The hemispherical power asymmetry (HPA) is the most persistent large-scale anomaly in the cosmic microwave background, detected above 3σ in temperature. Extending the measurement to E-mode polarisation requires correcting for E-to-B leakage introduced by incomplete sky coverage; existing approaches address this through inpainting of masked regions. We present a local bandpower estimator (LBE) that corrects for leakage deterministically through per-patch coupling matrix deconvolution, without inpainting. Applied to overlapping patches on the masked sky, the estimator yields simultaneous, deconvolved EE, BB, and EB bandpower estimates at a specified multipole range within each patch — a spectral decomposition not available from existing local estimators. We introduce half-missiondirectional consistency as a test statistic for spatial anisotropy, exploiting the independent noise realisations in Planck half-mission data splits. The LBE is independent of the test statistic: the deconvolved bandpower maps can be analysed with any statistical test. We demonstrate the framework on Planck PR3 data at ℓ = 15–40. The temperature HPA direction is recovered with sub-degree half-mission agreement. In E-mode polarisation, directions are consistent with Gimeno-Amo et al. (2023), with p-values of 3.0% for SEVEM and < 0.33% for Commander. Without deconvolution, the recovered directions shift to the galactic plane and the half-mission agreement becomes consistent with isotropy (p = 16.7%), confirming the necessity of leakage correction. The spectral decomposition yields EE coherence with null BB and EB, consistent with scalar primordial modulation. Signal injection tests validate the pipeline but establish that the directional consistency test is noise-limited at Planck sensitivity, motivating application to LiteBIRD.
Robert Lynch (Sun,) studied this question.