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

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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.