The two properties of the radial mass distribution of a gravitational lens that are well constrained by Einstein rings are the Einstein radius RE and ξ2 = REα″(RE)/(1 − κE), where α″(RE) and κE are the second derivative of the deflection profile and the convergence at RE, respectively. However, if there is a tight mathematical relationship between the radial mass profile and the angular structure, as is true of ellipsoids, an Einstein ring can appear to strongly distinguish radial mass distributions with the same ξ2. This problem is beautifully illustrated by the ellipsoidal models in Millon et al. When using Einstein rings to constrain the radial mass distribution, the angular structure of the models must contain all the degrees of freedom expected in nature (e.g. external shear, different ellipticities for the stars and the dark matter, modest deviations from elliptical structure, modest twists of the axes, modest ellipticity gradients, etc.) that work to decouple the radial and angular structures of the gravity. Models of Einstein rings with too few angular degrees of freedom will lead to strongly biased likelihood distinctions between radial mass distributions and very precise but inaccurate estimates of H0 based on gravitational lens time delays.

中文翻译：

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时间延迟透镜还原的过度约束模型：角尾如何摆动径向狗

受到爱因斯坦环约束的引力透镜径向质量分布的两个性质是爱因斯坦半径 RE 和 ξ2 = REα″(RE)/(1 − κE)，其中 α″(RE) 和 κE 是第二个偏转分布的导数和 RE 处的收敛，分别。然而，如果径向质量分布和角结构之间存在紧密的数学关系，就像椭球一样，爱因斯坦环似乎可以强烈区分具有相同 ξ2 的径向质量分布。Millon 等人的椭球模型很好地说明了这个问题。当使用爱因斯坦环来约束径向质量分布时，模型的角结构必须包含自然界中预期的所有自由度（例如外部剪切、恒星和暗物质的不同椭圆率、与椭圆结构的适度偏差、适度的轴扭曲、适度的椭圆度梯度等），这些都有助于解耦重力​​的径向和角结构。角度自由度太低的爱因斯坦环模型将导致径向质量分布之间的强烈偏差似然区别和基于引力透镜时间延迟的 H0 非常精确但不准确的估计。