Reconstruction of signals with unknown spectra in information field theory with parameter uncertainty

The optimal reconstruction of cosmic metric perturbations and other signals requires knowledge of their power spectra and other parameters. If these are not known a priori, they have to be measured simultaneously from the same data used for the signal reconstruction. We formulate the general problem of signal inference in the presence of unknown parameters within the framework of information field theory. We develop a generic parameter uncertainty renormalized estimation (PURE) technique and address the problem of reconstructing Gaussian signals with unknown power-spectrum with five different approaches: (i) separate maximum-a-posteriori power spectrum measurement and subsequent reconstruction, (ii) maximum-a-posteriori power reconstruction with marginalized power-spectrum, (iii) maximizing the joint posterior of signal and spectrum, (iv) guessing the spectrum from the variance in the Wiener filter map, and (v) renormalization flow analysis of the field theoretical problem providing the PURE filter. In all cases, the reconstruction can be described or approximated as Wiener filter operations with assumed signal spectra derived from the data according to the same recipe, but with differing coefficients. All of these filters, except the renormalized one, exhibit a perception threshold in case of a Jeffreys prior for the unknown spectrum. Data modes, with variance below this threshold do not affect the signal reconstruction at all. Filter (iv) seems to be similar to the so called Karhune-Loeve and Feldman-Kaiser-Peacock estimators for galaxy power spectra used in cosmology, which therefore should also exhibit a marginal perception threshold if correctly implemented. We present statistical performance tests and show that the PURE filter is superior to the others.Comment: 21 pages, 5 figures, accepted by PR

Temperature and Polarization Studies of the Cosmic Microwave Background

The cosmic microwave background (CMB) provides us with a wealth of information about the properties of our Universe. In this PhD work, we develop and apply new techniques for studying fundamental problems of cosmology using the CMB. Dark energy, if it exists, leaves a characteristic imprint in the CMB temperature fluctuations, the so-called integrated Sachs-Wolfe (ISW) effect. This small effect can be detected via its cross-correlation with the large-scale structure (LSS). We derive an optimal method for ISW detection using temperature and polarization data of the CMB which differs from that usually used in two fundamental ways: we keep the LSS distribution and a part of the primordial temperature fluctuations fixed, rather than averaging over different realisations as done in the standard method. For an ideal scenario, we obtain an overall enhancement of the detection significance of 23 per cent. For polarization data from the Planck Surveyor mission, this enhancement will be at least 10 per cent, where the limiting factor will be the contamination by Galactic foregrounds. The CMB is observed to be almost perfectly isotropic, which is considered strong evidence for the isotropy of the Universe. However, some anomalies have been found in the temperature map of the Wilkinson Microwave Anisotropy Probe (WMAP), which seem to question the statistical isotropy of the temperature fluctuations. In order to understand whether these are due to chance fluctuations or to a preferred direction intrinsic to the geometry of the primordial Universe, we compute the part of the WMAP polarization map which is uncorrelated with the temperature map, and use it as a statistically independent probe of the so-called axis of evil. The latter is an unusual alignment between the preferred directions of the quadrupole and the octopole in the temperature map. We find that the axis of the quadrupole of the uncorrelated polarization map aligns with the axis of evil, whereas the axis of the octopole does not. However, due to the high noise-level in the WMAP polarization map, we have an uncertainty of about 45 deg in our axes. With this uncertainty, the probability of at least one axis aligning by chance in an isotropic Universe is around 50 per cent. We therefore do not obtain evidence for or against a preferred direction intrinsic to the primordial Universe. For Planck, we expect the uncertainty in the axes to go down to 10-20 deg, again depending on how well the foregrounds can be removed from the map. Our technique applied to Planck data will thus serve as a powerful means to understand the origin of the CMB anomalies. Instead of studying particular features in the CMB maps as described above, we can also use the CMB to constrain several cosmological parameters simultaneously by sampling the parameter space. The parameter constraints obtained by WMAP marked the beginning of precision cosmology and were the biggest success of the mission. In such parameter sampling studies, the main bottleneck is usually the evaluation of the likelihood. We have thus implemented a sparse-grids based interpolation of the WMAP likelihood surface as a shortcut for the likelihood evaluation. This is orders of magnitude faster to compute than the original likelihood. Our method is a competitive alternative to other approaches for speeding up parameter sampling

The Kolmogorov-Smirnov test for the CMB

We investigate the statistics of the cosmic microwave background using the Kolmogorov-Smirnov test. We show that, when we correctly de-correlate the data, the partition function of the Kolmogorov stochasticity parameter is compatible with the Kolmogorov distribution and, contrary to previous claims, the CMB data are compatible with Gaussian fluctuations with the correlation function given by standard Lambda-CDM. We then use the Kolmogorov-Smirnov test to derive upper bounds on residual point source power in the CMB, and indicate the promise of this statistics for further datasets, especially Planck, to search for deviations from Gaussianity and for detecting point sources and Galactic foregrounds.Comment: Improved significance of the results (which remain unchanged) by using patches instead of ring segments in the analysis. Added sky maps of the Kolmogorov-parameter for original and de-correlated CMB ma

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

We develop information field theory (IFT) as a means of Bayesian 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, and the Boltzmann-Shannon information measure. The theory should be applicable in many fields. However, here, two cosmological signal recovery problems are discussed in their IFT-formulation. 1) Reconstruction of the cosmic large-scale structure matter distribution from discrete galaxy counts in incomplete galaxy surveys within a simple model of galaxy formation. We show that a Gaussian signal, which should resemble the initial density perturbations of the Universe, observed with a strongly non-linear, incomplete and Poissonian-noise affected response, as the processes of structure and galaxy formation and observations provide, can be reconstructed thanks to the virtue of a response-renormalization flow equation. 2) We design a filter to detect local non-linearities in the cosmic microwave background, which are predicted from some Early-Universe inflationary scenarios, and expected due to measurement imperfections. This filter is the optimal Bayes' estimator up to linear order in the non-linearity parameter and can be used even to construct sky maps of non-linearities in the data.Comment: 38 pages, 6 figures, LaTeX; version accepted by PR

Landscape, game and tourism

We search for an unusual alignment of the preferred axes of the quadrupole and octopole, the so-called axis of evil, in the CMB temperature and polarization data from WMAP. We use the part of the polarization map which is uncorrelated with the temperature map as a statistically independent probe of the axis of evil, which helps to assess whether the latter has a cosmological origin or if is a mere chance fluctuation in the temperature. Note, though, that for certain models creating a preferred axis in the temperature map, we would not expect to see the axis in the uncorrelated polarization map. We find that the axis of the quadrupole of the uncorrelated polarization map roughly aligns with the axis of evil within our measurement precision, whereas the axis of the octopole does not. However, with our measurement uncertainty, the probability of such a scenario to happen by chance in an isotropic universe is of the order of 50 per cent. We also find that the so-called cold spot present in the CMB temperature map is even colder in the part of the temperature map which is uncorrelated with the polarization, although there is still a large uncertainty in the latter. Therefore, our analysis of the axis of evil and a future analysis of the cold spot in the uncorrelated temperature data will strongly benefit from the polarization data expected from the Planck satellite.Comment: Version accepted by MNRAS, added some comments, corrected typo