Galactic Foreground Constraints on Primordial B-mode Detection for Ground-based Experiments

The component separation method used for our analysis is the CCA (Bonaldi et al. 2006; Ricciardi et al. 2010). This is a parametric foreground fitting algorithm that exploits a generalized least-squares method applied to second-order statistics. In its harmonic domain version, which we use in this work, it estimates both the frequency behavior of all the foregrounds, expressed as a function of a few parameters, and the auto- and cross-power spectra of the components. A parametric method is particularly useful when investigating the correlation between the goodness of foreground fitting and the intensity of the foregrounds, which is the goal of this analysis.

As with many other component separation techniques, CCA is a linear method. It assumes that the observed signal is a linear combination of the true signal from the different components in the sky plus some noise. By making a few other assumptions (that spectral and spatial features on the components are not correlated, and that the instrumental beam is constant within each passband), the signal can be written for each line of sight $\hat{{\bf{r}}}$ as

$\begin{eqnarray}&&{\boldsymbol{x}}(\hat{{\boldsymbol{r}}})=[{\boldsymbol{B}}* {\boldsymbol{Hs}}](\hat{{\boldsymbol{r}}})+{\boldsymbol{n}}(\hat{{\boldsymbol{r}}}),\end{eqnarray} \tag{ 3 }$

where x and n are vectors containing the observed signal and noise, respectively, B is a matrix containing the beam per frequency channel, s is the vector containing the true signal of each component, and the symbol * denotes convolution. H is the mixing matrix that contains the frequency spectra (SED) of all the components (three in our case: CMB, synchrotron, and thermal dust). If we transform to harmonic space, Equation (3) becomes

$\begin{eqnarray}&&\tilde{{\boldsymbol{x}}}=\tilde{{\boldsymbol{B}}}{\boldsymbol{H}}\tilde{{\boldsymbol{s}}}+\tilde{{\boldsymbol{n}}},\end{eqnarray} \tag{ 4 }$

where $\tilde{{\boldsymbol{x}}}$, $\tilde{{\boldsymbol{s}}}$, and $\tilde{{\boldsymbol{n}}}$ are the harmonic transforms of x , s , and n , respectively, and $\tilde{{\boldsymbol{B}}}$ is the harmonic transform of B .

The relation between the covariances is given by

$\begin{eqnarray}&&{\tilde{{\boldsymbol{C}}}}_{{\boldsymbol{x}}}=\tilde{{\boldsymbol{B}}}{\boldsymbol{H}}{\tilde{{\boldsymbol{C}}}}_{{\boldsymbol{s}}}{{\boldsymbol{H}}}^{T}{\tilde{{\boldsymbol{B}}}}^{\dagger }+{\tilde{{\boldsymbol{C}}}}_{{\boldsymbol{n}}},\end{eqnarray} \tag{ 5 }$

where the dagger superscript denotes the adjoint matrix, and the angular power spectra C for the observations x , for the true signal s and noise n . C _x and C _n are estimated from the data and noise properties, $\tilde{{\boldsymbol{B}}}$ is also known, while C _s and H are the unknowns.

To reduce the number of unknowns, the mixing matrix is expressed as a function of a few parameters H = H ( p ). We use the blackbody spectrum for the CMB (with no free parameters); Equation (2) for synchrotron, with the spectral index β_syn as a free parameter; and Equation (1) for thermal dust, where the spectral index β_dust and the temperature T_dust are, in general, both free parameters.

