Combining research methods from various areas of mathematics and physics, Probabilistic Models of Cosmic Backgrounds describes the isotropic random sections of certain fibre bundles and their applications to creating rigorous mathematical models of both discovered and hypothetical cosmic backgrounds. Previously scattered and hard-to-find mathematical and physical theories have been assembled from numerous textbooks, monographs, and research papers, and explained from different or even unexpected points of view. This consists of both classical and newly discovered results necessary for understanding a sophisticated problem of modelling cosmic backgrounds. The book contains a comprehensive description of mathematical and physical aspects of cosmic backgrounds with a clear focus on examples and explicit calculations. Its reader will bridge the gap of misunderstanding between the specialists in various theoretical and applied areas who speak different scientific languages. The audience of the book consists of scholars, students, and professional researchers. A scholar will find basic material for starting their own research. A student will use the book as supplementary material for various courses and modules. A professional mathematician will find a description of several physical phenomena at the rigorous mathematical level. A professional physicist will discover mathematical foundations for well-known physical theories.
Combining research methods from various areas of mathematics and physics, Probabilistic Models of Cosmic Backgrounds describes the isotropic random sections of certain fiber bundles and their applications to creating rigorous mathematical models of both discovered and hypothetical cosmic backgrounds. Previously scattered and hard-to-find mathematical and physical theories have been assembled from numerous textbooks, monographs, and research papers, and explained from different or even unexpected points of view. This consists of both classical and newly discovered results necessary for understanding a sophisticated problem of modelling cosmic backgrounds. The book contains a comprehensive description of mathematical and physical aspects of cosmic backgrounds with a clear focus on examples and explicit calculations. Its reader will bridge the gap of misunderstanding between the specialists in various theoretical and applied areas who speak different scientific languages. The audience of the book consists of scholars, students, and professional researchers. A scholar will find basic material for starting their own research. A student will use the book as supplementary material for various courses and modules. A professional mathematician will find a description of several physical phenomena at the rigorous mathematical level. A professional physicist will discover mathematical foundations for well-known physical theories.
The author describes the current state of the art in the theory of invariant random fields. This theory is based on several different areas of mathematics, including probability theory, differential geometry, harmonic analysis, and special functions. The present volume unifies many results scattered throughout the mathematical, physical, and engineering literature, as well as it introduces new results from this area first proved by the author. The book also presents many practical applications, in particular in such highly interesting areas as approximation theory, cosmology and earthquake engineering. It is intended for researchers and specialists working in the fields of stochastic processes, statistics, functional analysis, astronomy, and engineering.
From ancient soothsayers and astrologists to today’s pollsters and economists, probability theory has long been used to predict the future on the basis of past and present knowledge. Mathematical Models of Information and Stochastic Systems shows that the amount of knowledge about a system plays an important role in the mathematical models used to foretell the future of the system. It explains how this known quantity of information is used to derive a system’s probabilistic properties. After an introduction, the book presents several basic principles that are employed in the remainder of the text to develop useful examples of probability theory. It examines both discrete and continuous distribution functions and random variables, followed by a chapter on the average values, correlations, and covariances of functions of variables as well as the probabilistic mathematical model of quantum mechanics. The author then explores the concepts of randomness and entropy and derives various discrete probabilities and continuous probability density functions from what is known about a particular stochastic system. The final chapters discuss information of discrete and continuous systems, time-dependent stochastic processes, data analysis, and chaotic systems and fractals. By building a range of probability distributions based on prior knowledge of the problem, this classroom-tested text illustrates how to predict the behavior of diverse systems. A solutions manual is available for qualifying instructors.
