Cross-ILC Foreground Mitigation in CMB Analysis

• Cross-ILC Foreground Mitigation is a strategy that uses independent data splits and cross-spectra to effectively suppress noise bias and foreground systematics.
• It utilizes constrained and moment-augmented ILC methods, such as cILC, cMILC, and ocMILC, to precisely null targeted foreground emissions in CMB observations.
• This approach enhances the unbiased extraction of cosmological signals, facilitating robust detection of primordial B-modes, lensing effects, and spectral distortions.

Cross-ILC foreground mitigation refers to a class of data analysis strategies in cosmic microwave background (CMB) and related fields where internal linear combination (ILC) component separation is performed independently on two (or more) statistically independent data splits, and their cross-spectra (rather than auto-spectra) are used for science analysis. This approach suppresses both foreground systematics and the noise bias that can be problematic when mapmaking and spectral estimation are performed on the same data. Recent developments emphasize constrained and moment-augmented extensions to the basic ILC approach, enabling precise nulling of specific foreground emissions and their spectral variations, which is essential for unbiased extraction of primordial signals such as tensor B-modes, lensing, kSZ, or CMB spectral distortions even in the presence of complicated foreground environments (Remazeilles et al., 2018, McCarthy et al., 2023, Raghunathan et al., 2023, Remazeilles et al., 2020, Dou et al., 2023, Nakato et al., 9 Dec 2025, Carones et al., 27 Feb 2024, Hertig et al., 2 May 2024, Remazeilles et al., 2011, Darwish et al., 2021).

1. Theory and Motivation

Cross-ILC foreground mitigation builds upon several foundational concepts. The ILC estimator forms a minimum-variance linear combination of multi-frequency sky maps under the constraint that the target astrophysical component (e.g., CMB, $\mu$, $B$-mode, lensing) is preserved. Standard ILC does not in general null other foregrounds, leading to residual contamination that can bias cosmological results. Constrained ILC (cILC) introduces additional linear constraints to null specific foregrounds by enforcing, for instance, zero response to the thermal Sunyaev–Zeldovich (tSZ) effect or the cosmic infrared background (CIB) SEDs (Remazeilles et al., 2018, Darwish et al., 2021, Nakato et al., 9 Dec 2025).

However, auto-spectra computed from ILC maps are subject to a noise bias—an upward bias in the estimated angular power spectrum reflecting the noise realization that the optimal estimator is fit to suppress. Cross-ILC mitigates this by operating on two or more splits of the data (with independent noise realizations), building independent cleaned maps for each split, and using their cross-spectrum. This removes the leading noise bias and, when combined with cILC, enables high-fidelity cosmological parameter estimation even at the limit of instrumental noise. This framework has been generalized to include moment-based (cMILC) and partially-constrained ILCs (pcILC), which provide control over foreground suppression versus noise amplification (Remazeilles et al., 2020, Abylkairov et al., 2020, Carones et al., 27 Feb 2024).

2. Mathematical Formalism

The core of cross-ILC methods is the constrained optimization of multi-frequency linear weights to preserve the target SED while enforcing nulling of foreground SEDs. The general solution for the weight vector $w$ with $P$ constraints (target plus $P-1$ foregrounds) for a data vector $d$ and covariance $C$ is

$$w^\top = e^\top (A^\top C^{-1} A)^{-1} A^\top C^{-1}$$

where $A$ is the $N_\nu \times P$ mixing matrix ($N_\nu$: number of frequency channels), columns representing the SEDs to preserve and to null, and $e^\top$ picks out the preserved SED (typically $e^\top = (1, 0, ... 0)$). In more sophisticated approaches such as cMILC and ocMILC, moment expansions of SEDs, Taylor-expanded around pivot parameters, augment $A$ with higher-order derivatives to null not only average foregrounds but also their spatial/spectral variations (Remazeilles et al., 2020, Carones et al., 27 Feb 2024, Dou et al., 2023).

For the cross-ILC, two disjoint cleaned maps are processed independently with the same weight constraints and their cross-spectrum forms the estimator for the signal power spectrum, removing noise bias associated with shared noise between the splits (McCarthy et al., 2023, Remazeilles et al., 2020, Remazeilles et al., 2011). In cosmological parameter estimation (e.g., $f_{\rm NL}$, $r$), cross-power spectra of constraint-enforced maps (e.g., a $\mu$-distortion map free of temperature, a temperature map free of $\mu$) are used to derive unbiased constraints on primordial physics (Remazeilles et al., 2018, Hertig et al., 2 May 2024).

3. Implementation Details and Algorithmic Variants

Modern pipelines implement cross-ILC in harmonic, pixel, or needlet space. Needlet ILCs (NILC) are particularly powerful, offering simultaneous localization in scale and position, enabling adaptive weight computation that reflects local foreground complexity (McCarthy et al., 2023, Remazeilles et al., 2020, Carones et al., 27 Feb 2024, Remazeilles et al., 2011).