Finally, the data and source spectra are binned in multipoles to further reduce the number of unknowns. Then, in harmonic space, our model is the following equation:

$\begin{eqnarray}&&{{\boldsymbol{d}}}_{V}={{\boldsymbol{ \mathcal H }}}_{{kB}}{{\boldsymbol{c}}}_{V}+{\epsilon }_{V},\end{eqnarray} \tag{ 6 }$

where d _V are the noise-bias-subtracted auto- and cross-power spectra of the observations and arranged into a vector over the multipole bins $\hat{{\ell }}$. ${{\boldsymbol{ \mathcal H }}}_{{kB}}$ is a block-diagonal matrix that contains one block per multipole bin. Each block is a matrix given by ${{\boldsymbol{ \mathcal H }}}_{k}(\hat{{\ell }})=[\tilde{{\boldsymbol{B}}}(\hat{{\ell }}){\boldsymbol{H}}]\otimes [\tilde{{\boldsymbol{B}}}(\hat{{\ell }}){\boldsymbol{H}}]\,$, where ⨂ is the Kronecker product (hence the k subindex). This matrix, just like the mixing matrix, depends on the parameters p . c _V is the power spectra of the different source components in the model, again arranged into a vector over multipole bins, and _V represents the residuals of the power spectra, between the true sources in the sky and the modeled source components.

Ricciardi et al. (2010) shows that the mixing matrix parameters, p , and the binned power spectra of the components, c _V , can be found by minimizing the functional

$\begin{eqnarray}&&{\boldsymbol{\Phi }}({\boldsymbol{p}},{{\boldsymbol{c}}}_{V})={[{{\boldsymbol{d}}}_{V}-{{\boldsymbol{ \mathcal H }}}_{{kB}}({\boldsymbol{p}})\cdot {{\boldsymbol{c}}}_{V}]}^{T}{{\boldsymbol{ \mathcal N }}}_{\epsilon B}^{-1}[{{\boldsymbol{d}}}_{V}-{{\boldsymbol{ \mathcal H }}}_{{kB}}({\boldsymbol{p}})\cdot {{\boldsymbol{c}}}_{V}],\end{eqnarray} \tag{ 7 }$

where ${{\boldsymbol{ \mathcal N }}}_{\epsilon B}$ is a block-diagonal matrix, containing one block per multipole bin. Each block is the matrix ${{\bf{N}}}_{\epsilon }(\hat{{\ell }})$, which corresponds to the auto and cross terms (over the frequency channels) of the Gaussian covariance matrix for the noise power spectra.

For each value of the mixing matrix parameters p , the estimated source angular power spectra ${\bar{{\boldsymbol{c}}}}_{V}$ are obtained as a suitable linear mixture of the data power spectra, depending on the estimated mixing matrix ${{\boldsymbol{ \mathcal H }}}_{{kB}}({\boldsymbol{p}})$ and the noise covariance matrix ${{\boldsymbol{ \mathcal N }}}_{\epsilon B}$, given by

$\begin{eqnarray}&&{\bar{{\boldsymbol{c}}}}_{V}({\boldsymbol{p}})={\left[{{\boldsymbol{ \mathcal H }}}_{{kB}}^{T}({\boldsymbol{p}}){{\boldsymbol{ \mathcal N }}}_{\epsilon B}^{-1}{{\boldsymbol{ \mathcal H }}}_{{kB}}({\boldsymbol{p}})\right]}^{-1}{{\boldsymbol{ \mathcal H }}}_{{kB}}^{T}({\boldsymbol{p}}){{\boldsymbol{ \mathcal N }}}_{\epsilon B}^{-1}{{\boldsymbol{d}}}_{V}.\end{eqnarray} \tag{ 8 }$

The solution for both the mixing matrix and the source spectra yields the minimum value of Equation (7). The minimization in CCA is performed with simulated annealing over the foreground spectral parameters.

We have modified the CCA method slightly as compared to the one presented in Ricciardi et al. (2010). The estimation is done over the polarization spectra EE and BB, as opposed to working over the spectrum of any random scalar field, as before. We also used a more general noise covariance ${{\boldsymbol{N}}}_{\epsilon }(\hat{{\ell }})$ than the white-noise-only approximation used in Ricciardi et al. (2010). The namaster software (Alonso et al. 2019) has been used to estimate the angular power spectra of the data and the noise as well as the noise covariance matrix (whose estimation adapts the analytical estimate for a Gaussian covariance proposed in Efstathiou 2004; Couchot et al. 2017). We set the cross-power spectra to zero because we do not simulate noise cross-correlation among the frequency channels.

