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Gravitational Lensing in an Exact Locally Inhomogenous Cosmology

Allen Attard

Doctor of Philosophy 2005
Graduate Department of Astronomy and Astrophysics, University of Toronto

A Recursive Swiss-Cheese (RSC) cosmological model is an exact solution to Einstein's general relativistic field equations allowing for dramatic local density inhomogeneities while maintaining global homogeneity and isotropy. It is constructed by replacing spherical regions of an FRW background with higher density cores placed at the centre of a Schwarzschild vacuum, with each core itself potentially being given the same treatment and the process repeated to generate a range of multifractal structures.

Code was developed to tightly pack spheres into spaces of constant curvature in an efficient manner, and was used to develop libraries of packings with positive, negative, and zero curvature. Various projections are used to illustrate their structure, and means of measuring its dimensionality are discussed. A method by which these packings can be used as building blocks of an RSC model, along with a way of selecting parameters to define the model, is described, and a coordinate system allowing a relativistically consistent means of synchronizing its various components is developed.

Formulations of the optical scalar equations for the expansion and shear rates of a beam are considered, and a set suitable for numerical integration selected. The forms of the null geodesic beam trajectories in each region of the model are computed, and a parallel propagated shadow plane basis that can be consistently followed between the various model sections is established. This allowed the development of code using a fourth order, variable step size Runge-Kutta integration routine to compute the gravitational lensing effect within an RSC model by tracking the amplification and distortion of a series of beams that are propagated through it. The output generated allows the redshift evolution of these quantities to be plotted for each beam, and enables maps to be made of the "observed sky". The amplification signature produced by a single lens in the model is examined, and the form shown to be generally consistent with that found using a thin lens approximation, particularly when the lensing is weak. Distortion values are likewise shown to be reasonable, and results derived from propagating beams through a full RSC model are also presented.

Reproduced with permission. library@astro.utoronto.ca
February 5 2005