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• 1.
University of Lisbon, Portugal.
Free University Berlin, Germany. Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för ingenjörsvetenskap och fysik (from 2013).
Global dynamics and asymptotics for monomial scalar field potentials and perfect fluids2015Ingår i: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 32, nr 14, artikel-id 145005Artikel i tidskrift (Refereegranskat)

We consider a minimally coupled scalar field with a monomial potential and a perfect fluid in flat Friedmann-Lemaitre-Robertson-Walker cosmology. We apply local and global dynamical systems techniques to a new three-dimensional dynamical systems reformulation of the field equations on a compact state space. This leads to a visual global description of the solution space and asymptotic behavior. At late times we employ averaging techniques to prove statements about how the relationship between the equation of state of the fluid and the monomial exponent of the scalar field affects asymptotic source dominance and asymptotic manifest self-similarity breaking. We also situate the ’attractor’ solution in the three-dimensional state space and show that it corresponds to the one-dimensional unstable center manifold of a de Sitter fixed point, located on an unphysical boundary associated with the dynamics at early times. By deriving a center manifold expansion we obtain approximate expressions for the attractor solution. We subsequently improve the accuracy and range of the approximation by means of Pade approximants and compare with the slow-roll approximation.

• 2.
Technical University of Lisbon, Portugal.
Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för ingenjörsvetenskap och fysik (from 2013).
Global dynamics and inflationary center manifold and slow-roll approximants2015Ingår i: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 56, nr 1, artikel-id 012502Artikel i tidskrift (Refereegranskat)

We consider the familiar problem of a minimally coupled scalar field with quadratic potential in flat Friedmann-Lemaître-Robertson-Walker cosmology to illustrate a number of techniques and tools, which can be applied to a wide range of scalar field potentials and problems in, e.g., modified gravity. We present a global and regular dynamical systems description that yields a global understanding of the solution space, including asymptotic features. We introduce dynamical systems techniques such as center manifold expansions and use Padé approximants to obtain improved approximations for the “attractor solution” at early times. We also show that future asymptotic behavior is associated with a limit cycle, which shows that manifest self-similarity is asymptotically broken toward the future and gives approximate expressions for this behavior. We then combine these results to obtain global approximations for the attractor solution, which, e.g., might be used in the context of global measures. In addition, we elucidate the connection between slow-roll based approximations and the attractor solution, and compare these approximations with the center manifold based approximants.

• 3.
Center for Mathematical Analysis, Geometry and Dynamical Systems, Instituto Superior Técnico, Universidade de Lisboa, Portugal.
Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för ingenjörsvetenskap och fysik (from 2013).
Inflationary alpha-attractor cosmology: A global dynamical systems perspective2017Ingår i: Physical Review D: covering particles, fields, gravitation, and cosmology, ISSN 2470-0010, E-ISSN 2470-0029, Vol. 95, nr 8, artikel-id 083517Artikel i tidskrift (Refereegranskat)

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.

• 4.
Univ Lisbon, Inst Super Tecn, Ctr Math Anal Geometry & Dynam Syst, P-1049001 Lisbon, Portugal..
Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för ingenjörsvetenskap och fysik.
Scalar field deformations of Lambda CDM cosmology2015Ingår i: Physical Review D, ISSN 1550-7998, E-ISSN 1550-2368, Vol. 92, nr 10, artikel-id 103502Artikel i tidskrift (Refereegranskat)

This paper treats nonrelativistic matter and a scalar field phi with a monotonically decreasing potential minimally coupled to gravity in flat Friedmann-Lemaitre-Robertson-Walker cosmology. The field equations are reformulated as a three-dimensional dynamical system on an extended compact state space, complemented with cosmographic diagrams. A dynamical systems analysis provides global dynamical results describing possible asymptotic behavior. It is shown that one should impose global and asymptotic bounds on lambda = -V-1 dV/d phi to obtain viable cosmological models that continuously deform Lambda CDM cosmology. In particular we introduce a regularized inverse power-law potential as a simple specific example.

Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för fysik och elektroteknik.
Asymptotic silence of generic cosmological singularities2005Ingår i: Phys.Rev.Lett. 94 (2005) 051101Artikel i tidskrift (Refereegranskat)

In this letter we investigate the nature of generic cosmological singularities using the framework developed by Uggla et al. We do so by studying the past asymptotic dynamics of general vacuum G2 cosmologies, models that are expected to capture the singular behavior of generic cosmologies with no symmetries at all. In particular, our results indicate that asymptotic silence holds, i.e., that particle horizons along all timelines shrink to zero for generic solutions. Moreover, we provide evidence that spatial derivatives become dynamically insignificant along generic timelines, and that the evolution into the past along such timelines is governed by an asymptotic dynamical system which is associated with an invariant set -- the silent boundary. We also identify an attracting subset on the silent boundary that organizes the oscillatory dynamics of generic timelines in the singular regime. In addition, we discuss the dynamics associated with recurring spike formation

Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för fysik och elektroteknik.
Gowdy phenomenology in scale-invariant variables2004Ingår i: Class.Quant.Grav. 21 (2004) S29-S57Artikel i tidskrift (Refereegranskat)

The dynamics of Gowdy vacuum spacetimes is considered in terms of Hubble-normalized scale-invariant variables, using the timelike area temporal gauge. The resulting state space formulation provides for a simple mechanism for the formation of false'' and true spikes'' in the approach to the singularity, and a geometrical formulation for the local attractor

• 7. Carr,, B. J.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för fysik och elektroteknik.
The state space and physical interpretation of self-similar spherically symmetric perfect-fluid models2001Ingår i: Class.Quant.Grav. 18 (2001) 303-324Artikel i tidskrift (Refereegranskat)

The purpose of this paper is to further investigate the solution space of self-similar spherically symmetric perfect-fluid models and gain deeper understanding of the physical aspects of these solutions. We achieve this by combining the state space description of the homothetic approach with the use of the physically interesting quantities arising in the comoving approach. We focus on three types of models. First, we consider models that are natural inhomogeneous generalizations of the Friedmann Universe; such models are asymptotically Friedmann in their past and evolve fluctuations in the energy density at later times. Second, we consider so-called quasi-static models. This class includes models that undergo self-similar gravitational collapse and is important for studying the formation of naked singularities. If naked singularities do form, they have profound implications for the predictability of general relativity as a theory. Third, we consider a new class of asymptotically Minkowski self-similar spacetimes, emphasizing that some of them are associated with the self-similar solutions associated with the critical behaviour observed in recent gravitational collapse calculations

• 8. Fjällborg, Mikael
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för fysik och elektroteknik.
Self-gravitating stationary spherically symmetric systems in relativistic galactic dynamics2007Ingår i: Math. Proc. Cambridge Phil. Soc. 143 (2007) 731-752Artikel i tidskrift (Refereegranskat)

We study equilibrium states in relativistic galactic dynamics which are described by solutions of the Einstein-Vlasov system for collisionless matter. We recast the equations into a regular three-dimensional system of autonomous first order ordinary differential equations on a bounded state space. Based on a dynamical systems analysis we derive new theorems that guarantee that the steady state solutions have finite radii and masses

• 9. Heinzle, J. Mark
Karlstads universitet, Fakulteten för teknik- och naturvetenskap.
A new proof of the Bianchi type IX attractor theorem2009Ingår i: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 26, nr 7, s. 1-28Artikel i tidskrift (Refereegranskat)

We consider the dynamics towards the initial singularity of Bianchi type IX vacuum and orthogonal perfect fluid models with a linear equation of state. The Bianchi type IX attractor theorem' states that the past asymptotic behavior of generic type IX solutions is governed by Bianchi type I and II vacuum states (Mixmaster attractor). We give a comparatively short and self-contained new proof of this theorem. The proof we give is interesting in itself, but more importantly it illustrates and emphasizes that type IX is special, and to some extent misleading when one considers the broader context of generic models without symmetries

