Parameterizing Noise Covariance in Maximum-Likelihood Component Separation (2511.04546v1)

Published 6 Nov 2025 in astro-ph.CO

Abstract: We introduce a noise-aware extension to the parametric maximum-likelihood framework for component separation by modeling correlated $1/f^\alpha$ noise as a harmonic-space power law. This approach addresses a key limitation of existing implementations, for which a mismodelling of the statistical properties of the noise can lead to biases in the characterization of the spectral laws, and consequently biases in the recovered CMB maps. We propose a novel framework based on a modified ridge likelihood embedded in an ensemble-average pipeline and derive an analytic bias correction to control noise-induced foreground residuals. We discuss the practical applications of this approach in the absence of true noise information, leading to the choice of white noise as a realistic assumption. As a proof of concept, we apply this methodology to a set of simplified, idealized simulations inspired by the specifications of the proposed ECHO (CMB-Bh$\overline{a}$rat) mission, which features multi-frequency, large-format focal planes. We forecast the $95 \%$ upper limit on the tensor-to-scalar ratio, $r_{95}$, under a suite of realistic noise scenarios. Our results show that for an optimistic full sky observation, ECHO can achieve $r_{95}\leq 10^{-4}$ even in the presence of significant correlated noise, demonstrating the mission's capability to probe primordial gravitational waves with unprecedented sensitivity. Without degrading the statistical performance of the traditional component separation, this methodology offers a robust path toward next-generation B-mode searches and informs instrument design by quantifying the impact of noise correlations on cosmological parameter recovery.

Summary

The paper presents a noise-aware maximum-likelihood method that integrates a correlated 1/f^α noise model into CMB component separation.
It introduces a modified ridge likelihood with analytic bias correction to mitigate biases in key spectral parameters, particularly the dust spectral index.
The approach forecasts tensor-to-scalar ratio constraints for the ECHO mission, highlighting the impact of noise mis-modeling on primordial B-mode detection.

Parameterizing Noise Covariance in Maximum-Likelihood Component Separation

The paper "Parameterizing Noise Covariance in Maximum-Likelihood Component Separation" (2511.04546) presents a rigorous extension to standard parametric component separation for CMB polarization experiments. Its primary contribution is a noise-aware, maximum-likelihood framework that explicitly models correlated $1/f^\alpha$ noise within the harmonic domain, introduces a modified ridge likelihood with analytic bias correction, and demonstrates the impact of noise mis-modeling on both spectral parameter recovery and constraints on the tensor-to-scalar ratio $r$ . The methodology is validated in the context of the upcoming ECHO (CMB-Bhārat) mission, with detailed performance forecasts and design implications.

Motivation: CMB Polarization and Noise Modeling

In the effort to detect primordial gravitational waves through large-scale CMB B-modes, precise separation of cosmological and galactic signals is essential. Multi-frequency observations leverage differences in the spectral energy distributions of synchrotron, dust, and CMB, but component separation accuracy becomes dominated by systematic effects—including namely foreground mismodeling and instrumental noise mis-modeling—at the $r\sim 10^{-3}$ level.

Traditional parametric separation methods assume white noise, which neglects correlated (non-diagonal) structure arising from instrument drift, $1/f$ noise, and atmospheric or ground-based contaminants. The authors show that such neglect induces significant biases in spectral parameter recovery, notably in the dust spectral index $\beta_d$ , propagating into spurious B-mode power at the largest angular scales. Figure 1 illustrates bias in $\hat\beta_d$ due to correlated noise ignored in the likelihood.

Figure 1: Bias in the recovered dust spectral index $\hat\beta_{d}$ when correlated noise is ignored.

Methodology: Harmonic Power-Law Noise and Modified Ridge Likelihood

Spectral Parametric Formalism

The multi-frequency sky data are modeled as $\mathbf{d} = \mathbf{A}\,\mathbf{s} + \mathbf{n}$ , with $\mathbf{A}$ the mixing matrix encoding component spectral responses. Standard practice maximizes a spectral likelihood over the parameters $\beta$ describing each component's SED within a Gaussian noise model $\mathbf{N}$ , typically diagonal.

Correlated Noise Model

The extension presented treats instrument noise as a harmonic-domain power-law:

$\mathbf{N}_\ell^{(\nu)}(p) = \sigma_{\rm white}^2(\nu)\left[1+\left(\frac{\ell}{\ell_0(\nu)}\right)^{\alpha(\nu)}\right]$

with $\sigma^2_{\rm white}$ the white baseline, $\ell_0$ a transition multipole (knee), and $\alpha$ the low- $\ell$ power-law slope. This simple but flexible parameterization captures a broad class of correlated noise seen in actual CMB instruments.

Figure 2: Variation of the noise angular power spectra with changes in the noise parameters $p$ for fixed $\sigma_{\rm white}^2$ , showing influence of $\alpha$ and $\ell_0$ .

Modified Ridge Likelihood and Bias Correction

Direct maximization of the original spectral likelihood using a correlated noise model introduces systematic bias in the noise parameters, due to reduced effective degrees of freedom after profiling over the sky components. An analytic bias correction is derived, restoring the correct noise normalization by tracing out residual projections lost in profiling. The full "ridge likelihood" is cast as:

$-2\ln \mathcal{L}_\text{corrected}(\beta, p) = -2\ln \mathcal{L}_\text{ridge}(\beta, p) + \textrm{correction terms}$

Ensemble averaging over the likelihood removes the need for Monte Carlo sampling and allows efficient computation of parameter posteriors, critical for forecasting and optimization.

Figure 3: Joint posterior contours (68\% and 95\% confidence) for noise parameters $\alpha$ and $\ell_0$ marginalized over foregrounds, using the modified ridge likelihood. Strong bias arises without correction, as shown by the offset between recovered and true values.