"Mass tells spacetime how to curve, spacetime tells mass how to move". This famous quote by physicist John Archibald Wheeler succinctly summarizes General Relativity, the most successful theory that describes our universe at large scale. However, most of the mass that General Relativity describes, namely dark matter (DM), remains a mystery. We have solid evidence of the existence of DM from various observations, but we know little or nothing about the particle nature of DM and how DM particles interact with different particles. Completing this knowledge gap would improve or revolutionize our established cosmological model, the Lambda Cold-Dark Matter (CDM) model, and give directions to theories beyond the standard particle physics model.This work attempts to study DM by examining and extending existing modeling approaches of DM and its visible tracers in a probabilistic way. The single verified form of DM interaction is gravitational. Currently, the only way to infer the properties of DM is through visible tracers. Most of these indirect detections either have low signal-to-noise, sparse coverage, or missing variables. These limitations introduce additional modeling choices and uncertainties. A probabilistic approach allows us to propagate the uncertainties appropriately and marginalize any missing variables. There are two recurring types of visible tracers that my work uses. The first type of tracers are galaxies and observables in the overdense regions of DM. These tracers allow usto infer the macroscopic dynamical properties of DM distribution that we want to study. The second type of tracers, on the hand, are in the background, i.e. further away than the foreground dark matter, from us observers. The gravity of DM can bend spacetime such that the path of light traveling in the vicinity would also curve, leaving distortions in the galaxy images. The gravitational distortion of the images of the background galaxies is also known as gravitational lensing. In the introduction (first chapter) of this thesis, I will layout the technical history, terminology and the reasons behind choosing the various data sets and give an overview of the analysis methods for my thesis work. In chapter two, I will present the study based on the observational data of El Gordo, one of the most massive, most ancient, merging galaxy clusters. Under the extreme collision speeds during a merger of a galaxy cluster, it is more probable for DM particles in the cluster to manifest eects of self-interaction. Thus, if DM particles can interact with one another, some preliminary simulations have shown that large-scale spatial distribution of DM can show discrepancies from its galaxy-counterparts. This discrepancy is also known as the galaxy-DM offset, with a caveat. The long duration (millions of years) of a merger means that we cannot detect the direction of motions of the components directly to confirm the offset as a lag. My work on El Gordo was the first to show a quantitative method of estimating how likely the DM components of El Gordo are to be moving in a certain direction. This study was made possible by utilizing informative observables in various wavelengths, including a pair of radio shockwaves on the outer skirt of the cluster, enhanced X-ray emissivity and the decrement of the Sunyaev-Zel'Dovich effect for the infra-red observations. This comprehensive set of observables allowed us to formulate probabilistic constraints in our Monte Carlo simulation of El Gordo. Furthermore, the study also brought up several questions about the modeling choices for comparing the DM and the member-galaxy distributions of a cluster. For instance, do the DM maps and the galaxy maps have high enough resolution to show the delicate offset signal produced by the possible self-interaction of DM (SIDM)? To address my concerns from the study of El Gordo, I conducted a second investigation of galaxy clusters in a cosmological simulation, which is described in chapter 3. The dataset I chose was from the Illustris simulation. As this simulation assumes a Cold-Dark-Mattermodel (CDM) without requiring an SIDM model, any offset between DM and the member galaxies in a galaxy cluster provides an estimate of the variability of the galaxy-DM offset. My study shows that the variability in this setting is non-negligible compared to the small observed offsets, it is likely that random variation can account for the galaxy-DM offsets inobservations. The result weakens our belief that SIDM is the cause of the offsets. The fourth chapter of my dissertation builds on top of my previous experience with analyzing the weak lensing data for El Gordo. This time, I performed the weak lensing study for a dataset of a much larger spatial scale, such that, galaxy clusters look like parts of a homogeneous and isotropic DM web. At this scale, it is possible to compare the spatial distribution of DM to simulations to give competitive constraints on cosmological parameters. Using weak lensing signals for estimating cosmological parameters is also known as cosmic shear inference. While I used a parametric technique to estimate the mass of El Gordo in chapter 2, my work in chapter 4 introduces a new non-parametric model using a Gaussian Process. A Gaussian Process is a generalization of the multivariate normal distribution to higher dimensions. We can draw functional models from a Gaussian Process to describe our data. While the realizations are drawn from a multivariate normal distribution, we can specify the parameters and the functional structure of the covariance (kernel) matrix of the underlying distribution. This generative model gives us the ability to put probabilistic estimates of DM density in regions without any background galaxies. As I have built the lensing physics into the very core of the covariance kernel matrix, we can also simultaneously infer the several important lensing observables, such as shear and convergence, given some lensed galaxy shapes. More importantly, this technique relies on fewer assumptions about the photometric redshift than traditional cosmic shear analysis technique. This may reduce the bias towards a ducial cosmology and lead to interesting discoveries. However, this new technique is not without its challenges. Computationally, this technique requires an O(n3) runtime. Despite my best attempts to parallelize the computation, the algorithm takes longer for generating DM mass maps than traditional approaches. My work here marks the beginning of an alternative method for cosmic shear inference. Many promising approximation techniques have emerged to drastically speed up the runtime of doing inference with a Gaussian Process. Incorporating these approximations may make it possible to use this method to give tighter cosmological constraints from future sky surveys such as the Large Synoptic Survey Telescope. I conclude my work in Chapter 5 and discuss the implications of my work. This includes some future directions for analyzing DM by using simulations with different underlying DM models and real data.