• 10. Heinzle, J. Mark
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för fysik och elektroteknik.
Mixmaster: Fact and Belief2009Ingår i: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 26, nr 075016, s. 1-34Artikel i tidskrift (Refereegranskat)

We consider the dynamics towards the initial singularity of Bianchi type IX vacuum and orthogonal perfect fluid models with a linear equation of state. Surprisingly few facts are known about the Mixmaster' dynamics of these models, while at the same time most of the commonly held beliefs are rather vague. In this paper, we use Mixmaster facts as a base to build an infrastructure that makes it possible to sharpen the main Mixmaster beliefs. We formulate explicit conjectures concerning (i) the past asymptotic states of type IX solutions and (ii) the relevance of the Mixmaster/Kasner map for generic past asymptotic dynamics. The evidence for the conjectures is based on a study of the stochastic properties of this map in conjunction with dynamical systems techniques. We use a dynamical systems formulation, since this approach has so far been the only successful path to obtain theorems, but we also make comparisons with the metric' and Hamiltonian billiard' approaches

• 11. Heinzle, J Mark
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för fysik och elektroteknik.
Monotonic functions in Bianchi models: why they exist and how to find them2010Ingår i: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 27, nr 1Artikel i tidskrift (Refereegranskat)

All rigorous and detailed dynamical results in Bianchi cosmology rest upon the existence of a hierarchical structure of conserved quantities and monotonic functions. In this paper we uncover the underlying general mechanism and derive this hierarchical structure from the scale-automorphism group for an illustrative example, vacuum and diagonal class A perfect fluid models. First, kinematically, the scale-automorphism group leads to a reduced dynamical system that consists of a hierarchy of scale-automorphism invariant sets. Second, we show that, dynamically, the scale-automorphism group results in scale-automorphism invariant monotone functions and conserved quantities that restrict the flow of the reduced dynamical system

• 12.
University of Vienna, Austria.
Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för ingenjörsvetenskap och fysik (from 2013).
Spike Statistics2013Ingår i: General Relativity and Gravitation, ISSN 0001-7701, E-ISSN 1572-9532, Vol. 45, nr 5, s. 939-957Artikel i tidskrift (Refereegranskat)

In this paper we explore stochastical and statistical properties of so-called recurring spike induced Kasner sequences. Such sequences arise in recurring spike formation, which is needed together with the more familiar BKL scenario to yield a complete description of generic spacelike singularities. In particular we derive a probability distribution for recurring spike induced Kasner sequences, complementing similar available BKL results, which makes comparisons possible. As examples of applications, we derive results for so-called large and small curvature phases and the Hubble-normalized Weyl scalar.

• 13.
University of Vienna, Austria.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för fysik och elektroteknik. University of Waikato, New Zealand.
Spike oscillations2012Ingår i: Physical Review D, ISSN 1550-7998, E-ISSN 1550-2368, Vol. 86, s. 104049-104075Artikel i tidskrift (Refereegranskat)

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

• 14.
Faculty of Science and Technology, University of Stavanger, Norway.
Albert-Einstein-Institut, Am-Mûhlenberg, Germany. Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för fysik och elektroteknik. Karlstads universitet, Fakulteten för teknik- och naturvetenskap.
Future Asymptotics of Tilted Bianchi Type II Cosmologies2010Ingår i: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 27, s. 185006-Artikel i tidskrift (Refereegranskat)
• 15. Lim, Woei Chet
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för fysik och elektroteknik.
Asymptotic silence-breaking singularities2006Ingår i: Class. Quantum Grav. 23 (2006) 2607-2630Artikel i tidskrift (Refereegranskat)

We discuss three complementary aspects of scalar curvature singularities: asymptotic causal properties, asymptotic Ricci and Weyl curvature, and asymptotic spatial properties. We divide scalar curvature singularities into two classes: so-called asymptotically silent singularities and non-generic singularities that break asymptotic silence. The emphasis in this paper is on the latter class which have not been previously discussed. We illustrate the above aspects and concepts by describing the singularities of a number of representative explicit perfect fluid solutions

