Browsing by master's degree program "Master 's Programme in Theoretical and Computational Methods"

Many beyond the Standard Model theories include a first order phase transition in the early universe. A phase transition of this kind is presumed to be able to source gravitational waves that might be be observed with future detectors, such as the Laser Interferometer Space Antenna. A first order phase transition from a symmetric (metastable) minimum to the broken (stable) one causes the nucleation of broken phase bubbles. These bubbles expand and then collide. It is of importance to examine how the bubbles collide in depth, as the events during the collision affect the gravitational wave spectrum. We assume the field to interact very weakly or not at all with the particle fluid in the early universe. The universe also experiences fluctuations due to thermal or quantum effects. We look into how these background fluctuations affect the field evolution and bubble collisions during the phase transition in O(N) scalar field theory. Specifically, we numerically simulate two colliding bubbles nucleated on top of the background fluctuations, with the field being a N-dimensional vector in the O(N) group. Due to the symmetries present, the system can be examined in cylindrical coordinates, lowering the number of simulated spatial dimensions. In this thesis, we perform the calculation of initial state fluctuations and simulate them and two bubbles numerically. We present results of the simulation of the field, concentrating on the effects of fluctuations on the O(N) scalar field theory.

Sum-product networks (SPN) are graphical models capable of handling large amount of multi- dimensional data. Unlike many other graphical models, SPNs are tractable if certain structural requirements are fulfilled; a model is called tractable if probabilistic inference can be performed in a polynomial time with respect to the size of the model. The learning of SPNs can be separated into two modes, parameter and structure learning. Many earlier approaches to SPN learning have treated the two modes as separate, but it has been found that by alternating between these two modes, good results can be achieved. One example of this kind of algorithm was presented by Trapp et al. in an article Bayesian Learning of Sum-Product Networks (NeurIPS, 2019). This thesis discusses SPNs and a Bayesian learning algorithm developed based on the earlier men- tioned algorithm, differing in some of the used methods. The algorithm by Trapp et al. uses Gibbs sampling in the parameter learning phase, whereas here Metropolis-Hasting MCMC is used. The algorithm developed for this thesis was used in two experiments, with a small and simple SPN and with a larger and more complex SPN. Also, the effect of the data set size and the complexity of the data was explored. The results were compared to the results got from running the original algorithm developed by Trapp et al. The results show that having more data in the learning phase makes the results more accurate as it is easier for the model to spot patterns from a larger set of data. It was also shown that the model was able to learn the parameters in the experiments if the data were simple enough, in other words, if the dimensions of the data contained only one distribution per dimension. In the case of more complex data, where there were multiple distributions per dimension, the struggle of the computation was seen from the results.

We study a system of cold high-density matter consisting purely of quarks and gluons. The mathematical construction of Quantum Chromodynamics (QCD) introduces interactions between the fields, which modify the thermodynamic properties of the system. In the presence of interactions, we can not solve the thermodynamic properties of the system analytically. The method is to expand the result in a series in terms of the QCD coupling constant. This is referred to as the perturbation theory in the context of thermal field theory (TFT). The coupling constant describes the strength of the interaction. We introduce the basic calculation methods used in the QCD and the TFTs in general. We will also include the chemical potential associated with the number of quarks in the system in the calculation. In the case of zero temperature, quarks form a Fermi-sphere such that energy states lower than the chemical potential will be Pauli blocked. The resulting fermionic momentum integrals are modified as a consequence. We can split these integrals into two parts, referred to as the vacuum and matter parts. We can split the calculation of the pressure into two distinct contributions: one from skeleton diagrams and one from ring diagrams. The ring diagrams have unphysical IR divergences that we can not cancel using the counterterms. This is why hard thermal loop (HTL) effective field theory (EFT) is introduced. We will discuss this HTL framework, which requires the computation of the matter part of the gluon polarization tensor, which we will also evaluate in this thesis.

