We consider a dynamical systems formulation for models with an exponential scalar field and matter with a linear equation of state in a spatially flat and isotropic spacetime. In contrast to earlier work, which only considered linear hyperbolic fixed point analysis, we do a center manifold analysis of the non-hyperbolic fixed points associated with bifurcations. More importantly though, we construct monotonic functions and a Dulac function. Together with the complete local fixed point analysis this leads to proofs that describe the entire global dynamics of these models, thereby complementing previous local results in the literature. 

We study flat Friedmann-Lemaitre-Robertson-Walker alpha-attractor E- and T-models by introducing a dynamical systems framework that yields regularized unconstrained field equations on two-dimensional compact state spaces. This results in both illustrative figures and a complete description of the entire solution spaces of these models, including asymptotics. In particular, it is shown that observational viability, which requires a sufficient number of e-folds, is associated with a particular solution given by a one-dimensional center manifold of a past asymptotic de Sitter state, where the center manifold structure also explains why nearby solutions are attracted to this "inflationary attractor solution." A center manifold expansion yields a description of the inflationary regime with arbitrary analytic accuracy, where the slow-roll approximation asymptotically describes the tangency condition of the center manifold at the asymptotic de Sitter state.

The equations for quintessential alpha-attractor inflation with a single scalar field, radiation and matter in a spatially flat FLRW spacetime are recast into a regular dynamical system on a compact state space. This enables a complete description of the solution space of these models. The inflationary attractor solution is shown to correspond to the unstable center manifold of a de Sitter fixed point, and we describe connections between slow-roll and dynamical systems approximations for this solution, including Pade approximants. We also introduce a new method for systematically obtaining initial data for quintessence evolution by using dynamical systems properties; in particular, this method exploits that there exists a radiation dominated line of fixed points with an unstable quintessence attractor submanifold, which plays a role that is reminiscent of that of the inflationary attractor solution for inflation.

We derive a newregulardynamical system on a three-dimensionalcompactstate space describing linear scalar perturbations of spatially flat Robertson-Walker geometries for relativistic models with a minimally coupled scalar field with an exponential potential. This enables us to construct the global solution space, illustrated with figures, where known solutions are shown to reside on special invariant sets. We also use our dynamical systems approach to obtain new results about the comoving and uniform density curvature perturbations. Finally we show how to extend our approach to more general scalar field potentials. This leads to state spaces where the state space of the models with an exponential potential appears as invariant boundary sets, thereby illustrating their role as building blocks in a hierarchy of increasingly complex cosmological models.

We use dynamical systems methods to study quintessence models in a spatially flat and isotropic spacetime with matter and a scalar field with potentials for which lambda(v) = -V,v/V is bounded, thereby going beyond the exponential potential for which lambda(v) is constant. The scalar field equation of state parameter wv plays a central role when comparing quintessence models with observations, but with the dynamical systems used to date wv is an indeterminate, discontinuous, function on the state space in the asymptotically matter dominated regime. Our first main result is the introduction of new variables that lead to a regular dynamical system on a bounded three-dimensional state space on which wv is a regular function. The solution trajectories in the state space then provide a visualization of different types of quintessence evolution, and how initial conditions affect the transition between the matter and scalar field dominated epochs; this is complemented by graphs wv(N), where N is the e-fold time, which enables characterizing different types of quintessence evolution.

We study the mediation of supersymmetry breaking in string compactifications whose moduli are stabilized by nonperturbative effects. We begin with a critical review of arguments for sequestering in supergravity and in string theory. We then show that geometric isolation, even in a highly warped space, is insufficient to achieve sequestering: in type IIB compactifications, nonperturbative superpotentials involving the Kahler moduli introduce cross-couplings between well-separated visible and hidden sectors. The scale of the resulting soft terms depends on the moduli stabilization scenario. In the Large Volume Scenario, nonperturbative superpotential contributions to the soft trilinear A terms can introduce significant flavor violation, while in KKLT compactifications their effects are negligible. In both scenarios, the contributions to the mu and B mu parameters cannot be ignored in general. We conclude that sequestered supersymmetry breaking is possible in nonperturbatively-stabilized compactifications only if a mechanism in addition to bulk locality suppresses superpotential cross-couplings.

Children's conceptions are several about different natural phenomenon. One phenomenon that every child comes in contact with every day is the transition from day to night and vice versa. To be able to give children explanations about thing that happens and how they are connected to each other, it is important to listen to what conceptions they have and then explain on the basis from their conceptions. Individual semi constructed interviews have been made with pupils in 3rd grade to find out what conceptions they have. The results show that the pupils were certain that the sun, the moon, the earth and other planets are rotating out in space varied. The conclusion of this work is that the pupils conceptions about the phenomenon day and night can be connected with the scientific explanations and that they also had understanding about how different objects are rotating in space.