• 16. Lim, Woei Chet
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för fysik och elektroteknik.
Asymptotic isotropization in inhomogeneous cosmology2004Ingår i: Phys.Rev. D69 (2004) 103507Artikel i tidskrift (Refereegranskat)

In this paper we investigate asymptotic isotropization. We derive the asymptotic dynamics of spatially inhomogeneous cosmological models with a perfect fluid matter source and a positive cosmological constant near the de Sitter equilibrium state at late times, and near the flat FL equilibrium state at early times. Our results show that there exists an open set of solutions approaching the de Sitter state at late times, consistent with the cosmic no-hair conjecture. On the other hand, solutions that approach the flat FL state at early times are special and admit a so-called isotropic initial singularity. For both classes of models the asymptotic expansion of the line element contains an arbitrary spatial metric at leading order, indicating asymptotic spatial inhomogeneity. We show, however, that in the asymptotic regimes this spatial inhomogeneity is significant only at super-horizon scales

• 17. Mark Heinzle, J.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för fysik och elektroteknik.
Theory of Newtonian self-gravitating stationary spherically symmetric systems2006Ingår i: Math. Proc. Camb. Phil. Soc. 140 (2006) 177-192Artikel i tidskrift (Refereegranskat)

We investigate spherically symmetric equilibrium states of the Vlasov-Poisson system, relevant in galactic dynamics. We recast the equations into a regular three-dimensional system of autonomous first order ordinary differential equations on a region with compact closure.Based on a dynamical systems analysis we derive theorems that guarantee that the steady state solutions have finite mass and compact support

• 18. Mark Heinzle, J.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för fysik och elektroteknik.
Matter and dynamics in closed cosmologies2005Ingår i: Phys.Rev. D71 (2005) 083506Artikel i tidskrift (Refereegranskat)

To systematically analyze the dynamical implications of the matter content in cosmology, we generalize earlier dynamical systems approaches so that perfect fluids with a general barotropic equation of state can be treated. We focus on locally rotationally symmetric Bianchi type IX and Kantowski-Sachs orthogonal perfect fluid models, since such models exhibit a particularly rich dynamical structure and also illustrate typical features of more general cases. For these models, we recast Einstein's field equations into a regular system on a compact state space, which is the basis for our analysis. We prove that models expand from a singularity and recollapse to a singularity when the perfect fluid satisfies the strong energy condition. When the matter source admits Einstein's static model, we present a comprehensive dynamical description, which includes asymptotic behavior, of models in the neighborhood of the Einstein model; these results make earlier claims about homoclinic phenomena and chaos'' highly questionable. We also discuss aspects of the global asymptotic dynamics, in particular, we give criteria for the collapse to a singularity, and we describe when models expand forever to a state of infinite dilution; possible initial and final states are analyzed. Numerical investigations complement the analytical results

• 19. Mark Heinzle, J.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för fysik och elektroteknik.
Spherically symmetric relativistic stellar structures2003Ingår i: Class.Quant.Grav. 20 (2003) 4567-4586Artikel i tidskrift (Refereegranskat)

We investigate relativistic spherically symmetric static perfect fluid models in the framework of the theory of dynamical systems. The field equations are recast into a regular dynamical system on a 3-dimensional compact state space, thereby avoiding the non-regularity problems associated with the Tolman-Oppenheimer-Volkoff equation. The global picture of the solution space thus obtained is used to derive qualitative features and to prove theorems about mass-radius properties. The perfect fluids we discuss are described by barotropic equations of state that are asymptotically polytropic at low pressures and, for certain applications, asymptotically linear at high pressures. We employ dimensionless variables that are asymptotically homology invariant in the low pressure regime, and thus we generalize standard work on Newtonian polytropes to a relativistic setting and to a much larger class of equations of state. Our dynamical systems framework is particularly suited for numerical computations, as illustrated by several numerical examples, e.g., the ideal neutron gas and examples that involve phase transitions

• 20. Mark Heinzle, J.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för fysik och elektroteknik.
Homoclinic chaos and energy condition violation2006Ingår i: Phys. Rev. D74 (2006) 061502Artikel i tidskrift (Refereegranskat)

