Bias in low-multipole cosmic microwave background reconstructions

The large-angle, low-multipole cosmic microwave background (CMB) provides a unique view of the largest angular scales in the Universe. Study of these scales is hampered by the facts that we have only one Universe to observe, only a few independent samples of the underlying statistical distribution of these modes, and an incomplete sky to observe due to the interposing Galaxy. Techniques for reconstructing a full sky from partial-sky data are well known and have been applied to the large angular scales. In this work, we critically study the reconstruction process and show that, in practice, the reconstruction is biased due to leakage of information from the region obscured by foregrounds to the region used for the reconstruction. We conclude that, despite being suboptimal in a technical sense, using the unobscured region without reconstructing is the most robust measure of the true CMB sky. We also show that for noise-free data reconstructing using the usual optimal, unbiased estimator may be employed without smoothing, thus avoiding the leakage problem. Unfortunately, directly applying this to real data with noise and residual, unmasked foregrounds yields highly biased reconstructions requiring further care to apply this method successfully to the real-world CMB.