To translate the error on the BB power spectrum to an uncertainty on r, σ(r), we use the Fisher information matrix. This is given as

$\begin{eqnarray}&&{F}_{{ij}}=\sum _{\hat{{\ell }},\hat{{\ell }}^{\prime} =1}^{{N}_{b}}{{\boldsymbol{ \mathcal C }}}_{\hat{{\ell }},\hat{{\ell }}^{\prime} }^{-1}\displaystyle \frac{\partial {c}_{\hat{{\ell }}}}{\partial {p}_{i}}\displaystyle \frac{\partial {c}_{\hat{{\ell }}^{\prime} }}{\partial {p}_{j}},\end{eqnarray} \tag{ 9 }$

where N_b is the number of multipole bins, ${\boldsymbol{ \mathcal C }}$ is the covariance matrix over multipole bins of the observable, and $\partial {c}_{\hat{{\ell }}}/\partial {p}_{i}$ is the binned derivative of the fiducial model spectra with respect to the parameter p_i . The covariance matrix of the parameters is given by the inverse of F_ij . Therefore, the uncertainty of parameter i (marginalized over the rest of the parameters) is given by the square root of the ii element of the inverse of the Fisher information matrix. The off-diagonal elements will measure the correlation between parameters i and j.

In our case, the observable is the estimated CMB BB power spectrum ${C}_{{\ell }}^{{BB}}$ and the parameter we are interested in is the tensor to scalar ratio r. To correct for residual foreground contamination in the CMB BB power spectrum, we introduce a second parameter, ${A}_{\mathrm{fore}}$, which represents the residual foreground amplitude. We do not consider the foreground spectral parameters explicitly in the Fisher matrix, but they are included in the estimation of the ${{\boldsymbol{ \mathcal C }}}_{\hat{{\ell }},\hat{{\ell }}}$ covariance via the Monte Carlo simulation. Therefore, our fiducial model for the BB power spectrum is the following:

$\begin{eqnarray}\begin{array}{rcl}{c}_{{\ell }}(p) & = & {C}_{{\ell }}^{{BB},\mathrm{fiducial}}(r,{A}_{\mathrm{fore}})={{rC}}_{{\ell }}^{{BB},\mathrm{primordial}}(r=1)\\ & & +{C}_{{\ell }}^{{BB},\mathrm{lensing}}+{A}_{\mathrm{fore}}{C}_{{\ell }}^{{BB},\mathrm{foregrounds}}(p).\end{array}\end{eqnarray} \tag{ 10 }$

The covariance matrix of the observable ${\boldsymbol{ \mathcal C }}$ is easily calculated from 200 Monte Carlo iterations over the simulations. The derivative of the fiducial BB power spectra with respect to r and ${A}_{\mathrm{fore}}$ is calculated numerically with respect to mean values r = 0 and ${A}_{\mathrm{fore}}=1$.

To estimate ${C}_{{\ell }}^{{BB},\mathrm{foregrounds}}$, we produced companion simulations with only the foregrounds, without CMB and noise. This data set is mixed together with the same weights calculated by CCA, given in Equation (8). Finally, the power spectra ${C}_{{\ell }}^{{BB},\mathrm{foregrounds}}$ is given by the average over the 200 Monte Carlo simulations. This is labeled method A. We realize this procedure is not feasible with real observations, because pure foregrounds maps are not available. We also consider a second method, which is very pessimistic: Because thermal dust is the dominant foreground (as will be shown in Section 5.1), we produce 200 Gaussian realizations following a power-law angular power spectrum fitted in Planck Collaboration et al. (2020b) using a mask labeled Large Region (LR) 71, with f_sky = 0.71. We extrapolate this Gaussian realization map at 353 GHz to other frequencies using a standard MBB (Equation (1)) with the β_dust and T_dust maps from Planck Collaboration et al. (2016b). Note that because we are using the LR71 mask including the full Galactic emission, power will be grossly overestimated in the cleanest regions, but because we are using the Fisher information matrix, the derivative will cancel the absolute amplitude of the foreground residual spectrum. We mix the 200 power-law spectra Gaussian realizations with the same weights calculated by CCA, as described above. This is labeled method B. These two methods are the extremes in realism when accounting for foreground residuals in a full pipeline for the posterior of r, but in real observations, the data analysis will be closer to method A than B, because we could use foreground models like the ones from pysm, which is significantly more realistic than a Gaussian random field. We calculate all of our power spectra using D_ℓ = ℓ(ℓ + 1)C_ℓ /2π instead of C_ℓ .