We consider the Einstein equations coupled to an ultrastiff perfect fluid and prove the existence of a family of solutions with an initial singularity whose structure is that of explicit isotropic models. This family of solutions is ‘generic’ in the sense that it depends on as many free functions as a general solution, i.e., without imposing any symmetry assumptions, of the Einstein-Euler equations. The method we use is a that of a Fuchsian reduction.

According to Belinskii, Khalatnikov and Lifshitz (BKL), a generic spacelike singularity is characterized by asymptotic locality: Asymptotically, toward the singularity, each spatial point evolves independently from its neighbors, in an oscillatory manner that is represented by a sequence of Bianchi type I and II vacuum models. Recent investigations support a modified conjecture: The formation of spatial structures (`spikes') breaks asymptotic locality. The complete description of a generic spacelike singularity involves spike oscillations, which are described by sequences of Bianchi type I and certain inhomogeneous vacuum models. In this paper we describe how BKL and spike oscillations arise from concatenations of exact solutions in a Hubble-normalized state space setting, suggesting the existence of hidden symmetries and showing that the results of BKL are part of a greater picture

This thesis presents a brief review of gravitation and cosmology, and then gives an overview of the theory of cosmological perturbations; subsequently some applications are discussed, such as large-scale structure formation. Cosmological perturbations are here presented both in the Newtonian paradigm and in two di¤erent relativistic approaches. The relativistic approaches are (i) the metric approach, where small variations of the metric tensor are considered, and (ii) the covariant approach, which focusses on small variations of the curvature. Dealing with these two approaches also involves addressing the gauge problem –how to map an idealized world model into a more accurate world model.

In this thesis, after a general introduction, we first review some differential geometry to provide the mathematical background needed to derive the key equations in cosmology. Then we consider the Robertson-Walker geometry and its relationship to cosmography, i.e., how one makes measurements in cosmology. We finally connect the Robertson-Walker geometry to Einstein's field equation to obtain so-called cosmological Friedmann-Lemaître models. These models are subsequently studied by means of potential diagrams.

Abstract

Most children find astronomy an interesting subject and usually show an interest during lessons about the subject. The focus group in this paper is Swedish students in school year 4-6. In total there were 94 students from eight classes who answered the survey with simple drawings and explanations about for example how they thought the objects Sun–Earth–Moon orbit each other, which object they placed as the center in this system, what causes season and the reason behind day and night. Two teachers whom educate the students answered some questions about their teaching method, their thoughts about the subject in question and the students learning and understanding of astronomy.

In the study that are presented in this paper the same types of astronomical misconceptions can be seen amongst these students, that has been shown in similar studies of children all over the world. 20 % of the children in this study showed an earth centered Sun–Earth–Moon system. Other misconceptions that were shown was: the sun and the moon shares the same orbit around the earth, that the seasons are caused by the distance to the sun changing and that day and night happens because the earth orbits the sun. The use of concrete material and 3D-models in the education seemed to be important for the students understanding of the subject.

Keywords: Astronomy education – astronomy misconceptions – seasons – day/night – Celestial Motion in the Sun–Earth–Moon system

This thesis aims to give an introduction to quintessence models as well ascomparing those models with the currently accepted ΛCDM model. We willconsider quintessence in a spatially flat and isotropic Universe and useregular dynamical systems on a bounded three dimensional phase spacewhere we perform numerical calculations. This enables a globalunderstanding of these models and makes it possible to give thoroughdescriptions of the dynamics. We will discuss the three different types ofquintessence: Thawing, Scaling and Tracking quintessence and comparethem with the ΛCDM model. This thesis is based on several scientific papersand contains new numerical calculations and associated analysis.

This is a thesis on general relativity. It analyzes dynamical properties of Einstein's field equations in cosmology and in the vicinity of spacetime singularities in a number of different situations. Different techniques are used depending on the particular problem under study; dynamical systems methods are applied to cosmological models with spatial homogeneity; Hamiltonian methods are used in connection with dynamical systems to find global monotone quantities determining the asymptotic states; Fuchsian methods are used to quantify the structure of singularities in spacetimes without symmetries. All these separate methods of analysis provide insights about different facets of the structure of the equations, while at the same time they show the relationships between those facets when the different methods are used to analyze overlapping areas.

