Nonparametric inference has become an essential tool for studying the cosmos. This project consists of two intertwined components: (a) development of new theoretical tools and nonparametric methodologies that are inspired by problems in astrophysics but apply more broadly, and (b) application of these tools to two important astrophysical problems, which will frame the need for and guide the development of new statistical theory. Specifically, the investigators will focus on inference for the dark energy equation of state and on identifying filamentary structures from point process data such as that produced by galaxy surveys. The first problem gives rise to a challenging nonlinear inverse problem and demands a nonparametric approach, given what little is known about the dark energy equation of state. The investigators will develop new theory for nonlinear inverse problems that allow for accurate estimates and sharp confidence statements about the unknown function. These techniques will then be applied to Type Ia supernova data, possibly combined with other data sources, to make inferences about dark energy. The second problem gives rise to challenging spatial and inference problems. Current theory in the statistical literature applies to a single filament only, and techniques in the astronomical literature are not supported by theory. The investigators on this project will close that gap, developing theory for defining, identifying, and making inferences about the filamentary structures. The investigators will test this technique and apply it to galaxy survey data.
One of the most important problems in cosmology is understanding dark energy. The relationship between observable quantities and dark energy produces a challenging nonlinear inverse problem. With very little strong a priori information about the nature of dark energy, parametric approaches to the problem are limited and suboptimal. And with the promise of much larger data sets in the near future, there will be need and opportunity to extract fine-scale features of the dark energy equation of state. The investigators will develop new theory of inference for such problems, with a focus on estimation under shape constraints, sharp hypothesis testing, and accurate confidence sets. The goal is a substantial improvement in accuracy over the current best techniques. In particular, the investigators will focus on the problem of understanding dark energy and on identifying filamentary structures in distribution of matter. The former is one of the central problems in modern cosmology and demands state of the art statistical techniques to get the most from the data. The investigators will develop new statistical theory and methodologies that substantially improve the precision with which features of dark energy can be estimated from supernova data and other data sources. The latter problem is central to understanding the distribution of matter in the universe. Current statistical theory only applies to a limited version of the problem, and current astronomical methodologies do not have strong theoretical support. The investigators will close that gap and develop a method and corresponding theory that can handle realistic versions of the problem and give optimal or near-optimal performance.