Topological defects and solitons are nontrivial topological structures that can manifest as robust, nontrivial configurations of a physical field, and appear in many branches of physics, including condensed matter physics, quantum computing, and particle physics. A fruitful testbed for experimenting with these fascinating structures is provided by dilute Bose–Einstein condensates. Bose–Einstein condensation was first predicted in 1925, and Bose–Einstein condensation was finally achieved in a dilute atomic gas for the first time in 1995 in a breakthrough experiment. Since then, the study of Bose–Einstein condensates has expanded to the study of a variety of nontrivial topological structures in condensates of various atomic species. Bose–Einstein condensates with internal spin degrees of freedom may accommodate an especially rich variety of topological structures. Spinor condensates realized in optically trapped ultracold alkali atom gases can be conveniently controlled by external fields and afford an accurate mean-field description. In this thesis, we study the creation and evolution of a monopole-antimonopole pair in such a spin-1 Bose–Einstein condensate by numerically solving the Gross–Pitaevskii equation. The creation of Dirac monopole-antimonopole pairs in a spin-1 Bose–Einstein condensate was numerically demonstrated and a method for their creation was proposed in an earlier study. Our numerical results demonstrate that the proposed creation method can be used to create a pair of isolated monopoles with opposite topological charges in a spin-1 Bose–Einstein condensate. We found that the monopole-antimonopole pair created in the polar phase of the spin-1 condensate is unstable against decay into a pair of Alice rings with oscillating radii. As a result of a rapid polar-to-ferromagnetic transition, these Alice rings were observed to decay by expanding on a short timescale.

We will review techniques of perturbative thermal quantum chromodynamics (QCD) in the imaginary-time formalism (ITF). The Infrared (IR)-problems arising from the perturbative treatment of equilibrium thermodynamics of QCD and their phenomenological causes will be investigated in detail. We will also discuss the construction of two effective field theory (EFT) frameworks most often used in modern high precision calculations to overcome these. The EFTs are the dimensionally reduced theories EQCD and MQCD and Hard thermal loop effective theory (HTL). EQCD is three-dimensional Euclidean Yang-Mills theory coupled to an adjoint scalar field and MQCD is three-dimensional Euclidean pure Yang-Mills theory. The effective parameters in these theories are determined through matching calculations. HTL is based on resummation of hard thermal loops and uses effective propagators and vertex functions. We will also discuss the determination of the pressure of QCD perturbatively. In general, this thesis details calculations and the methodology.

The Smoluchowski coagulation equation is considered to be one of the most fundamental equations of the classical description of matter alongside with the Boltzman, Navier-Stokes and Euler equations. It has applications from physical chemistry to astronomy. In this thesis, a new existence result of measure valued solutions to the coagulation equation is proven. The proven existence result is stronger and more general than a previously claimed result. The proven result holds for a generic class of coagulation kernels, including various kernels used in applications. The coagulation equation models binary coagulation of objects characterized by a strictly positive real number called size, which often represents mass or volume in applications. In binary coagulation, two objects can merge together with a rate characterized by the so-called coagulation kernel. Time evolution of the size distribution is given by the coagulation equation. Traditionally the coagulation equation has two forms, discrete and continuous, which are referring to whether the objects sizes can take discrete or continuous values. A similar existence result to the one proven in this thesis has been obtained for the continuous coagulation equation, while the discrete coagulation equation is often favored in applications. Being able to study both discrete and continuous systems and their mixtures at the same time has motivated the study of measure valued solutions to the coagulation equation. After motivating the existence result proven in this thesis, its proof is organized into four Steps described at the end of the introduction. The needed mathematical tools and their connection to the four Steps are presented in chapter 2. The precise mathematical statement of the existence result is given in chapter 3 together with Step 1, where the coagulation equation will be regularized using a parameter ε ∈ (0, 1) into a more manageable regularized coagulation equation. Step 2 is done in chapter 4 and it consists of proving existence and uniqueness of a solution f_ε for each regularized coagulation equation. Step 3 and Step 4 are done in chapter 5. In Step 3, it will be proven that the regularized solutions {f_ε} have a converging subsequence in the topology of uniform convergence on compact sets. Step 4 finishes the existence proof by verifying that the subsequence’s limit satisfies the original coagulation equation. Possible improvements and future work are outlined in chapter 6.

Many extensions to the Standard Model of particle physics feature a first-order phase transition in the very early universe. This kind of a phase transition would source gravitational waves through the collision of nucleation bubbles. These in turn could be detected e.g. with the future space-based gravitational wave observatory LISA (Laser Interferometer Space Antenna). Cosmic strings, on the other hand, are line-like topological defects. In this work, we focus on global strings arising from the spontaneous breakdown of a global symmetry. One example of global strings are axionic strings, which are a popular research topic, owing to the role of the axion as a potential dark matter candidate and a solution to the strong CP problem. In this work, our aim is to combine these two sets of early-universe phenomena. We investigate the possibility of creating global strings through the bubble collisions of a first-order phase transition. We use a simplified model with a two-component scalar field to nucleate the bubbles and simulate their expansion, obtaining a short-lived network of global strings in the process. We present results for string lifetime, mean string separations corresponding to different mean bubble separations, and gravitational wave spectra.

