFI frequencies, we can neglect the contribution of the synchrotron, free-free, AME, and noise within our global mask. This reduces Eq. (A.3) to (A.4)The CMB contribution, α(c₃₅₃), is the CC coefficient obtained by correlating the CMB signal map with the 353 GHz template, which is independent of frequency. The dust emission at a given frequency is a scaled version of 353 GHz dust emission, , where is the mean dust SED over the given sky patch. The CC coefficient is then (A.5)where is proportional to the mean dust SED. The colour ratio is then given by (A.6)In the ratio the scaling between the and goes away. The colour ratio only depends on the dust spectral properties and not on the CMB signal. The extension of Eq. (A.5) in the presence of AME, synchrotron, and free-free emission at the WMAP and LFI frequencies is given by Eq. (8).

The CMB contribution, α(c₃₅₃), in the presence of inverse noise-weighting can be written as (A.7)where w_ν is a weighting factor given by . If the weighting factor depends on the frequency, the CMB contribution is not strictly constant in K_CMB units. This effect can reach up to 2%, as the weighting factors for the WMAP and Planck maps are quite different. This is not negligible compared to the dust emission at microwave frequencies. That is the reason why we do not use inverse noise-weighting in our minimization.

In Sect. 9.2.1 we compute the frequency dependence of synchrotron emission correlated with dust. The mean spectrum of this component is given by (A.8)assuming the CMB chance correlation term with synchrotron emission is zero over all the sky patches. We detect with high-significance in our analysis (Sect. 9.2.1), which cannot be just a chance correlation term. One would expect such a correlation, since synchrotron emission arises from the same ISM as dust emission.

Similarly for polarization, we minimize between the data and 353 GHz Stokes Q and U maps, as given by Eq. (13). The polarization CC coefficient is then given by (A.9)where N_P is the normalization factor for polarization. Following the same logic as described for intensity, the polarization CC coefficient at the HFI frequencies can be written as (A.10)

Assuming the dust polarization at a given frequency is a scaled version of 353 GHz dust polarization yields and . Putting this back to into Eq. (A.10) gives (A.11)The polarized colour ratio does not depend on the CMB like Eq. (A.5). The polarized CMB contribution α^P(c₃₅₃) is strictly constant in K_CMB units if we do not apply any noise weighting, similar to the intensity analysis. To deal with the noise, we first smooth all the maps to 1° resolution and then perform correlation over local patches on the sky. To compute the uncertainty on the CC coefficients, we rely on Monte Carlo simulations, as discussed in Appendix B.

Appendix B: Simulations

This appendix presents the simulations of the sky emission in intensity and polarization at HFI frequencies that we use to test the CC analysis. The intensity and polarization emission components are listed in Table B.1. The simulations use a simplified model of dust emission in intensity and polarization that is good enough to provide a realistic framework to test the CC analysis. They are computed on HEALPix pixels at N_side = 128 with a 1° Gaussian beam. The Monte Carlo simulations serve two specific purposes. First, we use them to check that the CC analysis does not introduce any bias on our estimations of the mean dust spectral indices in intensity and polarization. Second, they provide realistic uncertainties on the CC coefficients, which we use in the spectral fit to separate out the dust and the CMB emission (Sect. 5.3).

Appendix B.1: Intensity

At HFI frequencies, the main diffuse emission components are the thermal dust, free-free, CMB, and CIB emission. The simulations also include instrumental noise. We now describe how we simulate each of these components.

The Hi column density from the LAB survey (Kalberla et al. 2005) is taken as a proxy for thermal dust emission. We normalize the Hi data to a suitable amplitude to match the observed Planck data at 353 GHz and extrapolate to the other HFI frequencies using an MBB spectrum with a fixed spectral index β_d = 1.5 and temperature T_d = 19.6 K over the whole sky. The Hi data provide only a partial description of the thermal dust emission, as quoted in Planck Collaboration Int. XVII (2014). We include an additional dust component, spatially uncorrelated with the Hi data, to mimic the residuals present after adopting the IR-Hi correlation at 857 GHz. The additional dust-like emission is assumed to have an ℓ^-3 power spectrum, with a normalized amplitude of for ℓ = 2, where σ₈₅₇ is the residual at 857 GHz after applying the IR-Hi correlation and removing the CIB contribution (Planck Collaboration Int. XVII 2014). The amplitude of the uncorrelated Hi emission is normalized at 857 GHz, taken from Planck Collaboration Int. XVII (2014), and scaled to the HFI frequencies assuming β_R = 2.0 for a dust temperature of T_R = 19.6 K. We use the DDD Hα map as a proxy for free-free emission, which we compute at HFI frequencies for a spectral index β_f = −2.14 (in K_RJ units; Planck Collaboration Int. XIV 2014) and an electron temperature T_e = 7000 K (Dickinson et al. 2003). No dust extinction correction is applied to the DDD Hα map.