In this letter we discuss the connection between so-called homoclinic chaos and the violation of energy conditions in locally rotationally symmetric Bianchi type IX models, where the matter is assumed to be non-tilted dust and a positive cosmological constant. We show that homoclinic chaos in these models is an artifact of unphysical assumptions: it requires that there exist solutions with positive matter energy density $\rho>0$ that evolve through the singularity and beyond as solutions with negative matter energy density $\rho<0$. Homoclinic chaos is absent when it is assumed that the dust particles always retain their positive mass.In addition, we discuss more general models: for solutions that are not locally rotionally symmetric we demonstrate that the construction of extensions through the singularity, which is required for homoclinic chaos, is not possible in general

• 21. Mark Heinzle, J.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för fysik och elektroteknik.
Dynamics of the spatially homogeneous Bianchi type I Einstein-Vlasov equations2006Ingår i: Class. Quantum Grav. 23 (2006) 3463-3490Artikel i tidskrift (Refereegranskat)

We investigate the dynamics of spatially homogeneous solutions of the Einstein-Vlasov equations with Bianchi type I symmetry by using dynamical systems methods. All models are forever expanding and isotropize toward the future; toward the past there exists a singularity. We identify and describe all possible past asymptotic states; in particular, on the past attractor set we establish the existence of a heteroclinic network, which is a new type of feature in general relativity. This illustrates among other things that Vlasov matter can lead to quite different dynamics of cosmological models as compared to perfect fluids

• 22. Mark Heinzle, J.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för fysik och elektroteknik.
Newtonian Stellar Models2003Ingår i: Annals Phys. 308 (2003) 18-61Artikel i tidskrift (Refereegranskat)

We investigate static spherically symmetric perfect fluid models in Newtonian gravity for barotropic equations of state that are asymptotically polytropic at low and high pressures. This is done by casting the equations into a 3-dimensional regular dynamical system with bounded dependent variables. The low and high central pressure limits correspond to two 2-dimensional boundary subsets, described by homology invariant equations for exact polytropes. Thus the formulation naturally places work about polytropes in a more general context. The introduced framework yields a visual aid for obtaining qualitative information about the solution space and is also suitable for numerical investigations. Last, but not least, it makes a host of mathematical tools from dynamical systems theory available. This allows us to prove a number of theorems about the relationship between the equation of state and properties concerning total masses and radii

• 23. Nilsson, U. S.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för fysik och elektroteknik.
General Relativistic Stars : Polytropic Equations of State2001Ingår i: Annals Phys. 286 (2001) 292-319Artikel i tidskrift (Refereegranskat)

In this paper, the gravitational field equations for static spherically symmetric perfect fluid models with a polytropic equation of state, $p=k\rho^{1+1/n}$, are recast into two complementary 3-dimensional {\it regular} systems of ordinary differential equations on compact state spaces. The systems are analyzed numerically and qualitatively, using the theory of dynamical systems. Certain key solutions are shown to form building blocks which, to a large extent, determine the remaining solution structure. In one formulation, there exists a monotone function that forces the general relativistic solutions towards a part of the boundary of the state space that corresponds to the low pressure limit. The solutions on this boundary describe Newtonian models and thus the relationship to the Newtonian solution space is clearly displayed. It is numerically demonstrated that general relativistic models have finite radii when the polytropic index $n$ satisfies $0\leq n \lesssim 3.339$ and infinite radii when $n\geq 5$. When $3.339\lesssim n<5$, there exists a 1-parameter set of models with finite radii and a finite number, depending on $n$, with infinite radii

• 24. Nilsson, U. S.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för fysik och elektroteknik.
General Relativistic Stars: Linear Equations of State2001Ingår i: Annals Phys. 286 (2001) 278-291Artikel i tidskrift (Refereegranskat)