Key algorithmic elements include:

• Split-map generation: Data splits are constructed to be statistically independent (e.g., half-mission, half-ring, detector splits) (McCarthy et al., 2023, Dou et al., 2023).
• Local covariance estimation: Covariances are computed on domains (harmonic band, pixel patch, or needlet sphere) with smoothing kernels sized to retain statistical robustness while matching foreground complexity (McCarthy et al., 2023, Dou et al., 2023).
• Constraint management: In addition to basic cILC constraints, moment-based cMILC and optimized ocMILC dynamically determine the necessary set of moments and their pivot parameter values for each domain based on GNILC-like subspace diagnostics, balancing bias suppression and noise (Carones et al., 27 Feb 2024).
• Deprojection of multiple SEDs: Simultaneous nulling of e.g., tSZ, CIB, and foreground SED moments is implemented to ensure residuals lie below statistical noise (Remazeilles et al., 2020, Nakato et al., 9 Dec 2025, Raghunathan et al., 2023, Darwish et al., 2021).
• Noise-bias elimination via cross-spectra: The cross-spectral estimator ensures no noise auto-bias, with analytic corrections for residual higher-order effects (Remazeilles et al., 2020, McCarthy et al., 2023, Dou et al., 2023).
• Partial constraints (pcILC): Tradeoff between variance and bias is optimized via partial deprojections, enforcing only a fixed suppression of foregrounds to minimize noise inflation (Abylkairov et al., 2020, Carones et al., 27 Feb 2024).

4. Applications in CMB Science

Cross-ILC foreground mitigation now underpins a variety of key CMB analyses:

• Primordial B-mode detection: Implementation of cross-ILC in needlet or harmonic space with cMILC or ocMILC constraints allows recovery of $r$ with $r$-uncertainties down to $4 \times 10^{-4}$ for PICO-like missions, with controllable foreground and noise systematics (Carones et al., 27 Feb 2024, Remazeilles et al., 2020, Dou et al., 2023, Hertig et al., 2 May 2024).
• Kinematic SZ and lensing extraction: Cross-ILC using tSZ-free and CIB-free constrained maps suppresses bispectrum and trispectrum biases, enabling unbiased extraction of kSZ and lensing power for SPT-3G, SO, and CMB-S4—driving kSZ $S/N$ estimates up to $80\,\sigma$ and reducing lensing amplitude biases below $0.2\,\sigma$ (Raghunathan et al., 2023, Nakato et al., 9 Dec 2025, Darwish et al., 2021).
• Spectral distortion anisotropy ($\mu$) analyses: Constrained cross-ILC permits extraction of foreground-free $\mu$-distortion maps, yielding robust $f_{\rm NL}$ constraints on ultra-small scales for next-generation space missions (PIXIE, PICO) (Remazeilles et al., 2018).
• Component-separated mapmaking: pyilc and similar frameworks operationalize cross-ILC for Compton-$y$ and general tSZ cleaning, suppressing foreground cross-talk and providing validated products for cosmological cross-correlation analysis (McCarthy et al., 2023).

5. Performance, Limitations, and Optimization

Quantitative performance of cross-ILC mitigation strategies depends on the instrument configuration, sky coverage, and foreground complexity. Benchmarks established include:

• For CMB lensing ($L < 1000$), cross-ILC GMV estimators reduce the residual bias from $4\%$ (standard) to $<1\%$, corresponding to bias $<0.2\sigma$; additional geometric hardening (profile, point-source, or shear renormalization) can further suppress residuals (Nakato et al., 9 Dec 2025, Darwish et al., 2021).
• In B-mode analyses, cILC achieves foreground bias removal with modest ($<30\%$) increases in residual noise, while cMILC and ocMILC, by locally optimizing the set of deprojected moments, achieve further bias suppression with manageable noise penalty (Carones et al., 27 Feb 2024, Remazeilles et al., 2020).
• Partially constrained ILC (pcILC) methods interpolate between variance minimization and strict nulling, with simulations finding variance reduction by over a factor of two at $\ell>3000$ compared to cILC when a $20\%$ residual of standard ILC bias is tolerated (Abylkairov et al., 2020).
• Performance gains from extending frequency coverage, particularly below 40\,GHz and above 400\,GHz, are more significant than raw increases in instrumental sensitivity for foreground mitigation (Remazeilles et al., 2018).
• For small-aperture, small-sky experiments, noise-bias errors (NBE) from ILC weight estimation on limited data can become non-negligible, though typically subdominant to statistical uncertainties; cross-ILC effectively removes this bias (Dou et al., 2023).

Limitations remain:

• Highly complex or decorrelated foregrounds (e.g., CIB with non-rigid SEDs) prevent perfect nulling, but moment-based deprojections mitigate these effects.
• Increasing the number of constraints inevitably raises map noise; ocMILC provides an optimal and data-driven compromise.
• Systematic mismatches in filter or moment definitions between splits can reintroduce residuals; careful harmonization of pipeline settings is necessary.

6. Future Directions and Experimental Design Implications

Cross-ILC foreground mitigation is central to the design of future CMB experiments targeting primordial $B$-modes, spectral distortions, and CMB lensing at high precision. Key design requirements evidenced from current research include:

• Sufficiently broad frequency coverage to enable robust constraint-based foreground projection and conditioning of the mixing matrix (Remazeilles et al., 2018, McCarthy et al., 2023).
• Multiple independent data splits to fully realize cross-ILC noise-bias elimination (Remazeilles et al., 2020, McCarthy et al., 2023, Dou et al., 2023).
• High-frequency and low-frequency bands with well-characterized relative calibration to enable precise moment-based deprojection.
• Flexible data analysis pipelines (e.g., pyilc, ocMILC) capable of GNILC diagnosis and adaptive constraint management (Carones et al., 27 Feb 2024, McCarthy et al., 2023).
• For lensing and cross-correlation cosmology, incorporation of geometric hardening and cross-ILC forms in minimum-variance estimators, and careful bias diagnostics validated on realistic non-Gaussian simulations (Nakato et al., 9 Dec 2025, Darwish et al., 2021, Raghunathan et al., 2023).

Continued development of these methods will enable unbiased cosmological inference in the presence of increasingly complex, spatially varying, and non-Gaussian foregrounds, supporting the scientific goals of current and next-generation CMB and large-scale structure surveys.