Master of Science / Department of Physics / Bharat Ratra / In this report, we discuss the four standard energy conditions of General Relativity (null, weak, dominant, and strong) and investigate their cosmological consequences. We note that these energy conditions can be compatible with cosmic acceleration provided that a repulsive cosmological constant exists and the acceleration stays within certain bounds. Scalar fields and dark energy, and their relationships to the energy conditions, are also discussed. Special attention is paid to the 1988 Ratra-Peebles scalar field model, which is notable in that it provides a physical self-consistent framework for the phenomenology of dark energy. Appendix B, which is part of joint-research with Anatoly Pavlov, Khaled Saaidi, and Bharat Ratra, reports on the existence of the Ratra-Peebles scalar field tracker solution in a curvature-dominated universe, and discusses the problem of investigating the evolution of long-wavelength inhomogeneities in this solution while taking into account the gravitational back-reaction (in the linear perturbative approximation).

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Submitted by Emanuel Varela Cardoso (emanuel.varela@ufcg.edu.br) on 2018-11-07T16:46:32Z No. of bitstreams: 1 DOUGLAS ALVES FERREIRA – DISSERTAÇÃO (PPGFísica) 2016.pdf: 1222935 bytes, checksum: 98d8066b80898d8ad6f6e489bca48723 (MD5) / Made available in DSpace on 2018-11-07T16:46:32Z (GMT). No. of bitstreams: 1 DOUGLAS ALVES FERREIRA – DISSERTAÇÃO (PPGFísica) 2016.pdf: 1222935 bytes, checksum: 98d8066b80898d8ad6f6e489bca48723 (MD5) Previous issue date: 2016-07 / Capes / Neste trabalho estudamos o comportamento de um campo escalar real, defeitos topológicos e não-topológicos. Para tanto, utilizamos o método proposto por Bogomol’nyi- Prasa de Sommerfield, o qual permite encontrar as soluções das equações de movimento de uma teoria clássica de campos, por meio de equações diferenciais de primeira ordem provenientes da minimização da energia. Estas soluções são chamadas de soluções BPS. Revisamos também a aplicabilidade do método BPS para modelos envolvendo dois campos escalares reais. Além disso, estudamos em detalhes os chamados métodos de deformação e de extensão de modelos. O método de extensão de modelos que até então era aplicado em teorias descritas por dois campos escalares, neste trabalho é ampliado para descrever modelos de três campos escalares com soluções analíticas. Outro ponto fundamental desse trabalho foi construir um novo procedimento, baseado nos métodos de deformação e de extensão, para gerar uma série de novos modelos analíticos. Este procedimento nos permitiu generalizar um sistema de dois campos escalares que envolve termos quebra de simetria de Lorentz. / In this work we study the behavior of a real scalarfield, topological defects and non-topological. Therefore we use the Bogomol’nyi-Prasad-Sommerfield method, which allows us to find the solutions for thee quations of motion of a classical field theory, by using first order differential equations related which the minimal energy of the system. These solutions are called BPS solutions. Here we also review the applicability of the BPS method in a two scalar field theory. More over, we discuss in details the deformation and the extension methods. The extension method which was applied to construct two scalar fields models up to now, is improved to generate new three scalar fields models with analytical solutions. Another key point of this work is the construction a new procedure based on the deformation and the extension methods, in order to generate new analytical models. Such a procedure allowed us to generalize a two scalar fields system involving Lorentz symmetry breaking terms.