For the CMB, we compute Gaussian realizations of the CMB sky from the theoretical power spectrum of the Planck best-fit model (Planck Collaboration XV 2014). The CIB emission is generated using the best-fit model of CIB anisotropies at 353 GHz obtained directly from the Planck data (Planck Collaboration XXX 2014). We assume 100% correlated CIB across all the HFI frequencies, assuming an MBB spectrum with β_CIB = 1.3 and T_CIB = 18.4 K. The Gaussian realizations of the instrumental noise are obtained at each frequency, using the noise variance maps (Planck Collaboration VI 2014). The noise realizations are simulated at the full resolution of the Planck data, before smoothing to 1° resolution and reducing the pixelization from N_side = 2048 to 128.

We compute 1000 realizations of sky maps of the additional dust component, together with the CMB and CIB anisotropies. Independent realizations of the instrumental noise are generated for each sky simulation at a given frequency. The dust component computed from the Hi map and the free-free emission traced by the Hα map are kept fixed.

We analyse the 1000 simulated maps with the CC method applied to the Planck intensity data. We compute the mean and standard deviation of the values for each sky patch. Both are plotted in Fig. B.1 versus the local dispersion of the 353 GHz template, . There is no bias on the estimation of the mean dust spectral index . We recover a mean value equal to the index of 1.5 we used for the main Hi-correlated dust component. The uncertainties on , and hence on , are associated with noise, CIB anisotropies, free-free emission, and the additional dust component. The 1σ dispersion of across sky patches for a given Monte Carlo realization is 0.02. This is smaller than the scatter of 0.07 measured for the Planck data. We interpret the difference as evidence for a small intrinsic dispersion in the spectral index of the dust emission.

Appendix B.2: Polarization

The simulations of the polarized sky at HFI frequencies include polarized CMB, thermal dust emission, and noise. We compute 1000 realizations of the CMB Stokes Q and U maps using the best-fit Planck model (Planck Collaboration XV 2014), smoothed to 1° resolution at HEALPix resolution N_side = 128. Random realizations of Gaussian noise Q and U maps are generated at each pixel using the 3 × 3 noise covariance matrix defined at N_side = 2048. The noise maps are then smoothed to 1° resolution and projected on to a HEALPix map at N_side = 128. We generate independent realizations of the instrumental noise to mimic the detector sets at 353 GHz (Sect. 2.1.2).

	Fig. B.1 R100(353,217) colour ratios from the Monte Carlo simulations for intensity (top) and polarization (bottom). The two plots show that the CC analysis does not introduce any bias on the estimation of βd,mm.
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For polarized thermal dust emission, we use the following model: (B.1)Here p_d, , and ψ_d are the polarization fraction, the dust intensity at reference frequency, and the polarization angle, respectively. The reference frequency is ν_ref = 353 GHz. We fix p_d to a constant value of 10% over the whole sky. The map is that obtained by Planck Collaboration XI (2014) from the spectral fit to the high frequency Planck and IRAS 100 μm data, with an MBB model. For the dust polarization, we use an MBB spectrum with and T_d = 19.6 K, constant over the whole sky. We derive ψ_d from the 1° smoothed Planck Stokes maps using the relation (B.2)We choose ν₁ and ν₂ as the 143 and 353 GHz, respectively. The difference between the two frequencies removes the CMB contribution.

We analyse 1000 polarized simulated maps using the CC analysis as applied to the Planck data. We compute the mean and the standard deviation of the for each sky patch. The plot of versus is shown in Fig. B.1. We find no bias in the estimation of and hence in the measurement of . The 1σ dispersion of across sky patches for a given simulation is 0.07. The 1σ dispersion of from the simulations is smaller compared than that measured from the Planck data, because we use a simplified white noise model. However, some of the dispersion may come from the intrinsic dispersion of the polarized dust spectral index, and also additional Galactic polarized emission components, which we neglect in the simulations.