In this paper Einstein's field equations, for static spherically symmetric perfect fluid models with a linear barotropic equation of state, are recast into a 3-dimensional regular system of ordinary differential equations on a compact state space. The system is analyzed qualitatively, using the theory of dynamical systems, and numerically. It is shown that certain special solutions play important roles as building blocks for the solution structure in general. In particular, these special solutions determine many of the features exhibited by solutions with a regular center and large central pressure. It is also shown that the present approach can be applied to more general classes of barotropic equations of state

• 25. Röhr, Niklas
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för fysik och elektroteknik.
Conformal regularization of Einstein's field equations2005Ingår i: Class. Quantum Grav. 22 (2005) 3775-3787Artikel i tidskrift (Refereegranskat)

To study asymptotic structures, we regularize Einstein's field equations by means of conformal transformations. The conformal factor is chosen so that it carries a dimensional scale that captures crucial asymptotic features. By choosing a conformal orthonormal frame we obtain a coupled system of differential equations for a set of dimensionless variables, associated with the conformal dimensionless metric, where the variables describe ratios with respect to the chosen asymptotic scale structure. As examples, we describe some explicit choices of conformal factors and coordinates appropriate for the situation of a timelike congruence approaching a singularity. One choice is shown to just slightly modify the so-called Hubble-normalized approach, and one leads to dimensionless first order symmetric hyperbolic equations. We also discuss differences and similarities with other conformal approaches in the literature, as regards, e.g., isotropic singularities

• 26.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för fysik och elektroteknik.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för fysik och elektroteknik.
Bianchi Type I Models with Two Tilted Fluids2008Ingår i: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 25, nr 22, s. 1-23Artikel i tidskrift (Refereegranskat)
• 27.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för fysik och elektroteknik.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för fysik och elektroteknik.
Perfect Fluids and Generic Spacelike Singularities2010Ingår i: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 27, nr 2, s. 025013-Artikel i tidskrift (Refereegranskat)

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.

• 28.
Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för ingenjörsvetenskap och fysik (from 2013).
Global cosmological dynamics for the scalar field representation of the modified Chaplygin gas2013Ingår i: Physical Review D, ISSN 1550-7998, E-ISSN 1550-2368, Vol. D88, s. 064040-1-064040-7Artikel i tidskrift (Refereegranskat)

In this paper we investigate the global dynamics for the minimally coupled scalar field representation of the modified Chaplygin gas in the context of flat Friedmann-Lemaître-Robertson Walker cosmology. The tool for doing this is a new set of bounded variables that lead to a regular dynamical system. It is shown that the exact modified Chaplygin gas perfect fluid solution appears as a straight line in the associated phase plane. It is also shown that no other solutions stay close to this solution during their entire temporal evolution, but that there exists an open subset of solutions that stay arbitrarily close during an intermediate time interval, and into the future in the case when the scalar field potential exhibits a global minimum.

• 29.
Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för ingenjörsvetenskap och fysik (from 2013).
Recent developments concerning generic spacelike singularities2013Ingår i: General Relativity and Gravitation, ISSN 0001-7701, E-ISSN 1572-9532, Vol. 45, s. 1669-1710Artikel i tidskrift (Refereegranskat)

This is a review of recent progress concerning generic spacelikesingularities in general relativity. For brevity the main focus is on singularities in vacuum spacetimes, although the connection with, and the role of, matter for generic singularity formation is also commented on. The paper describes recent developments in two areas and show how these are connected within the context of the conformally Hubble-normalized state space approach. The first area is oscillatory singularities in spatially homogeneous cosmology and the connection between asymptotic behaviour and heteroclinic chains. The second area concerns oscillatory singularities in inhomogeneous models, especially spike chains and recurring spikes. The review also outlines some underlying reasons for why the structures that are the foundation for generic oscillatory behaviour exists at all, which entails discussing how underlying physical principles and applications of solutiongenerating techniques yield hierarchical structures and connections between them. Finally, it is pointed out that recent progress concerning generic singularities motivates some speculations that suggest that a paradigm shift concerning their physical role, and what mathematical issues to address,might be in order.