Pendant les deux dernières décennies l'idée d'une nature holographique de la gravité a pris forme à travers la correspondance AdS/CFT, aussi connue sous le nom de dualité jauge/gravité. CFT correspond à « conformal field theory », théorie conforme des champs, et dans la dualité il s'agit d'une théorie de jauge dans la limite de grand N 1. AdS représente l'espace d'anti-de Sitter, une solution maximalement symétrique des équations d'Einstein avec une constante cosmologique négative, et correspond au côté gravitationnel de la dualité. Dans certaines limites, des théories sur AdS avec de la gravité en d+1 dimensions peuvent être associées à des CFTs sans gravité en d dimensions, d'où le nom « dualité ». Cette dualité est aussi dite « holographique » par analogie avec le concept optique homonyme qui indique la possibilité de générer une image tridimensionnelle comme la projection d'un écran ou d'un film bidimensionnel. Le terme holographie vient des mots grecs holos, « en entier », et graphe, « écriture. Une telle projection, malgré le fait que l'information est stockée en 2 dimensions, contiendrait toute l'information pour reconstruire l'image tridimensionnelle. Dans la dualité jauge/gravité, la théorie de jauge se comporte comme un film d-dimensionnel qui contient la même information que l'image gravitationnelle (d+1)-dimensionnelle. Cette dualité relie la théorie gravitationnelle à la théorie quantique de champs (TQC) dual à travers des conditions aux limites sur des champs qui vivent dans AdS. Dans ce sens-là, la théorie de jauge peut être considérée comme définie sur le bord d'AdS, ce qui renforce l'analogie optique et, pour cela, la dualité est aussi connue comme la correspondance « bulk/boundary » ou « intérieur/bord ». Une de ses principales propriétés est l'association d'une TQC fortement couplée à une théorie gravitationnelle faiblement couplée et vice-versa. Pour cette raison, dans cette thèse j'utilise un intérieur faiblement couplé pour explorer et identifier des propriétés non-perturbatives de TQCs dans la limite de couplage fort. Cette thèse explore l'holographie à température nulle et finie. Nos objets d'intérêt sont des TQCs générées par la brisure de l'invariance d'échelle de CFTs et qui peuvent être étudiées à travers le groupe de renormalisation (GR). Le profil des champs au long de la dimension supplémentaire à l'intérieur est dual à des flots du GR sur la TQC vivant sur le bord, car la dimension supplémentaire est en correspondance avec l'échelle d'énergie. La correspondance va plus loin en identifiant les champs de l'intérieur comme duaux aux couplages renormalisés de la TQC, ce qui mène au concept du GR holographique. Avec le GR holographique, dans cette thèse je vais explorer des comportements qui sont d'une nature intrinsèquement non-perturbatifs du point de vue de la QFT. Les principaux résultats sont les suivants. A température nulle, pour un seul couplage, nous avons classifié toutes les solutions de notre système et identifié trois types de flots exotiques correspondant à des solutions qui inversent leur direction au long du flot, d'autres qui sautent des points fixes et des flots qui interpolent entre des minima du potentiel. Ces résultats ont été généralisés à plusieurs couplages à température nulle. Je présente également la relation entre la fonction principale de Hamilton et la nature du champ de vitesses des couplages: gradient ou non. À température finie nous avons considéré un seul couplage et exploré la thermodynamique des trois types de solutions exotiques mentionnées ci-dessus. Nous avons identifié une transition de phase entre des solutions qui sautent et qui ne sautent pas des points fixes, une discontinuité de l'énergie libre pour un potentiel admettant des solutions qui inversent le sens du flot à température nulle et la non-existence de solutions à température finie associées à un flot entre minima pour un potentiel qui admet une telle solution à température nulle. / Over the past twenty years the idea that gravity is holographic has become progressively concrete, materialised through the AdS/CFT correspondence, also known as the gauge/gravity duality. CFT stands for conformal field theory and in the correspondence it is a gauge-theory in the large N limit1. AdS stands for anti-de Sitter space-time, a maximally symmetric solution of Einstein’s equations with negative cosmological constant, it corresponds to the gravitational side of the duality. In some limits, theories on AdS with gravity in d + 1 dimensions can be mapped to CFTs without gravity in d dimensions and vice-versa, hence the name “duality”. Another term for the gauge/gravity duality is holographic duality. The term holography comes from the Greek words holos, “whole”, and graphe, “writing” or “drawing”. In physics, the term holography originates in optics, referring to the possibility of generating a 3-dimensional image as a projection from a bi- dimensional screen or film. In such a projection, despite of the fact that the film has one spatial dimension less than the projection, the film would contain all the information to recover the three-dimensional image. In the gauge/gravity duality, the gauge-theory behaves as a d-dimensional film which contains the same information as the (d + 1)-dimensional gravitational image. This analogy is reinforced by the fact that the duality relates the gravitational theory to the dual resulting quantum field theory (QFT) via boundary conditions of the fields living in the AdS bulk. In this sense, the gauge theory can be thought of as living at the boundary of AdS and the duality is also know as the bulk/boundary correspondence. One of the most important features of the correspondence is the mapping of a strongly coupled QFT into a weakly coupled gravitational theory and vice-versa. For this reason, in this thesis I will use a weakly coupled bulk theory to explore and identify non-perturbative features of QFT in the strong coupling regime. This thesis explores holography at zero and finite temperature. Our main concern are the CFTs in which scale invariance is either spontaneously or explicitly broken and the resulting QFT can be studied via the renormalisation group (RG). The profile of fields along the extra-dimension in the bulk is dual to renormalisation group flows in the QFT side (boundary), as the extra-dimension can be mapped to an energy scale. The mapping goes further by identifying bulk fields as dual to QFT running couplings, leading to the so-called holographic renormalisation group. With the holographic RG in what follows I will explore behaviours that are of an intrinsically non-perturbative nature from the QFT standpoint. The main results are as follows. At zero temperature, for a single coupling, we classified all possible solutions in our setup and identified three kinds of exotic flows corresponding to solutions reversing direction along the flow (bounces), flows skipping fixed points and solutions interpolating between minima of the potential. These results are generalised to many couplings at zero temperature. I also present a complete map between forms of the Hamilton's principal function and the gradient or non-gradient nature of the solutions. At finite temperature we considered a single coupling setup and explored the thermodynamics of the three kinds of above-mentioned exotic flows. We identified a phase transition between skipping and non-skipping solutions, a discontinuous free energy for a bouncing potential and the non-existence of a finite-temperature solutions for a chosen potential admitting a minimum-to-minimum solution.