Implementation Pipeline and Recovery Performance

Ensemble-Average Likelihood and Weight Derivation

The paper shows how, in the limit of infinite realizations, the sample covariance in harmonic space can be replaced by traces over theoretical covariance matrices ( $\mathbf{C}_\ell^\mathrm{CMB}$ , $\mathbf{C}_\ell^\mathrm{FG}$ , $\mathbf{N}_\ell$ ). This drastically reduces computational costs while maintaining statistical consistency for predictions and parameter inference.

Posterior sampling over $(\beta, p)$ is performed with standard MCMC (emcee), using the derived bias-corrected likelihood. The optimal component separation weights $\widehat{\mathbf{W}}$ are constructed from the recovered mixing matrix and noise covariance per multipole. The residual noise and statistical foreground biases in the recovered CMB power spectrum are analytically separable.

Practical Considerations

In actual data, the true per-mode noise realizations $\mathbf{n}_{\ell m}$ are not known and are replaced by either model-predicted covariances (assumed white, for lack of better knowledge) or, if possible, a measured noise PSD.
The impact of beam convolution is included by simulating frequency-dependent Gaussian beams, then smoothing all maps to a common resolution. This mitigates bandpass mismatch and further non-idealities.
No simulation-based realization averaging is needed: all forecasts are direct from analytic ensemble-averaging, facilitating rapid assessment of instrument design impacts on $r$ constraints.

Results: Biases, Parameter Recovery, and $r_{95}$ Forecasts

Influence of Noise Parameterization

Comprehensive simulation studies demonstrate that:

As noise slope $\alpha$ becomes more negative (e.g., from white to pink to red), residual noise power at low multipole ( $\ell$ ) increases greatly, contaminating the regime critical to $B$ -mode detection. See Figure 4 for joint posterior recovery.
Larger $\ell_0$ values extend correlated noise to higher angular scales, similarly degrading sensitivity.
Figure 4: Joint posterior contours (68\% and 95\%) for noise parameters $(\alpha, \ell_0)$ , showing parameter recovery variance under different bias corrections.

The marginalized posteriors for the spectral parameters (e.g., dust index $\beta_d$ ) remain unimodal and centered on the true value for moderate noise models, but undergo significant broadening and bias with increasing noise complexity or when noise mis-modeling is present.

Figure 5: Recovered CMB angular power spectra $\widehat{\mathbf{C}}_\ell$ across noise models, illustrating increased residuals at low $\ell$ for more complicated correlated noise.

Recovery under Generalized Noise & Beam

Combining both frequency-dependent $\alpha$ and $\ell_0$ with realistic beam smoothing, the framework remains robust: spectral recovery remains unbiased, with only modest broadening of posteriors. However, the low- $\ell$ CMB power spectrum can still be dominated by residual noise under pessimistic scenarios.

Figure 6: Recovered CMB angular power spectra $\widehat{\mathbf{C}}_\ell$ in the most complex noise and beam scenario, showing excellent agreement at moderate $\ell$ but persistent noise bias at the lowest multipoles.

Figure 7: Marginalized posterior distributions for $\beta_d$ in the generalized noise+beam setting; distributions remain centered though slightly broadened compared to simpler noise models.

Impact on Tensor-to-Scalar Ratio $r$

Forecasted constraints for the ECHO mission under various noise models show that, even without delensing and with optimistic full-sky coverage, $r_{95}$ ranges from $8 \times 10^{-5}$ (ideal white noise) to $5 \times 10^{-4}$ (most complex correlated and beam-convolved noise). The degradation is most sensitive to $\alpha$ ; increases in $\ell_0$ are less detrimental but still notable. Reducing sky fraction to $f_\text{sky} = 0.5$ increases $r_{95}$ by roughly $\sqrt{2}$ .

The authors cross-validate their forecasts with Fisher-matrix error propagation, confirming the consistency of the analytic formalism.

Theoretical and Practical Implications

This work substantiates several important points for current and next-generation CMB B-mode experiments:

Correlated Noise Biases Recovery: Any mismatch between assumed and actual noise covariance, especially when this mismatch is frequency-dependent, can bias foreground spectral parameter estimates, thereby biasing the CMB signal extraction in ways that mimic or mask a primordial $B$ -mode signal.
Analytic Bias Correction is Feasible and Required: The modified ridge likelihood, coupled with explicit bias correction, ensures unbiased spectral parameter recovery even with complex noise models—all while keeping the analysis tractable without laborious Monte Carlo pipelines.
Forecast Relevance for Instrument Design: Sensitivity forecasts for $r$ must fully account for correlated noise. The analytic machinery developed here enables fast, robust surveying of design parameter space and systematics, providing direct input to mission configuration decisions (sampling, frequency selection, beam width, scan strategy).
Scalability to Realistic Missions: The methodology is directly portable to more realistic scenarios including spatially-varying foregrounds, partial sky masks, and more sophisticated noise models, as foreseen for implementation in future work.

Conclusions

The paper introduces a statistically sound, computationally efficient method for incorporating correlated noise models into parametric component separation for CMB experiments. By capturing noise-induced residuals and their biasing effect analytically, it provides a robust framework for accurate inference of both foreground and cosmological parameters—demonstrated here through forecasts for the ECHO mission, but applicable broadly. The critical finding is that correlated noise, when properly modeled and analytically corrected for, need not fundamentally limit the tensor-to-scalar ratio sensitivity, but unmodeled systematics can quickly dominate final performance.

Future development should extend this analytic approach to more complex foregrounds, spatially varying parameters, hybrid blind/parametric separation algorithms, and non-Gaussian noise properties. Additionally, the formalism offers a template for similar analyses in other scientific fields where component separation under correlated noise is essential.