Appendix C: Mean dust SED with dust extinction and scattering correction on the Hα template

The mean dust SED for intensity presented in this paper is obtained using the three-template fit with no extinction and dust scattering correction from the DDD Hα template. The effect of dust extinction (f_d) on the Hα template is described in Eq. (3) of Dickinson et al. (2003), whereas the effect of dust scattering (s_d) on the Hα template is described in Eq. (26) of Bennett et al. (2013). The mean measured value of s_d is 0.11 R( MJy sr^-1)^-1 in high Galactic latitude regions (Lehtinen et al. 2010; Witt et al. 2010; Seon & Witt 2012; Brandt & Draine 2012; Bennett et al. 2013). To check the impact of the f_d and s_d corrected Hα template on the mean dust SED, we repeat the analysis with different combinations of f_d and s_d. The three different combinations of f_d and s_d corrected Hα templates we choose are: f_d = 0.3 and s_d = 0.0 R( MJy sr^-1)^-1; f_d = 0.0 and s_d = 0.11 R( MJy sr^-1)^-1; and f_d = 0.3 and s_d = 0.11 R( MJy sr^-1)^-1. The fractional change in the mean dust SED with respect to the reference dust SED (f_d = 0.0 and s_d = 0.0 R( MJy sr^-1)^-1) is presented in Fig. C.1. At higher frequencies (ν ≥ 100 GHz), the impact of both dust extinction and scattering is negligible. However at frequencies ν ≤ 50 GHz, the fractional change on the mean SED can go as high as ±4%. The f_d and s_d parameters are degenerate, although their effect on the derived best-fit parameter of models DI+AI and DI+AII, listed in Table 4, is very small.

	Fig. C.1 Fractional change in the mean dust SED with respect to the reference dust SED presented in this paper, for different combinations of fd and sd corrections on the DDD Hα template.
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Appendix D: Power spectra of the templates

	Fig. D.1 Power spectra of the 408 MHz, DDD Hα, and SMICAsubtracted 353 GHz templates, smoothed to 1° resolution for different Galactic masks (or fsky).
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In this section, we compute the temperature power spectra of the three templates at 1° resolution: 408 MHz; DDD Hα; and 353 GHz dust template. The Planck353 GHz map contains a significant component of CMB anisotropies. Taking the SMICA map (Planck Collaboration XII 2014) as a proxy for the CMB map, we remove its contribution from the 353 GHz total map. These spectra are combined with the SEDs from this paper to compute the contributions of each emission component to the microwave sky emission as a function of angular scales in Planck Collaboration I (2014, see their Figs. 27 and 28).

For the computation of the power spectra, we consider the four diffuse Galactic masks based on the percentage of the sky retained (f_sky), i.e., G40, G60, G70 and G80 (Planck Collaboration XV 2014). The same set of masks have been used in the likelihood analysis of the 2013 Planck data release (Planck Collaboration XV 2014). The power spectra are computed only at low multipoles (ℓ< 100) with PolSpice v2.9.0 (Chon et al. 2004), corrected for the masking, beam, and pixel window effect. Figure D.1 presents binned power spectra of the three templates: 408 MHz, DDD Hα, and SMICA-subtracted 353 GHz maps as a function of the Galactic masks. The uncertainties on the binned power spectra include only the statistical variance and not the cosmic variance.

At low multipoles, ℓ< 100, the three power spectra are well-fit with a power-law model. Using this assumption, the measured power spectra are written as C_ℓ = A × (ℓ/ 100)^α. Here A represents the normalized amplitude at ℓ = 100 and α represents the slope of the power-law for a given template. We fix α based on the measured spectra and only fit for the amplitudes as a function of the Galactic masks. We find that the slope of the 408 MHz spectra over all the Galactic masks is consistent with − 2.5. In case of DDD Hα template is − 2.2 over the masks and the same for the SMICA-subtracted 353 GHz template is − 2.4. The results of the power-law fit for the three templates and different Galactic masks are shown as a dashed lines in Fig. D.1. The amplitudes of each of the templates as a function of the Galactic masks (or f_sky) are listed in Table D.1.

The amplitudes of the given templates vary nonlinearly as a function of f_sky. They can be fitted with a second-order polynomial in a log A – log f_sky plane. Combining the ν, ℓ, and f_sky dependence, we analytically model the power spectra of the diffuse synchrotron, free-free, and dust emission components for intensity. For amplitude normalization, we made an assumption on the nature of the synchrotron and free-free emission. We assume a single power-law model for the synchrotron emission from 408 MHz to microwave frequencies, ν ≤ 353 GHz. For free-free emission, we assume a single power-law model at microwave frequencies, with a mean electron temperature of 7000 K (Dickinson et al. 2003). The power spectra of the three diffuse emission components, in units, are where ν_b = 23 GHz, ν_c = 0.408 GHz, is a spectral model of the dust emission given by one of the two models presented in Eqs. (26), and (27). The derived analytical model of these power spectra are valid in the frequency range 20 to 353 GHz, and for f_sky between 0.4 and 0.8.

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