This thesis is about visualizing a kind of data that is trivial to process by computers but difficult to imagine by humans because nature does not allow for intuition with this type of information: high-dimensional data. Such data often result from representing observations of objects under various aspects or with different properties. In many applications, a typical, laborious task is to find related objects or to group those that are similar to each other. One classic solution for this task is to imagine the data as vectors in a Euclidean space with object variables as dimensions. Utilizing Euclidean distance as a measure of similarity, objects with similar properties and values accumulate to groups, so-called clusters, that are exposed by cluster analysis on the high-dimensional point cloud. Because similar vectors can be thought of as objects that are alike in terms of their attributes, the point cloud\'s structure and individual cluster properties, like their size or compactness, summarize data categories and their relative importance. The contribution of this thesis is a novel analysis approach for visual exploration of high-dimensional point clouds without suffering from structural occlusion. The work is based on implementing two key concepts: The first idea is to discard those geometric properties that cannot be preserved and, thus, lead to the typical artifacts. Topological concepts are used instead to shift away the focus from a point-centered view on the data to a more structure-centered perspective. The advantage is that topology-driven clustering information can be extracted in the data\'s original domain and be preserved without loss in low dimensions. The second idea is to split the analysis into a topology-based global overview and a subsequent geometric local refinement. The occlusion-free overview enables the analyst to identify features and to link them to other visualizations that permit analysis of those properties not captured by the topological abstraction, e.g. cluster shape or value distributions in particular dimensions or subspaces. The advantage of separating structure from data point analysis is that restricting local analysis only to data subsets significantly reduces artifacts and the visual complexity of standard techniques. That is, the additional topological layer enables the analyst to identify structure that was hidden before and to focus on particular features by suppressing irrelevant points during local feature analysis. This thesis addresses the topology-based visual analysis of high-dimensional point clouds for both the time-invariant and the time-varying case. Time-invariant means that the points do not change in their number or positions. That is, the analyst explores the clustering of a fixed and constant set of points. The extension to the time-varying case implies the analysis of a varying clustering, where clusters appear as new, merge or split, or vanish. Especially for high-dimensional data, both tracking---which means to relate features over time---but also visualizing changing structure are difficult problems to solve.

Scientific phenomena are often studied through collections of related scalar fields such as data generated by simulation experiments that are parameter or time dependent . Exploration of such data requires robust measures to compare them in a feature aware and intuitive manner. Topological data analysis is a growing area that has had success in analyzing and visualizing scalar fields in a feature aware manner based on the topological features. Various data structures such as contour and merge trees, Morse-Smale complexes and extremum graphs have been developed to study scalar fields. The extremum graph is a topological data structure based on either the maxima or the minima of a scalar field. It preserves local geometrical structure by maintaining relative locations of extrema and their neighborhoods. It provides a suitable abstraction to study a collection of datasets where features are expressed by descending or ascending manifolds and their proximity is of importance. In this thesis, we design a measure to understand the similarity between scalar fields based on the extremum graph abstraction. We propose a topological structure called the complete extremum graph and define a distance measure on it that compares scalar fields in a feature aware manner. We design an algorithm for computing the distance and show its applications in analyzing time varying data such as understanding periodicity, feature correspondence and tracking, and identifying key frames.

