Parameterizing Noise Covariance in Maximum-Likelihood Component Separation

TL;DR

The paper addresses biases in maximum-likelihood component separation caused by mis-modelled noise correlations in CMB data. It introduces a harmonic-space power-law noise model $N_\ell= \sigma_{\rm white}^2 \left[1+( \ell/\ell_0)^{\alpha}\right]$, embeds it in a bias-corrected ridge likelihood, and uses an ensemble-average pipeline to jointly infer foreground and noise parameters without Monte Carlo simulations. The authors forecast the 95% upper limit on the tensor-to-scalar ratio $r_{95}$ for the ECHO mission across diverse noise scenarios and show that correlated noise can degrade $r_{95}$ by up to an order of magnitude, while still achieving sensitivities near $10^{-4}$ under optimistic conditions. The framework informs instrument design and demonstrates a robust path toward detecting primordial B-modes with next-generation CMB experiments.

Abstract

We introduce a noise-aware extension to the parametric maximum-likelihood framework for component separation by modeling correlated $1/f^α$ 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.

Figures (10)

• Figure 1: Bias in the recovered dust spectral index $\hat{\beta}_{d}$ when correlated noise is ignored, considering only white noise in the evaluation of the spectral likelihood, Equation \ref{['eq:spec_likelihood']}. The input noise follows Equation \ref{['eq:noise_aps']} with $\alpha=-6$, and $\ell_{0}$ is drawn $100\times n_{f}$ times from a uniform interval $[\ell_{\min}=2,\ell_{\max}=L]$. As $L$ increases, the noise model's complexity grows and the bias on $\hat{\beta}_{d}$ increases when using a white-noise likelihood. The blue curve represents the average bias over the 100 simulations.
• Figure 2: Joint posterior contours (68 % and 95 % confidence levels) for the noise parameters $\alpha$ and $\ell_{0}$ marginalized over the foreground spectral parameters ($\beta_{d}$, $T_{d}$, $\beta_{s}$) obtained by sampling the ensemble averaged version of the modified ridge likelihood (Equation \ref{['eq:RidgeLikelihood']}). Dashed lines mark the recovered values --- $\alpha$ and $\ell_0$ have bias of $\sim 50 \sigma$ and $\sim 200 \sigma$ respectively, when compared to their true values of $\alpha = -1, \;\ell_0 = \ell_{\max}/2 = 128$.
• Figure 3: Variation of the noise angular power spectra with changes in the noise parameters $p$ for fixed $\sigma_{\rm white}^{2}=1$. Top: Changes observed in noise power when the slope $\alpha$ is varied at constant $\ell_{0}=128$. Steeper (more negative) $\alpha$ values boost power at low multipoles, then converge to the white-noise level for $\ell>\ell_{0}$. Bottom: Changes observed for different knee scales $\ell_{0}$, with fixed $\alpha=-1$. Increasing $\ell_{0}$ extends the correlated-noise regime to higher multipoles, reducing the signal-to-noise ratio over those scales.
• Figure 4: Joint posterior contours (68 % and 95 % confidence levels) for the noise parameters $(\alpha,\,\ell_{0})$ and foreground spectral parameters ($\beta_{d}$, $T_{d}$, $\beta_{s}$) obtained by sampling different likelihoods -- the ensemble-averaged version of the original ridge likelihood (Equation \ref{['eq:RidgeLikelihood']}) without bias correction(red), the generalized bias-corrected ridge likelihood (Equation \ref{['eq:final_eq']}) assuming white noise for bias correction as per Equation \ref{['white_noise_th']} (green), and the generalized bias-corrected ridge likelihood (Equation \ref{['eq:final_eq']}) assuming exact knowledge of input noise as per Equation \ref{['eq:pink_noise_th']} (blue). Dashed lines mark the true input values: $\alpha=-1$, $\ell_{0}=128$ for the noise model, and $\beta_{d}=1.54$, $T_{d}=20\text{ K}$, $\beta_{s}=-3$ for the foregrounds. The spectral parameters are recovered identically, while the performance on the noise parameters differs, with a reduction in bias seen when more information about the noise is given.
• Figure 5: Recovered CMB angular power spectra $\mathbf{\widehat{C}_{\ell}^{\rm CMB}}$ for each noise slope $\alpha$ at a constant knee multipole $\ell_0 = \ell_{\rm max}/2$, where $\ell_{\rm max} = 256$. The black dashed line shows the theoretical CMB spectrum, and colored lines correspond to $\alpha=0$, $-1$, $-2$, and a channel-dependent $\alpha$ spanning $-1$ to $-5$. Larger $|\alpha|$ values increase residual noise power at low multipoles, causing greater deviations from the theoretical curve.