Signal inference in Galactic astrophysics

In this thesis we present a combination of methodological work with very wide focus and some specific astrophysical applications. We advance the knowledge on the Galactic interstellar medium by studying new ways of inferring select properties related to its magnetic field. We derive the statistical tools needed in a rigorous way from probabilistic considerations. One quantity describing the statistical properties of magnetic fields is their helicity. We apply a recently developed technique to detect magnetic helicity from astronomical observations to data from the Milky Way. No indications of helicity are found. Using a series of simulations in the Galactic setting, we are able to show that the technique fails to detect helicity in cases in which the underlying electron density varies too strongly. Thus, we are able to conclude that either this is the case in the Milky Way or the Galactic magnetic field is non-helical. We further develop a technique, needed among other things to enable the correct application of the helicity test, to reconstruct continuous signals from noisy data for which the noise level is unknown. To do this, we make use of the statistical correlation structure of the signal which we reconstruct from the same data set in a self-consistent way. Fluctuations in the data that are inconsistent with this correlation structure are then assigned to the data's error budget. The development of this technique is partly motivated by the goal of creating an all-sky map of the Galactic contribution to the astronomical Faraday rotation effect, which probes both the Galactic magnetic field and the density of free thermal electrons. Since the quantity that is observed is the Faraday rotation of a radio source, influenced by all magnetic fields between the source and the observer, extragalactic contributions need to be filtered out. We use the technique for reconstructions in the presence of an uncertain degree of noisiness to assign the extragalactic contributions - as well as some other observational effects - to the error budget of the data and thus single out the Galactic contribution. The resulting map is the most detailed and precise map of its kind and the only one in which the extragalactic contributions have been filtered out. Finally, we develop a method to reconstruct log-normal signal fields, i.e. strictly positive signal fields for which the strengths of the fluctuations vary over several orders of magnitude. This is done with a view to reconstructions of emission maps due to different processes in the Galactic interstellar medium

Diagnostics for insufficiencies of posterior calculations in Bayesian signal inference

We present an error-diagnostic validation method for posterior distributions in Bayesian signal inference, an advancement of a previous work. It transfers deviations from the correct posterior into characteristic deviations from a uniform distribution of a quantity constructed for this purpose. We show that this method is able to reveal and discriminate several kinds of numerical and approximation errors, as well as their impact on the posterior distribution. For this we present four typical analytical examples of posteriors with incorrect variance, skewness, position of the maximum, or normalization. We show further how this test can be applied to multidimensional signals

The Denoised, Deconvolved, and Decomposed Fermi γ\gamma-ray sky - An application of the D3^3PO algorithm

We analyze the 6.5yr all-sky data from the Fermi LAT restricted to gamma-ray photons with energies between 0.6-307.2GeV. Raw count maps show a superposition of diffuse and point-like emission structures and are subject to shot noise and instrumental artifacts. Using the D3PO inference algorithm, we model the observed photon counts as the sum of a diffuse and a point-like photon flux, convolved with the instrumental beam and subject to Poissonian shot noise. D3PO performs a Bayesian inference in this setting without the use of spatial or spectral templates;i.e., it removes the shot noise, deconvolves the instrumental response, and yields estimates for the two flux components separately. The non-parametric reconstruction uncovers the morphology of the diffuse photon flux up to several hundred GeV. We present an all-sky spectral index map for the diffuse component. We show that the diffuse gamma-ray flux can be described phenomenologically by only two distinct components: a soft component, presumably dominated by hadronic processes, tracing the dense, cold interstellar medium and a hard component, presumably dominated by leptonic interactions, following the hot and dilute medium and outflows such as the Fermi bubbles. A comparison of the soft component with the Galactic dust emission indicates that the dust-to-soft-gamma ratio in the interstellar medium decreases with latitude. The spectrally hard component exists in a thick Galactic disk and tends to flow out of the Galaxy at some locations. Furthermore, we find the angular power spectrum of the diffuse flux to roughly follow a power law with an index of 2.47 on large scales, independent of energy. Our first catalog of source candidates includes 3106 candidates of which we associate 1381(1897) with known sources from the 2nd(3rd) Fermi catalog. We observe gamma-ray emission in the direction of a few galaxy clusters hosting radio halos.Comment: re-submission after referee report (A&A); 17 pages, many colorful figures, 4 tables; bug fixed, flux scale now consistent with Fermi, even lower residual level, pDF -> 1DF source catalog, tentative detection of a few clusters of galaxies, online material http://www.mpa-garching.mpg.de/ift/fermi

