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

We develop information field theory (IFT) as a means of Bayesian, data based inference on spatially distributed signals, the information fields. A didactical approach is attempted. Starting from general considerations on the nature of measurements, signals, noise, and their relation to a physical reality, we derive the information Hamiltonian, the source field, propagator, and interaction terms. Free IFT reproduces the well known Wiener-filter theory. Interacting IFT can be diagrammatically expanded, for which we provide the Feynman rules in position-, Fourier-, and spherical harmonics space. Furthermore, we provide the Boltzmann-Shannon information measure of IFT based on the Helmholtz free energy, in order to show how to optimize observational strategies for maximal information retrieval.The theory should be applicable in many fields. However, here, two cosmological signal recovery problems are discussed in detail in their IFT-formulation. 1) Reconstruction of the cosmic large-scale structure matter distribution from discrete galaxy counts in incomplete galaxy surveys. 2) Optimal filtering to detect local non-Gaussianities in the cosmic microwave background, which are predicted from some Early-Universe inflationary scenarios. The derived Bayesian filter can be used even to construct sky maps of non-Gaussianies, and takes into account that different data realizations are differently well suited to unravel non-Gaussianities.