Nesta dissertação apresentamos um método de quantização matemática e conceitualmente rigoroso para o campo escalar livre de interações. Trazemos de início alguns aspéctos importantes da Teoria de Distribuições e colocamos alguns pontos de geometria Lorentziana. O restante do trabalho é dividido em duas partes: na primeira, estudamos equações de onda em variedades Lorentzianas globalmente hiperbólicas e apresentamos o conceito de soluções fundamentais no contexto de equações locais. Em seguida, progressivamente construímos soluções fundamentais para o operador de onda a partir da distribuição de Riesz. Uma vez estabelecida uma solução para a equação de onda em uma vizinhança de um ponto da variedade, tratamos de construir uma solução global a partir da extensão do problema de Cauchy a toda a variedade, donde as soluções fundamentais dão lugar aos operadores de Green a partir da introdução de uma condição de contorno. Na última parte do trabalho, apresentamos um mínimo da Teoria de Categorias e Funtores para utilizar esse formalismo na contrução de um funtor de segunda quantização entre a categoria de variedades Lorentzianas globalmente hiperbólicas e a categoria de redes de álgebras C* satisfazendo os axiomas de Haag-Kastler. Ao fim, retomamos o caso particular do campo escalar quântico livre. / In this thesis we present a both mathematical and conceptually rigorous quantization method for the neutral scalar field free of interactions. Initially, we introduce some aspects of the Theory of Distributions and we establish some points of Lorentzian geometry. The rest of the work is divided in two parts: in the first one, we study wave equations on globally hyperbolic Lorentzian manifolds, hence presenting the concept of fundamental solutions within the context of locally defined wave equations. Next, we progressively construct fundamental solutions for the wave operator from the Riesz distribution. Once established a solution to the wave equation in a neighbourhood of a point of the manifold, we move forward to produce a global solution from the extension of the Cauchy problem to the whole manifold. At this stage, fundamental solutions are replaced by Green\'s operators by the imposition of appropriate boundary conditions. In the last part, we present a minimum on the Theory of Categories and Functors. This is followed by the use of this formalism in the development of a second-quantization functor between the category of Lorentzian globally hyperbolic manifolds and the category of nets of C*-algebras obeying Haag-Kastler axioms. Finally, we turn our attention to the particular case of the quantum free scalar field.

Made available in DSpace on 2015-05-14T12:14:12Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 3660633 bytes, checksum: 7309d28729d29dd071bc87f7c5609ebc (MD5) Previous issue date: 2013-11-20 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The non-linear science is a central topic covering several investigation areas, such as biology, chemistry, mathematics and physics. In the first part of this thesis, we studied the non-linearity in the scope of classical field theory. The discussions are based on static solutions in (1, 1) space-time dimensions, and they are focused on kinks and lumps defects. In the related procedures, we show several techniques which allowed us to determine new models with their respective analytical solutions. The main mathematical tool to obtain these results is the so called deformation method, which was also an essential piece in the construction of a new extension method. This method presents the determination of new two scalar fields models from the coupling between two one scalar field systems. The method was analyzed carefully, as well as the linear stability, the zero modes, the total energy and the superpotentials, related with the new families of potentials. Furthermore, in the second part we presented the basics concepts about the Brownian Motion, where we analised the features of the solution of the Langevin Equation, and we also introduced a path integral approach to this problem in a quantum field theory way. / A ciência não-linear é tema central de diversas linhas de investigação, cobrindo áreas como a biologia, a física, a matemática e a química. Nossa primeira vertente de trabalho nesta tese, consiste no estudo de não-linearidades via abordagem de teoria clássica de campos. As discussões estão baseadas em soluções estáticas em (1, 1) dimensões, com destaque para o chamados defeitos tipo kink e lump. Nos procedimentos relatados, discorremos a respeito de diversas técnicas para a determinação de novos modelos com suas respectivas soluções analíticas. Um ferramental fundamental para a obtenção desses resultados é o chamado método de deformação, o qual também foi parte essencial para a criação de um método de extensão de modelos, onde visamos a construção de modelos de dois campos reais a partir do acoplamento entre dois modelos de um campo. Tal método também foi exposto em detalhes, bem como as análises sobre estabilidade linear, cálculo de modos zeros, determinação da energia total e dos superpotenciais, relativos às novas famílias de potenciais. Já a segunda linha de pesquisa, refere-se aos conceitos básicos do movimento browniano, onde analisamos as propriedades da solução da equação de Langevin, e na introdução de uma abordagem via integrais de trajetória para descrevê-lo nos moldes de teoria de quântica de campos.

