We introduce a new mathematical tool (a direction-dependent probe) to analyze the randomness of purported isotropic Gaussian random fields on the sphere. If the field is isotropic and Gaussian then the probe coefficients for a given direction should be realizations of uncorrelated scalar Gaussian random variables. To study the randomness of a field, we use the autocorrelation of the sequence of probe coefficients (which are just the Fourier coefficients aℓ,0 if the z-axis is taken in the probe direction). We introduce a particular function on the sphere (called the AC discrepancy) that accentuates the departure from Gaussianity and isotropy. We apply the probe to assess the full-sky cosmic microwave background (CMB) temperature maps produced by the Planck collaboration (PR2 2015 and PR3 2018), with special attention to the inpainted maps. We find that for some of the maps, there are many directions for which the departures are significant, especially near the galactic plane. We also look briefly at the noninpainted Planck maps, for which the computed AC discrepancy maps have a very different character, with features that are global rather than local.

中文翻译：

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应用于宇宙微波背景图的高斯性和各向同性新探测

我们引入了一种新的数学工具（与方向相关的探针）来分析球体上所谓的各向同性高斯随机场的随机性。如果场是各向同性的和高斯的，那么给定方向的探测系数应该是不相关的标量高斯随机变量的实现。为了研究场的随机性，我们使用探测系数序列的自相关（这只是傅立叶系数一种ℓ,0如果z-轴取探针方向）。我们在球体上引入一个特定的函数（称为交流差异) 这突出了与高斯性和各向同性的背离。我们应用探针来评估由普朗克合作（PR2 2015 和 PR3 2018），特别注意内绘地图。我们发现，对于某些地图，有许多方向的偏离很重要，尤其是在银河平面附近。我们还简要介绍了未绘制的普朗克图，其中计算的 AC 差异图具有非常不同的特征，具有全局而非局部特征。