Partially constrained internal linear combination: A method for low-noise CMB foreground mitigation

Internal linear combination (ILC) methods are some of the most widely used multifrequency cleaning techniques employed in cosmic microwave background (CMB) data analysis. These methods reduce foregrounds by minimizing the total variance in the coadded map (subject to a signal-preservation constraint), although often significant foreground residuals or biases remain. A modification to the ILC method is the constrained ILC, which explicitly nulls certain foreground components; however, this foreground nulling often comes at a high price for ground-based CMB datasets, with the map noise increasing significantly on small scales. In this paper we explore a new method, the partially constrained ILC, which allows us to optimize the tradeoff between foreground bias and variance in ILC methods. In particular, this method allows us to minimize the variance subject to an inequality constraint requiring that the constrained foregrounds are reduced by at least a fixed factor, which can be chosen based on the foreground sensitivity of the intended application. We test our method on simulated sky maps for a Simons Observatory–like experiment; we find that for cleaning thermal Sunyaev-Zel’dovich contamination at $\ell \in [3000,4800]$, if a small thermal Sunyaev-Zel’dovich residual of 20% of the standard ILC residual can be tolerated, the variance of the CMB temperature map is reduced by at least 50% over the constrained ILC value. We also demonstrate an application of this method to reduce noise in CMB lensing reconstruction.

Figure 1

Partial deprojection of the TSZ component. Results from our new PCILC method applied to simulated sky maps with various foreground bias threshold values as defined in Eq. (17) (blue and black curves) compared to the standard ILC (red) and the constrained ILC, or CILC, (green) results. Upper left: total power spectra of the reconstructed (PC)ILC CMB maps. Upper right: the ratio of the power spectra to the total CMB power spectrum obtained with the standard ILC. Lower left and lower right: the residual TSZ power (left) and CIB power (right) in the coadded maps, measured relative to the power of the TSZ or CIB at 145 GHz [see Eq. (17)]. It can be seen that, if a small bias can be tolerated, the PCILC method provides a significant variance reduction when compared with the CILC.

Figure 2

Complementary results to Fig. 1. Expected noise power (left) and residual foreground power (right).

Figure 3

Partial deprojection of the CIB component. The same configuration of plots as in Fig. 1, but now with the CIB foreground component deprojected or partially deprojected, as labeled in the plot legends. The CIB SED is taken to be a modified blackbody [see Eq. (16)] in the ILC constraints, but the sky simulations are constructed with a realistic model that produces decorrelation and a nonrigid SED that varies with frequency and sky position. This is why the residual CIB bias is slightly nonzero even when we set $B_{\mathrm{CIB}}=0$ (see the green curve in the lower right panel).

Figure 4

Complementary results to Fig. 3. Expected noise power (left) and residual foreground power (right).

Figure 5

Partial deprojection of both TSZ and CIB components. The same configuration of plots as in Fig. 1, but with simultaneous deprojection or partial deprojection of both the TSZ and CIB foreground components, as labeled in the plot legends. Again, it can be seen in the top right panel that, if a small bias can be tolerated, the PCILC method provides a significant variance reduction when compared with the CILC. However, the bottom right panel shows that the CIB bias remains significantly nonzero even when we attempt to fully deproject this component. This situation can arise when (partially) deprojecting multiple foregrounds with a small number of frequency channels, such that even small inaccuracies in the modeling lead to non-negligible residual foreground biases (see text for further discussion).

Figure 6

Complementary results to Fig. 5. Expected noise power (left) and residual foreground power (right).

Figure 7

Partial deprojection of both TSZ and CIB components. The residual CIB power in the coadded maps, measured relative to the power of the CIB at 145 GHz. The results shown here are obtained for the simplified sky simulations described in Sec. 3e, in which the CIB field is comprised of a single component that is simply rescaled in frequency using Eq. (16). In contrast to Fig. 5, the CIB bias now behaves as expected, which demonstrates that the behavior seen previously was due to CIB decorrelation and SED variations.

Figure 8

Contour plot of the ratio of the total CMB map power spectra obtained with PCILC for two components (TSZ and CIB) to the spectrum of the total CMB obtained with standard ILC, at $\ell =3500$ as a function of $B_{\mathrm{TSZ}}^{\mathrm{th}}$ and $B_{\mathrm{CIB}}^{\mathrm{th}}$.

Figure 9

(Partial) deprojection of both TSZ and CIB in CMB maps used for lensing reconstruction, using the temperature CMB noise curves in Fig. 5. In both panels (a) and (b) the lensing noise curves are shown for four cases: standard ILC CMB maps used in a quadratic lensing estimator; a CMB map cleaned with CILC, used in a quadratic lensing estimator; and PCILC-cleaned CMB maps, again used in a quadratic lensing estimator. The multifrequency-cleaned maps are labeled with a number that represents the average increase in noise with respect to lensing reconstruction using the standard ILC CMB map. It can be seen that the PCILC gives a significant noise reduction on large scales of around 30%, when compared with CILC foreground mitigation methods.

Figure 10

As for Fig. 9, but (partially) deprojecting only CIB, using the temperature CMB noise curves in Fig. 3. In this case there are not relevant CMB lensing noise improvements between CILC and PCILC, as CIB deprojection does not lead to a huge blowing up in CMB temperature noise for the CMB scales of lensing reconstruction.

Figure 11

As for Fig. 9, but (partially) deprojecting only TSZ, using the temperature CMB noise curves in Fig. 1. On large scales, the PCILC derived CMB lensing noise performs better than CILC one at around 10%.

Figure 12

Effect of decorrelation between frequency maps (right) and suboptimal CIB spectral response model (left) on CIB bias.

Figure 13

Weight values for each frequency channel. Left: weights obtained from CILC for two-component (CIB and TSZ) deprojection. Right: weights obtained for CIB deprojection.