We assess the impact of nonideal, continuously rotating half-wave plates (HWPs) on cosmic microwave background (CMB) polarization measurements targeting large angular scale signal. Such hardware solutions are used in or planned for multiple modern CMB efforts, both ground-based, for instance, small aperture telescopes of simons observatory or satellite borne, such as litebird. Using a frequency-dependent parametric model based on the Mueller matrix formalism, we characterize the induced mixing of Stokes parameters. Through end-to-end simulations, we propagate these effects from time-ordered data to cosmology via map-making and component-separation stages, quantifying their impact on the $B$-mode power spectrum and the tensor-to-scalar ratio, $r$. Our analysis shows that neglecting the frequency dependence of a three-layer HWP gives rise to significant polarization leakage, biases foreground spectral parameters, and thus leads to residual contamination in the recovered CMB maps. To mitigate these effects, we investigate multiple analysis strategies progressively incorporating a more complete description of the instrumental response. At the map-making level, this requires generalizing the standard pointing matrix, to account not only for the scanning strategy but also for the full time- and frequency-dependent instrumental response. We find that two standard HWP models, referred to as effective and stack HWP models, reduce the biases only down to $r\sim {10}^{-2}$; however, a more advanced approach based on a generalization of both map-making and component-separation procedures, implemented using jax, can suppress it down to $r\sim 7\times {10}^{-4}$. Finally, we extend this approach to a time-domain component-separation framework, enabling a statistically consistent treatment of instrumental response in the presence of time-domain features such correlated noise. We demonstrate its feasibility and validate it by performing a full end-to-end analysis, recovering results in good agreement with the map-based ones. This sets the stage for the full exploitation of this approach’s capability in the future.

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• Cosmic microwave background
• Astrophysical & cosmological simulations