Studies in exact solutions and inhomogeneous cosmological models

Current problems in modern cosmology invite us to consider some exact solutions to the Einstein Field Equations (EFE) that represent more general cosmological models than the Friedmann Lemaitre Robertson Walker (FLRW) models. Studies in exact solutions to the EFE are further motivated by applications in the study of neutron stars and other compact objects. Since General Relativity was proposed, about 1500 exact solutions, including around 300 cosmological models, have been discovered. Much of the work of discovering these solutions was carried out by hand, and numerous typos or errors have already been found. In addition, the physical interpretations of many of these spacetimes have not been fully explored. To solve this problem, we consider an inverse approach to the EFE by asking what types of fluid sources could generate a given spacetime metric. The work described here provides a theoretical framework to validate many existing exact solutions. This approach removes the difficulty of trying to explicitly solve the nonlinear differential equations that constitute the EFE, but still allows one to verify exact solutions. The idea is to verify the relationships between components of the Einstein Tensor and properties of the fluid source. By considering the field equations with this inverse approach, we avoid the time-consuming and difficult task of re-deriving known solutions. While previous work has considered the inverse approach for fluids with zero energy flux in warped product type B 1 spacetimes, we extend the work by considering imperfect fluids with isotropic pressure. We do this in the three canonical types of coordinates: double null, null, and diagonal. We apply the work to example spacetimes, and then we further extend the work by directly considering spacetimes with perfect fluid sources. Applying our work to modern cosmology, we develop the inverse approach for the quasispherical Szekeres models with perfect fluid sources and consider these models as possible cosmological models. We explore a specific quasispherical Szekeres model. We numerically calculate the magnitude-redshift curve and compare the results to supernovae data and the standard model. The developed approach is algorithmic and will be useful for studying exact solutions with computer algebra programs.