Our estimation of σ(r) (marginalized over ${A}_{\mathrm{fore}}$ for method A, using the pure foregrounds to construct the foreground residual spectrum) over the 192 f_sky = 0.1 circular regions is shown in Figure 3, for the SO-SAT fiducial baseline noise configuration. Each dot is located at the center of the region and its color represents the estimated σ(r) value according to the color scale. In the background, we plot the polarization intensity $P=\sqrt{{Q}^{2}+{U}^{2}}$ for the Planck 353 GHz frequency map (Planck Collaboration et al. 2020c). The light-green contours show the SO-SAT mask footprint. We also show the histogram of σ(r) over the masked regions in the lower-left corner.

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In general, the σ(r) values are of the order of a few 10⁻² for regions in the Galactic plane, due to foreground contamination. The regions with the lowest σ(r) are close to the north and south Galactic poles. Most of them coincide with the location of the SO-SAT mask (shown as green contours in Figure 3), with some few exceptions of regions with high decl., not observable from the SO site in the Atacama desert in Chile, above the SO-SAT mask contours at the right and left edges. The estimation over the SO-SAT mask yields σ(r) = 4.9 × 10⁻³ (2.7 × 10⁻³ before marginalization over ${A}_{\mathrm{fore}}$).

Table 2 lists the σ(r) constraints for both the SO-SAT mask and the best result (for method A) of the 192 masked regions considered in this work, as well as the result for method B for the same masks, for the SO-SAT fiducial baseline noise configuration.

Table 2. Results for the σ(r) Forecast in Units of 10⁻³, Both in the SO-SAT Mask as Well as the Best Result from the 192 Masked Regions over the Full Sky, for Both the Somewhat Optimistic Method A and the Very Pessimistic Method B

Instrumental	σ(r) SO-SAT	σ(r) SO-SAT	σ(r) Best	σ(r) Best	Center of Best
Configuration	Mask A	Mask B	Masked Region A	Masked Region B	Masked Region
SO-SAT fiducial, baseline noise	4.9	5.2	3.9	6.8	l = 2250 (b = −783
SO-SAT fiducial, goal noise	3.3	3.4	3.1	4.9	l = 450 (b = −783
SO-SAT extended, goal noise	3.1	3.5	3.0	4.9	l = 450 (b = −783

Note. Fiducial corresponds to the six SO-SAT frequency channels. Extended corresponds to the fiducial plus three extra high-frequency CCAT-p channels.