• 30.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för fysik och elektroteknik.
Spacetime singularities: Recent developments2013Ingår i: International Journal of Modern Physics D, ISSN 0218-2718, Vol. 22, nr 3, s. 1330002-1330022Artikel i tidskrift (Refereegranskat)

Recent developments concerning oscillatory spacelike singularities in general relativity are taking place on two fronts. The first treats generic singularities in spatially homogeneous cosmology, most notably Bianchi types VIII and IX. The second deals with generic oscillatory singularities in inhomogeneous cosmologies, especially those with two commuting spacelike Killing vectors. This paper describes recent progress in these two areas: in the spatially homogeneous case, focus is on mathematically rigorous results, while analytical and numerical results concerning generic behavior and so-called recurring spike formation are the main topics in the inhomogeneous case. Unifying themes are connections between asymptotic behavior, hierarchical structures and solution generating techniques, which provide hints for a link between the nature of generic singularities and a hierarchy of hidden asymptotic symmetries.

• 31.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för fysik och elektroteknik.
The Nature of Generic Cosmological Singularities2006Ingår i: Plenariebidrag "11th Marcel Grossmann Meeting on Recent Developments in General Relativity" (2006); även arXiv:0706.0463Artikel i tidskrift (Refereegranskat)

The existence of a singularity by definition implies a preferred scale--the affine parameter distance from/to the singularity of a causal geodesic that is used to define it. However, this variable scale is also captured by the expansion along the geodesic, and this can be used to obtain a regularized state space picture by means of a conformal transformation that factors out the expansion. This leads to the conformal Hubble-normalized' orthonormal frame approach which allows one to translate methods and results concerning spatially homogeneous models into the generic inhomogeneous context, which in turn enables one to derive the dynamical nature of generic cosmological singularities. Here we describe this approach and outline the derivation of the cosmological billiard attractor,' which describes the generic dynamical asymptotic behavior towards a generic spacelike singularity. We also compare the dynamical systems picture' resulting from this approach with other work on generic spacelike singularities: the metric approach of Belinskii, Lifschitz, and Khalatnikov, and the recent Iwasawa based Hamiltonian method used by Damour, Henneaux, and Nicolai; in particular we show that the cosmological billiards obtained by the latter and the cosmological billiard attractor form complementary dual' descriptions of the generic asymptotic dynamics of generic spacelike singularities

• 32.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för fysik och elektroteknik.
Vad händer då rum och tid upphör?2005Ingår i: Kosmos (2005) 65-91Artikel i tidskrift (Övrig (populärvetenskap, debatt, mm))
• 33.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för fysik och elektroteknik.
The cosmological billiard attractor2009Ingår i: Advances in Theoretical and Mathematical Physics, ISSN 1095-0761, E-ISSN 1095-0753, Vol. 13, nr 2, s. 293-407Artikel i tidskrift (Refereegranskat)

This article is devoted to a study of the asymptotic dynamics of generic solutions of the Einstein vacuum equations toward a generic spacelike singularity. Starting from fundamental assumptions about the nature of generic spacelike singularities we derive in a step-by-step manner the cosmological billiard conjecture: we show that the generic asymptotic dynamics of solutions is represented by (randomized) sequences of heteroclinic orbits on the billiard attractor'. Our analysis rests on two pillars: (i) a dynamical systems formulation based on the conformal Hubble-normalized orthonormal frame approach expressed in an Iwasawa frame; (ii) stochastic methods and the interplay between genericity and stochasticity. Our work generalizes and improves the level of rigor of previous work by Belinskii, Khalatnikov, and Lifshitz; furthermore, we establish that our approach and the Hamiltonian approach to cosmological billiards', as elaborated by Damour, Hennaux, and Nicolai, can be viewed as yielding dual' representations of the asymptotic dynamics

• 34.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för fysik och elektroteknik.
The past attractor in inhomogeneous cosmology2003Ingår i: Phys.Rev. D68 (2003) 103502Artikel i tidskrift (Refereegranskat)

