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  • 1.
    Alho, Artur
    et al.
    Portugal.
    Hell, Juliette
    Germany.
    Uggla, Claes
    Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Engineering and Physics.
    Global dynamics and asymptotics for monomial scalar field potentials and perfect fluids2015In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 32, no 14, article id 145005Article in journal (Refereed)
    Abstract [en]

    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. Heinzle, J. Mark
    et al.
    Uggla, Claes
    Karlstad University, Faculty of Technology and Science.
    A new proof of the Bianchi type IX attractor theorem2009In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 26, no 7, p. 1-28Article in journal (Refereed)
    Abstract [en]

    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

  • 3. Heinzle, J. Mark
    et al.
    Uggla, Claes
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Mixmaster: Fact and Belief2009In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 26, no 075016, p. 1-34Article in journal (Refereed)
    Abstract [en]

    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

  • 4. Heinzle, J Mark
    et al.
    Uggla, Claes
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Monotonic functions in Bianchi models: why they exist and how to find them2010In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 27, no 1Article in journal (Refereed)
    Abstract [en]

    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

  • 5.
    Hervik, Sigbjørn
    et al.
    Faculty of Science and Technology, University of Stavanger, Norway.
    Lim, Woei Chet
    Albert-Einstein-Institut, Am-Mûhlenberg, Germany.
    Sandin, Patrik
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Uggla, Claes
    Karlstad University, Faculty of Technology and Science.
    Future Asymptotics of Tilted Bianchi Type II Cosmologies2010In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 27, p. 185006-Article in journal (Refereed)
  • 6.
    Sandin, Patrik
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Uggla, Claes
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Bianchi Type I Models with Two Tilted Fluids2008In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 25, no 22, p. 1-23Article in journal (Refereed)
  • 7.
    Sandin, Patrik
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Uggla, Claes
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Perfect Fluids and Generic Spacelike Singularities2010In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 27, no 2, p. 025013-Article in journal (Refereed)
    Abstract [en]

    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.

  • 8.
    Uggla, Claes
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Wainwright, John
    Cosmological perturbation theory revisited2011In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 28, no 17, p. 175017-175043Article in journal (Refereed)
    Abstract [en]

    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

  • 9.
    Uggla, Claes
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Physics and Electrical Engineering.
    Wainwright, John
    Scalar cosmological perturbations2012In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 29, no 10, p. 105002-105029Article in journal (Refereed)
    Abstract [en]

    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

  • 10.
    Uggla, Claes
    et al.
    Karlstad University, Faculty of Technology and Science. Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Engineering and Physics.
    Wainwright, John
    University of Waterloo.
    Simple expressions for second order density perturbations in standard cosmology2014In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 31, no 10, p. 105008-Article in journal (Refereed)
    Abstract [en]

    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.

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