Flares are short, high-energy magnetic events on stars, including the Sun. Observations of young stars and red dwarfs regularly show the occurrence of flare events multiple orders of magnitude more energetic than even the fiercest solar storms ever recorded. As our technology remains vulnerable to disruptions due to space weather, the study of flares and other stellar magnetic activity is crucial. Until recently, the detection of extrasolar flares has required much manual work and observation resources. This work presents a mostly automatic pipeline to detect and estimate the energies of extrasolar flare events from optical light curves. To model and remove the star's background radiation in spite of complex periodicity, short windows of nonlinear support vector regression are used to form a multi-model consensus. Outliers above the background are flagged as likely flare events, and a template model is fitted to the flux residual to estimate the energy. This approach is tested on light curves collected from the stars AB Doradus and EK Draconis by the Transiting Exoplanet Survey Satellite, and dozens of flare events are found. The results are consistent with recent literature, and the method is generalizable for further observations with different telescopes and different stars. Challenges remain regarding edge cases, uncertainties, and reliance on user input.

The purpose of this work is to investigate the scaling of ’t Hooft-Polyakov monopoles in the early universe. These monopoles are a general prediction of a grand unified theory phase transition in the early universe. Understanding the behavior of monopoles in the early universe is thus important. We tentatively find a scaling for monopole separation which predicts that the fraction of the universe’s energy in monopoles remains constant in the radiation era, regardless of initial monopole density. We perform lattice simulations on an expanding lattice with a cosmological background. We use the simplest fields which produce ’t Hooft-Polyakov monopoles, namely the SU(2) gauge fields and a Higgs field in the adjoint representation. We initialize the fields such that we can control the initial monopole density. At the beginning of the simulations, a damping phase is performed to suppress nonphysical fluctuations in the fields, which are remnants from the initialization. The fields are then evolved according to the discretized field equations. Among other things, the number of monopoles is counted periodically during the simulation. To extend the dynamical range of the runs, the Press-Spergel-Ryden method is used to first grow the monopole size before the main evolution phase. There are different ways to estimate the average separation between monopoles in a monopole network, as well as to estimate the root mean square velocity of the monopoles. We use these estimators to find out how the average separation and velocity evolve during the runs. To find the scaling solution of the system, we fit the separation estimate on a function of conformal time. This way we find that the average separation ξ depends on conformal time η as ξ ∝ η^(1/3) , which indicates that the monopole density scales in conformal time the same way as the critical energy density of the universe. We additionally find that the velocity measured with the velocity estimators depends on the separation as approximately v ∝ dξ/dη. It’s been shown that a possible grand unified phase transition would produce an abundance of ’t Hooft-Polyakov monopoles and that some of these would survive to the present day and begin to dominate the energy density of the universe. Our result seemingly disagrees with this prediction, though there are several reasons why the predictions might not be compatible with the model we simulate. For one, in our model the monopoles do not move with thermal velocities, unlike what most of the predictions assume happens in the early universe. Thus future work of simulations with thermal velocities added would be needed. Additionally we ran simulations only in the radiation dominated era of the universe. During the matter domination era, the monopoles might behave differently.

This thesis reviews state-of-the-art top-down holographic methods used for modeling dense matter in neutron stars. This is done with the help of the Witten-Sakai-Sugimoto (WSS) model, which attempts to construct a holographic version of quantum chromodynamics (QCD) to mimic its features. As a starting chapter, string theory is reviewed in a quick fashion for the reader to understand some of the (historical) developments behind this construction. Bosonic and superstrings are reviewed along conformal field theory, and focus is put on Dp-branes and compactifications of spacetime. This chapter will also explain much of the jargon used in the thesis, which otherwise easily obstructs the main message. After a sufficient understanding of string theory has been achieved, we will move on to holography and holographic dualities in the next chapter, focusing on AdS/CFT and actual computations using holography. Matching of theories is discussed to set up a holographic dictionary. After this, we need to choose either a top-down or a bottom-up approach, from which we will use the former since we are going to use the WSS model. After this comes a brief review of QCD and its central features to be reproduced in holographic QCD. Immediately following this, we will review the Witten-Sakai-Sugimoto model, which is qualitatively and sometimes also quantitatively a reasonable holographic version of QCD. We will discuss WSS’s successes and room for improvement, especially in places that might affect the analysis that we are about to perform on neutron stars. Finally, after all this theoretical development, we will delve into the world of neutron stars. A quick review of the basic features and astrophysical constraints of neutron stars, along with difficulties in modeling them, is given. After this, we will discuss two models of neutron stars, the first one being a toy model with simplified physics and the other a more realistic one. The basic workflow that is required to get to the equation of state data and other relevant observables from a string theoretic action is given step-by-step, and many recent results using this model are reviewed. In the end, the future of the development of the holographic duality, constructing models with it, and modeling of neutron stars is discussed.