CMB component-separated power spectrum estimation by Spectral Internal Linear Combination (SpILC)

Abstract

Component separation methods mitigate the cross-contamination between different extragalactic and galactic contributions to cosmic microwave background (CMB) data. This is often done by linearly combining CMB maps from different frequency channels using internal linear combination (ILC) methods. We demonstrate that deriving power spectrum estimators directly by linearly combining auto- and cross-spectra instead of maps allows us to obtain a different constrained-optimization problem that allows fewer (deprojection) constraint equations than combining at map level using the constrained ILC method. Through simulations, we show that our Spectral internal linear combination (SpILC) produces CMB power spectrum estimators with more than 7 times smaller errorbars than constrained ILC (with thermal Sunyaev-Zel'dovich and cosmic infrared background deprojections) at $\ell\gtrsim 4000$ for Simons Observatory-like observations. Spectral ILC outperforms constrained ILC methods when some modeled components are spatially uncorrelated, e.g. the primary CMB is uncorrelated with foregrounds, and the difference in performance is most significant at noise-dominated scales. More generally, our work shows that component-separated maps with foreground deprojections do not necessarily produce minimum-variance two-or-higher-point estimators.

Figures (5)

• Figure 1: Auto and cross spectra $\mathcal{D}_\ell \equiv\ell(\ell+1)C_\ell/2\pi$ for different components of the simulated maps at 39 GHz (top), 93 GHz (middle), and 225 GHz (bottom). The simulated data are beam-deconvolved with beams and noise profiles corresponding to SO LAT goal level.
• Figure 2: Left: Errorbar size ratios of various SpILC (solid) and cILC estimators (dashed) of the CMB+kSZ power spectrum compared to the ILC (no deprojection) power spectrum. The line colors denote the corresponding deprojections: tSZ$\times$tSZ + tSZ$\times$CIB + CIB$\times$CIB (blue), CIB$\times$CIB + tSZ$\times$CIB (orange), tSZ$\times$tSZ + tSZ$\times$CIB (green), and no deprojections (purple) where cILC reduces to standard ILC. At small scales $\ell\gtrsim 4000$, the CIB and tSZ deprojected SpILC (cILC) errorbar $\sigma(\hat{K}_\ell)$ is more than 5 (40) times larger than the ILC errorbar size $\sigma(\hat{C}_\ell^\text{ILC})$, and in turn the cILC estimator has $\gtrsim 8$ times the errorbar size of the SpILC estimator. Right: Similar to left, but for the tSZ power spectrum, where the fully constrained case deprojects CMB+kSZ and CIB.
• Figure 3: The true (black) CMB power spectrum $\mathcal{D}_\ell^{\text{CMB+kSZ}}\equiv \ell(\ell+1)C^{\text{CMB+kSZ}}_\ell/2\pi$ (left panel) and tSZ power spectrum $\mathcal{D}_\ell^{yy}$ (right panel) are shown along with the $\pm1$$\sigma$ regions of the estimator $\hat{K}_{\ell}^\text{1real}$ for the constrained ILC (cILC, blue), constrained Spectral ILC (SpILC, orange), the data-split constrained ILC (cILCsplit, red) and the data-split constrained Spectral ILC (SpILCsplit, green). The deprojected components are the tSZ and CIB for the CMB spectrum (left), and CMB+kSZ and CIB for the tSZ spectrum (right). At each multipole moment, the measured spectra $\hat{C}^{ij}_\ell$ are band-averaged with a band-width of $\Delta\ell=30$ as in Eq. (\ref{['eq:bandavg']}). The unbiased estimator with the smallest variance is the SpILCsplit, and it has significantly lower variance than the analagous unbiased map-version, the cILCsplit (as quantified in Figs. \ref{['fig:var_comparison']} and \ref{['fig:one_real_6noise']}). The large biases for cILC and SpILC are due to noise bias, as the split estimators are unbiased.
• Figure 4: Left: Biases of (CMB+kSZ)$\times$(CMB+kSZ) power spectrum estimated at SO goal noise levels for cILCsplit and SpILCsplit, which eliminates instrumental noise bias. The errorbars are $\pm1$$\sigma$ about the mean error $\langle \hat{\varepsilon}_\ell \rangle$ over 100 realizations, where the error is defined as the difference of the estimator value from the true spectra. For each estimator, the weights are estimated using two methods: 1. weights $W_{\ell}^{ij}$ (blue for cILCsplit, orange for SpILCsplit from the realization-averaged spectra $\langle \hat{C}_\ell^{ij} \rangle$, whereas 2. weights $\hat{W}^{\text{1real}}_{\ell,ij}$ are derived for each realization using data from that realization (green for cILCsplit, red for SpILCsplit). The tSZ and CIB are deprojected, and the measured spectra for each realization are band-averaged with a band-width of $\Delta\ell=30$. Right: Similar to the left panel, but for the tSZ power spectrum. The CMB+kSZ and CIB are deprojected.
• Figure 5: Weight matrix $\hat{W}^{ij}_\ell$ visualized. The four squares from the left are the SpILC (CMB+kSZ)$\times$(CMB+kSZ) weights corresponding to the results in Fig. \ref{['fig:temp_3comp_12noise_19real_ensemble']} at $\ell=1500,2500,3500,4500$ respectively; the rightmost square is the SpILC (CMB+kSZ)$\times$(CMB+kSZ) weight at $\ell=4500$ for instrumental-noise-only input spectra $C^{ij}_\ell= N^{ij}_\ell$. The tSZ and CIB are deprojected, and the measured spectra are band-averaged with a band-width of $\Delta\ell=30$.