Nesta dissertação apresentamos um método de quantização matemática e conceitualmente rigoroso para o campo escalar livre de interações. Trazemos de início alguns aspéctos importantes da Teoria de Distribuições e colocamos alguns pontos de geometria Lorentziana. O restante do trabalho é dividido em duas partes: na primeira, estudamos equações de onda em variedades Lorentzianas globalmente hiperbólicas e apresentamos o conceito de soluções fundamentais no contexto de equações locais. Em seguida, progressivamente construímos soluções fundamentais para o operador de onda a partir da distribuição de Riesz. Uma vez estabelecida uma solução para a equação de onda em uma vizinhança de um ponto da variedade, tratamos de construir uma solução global a partir da extensão do problema de Cauchy a toda a variedade, donde as soluções fundamentais dão lugar aos operadores de Green a partir da introdução de uma condição de contorno. Na última parte do trabalho, apresentamos um mínimo da Teoria de Categorias e Funtores para utilizar esse formalismo na contrução de um funtor de segunda quantização entre a categoria de variedades Lorentzianas globalmente hiperbólicas e a categoria de redes de álgebras C* satisfazendo os axiomas de Haag-Kastler. Ao fim, retomamos o caso particular do campo escalar quântico livre. / In this thesis we present a both mathematical and conceptually rigorous quantization method for the neutral scalar field free of interactions. Initially, we introduce some aspects of the Theory of Distributions and we establish some points of Lorentzian geometry. The rest of the work is divided in two parts: in the first one, we study wave equations on globally hyperbolic Lorentzian manifolds, hence presenting the concept of fundamental solutions within the context of locally defined wave equations. Next, we progressively construct fundamental solutions for the wave operator from the Riesz distribution. Once established a solution to the wave equation in a neighbourhood of a point of the manifold, we move forward to produce a global solution from the extension of the Cauchy problem to the whole manifold. At this stage, fundamental solutions are replaced by Green\'s operators by the imposition of appropriate boundary conditions. In the last part, we present a minimum on the Theory of Categories and Functors. This is followed by the use of this formalism in the development of a second-quantization functor between the category of Lorentzian globally hyperbolic manifolds and the category of nets of C*-algebras obeying Haag-Kastler axioms. Finally, we turn our attention to the particular case of the quantum free scalar field.

Data from present day scientific simulations and observations of physical processes often consist of multiple scalar fields. It is important to study the interactions between the fields to understand the underlying phenomena. A visual representation of these interactions would assist the scientist by providing quick insights into complex relationships that exist between the fields. We describe new techniques for visual analysis of multifield scalar data where the relationships can be quantified by the gradients of the individual scalar fields and their mutual alignment. Empirically, gradients along with their mutual alignment have been shown to be a good indicator of the relationships between the different scalar variables. The Jacobi set, defined as the set of points where the gradients are linearly dependent, describes the relationship between the gradient fields. The Jacobi set of two piecewise linear functions may contain several components indicative of noisy or a feature-rich dataset. For two dimensional domains, we pose the problem of simplification as the extraction of level sets and offset contours and describe a robust technique to simplify and create a multi-resolution representation of the Jacobi set. Existing isosurface-based techniques for scalar data exploration like Reeb graphs, contour spectra, isosurface statistics, etc., study a scalar field in isolation. We argue that the identification of interesting isovalues in a multifield data set should necessarily be based on the interaction between the different fields. We introduce a variation density function that profiles the relationship between multiple scalar fields over isosurfaces of a given scalar field. This profile serves as a valuable tool for multifield data exploration because it provides the user with cues to identify interesting isovalues of scalar fields. Finally, we introduce a new multifield comparison measure to capture relationships between scalar variables. We also show that our measure is insensitive to noise in the scalar fields and to noise in their gradients. Further, it can be computed robustly and efficiently. The comparison measure can be used to identify regions of interest in the domain where interactions between the scalar fields are significant. Subsequent visualization of the data focuses on these regions of interest leading to effective visual analysis. We demonstrate the effectiveness of our techniques by applying them to real world data from different domains like combustion studies, climate sciences and computer graphics.