Information field theory for cosmological perturbation reconstruction and non-linear signal analysis

IFT and its application. The concepts of IFT are introduced in Sec. II, where Bayesian methodology, the distinction of physical and information fields, the definition of signal response and noise, as well the design of signal spaces are discussed. The basic IFT formalism including the free theory is introduced in Sec. III, which, according to our judgement, summarizes and unifies the previous knowledge on IFT before this paper. An impatient reader, only interested in applying IFT and not worrying about concepts, may start reading in Sec. III. From Sec. IV on the new results of this work are presented, starting with the discussion of interacting information fields, their Hamiltonians and Feynman rules, and the Boltzmann-Shannon information measure. The normalisability of sensibly constructed IFTs is shown, as well the classical information field equation is presented there. A step-by-step recipe of how to derive and implement an IFT algorithms is also provided. Details of the notation can be found, if not defined in the main text, in Appendix A. Applications of the theory are provided in the following two sections, which can be skipped by a reader interested only in the general theoretical framework. Although specific inference problems are addressed, they should serve as a blueprint for the tackling of similar problems. In Sec. V the problem of the reconstruction of the cosmic matter distribution from galaxy surveys is analyzed in terms of a Poissonain data model. In Sec. VI we derive an optimal estimator for non-Gaussianity in the CMB, and show how it can be generalized to map potential non-Gaussianities in the CMB sky. Our summary and outlook can be found in Sec. VII.