Random Gaussian Curvatures, Image Centroids and Caustic Surfaces in Gravitational Lensing

摘要：NSF提案，DMS-0302812 （Petters）该提案提出了一个由两部分组成的研究计划，该计划在数学和天体物理学之间建立了协同互动。程序1旨在发展一种数学理论，以确定随机时滞函数的高斯曲率临界点的概率分布。本研究项目开辟了新的数学方向，将奇点理论与概率论（如随机多项式的零和瑞利-列维统计中的极限定理）联系起来。此外，项目1还直接应用于通过星系透镜特征中的通量比异常来理解星系中暗物质的本质。项目2旨在将薄屏弱场微透镜中焦散点附近光源的光曲线和像心的分类扩展到克尔时空的强引力场。这涉及到研究一个时空中飞行焦散曲面的整体拓扑、几何和奇异理论结构。程序2指出了与光焦性表面的几何形状和引力透镜的物质奇点的拓扑不变量有关的新数学问题。这个项目也可能对爱因斯坦的广义相对论进行检验，特别是当它与据信位于银河系核心的大质量黑洞有关时。星系晕中的暗物质和银河系中心的黑洞是天体物理学中两个重要而紧迫的课题。这些问题直接受到几何分析的强大数学方法的影响，因为引力透镜既是探测暗物质分布和黑洞本质的独特工具，又是建立在几何、分析和概率概念之上的理论。暗物质研究需要发展一种数学理论，使人们能够区分一般的暗物质透镜特征和用于暗物质的特定选择的简单透镜模型的特定特征。该建议的第一部分讨论如何形成这样一个数学理论。这项工作将开辟新的数学方向，将奇点理论的几何方面与概率论的几个领域（瑞利-利维统计，随机多项式的零等）联系起来。该提案的第二部分集中在爱因斯坦广义相对论的一个基本预测上，即黑洞的存在。虽然黑洞不能被直接看到，但它极强的引力场扭曲了黑洞周围的时空，导致设法穿过该区域的光盘被弯曲成令人印象深刻的大角度（例如，光线在到达观察者之前可以绕黑洞无数次）。对这种极端重力的探测将是对我们对空间和时间本质理解的一次关键考验。该提议从数学上探索了由一般认为位于银河系中心的大质量黑洞引起的强引力场的透镜特征。这应该借鉴微分几何和奇点理论的技术，并在这些数学主题和黑洞物理学之间创造协同效应。

ABSTRACT: NSF Proposal, DMS-0302812 (Petters)This proposal presents a two part research program thatcreates a synergistic interaction between mathematics and astrophysics.Program 1 aims at developing a mathematical theory to determine theprobability distributions of Gaussian curvaturesat critical points of random time-delay functions. This research programopens up new mathematical directions that bridge singularity theory withprobability theory (e.g., zeros of random polynomials andlimit theorems in Rayleigh-Levy statistics). In addition, Program 1 hasdirect applications to understanding the nature of dark matter in galactichalos via the flux ratio anomalies in the lensing signatures of galaxies. Program 2 seeks to extend the classification of light curves and imagecentroids of sources near caustics in thin-screen, weak-field microlensingto the strong gravitational field of a Kerr spacetime. This involves studying the global topology, geometry, and singular theoretic structure oflight caustic surfaces in a spacetime. Program 2 points to new mathematicalissues relating the geometry of light caustic surfaces to topologicalinvariants of the matter singularities of gravitational lenses. This programmay also yield tests of Einstein's General Theory of Relativity, especially asit pertains to the massive black hole believed to lie in the nucleus of ourgalaxy.The dark matter in galactic halos and the black hole inthe center of our galaxy are two central and pressing topics in astrophysics.These issues are directly impacted by the powerful mathematical methods ofgeometric analysis because gravitational lensing is simultaneously a uniquetool for probing the nature of dark matter distributions and black holes, and a theory that is built on geometric, analytical, and probabilistic concepts.The dark matter study requires developing a mathematical theory that allowsone to differentiate generic dark matter lensing signatures from features thatare specific to the particular choice simple lens model used for dark matter. The first part of the proposal deals with formulating such a mathematicaltheory. This work would open up new mathematical directions that bridge thegeometric aspects of singularity theory with several areas in probabilitytheory (Rayleigh-Levy statistics, zeros of random polynomials, etc.). Thesecond part of the proposal focuses on one of the fundamental predictions ofEinstein's General Theory of Relativity, namely,the existence of black holes. Though a black hole cannot be seen directly, its extremely stronggravitational field warps the spacetime about the black hole causing lightrays that managed to get through the region to be bent with impressively largeangles (e.g., rays can loop around a black hole numerous times before arrivingat the observer). The probing of such extremes of gravity would be a criticaltest of our understanding of the nature of space and time. The proposalexplores mathematically the lensing signatures of the strong gravitationalfield due to the massive black hole generally believed to be at the center ofour galaxy. This should draw upon techniques from differential geometry andsingularity theory, and create synergies between these mathematicaltopics and the physics of black holes.