### 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.