Recovering a Dipolar Modulation from a Masked Sky: Bias and Variance of the Hemispherical Power Asymmetry Estimator at Planck Amplitude

Planck's isotropy analyses report a dipolar modulation of the low-multipole sky, an amplitude of about 0.07 in the local power, equivalent to a hemispherical power asymmetry. This paper asks how faithfully the standard hemispherical estimator recovers such a modulation, using three hundred synthetic skies at each of four injected amplitudes. At the true axis the estimator is unbiased in slope, recovering injected amplitudes with a response slope of 0.966, but its per-sky standard deviation at the Planck amplitude is 0.2309, comparable to the signal itself; and when the axis is chosen by maximising the same statistic, as real analyses must, the null skies alone return a mean apparent amplitude of 0.3702. An apparent asymmetry at the Planck level is therefore an expected product of the estimation procedure on a fraction of isotropic skies, and any claim built on it must be calibrated against the best-axis null, not the true-axis one. The study is simulated, not measured.