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The σ(r) estimated in the forecast done by SO (Ade et al. 2019) finds σ(r) = 1.9 × 10⁻³ in the Fisher matrix configuration. Our result is ∼2.5 times that value. We can attribute this difference to several factors. First, the SO Fisher forecast is a different method from what we do in this work. Their approach is a direct likelihood Monte Carlo Markov Chain of the simulated observations' auto- and cross-spectra, modeling the foregrounds and the CMB at the same time, skipping component separation, an approach followed in, e.g., BICEP2/Keck Collaboration et al. (2015). The foregrounds are modeled as power laws in multipole space, as well as an MBB for thermal dust and a power law for synchrotron in frequency space. Having the former constraint as a stronger prior would lead to improved agreement between the simulated and (likelihood) modeled foreground components, and in turn to an improved estimate of the CMB signal. This is a constraint we do not impose in our component separation modeling (we only impose the SEDs in frequency space). Also, the fact that we perform component separation in a previous and separate step does amplify the noise to some level when inverting matrices at the point of reconstructing the power spectra of the components, which would explain some of the increased scatter of the ${D}_{{\ell }}^{{BB}}$ estimates, which translate into a larger σ(r) in the Fisher forecasts. Another difference between our modeling and the one done in Ade et al. (2019) is that the Fisher matrix of the ${D}_{{\ell }}^{{BB}}$ is calculated as second derivatives of the likelihood around the fiducial parameters, while in our case this is calculated empirically over 200 Monte Carlo iterations. Finally, the effect of the ${A}_{\mathrm{fore}}$ parameter does increase σ(r) in our approach when marginalizing over it, as it does to other component separation methods in Ade et al. (2019) that do marginalize over foreground residuals, such as xForecast, BFoRe, and ILC. These methods see an increment in the error bar to σ(r) ∼ 3.4 × 10⁻³.

In Figure 4 (top), we show some examples of the correlation between r and ${A}_{\mathrm{fore}}$ for method A. ${A}_{\mathrm{fore}}$ is better constrained in regions with high foreground contamination. At the same time, r and ${A}_{\mathrm{fore}}$ are anticorrelated in these same regions and that anticorrelation diminishes toward zero as we move away from the GC toward the foreground-free Galactic pole regions. Figure 4 (top) shows this for four representative regions: one close to the north Galactic pole, one close to the south Galactic pole, one close to the GC, and the SO-SAT mask. This correlation allows us to quantify the bias introduced by the foreground contamination.

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In Figure 4 (bottom) we compare the σ(r) constraint in all 192 circular masked regions with and without marginalization over the ${A}_{\mathrm{fore}}$ parameter for method A. ⁵ Δσ(r) is the difference between the marginalized and unmarginalized σ(r) error bar. We can see that the foreground contaminated regions along the Galactic plane have a much larger Δσ(r). Weak polarized foreground regions in the Galactic poles have Δσ(r) of a few 10⁻³, while strong polarized foreground regions in the Galactic plane have Δσ(r) of a few 10⁻². In practice, this shows the effect of marginalizing over ${A}_{\mathrm{fore}}$ and how it leads to a σ(r) parameter that is reflective of both scatter as well as bias due to strong foreground contamination.

Besides the fiducial SO-SAT instrumental configuration with baseline noise level, we consider two extra instrumental configurations: the fiducial SO-SAT channels but with the lower noise goal performance and the fiducial SO-SAT channels with the goal noise performance supplemented with the three CCAT-prime channels at 350, 410, and 850 GHz, labeled the extended configuration. The results for these configurations are listed in Table 2.

The improvement introduced by the lower noise level is evident. The σ(r) constraint over the SO-SAT mask improves from σ(r) = 4.9 × 10⁻³ to σ(r) = 3.3 × 10⁻³ when comparing baseline and goal levels. Figure 5 (top) shows the confidence contours and posterior probabilities for three representative masked regions for this instrumental configuration, analogous to Figure 4 (top).

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The resulting forecast for the extended frequency configuration on the SO-SAT mask is σ(r) = 3.1 × 10⁻³, which is slightly better than the constraint achieved with only the six fiducial SO-SAT channels. For this configuration, the spectral parameter T_dust was still kept fixed to 19 K. In Section 5.1, we discuss why our σ(r) forecast is about the same (or improves a little) even though in this case we are including additional information from three extra frequency channels to constrain the thermal dust. Figure 5 (bottom) shows the confidence contours and posterior probabilities for the same three representative regions for this extended instrumental configuration.