We present a general framework for analyzing spatially inhomogeneous cosmological dynamics. It employs Hubble-normalized scale-invariant variables which are defined within the orthonormal frame formalism, and leads to the formulation of Einstein's field equations with a perfect fluid matter source as an autonomous system of evolution equations and constraints. This framework incorporates spatially homogeneous dynamics in a natural way as a special case, thereby placing earlier work on spatially homogeneous cosmology in a broader context, and allows us to draw on experience gained in that field using dynamical systems methods. One of our goals is to provide a precise formulation of the approach to the spacelike initial singularity in cosmological models, described heuristically by Belinski\v{\i}, Khalatnikov and Lifshitz. Specifically, we construct an invariant set which we conjecture forms the local past attractor for the evolution equations. We anticipate that this new formulation will provide the basis for proving rigorous theorems concerning the asymptotic behavior of spatially inhomogeneous cosmological models

• 35.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för fysik och elektroteknik.
A simplified structure for second order cosmological perturbation equations2013Ingår i: General Relativity and Gravitation, ISSN 0001-7701, E-ISSN 1572-9532, Vol. 45, nr 3, s. 643-674Artikel i tidskrift (Refereegranskat)

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.

• 36.
Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för ingenjörsvetenskap och fysik (from 2013).
Asymptotic analysis of perturbed dust cosmologies to second order2013Ingår i: General Relativity and Gravitation, ISSN 0001-7701, E-ISSN 1572-9532, Vol. 45, nr 8, s. 1467-1492Artikel i tidskrift (Refereegranskat)

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.

• 37.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för fysik och elektroteknik.
Cosmological perturbation theory revisited2011Ingår i: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 28, nr 17, s. 175017-175043Artikel i tidskrift (Refereegranskat)

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

• 38.
Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för ingenjörsvetenskap och fysik (from 2013).
Univ Waterloo, Dept Appl Math, Waterloo, Canada.
Dynamics of cosmological perturbations at first and second order2018Ingår i: Physical Review D: covering particles, fields, gravitation, and cosmology, ISSN 2470-0010, E-ISSN 2470-0029, Vol. 98, nr 10, artikel-id 103534Artikel i tidskrift (Refereegranskat)

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.

• 39.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för fysik och elektroteknik.
Scalar cosmological perturbations2012Ingår i: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 29, nr 10, s. 105002-105029Artikel i tidskrift (Refereegranskat)

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

• 40.
Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för ingenjörsvetenskap och fysik (from 2013).
Second order cosmological perturbations: simplified gauge change formulas2019Ingår i: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 36, nr 3, s. 1-19, artikel-id 035004Artikel i tidskrift (Refereegranskat)

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.

• 41.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap. Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för ingenjörsvetenskap och fysik.
University of Waterloo.
Second order density perturbations for dust cosmologies2014Ingår i: Physical Review D, ISSN 1550-7998, E-ISSN 1550-2368, Vol. D90, s. 043511-Artikel i tidskrift (Refereegranskat)

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.

• 42.
Karlstads universitet, Fakulteten för teknik- och naturvetenskap. Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för ingenjörsvetenskap och fysik.
University of Waterloo.
Simple expressions for second order density perturbations in standard cosmology2014Ingår i: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 31, nr 10, s. 105008-Artikel i tidskrift (Refereegranskat)

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.

• 43. van Elst, Henk
Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för fysik och elektroteknik.
Dynamical systems approach to G2 cosmology2002Ingår i: Class.Quant.Grav. 19 (2002) 51-82Artikel i tidskrift (Refereegranskat)

In this paper we present a new approach for studying the dynamics of spatially inhomogeneous cosmological models with one spatial degree of freedom. By introducing suitable scale-invariant dependent variables we write the evolution equations of the Einstein field equations as a system of autonomous partial differential equations in first-order symmetric hyperbolic format, whose explicit form depends on the choice of gauge. As a first application, we show that the asymptotic behaviour near the cosmological initial singularity can be given a simple geometrical description in terms of the local past attractor on the boundary of the scale-invariant dynamical state space. The analysis suggests the name asymptotic silence'' to describe the evolution of the gravitational field near the cosmological initial singularity

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