Towards an Optimal Cosmological Detection of Neutrino Mass with Bayesian Inference

High-precision measurements of large-scale cosmic structure are expected to revolutionize our understanding of fundamental physics, for example by the quantifying neutrino mass and elucidating the nature of dark energy. This dissertation tackles various of the challenges that must be faced in order to optimally extract information from cosmological surveys, taking a particular interest in neutrino mass.Massive neutrinos suppress the growth of cosmic structure on small scales, where gravity is nonlinear. It is currently an urgent task to determine how to maximally retrieve information in the nonlinear regime, as a traditional power spectrum analysis is no longer optimal. We start by using simulations to investigate the amount of information regarding neutrino mass present in cosmic structure. We find that while there is in principle a lot of information, only a small fraction will be measurable by upcoming surveys such as DESI and LSST. This motivates the need to perform a combined analysis with other cosmological tracers, such as the cosmic microwave background, and galaxy peculiar velocities. We thus develop a Bayesian forward modeling framework to combine field-level inference with galaxy peculiar velocities as a means to optimally extract information.Additionally, various numerical challenges arise due to the nonlinear effects of gravity. For example, computing the covariance matrices required for likelihood-based analyses becomes challenging, in particular due to the super-sample covariance effect. Moreover, the inference often involves non-trivial posterior surfaces which are plagued by volume effects such as the Look-Elsewhere Effect. This is particularly prevalent when searching for evidence of exotic new models. In the latter chapters of the dissertation we provide methods to solve both of these problems.All put together, this dissertation provides the ingredients for the cosmology community to move closer to an optimal measurement of neutrino mass and other