Bayesian Methods for Analyzing the Large Scale Structure of the Universe

TLDR: In this paper, two Bayesian data analysis computer algorithms are developed for estimating the 3D density field and its power-spectrum from galaxy observations, which can provide a complete statistical characterization and a detailed cosmographic description of the large scale structure in our universe.

Abstract: The cosmic large scale structure is of special relevance for testing current cosmological theories about the origin and evolution of the Universe. Throughout cosmic history, it evolved from tiny quantum fluctuations, generated during the early epoch of inflation, to the filamentary cosmic web presently observed by our telescopes. Observations and analyses of this large scale structure will hence test this picture, and will provide valuable information on the processes of cosmic structure formation as well as they will reveal the cosmological parameters governing the dynamics of the Universe. Beside measurements of the cosmic microwave backround, galaxy observations are of particular interest to modern precision cosmology. They are complementary to many other sources of information, such as cosmic microwave background experiments, since they probe a different epoch. Galaxies report the cosmic evolution over an enormous period ranging from the end of the epoch of reionization, when luminous objects first appeared, till today. For this reason, galaxy surveys are excellent probes of the dynamics and evolution of the Universe. Especially the Sloan Digital Sky Survey is one of the most ambitious surveys in the history of astronomy. It provides measurements of 930,000 galaxy spectra as well as the according angular and redshift positions of galaxies over an area which covers more than a quarter of the sky. This enormous amount of precise data allows for an unprecedented access to the three dimensional cosmic matter distribution and its evolution. However, observables, such as positions and properties of galaxies, provide only an inaccurate picture of the cosmic large scale structure due to a variety of statistical and systematic observational uncertainties. In particular, the continuous cosmic density field is only traced by a set of discrete galaxies introducing statistical uncertainties in the form of Poisson distributed noise. Further, galaxy surveys are subject to a variety of complications such as instrumental limitations or the nature of the observation itself. The solution to the underlying problem of characterizing the large scale structure in the Universe therefore requires a statistical approach. The main theme of this PhD-thesis is the development of new Bayesian data analysis methods which provide a complete statistical characterization and a detailed cosmographic description of the large scale structure in our Universe. The required epistemological concepts, the mathematical framework of Bayesian statistics as well as numerical considerations are thoroughly discussed. On this basis two Bayesian data analysis computer algorithms are developed. The first of which is called ARES (Algorithm for REconstruction and Sampling). It aims at the joint inference of the three dimensional density field and its power-spectrum from galaxy observations. The ARES algorithm accurately treats many observational systematics and statistical uncertainties, such as the survey geometry, galaxy selection effects, blurring effects and noise. Further, ARES provides a full statistical characterization of the three dimensional density field, the power-spectrum and their joint uncertainties by exploring the high dimensional space of their joint posterior via a very efficient Gibbs sampling scheme. The posterior is the probability of the model given the observations and all other available informations. As a result, ARES provides a sampled representation of the joint posterior, which conclusively characterizes many of the statistical properties of the large scale structure. This probability distribution allows for a variety of scientific applications, such as reporting any desired statistical summary or testing of cosmological models via Bayesian model comparison or Bayesian odds factors. The second computer algorithm, HADES (Hamiltonian Density Estimation and Sampling), is specifically designed to infer the fully evolved cosmic density field deep into the non-linear regime. In particular, HADES accurately treats the non-linear relationship between the observed galaxy distribution and the underlying continuous density field by correctly accounting for the Poissonian nature of the observables. This allows for very precise recovery of the density field even in sparsely sampled regions. HADES also provides a complete statistical description of the non-linear cosmic density field in the form of a sampled representation of a cosmic density posterior. Beside the possibility of reporting any desired statistical summary of the density field or power-spectrum, such representations of the according posterior distributions also allow for simple non-linear and non-Gaussian error propagation to any quantity finally inferred from the analysis results. The application of HADES to the latest Sloan Digital Sky Survey data denotes the first fully Bayesian non-linear density inference conducted so far. The results obtained from this procedure represent the filamentary structure of our cosmic neighborhood in unprecedented accuracy.