We have verified that the most important contributor to the polarized foreground contamination in the fiducial six-SO-SAT-frequency channel configuration is the thermal dust. We have estimated the dust and synchrotron residuals separately by mixing simulations that only contain each corresponding foreground with the CCA-estimated weights. In these, the synchrotron residual is significantly smaller than the thermal dust one, so from this point forward, we focus our discussion on the effect of thermal dust. However, a similar analysis can be repeated for synchrotron and how its residual can be mitigated.

The high-frequency channels from CCAT-prime are highly dust dominated, and as such, they can measure the thermal dust precisely. They also have the potential to introduce strong foreground contamination to the estimated CMB signal if we do not model the SED perfectly. Whether adding such channels gives better or worse results in terms of CMB recovery critically depends on the complexity of the foreground sky in the considered region and the adequacy of the adopted component separation model to represent it. In our case, the balance is not quite so favorable, as the simulated thermal dust has a spatially variable T_dust and β_dust, while our model adopts constant values within a given patch. We note that the small improvement in σ(r) when adding the CCAT-prime high-frequency channels is a consequence of dust residuals contaminating the primordial BB, which is due to this mismatch between the simulated model and component separation modeling.

First, we have checked that if we consider simpler simulations, e.g., a spatially constant MBB SED, the agreement between the simulation and the component separation modeling is much closer for thermal dust, therefore the thermal dust residuals are smaller and there is a noticeable improvement in σ(r) when using the extra information from the high-frequency CCAT-prime channels.

To illustrate the issue in our simulation with spatially variable spectral parameters, we can look at the weights that are used to mix the observed data power spectra d _V , described in Equation (8). Figure 6 shows the product of the weights and the data vector ${{\boldsymbol{d}}}_{V}={\tilde{C}}_{x}(\hat{{\ell }})-{\tilde{C}}_{n}(\hat{{\ell }})$ for all multipole bins $\hat{{\ell }}$, where d _V is the noise-bias-subtracted full-mix signal. This is for the CMB ${D}_{{\ell }}^{{BB}}$ within the SO-SAT mask with the extended configuration and goal noise levels. For clarity, this figure shows only the contributions for the auto-power spectra for each of the 9 frequency channels, even though the CCA method uses all 45 auto- and cross-spectra between the frequency channels. This figure illustrates the relative contribution from each channel to reconstruct the CMB angular power spectrum ${D}_{{\ell }}^{{BB}}$. For example, the 93, 145, and 225 GHz channels contribute the most to the CMB, which makes sense because these are the frequencies where the CMB emission is strongest. Conversely, the low and high frequencies of 27, 39, 350, 410, and 850 GHz have the least relative contribution, which also makes sense. Ideally the CCAT-prime channels weights would be small, but just enough to subtract the dust out of the the channels where the CMB is strong. For the thermal dust, the mismatch in the simulated SEDs and the component separation modeled SEDs in CCA will result in errors in the weights, with the effect of high-frequency channels contaminating the CMB reconstruction with thermal dust residuals. If the weights for the high-frequency channels are small, but they are multiplied by a very strong thermal dust power spectra, such as the one present in the 350, 410, or 850 GHz channels, they will introduce some dust residual in the CMB reconstruction, which will contaminate the B-modes in a significant way.

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As another example, we can look at the BB thermal dust residuals directly. In Figure 7, we plot the reconstructed CMB BB spectrum as triangles in the SO-SAT mask for the fiducial (blue) versus extended (red) configurations, both with goal noise levels. Shown in the plot is the mean over the 200 Monte Carlo simulations, while the error bar corresponds to the standard deviation of that. We also plot the BB thermal dust residuals (squares) for the same two configurations. These are calculated with the maps of only the simulated thermal dust, mixed with the CCA-calculated weights used for the full simulation for the corresponding case. This is done in the exact same manner as how we calculate ${C}_{{\ell }}^{{BB},\mathrm{foregrounds}}$ in Equation (10). Both configurations have roughly the same error bar in their estimation of the CMB, as well as roughly the same dust residual. The nine-channel extended configuration has a slightly smaller nondiagonal term in the Fisher matrix, which means that r and ${A}_{\mathrm{fore}}$ are slightly less correlated, and this configuration has a slight edge over the six-channel fiducial configuration. We do not see the dust residuals diminishing substantially when adding the extra information from three high-frequency channels, which suggests that our thermal dust component separation modeling is not detailed enough.

