CosmoForge II. An exact treatment of arbitrary linear filters in quadratic maximum-likelihood power spectrum estimation

Ground-based cosmic microwave background (CMB) experiments filter their time-ordered data to remove $1\mathrm{/}f$ noise and ground pickup, which removes or mixes the large angular scales where primordial $B$ modes peak. A common remedy is a pseudo-$C_{\mathrm{\ell }}$ estimator with harmonic $B$-purification and a transfer function measured from simulations. Another applies it after a matrix-based $E\mathrm{/}B$ purification of the maps. We extend the quadratic maximum-likelihood (QML) estimator of CosmoForge to filtered maps, treating the filter exactly: a linear filter conjugates the estimator’s ingredients without altering its structure. The same holds for the pixel-based likelihood of CosmoForge, which gives the smallest error bar achievable on the same maps. We compare all methods in a simplified setup under several map-level filters through the error bars on bandpowers and on the tensor-to-scalar ratio $r$. QML sits $2$ to $4\mathrm{\%}$ above the pixel-likelihood bound on $\sigma (r)$ in every filter state. At low noise, the pure pseudo-$C_{\mathrm{\ell }}$ $\sigma (r)$ is $1.21$ times QML’s without filtering, $3.0$ times under a scan filter along stripes of constant declination, and $3.7$ times under one from a toy constant-elevation scan. The matrix-purified pseudo-$C_{\mathrm{\ell }}$ $\sigma (r)$ is at most $1.03$ times QML’s except under the constant-elevation filters, where it reaches $1.10$. These factors shrink towards unity at higher noise. The pure pseudo-$C_{\mathrm{\ell }}$ gap comes from apodisation and from harmonic $E\mathrm{/}B$ separation; pixel-space separation closes it, except under the constant-elevation filters, which attenuate modes rather than remove them and widen the gap. QML stays within a few per cent of the optimal error bars without the tuning and simulated transfer functions of the pseudo-$C_{\mathrm{\ell }}$ pipelines. Thus, CosmoForge provides a robust framework for power-spectrum estimation and inference on filtered maps. (Abridged)