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arXiv 0912.0490astro-ph.GAastro-ph.COastro-ph.IM
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引力透镜的数学:多重成像和放大

Mathematics of Gravitational Lensing: Multiple Imaging and Magnification

A. O. Petters, M. C. Werner

摘要:

引力透镜的数学理论揭示了引力透镜的许多普遍性和全局性。从多重成像开始,我们回顾了莫尔斯理论的图像计数公式和下限结果,以及在单透镜和多透镜平面的情况下的复代数上限。我们讨论了随机透镜的数学的最新进展,讨论了最小透镜图像的全球预期数量的一般公式以及微透镜随机时间延迟函数,随机透镜映射和随机剪切的概率密度的渐近公式,以及微极小值的全球预期数量的渐近表达式。多重成像在光学几何和时空设置处理。我们回顾了全球放大率关系的结果模型依赖的情况下,并涵盖了最近的发展普遍的局部放大率关系高阶焦散。

英文摘要:

The mathematical theory of gravitational lensing has revealed many generic and global properties. Beginning with multiple imaging, we review Morse-theoretic image counting formulas and lower bound results, and complex-algebraic upper bounds in the case of single and multiple lens planes. We discuss recent advances in the mathematics of stochastic lensing, discussing a general formula for the global expected number of minimum lensed images as well as asymptotic formulas for the probability densities of the microlensing random time delay functions, random lensing maps, and random shear, and an asymptotic expression for the global expected number of micro-minima. Multiple imaging in optical geometry and a spacetime setting are treated. We review global magnification relation results for model-dependent scenarios and cover recent developments on universal local magnification relations for higher order caustics.

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