The thesis consists of two parts. Part I reviews the areas of mathematics and cosmology necessary to understand the material in part II, which consists of five papers. The first two of those papers uses dynamical systems methods to analyze the simplest possible homogeneous model with two tilted perfect fluids with a linear equation of state. The third paper investigates the past asymptotic dynamics of barotropic multi-fluid models that approach a `silent and local' space-like singularity to the past. The fourth paper uses Hamiltonian methods to derive new monotone functions for the tilted Bianchi type II model that can be used to completely characterize the future asymptotic states globally. The last paper proves that there exists a full set of solutions to Einstein's field equations coupled to an ultra-stiff perfect fluid that has an initial singularity that is very much like the singularity in Friedman models in a precisely defined way.

We present the 1+3 Hubble-normalized conformal orthonormal frame approach to Einstein field equations, and specialize it to a source that consists of perfect fluids with general barotropic equations of state. We use this framework to give specific mathematical content to conjectures about generic spacelike singularities that were originally introduced by Belinskii, Khalatnikov and Lifshitz. Assuming that the conjectures hold, we derive results about how the properties of fluids and generic spacelike singularities affect each other.

Increasingly accurate observations of the cosmic microwave background and the large scale distribution of galaxies necessitate the study of nonlinear perturbations of Friedmann–Lemaitre cosmologies, whose equations are notoriously complicated. In this paper we present a new derivation of the governing equations for second order perturbations within the framework of the metric-based approach that is minimal, as regards amount of calculation and length of expressions, and flexible, as regards choice of gauge and stress–energy tensor. Because of their generality and the simplicity of their structure our equations provide a convenient starting point for determining the behaviour of nonlinear perturbations of FL cosmologies with any given stress–energy content, using either the Poisson gauge or the uniform curvature gauge.

Nonlinear perturbations of Friedmann-Lemaitre cosmologies with dust and a cosmological constant Lambda>0 have recently attracted considerable attention. In this paper our first goal is to compare the evolution of the first and second order perturbations by determining their asymptotic behaviour at late times in ever-expanding models. We show that in the presence of spatial curvature K or a cosmological constant, the density perturbation approaches a finite limit both to first and second order, but the rate of approach depends on the model, being power law in the scale factor if Lambda>0 but logarithmic if Lambda=0 and K<0. Scalar perturbations in general contain a growing and a decaying mode. We find, somewhat surprisingly, that if Lambda>0 the decaying mode does not die a way, i.e.  it contributes on an equal footing as the growing mode to the asymptotic expression for the density perturbation. On the other hand, the future asymptotic regime of the Einstein-de Sitter universe (K=Lambda=0) is completely different, as exemplified by the density perturbation which diverges; moreover, the second order perturbation diverges faster than the first order perturbation, which suggests that the Einstein-de Sitter universe is unstable to perturbations, and that the perturbation series do not converge towards the future. We conclude that the presence of spatial curvature or a cosmological constant stabilizes the perturbations. Our second goal is to derive an explicit expression for the second order density perturbation that can be used to study the effects of including a cosmological constant and spatial curvature.

Increasingly accurate observations are driving theoretical cosmology towards the use of more sophisticated descriptions of matter and the study of nonlinear perturbations of Friedmann–Lemaitre cosmologies, whose governing equations are notoriously complicated. Our goal in this paper is to formulate the governing equations for linear perturbation theory in a particularly simple and concise form in order to facilitate the extension to nonlinear perturbations. Our approach has several novel features. We show that the use of so-called intrinsic gauge invariants has two advantages. It naturally leads to (i) a physically motivated choice of a gauge invariant associated with the matter density, and (ii) two distinct and complementary ways of formulating the evolution equations for scalar perturbations, associated with the work of Bardeen and of Kodama and Sasaki. In the first case, the perturbed Einstein tensor gives rise to a second-order (in time) linear differential operator, and in the second case to a pair of coupled first-order (in time) linear differential operators. These operators are of fundamental importance in cosmological perturbation theory, since they provide the leading order terms in the governing equations for nonlinear perturbations

In this paper we give five gauge-invariant systems of governing equations for first and second order scalar perturbations of flat Friedmann-Lemaitre universes that are minimal in the sense that they contain no redundant equations or variables. We normalize the variables so that they are dimensionless, which leads to systems of equations that are simple and ready-to-use. We compare the properties and utility of the different systems. For example, they serve as a starting point for finding explicit solutions for two benchmark problems in cosmological perturbation theory at second order: adiabatic perturbations in the superhorizon regime (the long wavelength limit) and perturbations of ACDM universes. However, our framework has much wider applicability and serves as a reference for future work in the field.

