Page:Cbass template fitting.pdf/5

5904S. E. Harper et al. et al. (1998), which we will refer to as FDS8. The FDS8 map is an extrapolation of the 100 and 240-μm IRAS maps calibrated to the COBE-FIRAS spectral data. For many previous studies of dust correlated AME, this map has been used, however, it has been superseded by direct observations of these frequencies by the Planck mission.

3.4 WMAP data

We use the Wilkinson Microwave Anisotropy Probe (WMAP) 9year data release (Bennett et al. 2013) convolved to a Gaussian 1° resolution beam obtained from the LAMBDA website. The WMAP satellite observations were made using ten differencing assemblies at five frequencies from 23 to 94 GHz. The original FWHM resolution of the instrument was 0°.93 to 0°.23. For the calibration uncertainty of the WMAP maps, we assign a 3 per cent uncertainty to account for colour corrections and residual beam asymmetries. We convert all the maps to brightness temperature units from thermodynamic units relative to the CMB.

3.5 Planck data

We use the 2018 release of the Planck all-sky maps (Planck Collaboration I 2020). The Planck low-frequency instrument (LFI) was a pseudo-correlation radiometer that operated between 30 and 70 GHz, with FWHM resolutions of 33 to 13 arcmin; and the high-frequency instrument (HFI) used bolometers that spanned a frequency range of 100–857 GHz, with resolutions of 7.2–4.5 arcmin FWHM. We adopt calibration uncertainties for the LFI of 3 per cent (Planck Collaboration Int. XV 2014) and HFI of 5 per cent (Planck Collaboration I 2020); these uncertainties account for colour corrections, residual beam asymmetries, and other systematics. Units were converted from thermodynamic units relative into the CMB to brightness temperature units.

4 TEMPLATE FITTING

4.1 Method

To estimate the contribution of a given emission component to each frequency map, we use a template fitting method (also referred to as a correlation analysis); a technique that has been well established for studying Galactic foregrounds (Kogut et al. 1996; de Oliveira-Costa et al. 1997; Davies et al. 2006; Planck Collaboration XXI 2011; Ghosh et al. 2012; Peel et al. 2012; Planck Collaboration Int. XXII 2015). The method assumes that the sky signal can be decomposed into a linear combination of NZ template maps, where each template is chosen to trace a single emission component. Template fitting is performed by solving for the template coefficients a in

$$\mathbf{d} = \mathbf{Z}\mathbf{a} + n,$$ (3)

where each column of Z (Npix × NZ) contains the Npix pixel intensities of the templates used to decompose the sky vector d (Npix). We want to solve for the template coefficients vector a, where each coefficient describes the radio brightness of a given emission component at a given frequency per unit template. Least-squares solution gives

$$\hat{\mathbf{a}} = \left(\mathbf{Z}^T\mathbf{N}^{-1}\mathbf{Z}\right)^{-1} \mathbf{Z}^{T}\mathbf{N}^{-1}\mathbf{d},$$ (4)

where N−1 is the pixel noise covariance matrix. The details of the noise covariance matrix are described in Section 4.5. We fit for the synchrotron, dust, and free–free emission components, as well as an arbitrary offset. Before we perform the template fit we first subtract the mean offset of each template. Subtracting an offset from each template does not impact the fitted coefficients but does improve convergence and removes arbitrary correlations between templates and the data.

Coefficient uncertainties can be calculated using the covariance of the coefficients as

$$C_a = \mathbf{Z}^T\mathbf{N}^{-1}\mathbf{Z}~,$$ (5)

where each parameter is as defined in equation (4). However, estimating the coefficient uncertainties in this manner requires both the templates to be perfect representations of the underlying emission, and for the noise to be Gaussian distributed in each data set. Therefore we instead estimate uncertainties and the correlation between coefficients using the bootstrapping method (Efron 1979; Efron & Tibshirani 1986). The bootstrapping method gives unbiased estimates of the uncertainties in the coefficients by randomly resampling with replacement the pixels in each region. Resampling is done 1000 times for each region resulting in uncertainties in the bootstrapped coefficient uncertainties of ≈3 per cent. We estimate the coefficient covariance matrix by averaging over the outerproduct of all the estimates of a,

$$C_a = \left \langle \hat{\mathbf{a}}\hat{\mathbf{a}}^T\right \rangle.$$ (6)

We find that for regions far from bright sources of Galactic emission the differences between the coefficient uncertainties estimated using equations (5) and (6) were small – the bootstrapped uncertainties were 10–20 per cent larger for data between 5 and 60 GHz. However, for regions nearer to the Galactic plane, or coincident with bright features (i.e. Eridanus/Orion – regions 69, 82, 83, and 97 in Fig. 2) the bootstrapped uncertainties can be as much as an order-of-magnitude larger. For the higher frequencies (ν > 143 GHz), the bootstrapped uncertainties were systematically larger in all regions by as much as an order-of-magnitude, which is likely due to not including any additional sources of noise at high frequency other than the instrument noise. Therefore, it is clear that the bootstrapped uncertainties are more reliable and representative of the data.

4.2 Free-free emission removal

The C-BASS map contains contributions from both synchrotron and free–free emission, the latter from the warm-ionized medium (WIM). When using the 4.76-GHz C-BASS map as a synchrotron template we first subtract a global estimate of the free–free emission using the Hα data. We use the best-fitting Hα coefficient at 4.76 GHz of Tbff /IHα = (195 ± 5) μK/R (see Section 5.3 for derivation of this value) to scale the Hα data, and then subtract this from the C-BASS map.

It is possible that subtracting the free–free emission in the CBASS map may result in systematic biases in later results (since we are using the same Hα map to trace free–free emission at higher frequencies). Therefore, we subtracted free–free templates from the C-BASS data using fixed electron temperature values of 5000, 6000, 7000, and 8000 K using equation (15). Changing the amplitude of the subtracted free–free component resulted in no significant change in the fitted coefficients discussed in Section 5, which is not unexpected since free–free emission contributes less than 20 per cent of the total emission at 4.76 GHz at high Galactic latitudes.

4.3 Mask

Template fitting is only as effective as the templates that trace the underlying emission components at a given frequency. In order to MNRAS 513, 5900–5919 (2022)

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