The Galaxy in circular polarization: all-sky radio prediction, detection strategy, and the charge of the leptonic cosmic rays

The diffuse Galactic synchrotron emission should exhibit a low level of diffuse circular polarization (CP) due to the circular motions of the emitting relativistic electrons. This probes the Galactic magnetic field in a similar way as the product of total Galactic synchrotron intensity times Faraday depth. We use this to construct an all sky prediction of the so far unexplored Galactic CP from existing measurements. This map can be used to search for this CP signal in low frequency radio data even prior to imaging. If detected as predicted, it would confirm the expectation that relativistic electrons, and not positrons, are responsible for the Galactic radio emission. Furthermore, the strength of real to predicted circular polarization would provide statistical information on magnetic structures along the line-of-sights.Comment: 11 pages, 5 figures, revise

Fast and precise way to calculate the posterior for the local non-Gaussianity parameter fnlf_\text{nl} from cosmic microwave background observations

We present an approximate calculation of the full Bayesian posterior probability distribution for the local non-Gaussianity parameter $f_{\text{nl}}$ from observations of cosmic microwave background anisotropies within the framework of information field theory. The approximation that we introduce allows us to dispense with numerically expensive sampling techniques. We use a novel posterior validation method (DIP test) in cosmology to test the precision of our method. It transfers inaccuracies of the calculated posterior into deviations from a uniform distribution for a specially constructed test quantity. For this procedure we study toy cases that use one- and two-dimensional flat skies, as well as the full spherical sky. We find that we are able to calculate the posterior precisely under a flat-sky approximation, albeit not in the spherical case. We argue that this is most likely due to an insufficient precision of the used numerical implementation of the spherical harmonic transform, which might affect other non-Gaussianity estimators as well. Furthermore, we present how a nonlinear reconstruction of the primordial gravitational potential on the full spherical sky can be obtained in principle. Using the flat-sky approximation, we find deviations for the posterior of $f_{\text{nl}}$ from a Gaussian shape that become more significant for larger values of the underlying true $f_{\text{nl}}$. We also perform a comparison to the well-known estimator of Komatsu et al. [Astrophys. J. 634, 14 (2005)] and finally derive the posterior for the local non-Gaussianity parameter $g_{\text{nl}}$ as an example of how to extend the introduced formalism to higher orders of non-Gaussianity

Reconstructing signals from noisy data with unknown signal and noise covariance

We derive a method to reconstruct Gaussian signals from linear measurements with Gaussian noise. This new algorithm is intended for applications in astrophysics and other sciences. The starting point of our considerations is the principle of minimum Gibbs free energy which was previously used to derive a signal reconstruction algorithm handling uncertainties in the signal covariance. We extend this algorithm to simultaneously uncertain noise and signal covariances using the same principles in the derivation. The resulting equations are general enough to be applied in many different contexts. We demonstrate the performance of the algorithm by applying it to specific example situations and compare it to algorithms not allowing for uncertainties in the noise covariance. The results show that the method we suggest performs very well under a variety of circumstances and is indeed qualitatively superior to the other methods in cases where uncertainty in the noise covariance is present.Comment: 12 pages, 6 figures; 1D example added; accepted for publication in Phys. Rev.

Improving stochastic estimates with inference methods: calculating matrix diagonals

Estimating the diagonal entries of a matrix, that is not directly accessible but only available as a linear operator in the form of a computer routine, is a common necessity in many computational applications, especially in image reconstruction and statistical inference. Here, methods of statistical inference are used to improve the accuracy or the computational costs of matrix probing methods to estimate matrix diagonals. In particular, the generalized Wiener filter methodology, as developed within information field theory, is shown to significantly improve estimates based on only a few sampling probes, in cases in which some form of continuity of the solution can be assumed. The strength, length scale, and precise functional form of the exploited autocorrelation function of the matrix diagonal is determined from the probes themselves. The developed algorithm is successfully applied to mock and real world problems. These performance tests show that, in situations where a matrix diagonal has to be calculated from only a small number of computationally expensive probes, a speedup by a factor of 2 to 10 is possible with the proposed method.Comment: 9 pages, 6 figures, accepted by Phys. Rev. E; introduction revised, results unchanged; page proofs implemente