Optimal Reconstruction of Cosmological Density Fields

A key objective of modern cosmology is to determine the composition and distribution of matter in the universe. While current observations seem to match the standard cosmological model with remarkable precision, there remains tensions between observations as well as mysteries relating to the true nature of dark matter and dark energy. Despite the recent increased availability of cosmological data across a wide redshift, these tensions have remained or been further worsened. With the explosion of astronomical data in the coming decade, it has become increasingly critical to extract the maximum possible amount of information available across all available scales. As the available volume for analysis increases, we are no longer sample variance limited and existing summary statistics (as well as related estimators) need to be re-examined. Fortunately, parallel with the construction of these surveys there is significant development in the computational techniques used to analyze that data. Algorithmic developments over the past decade and expansion of computational resources allow large cosmological simulations to be run with relative simplicity and parallel theoretical developments motivate increased interest in recovering the underlying large scale structure of the universe beyond the power spectra.The detailed study of this large scale structure has the potential to shed light on various unanswered questions and under-constrained physical models for the dark sector and the nature of gravity. As we reach higher redshifts with statistically significant samples, the large scale structure can serve as a link between local observations and the cosmic microwave background. These surveys rely on a variety of biased probes, including the lensing and distribution of galaxies, imprints of large scale structure in secondary anisotropies of the CMB, and absorption lines in the spectra of high redshift quasars. These observations are complementary; they probe different scales, have different sources of astrophysical and observational uncertainties, have unique degenercies in parameter space, and require their own methods to extract cosmological parameters from.In this thesis, I discuss a number of new developments in the analysis of these diverse cosmological datasets. After introductory material, I discuss work re-examining the lensing of the Cosmic Microwave Background by cluster-sized objects and implement techniques for accurate mass estimation. I demonstrate that this analysis is optimal in the low noise, small scale limit. In the second part, I develop a maximum likelihood formalism for linear density fields, applicable for reconstructing underlying signal from a variety of cosmological probes including projected galaxy fields and cosmic shear, showing that effects of anisotropic noise and masking can be mitigated. Finally, I extend this work to nonlinear observables by using a forward modeling approach for Lyman Alpha forest tomography, finding more accurate cosmic web reconstruction verses existing techniques. The unifying theme of all these works is revisiting existing matter density reconstruction techniques with a critical eye and using new statistical and computational techniques to efficiently perform an unbiased, lower variance, estimate. Included is discussion of the possible impacts of these methods to improve constraints of cosmological parameters and/or astrophysical processes