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We know that parametric methods might not have enough complexity to model the dust residuals that will bias r (Hensley & Bull 2018). We can look at examples in the literature that increase the modeling complexity for the thermal dust SED by, for example, using an improved pixel-based foreground parameterization component separation, which allows for the partition of the sky into disjoint patches that can be modeled independently (Errard & Stompor 2019). In recent years the moment expansion has shown promise when performing thermal dust component separation (Chluba et al. 2017b; Mangilli et al. 2021; Remazeilles et al. 2021; Azzoni et al. 2021). In these methods, the dust spectral parameters are modeled as some value with a small perturbation, so we can account for a higher-order expansion when describing the spatial variability of the dust SED. This is also handy when describing deviations from a perfect MBB spectrum, as we should expect from the measured dust frequency decorrelation (Planck Collaboration et al. 2017; Clark et al. 2021). We can always increase the complexity of the modeling by adding more parameters to the model, but this will increase the degrees of freedom and ultimately dilute the statistical certainty of the estimates. This analysis stresses the importance of careful modeling in the component separation. Dust-monitor channels can be included for improving the dust modeling (e.g., fitting T_dust), but our analysis suggests that maybe they should not be used naively in the CMB reconstruction itself. We are not saying that using the extra high-frequency information is not useful but rather that our component separation modeling is not good enough in dealing with the residuals that arise by modeling as spatially constant something that is actually spatially variable. Extra care is warranted when modeling components and minimizing the mismatch between observations and models, especially if one wishes to include such high-frequency channels in the component separation analysis.

Given our analysis, we now attempt to answer the question that we posed at the beginning of this work: Is the region of sky where the foregrounds are at a minimum also the region that yields the smallest possible σ(r)?

In Figure 8, we plot the σ(r) forecast (for method A) versus the polarized intensity of the thermal dust for each of the 192 masked regions we consider in this work. The polarized thermal dust intensity is measured by calculating the BB angular power spectrum of the 2018 data release Planck 353 GHz frequency map (Planck Collaboration et al. 2020c) as prescribed in Planck Collaboration et al. (2020b) and fitting a power law to it in the multipole range ℓ = 40−599, given by ${{ \mathcal D }}_{{\ell }}^{{BB}}={A}_{\mathrm{dust}}^{{BB}}{({\ell }/80)}^{{\alpha }_{{BB}}+2}$. In the figure, we plot the ${A}_{\mathrm{dust}}^{{BB}}$ amplitude. In blue, we show the fiducial configuration with baseline noise levels, the red points are the same as the goal noise level, and the green points are the extended configuration with the goal noise levels.

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This figure shows that when polarized thermal dust is weak (roughly when ${A}_{\mathrm{dust}}^{{BB}}\lt {10}^{2}$ μK²), the forecasted σ(r) is roughly constant, meaning that in the cleanest regions of the sky, we can obtain a similar BB detection performance, regardless of the exact foreground contamination. Then, for regions with stronger foregrounds, σ(r) increases slightly with ${A}_{\mathrm{dust}}^{{BB}}$, until we reach regions where σ(r) increases exponentially, roughly where ${A}_{\mathrm{dust}}^{{BB}}\gt {10}^{2}$ μK³. Regarding the difference in σ(r) for regions with similar dust contamination, we have checked that this is an effect due to the uncertainty on ${A}_{\mathrm{fore}}$, either through a higher synchrotron contamination or more spatial complexity of the dust spectral parameters in the region.