Scalar perturbations of Friedmann–Lemaitre cosmologies can be analyzed in a variety of ways using Einstein's field equations, the Ricci and Bianchi identities, or the conservation equations for the stress–energy tensor, and possibly introducing a timelike reference congruence. The common ground is the use of gauge invariants derived from the metric tensor, the stress–energy tensor, or from vectors associated with a reference congruence, as basic variables. Although there is a complication in that there is no unique choice of gauge invariants, we will show that this can be used to advantage. With this in mind our first goal is to present an efficient way of constructing dimensionless gauge invariants associated with the tensors that are involved, and of determining their inter-relationships. Our second goal is to give a unified treatment of the various ways of writing the governing equations in dimensionless form using gauge-invariant variables, showing how simplicity can be achieved by a suitable choice of variables and normalization factors. Our third goal is to elucidate the connection between the metric-based approach and the so-called 1 + 3 gauge-invariant approach to cosmological perturbations. We restrict our considerations to linear perturbations, but our intent is to set the stage for the extension to second-order perturbations

In this paper we present a new formulation of the change of gauge formulas in second order cosmological perturbation theory which unifies and simplifies known results. Our approach is based on defining new second order scalar perturbation variables by adding a multiple of the square of the corresponding first order variables to each second order variable. A bonus is that these new perturbation variables are of broader significance in that they also simplify the analysis of second order scalar perturbations in the super-horizon regime in a number of ways, and lead to new conserved quantities.

We present simple expressions for the relativistic first and second order fractional density perturbations for FL cosmologies with dust, in four different gauges: the Poisson, uniform curvature, total matter and synchronous gauges. We include a cosmological constant and arbitrary spatial curvature in the background. A distinctive feature of our approach is our description of the spatial dependence of the perturbations using a canonical set of quadratic differential expressions involving an arbitrary spatial function that arises as a conserved quantity. This enables us to unify, simplify and extend previous seemingly disparate results. We use the primordial matter and metric perturbations that emerge at the end of the inflationary epoch to determine the additional arbitrary spatial function that arises when integrating the second order perturbation equations. This introduces a non-Gaussianity parameter into the expressions for the second order density perturbation. In the special case of zero spatial curvature we show that the time evolution simplifies significantly, and requires the use of only two non-elementary functions, the so-called growth supression factor at the linear level, and one new function at the second order level. We expect that the results will be useful in applications, for example, studying the effects of primordial non-Gaussianity on the large scale structure of the universe.

The study of long-wavelength scalar perturbations, in particular the existence of conserved quantities when the perturbations are adiabatic, plays an important role in e.g., inflationary cosmology. In this paper we present some new conserved quantities at second order and relate them to the curvature perturbation in the uniform density gauge, zeta, and the comoving curvature perturbation, R. We also, for the first time, derive the general solution of the perturbed Einstein equations at second order, which thereby contains both growing and decaying modes, for adiabatic long-wavelength perturbations for a stress-energy tensor with zero anisotropic stresses and zero heat flux. The derivation uses the total matter gauge, but results are subsequently translated to the uniform curvature and Poisson (longitudinal, zero shear) gauges.

In this paper we present four simple expressions for the relativistic first and second order fractional density perturbations for ΛCDM cosmologies in different gauges: the Poisson, uniform curvature, total matter and synchronous gauges. A distinctive feature of our approach is the use of a canonical set of quadratic differential expressions involving an arbitrary spatial function, the so-called comoving curvature perturbation, to describe the spatial dependence, which enables us to unify, simplify and extend previous seemingly disparate results. The simple structure of the expressions makes the evolution of the density perturbations completely transparent and clearly displays the effect of the cosmological constant on the dynamics, namely that it stabilizes the perturbations. We expect that the results will be useful in applications, for example, studying the effects of primordial non-Gaussianity on the large scale structure of the universe.

In this paper we consider second order perturbations of a flat Friedmann-Lemaitre universe whose stress-energy content is a single minimally coupled scalar field with an arbitrary potential. We derive the general solution of the perturbed Einstein equations in explicit form for this class of models when the perturbations are in the super-horizon regime. As a by-product we obtain a new conserved quantity for long wavelength perturbations of a single scalar field at second order.

Experimental data suggest that the universe is homogeneous and isotropic on sufficiently large scales. An exact solution of the Einstein field equations exists for a homogeneous and isotropic universe, also known as a Friedmann-Lemaître-Robertson-Walker (FLRW) universe. However, this model is only a first approximation since we know that, locally, the universe has anisotropic and inhomogeneous structures such as galaxies and clusters of galaxies. In order to successfully introduce inhomogeneities and anisotropies to the model one uses perturbative methods. In cosmological perturbations the FLRW universe is considered the zeroth order term in a perturbation expansion and perturbation theory is used to derive higher order terms which one tries to match with observations. In this thesis I present a review of the main concepts of general relativity, discuss the 3+1 formalism which gives us the Einstein field equations in a useful form for the perturbative analysis, and lastly, I derive the first order scalar perturbations of the Einstein field equations.