Parameterizing Noise Covariance in Maximum-Likelihood Component Separation

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.