| 1. |
- Alho, Artur, et al.
(författare)
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Cosmological global dynamical systems analysis
- 2022
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Ingår i: Classical and quantum gravity. - : Institute of Physics (IOP). - 0264-9381 .- 1361-6382. ; 39:14, s. 1-14
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 2. |
- Alho, Artur, et al.
(författare)
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Dynamical systems in perturbative scalar field cosmology
- 2020
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Ingår i: Classical and quantum gravity. - : American Institute of Physics (AIP). - 0264-9381 .- 1361-6382. ; 37:22
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 3. |
- Alho, Artur, et al.
(författare)
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Dynamics of Robertson-Walker spacetimes with diffusion
- 2015
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Ingår i: Annals of Physics. - : Elsevier BV. - 0003-4916 .- 1096-035X. ; 354, s. 475-488
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Tidskriftsartikel (refereegranskat)abstract
- We study the dynamics of spatially homogeneous and isotropic spacetimes containing a fluid undergoing microscopic velocity diffusion in a cosmological scalar field. After deriving a few exact solutions of the equations, we continue by analyzing the qualitative behavior of general solutions. To this purpose we recast the equations in the form of a two dimensional dynamical system and perform a global analysis of the flow. Among the admissible behaviors, we find solutions that are asymptotically de-Sitter both in the past and future time directions and which undergo accelerated expansion at all times.
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| 4. |
- Alho, Artur, et al.
(författare)
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Global dynamics and asymptotics for monomial scalar field potentials and perfect fluids
- 2015
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Ingår i: Classical and quantum gravity. - : Institute of Physics (IOP). - 0264-9381 .- 1361-6382. ; 32:14
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 5. |
- Alho, Artur, et al.
(författare)
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Global dynamics and inflationary center manifold and slow-roll approximants
- 2015
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Ingår i: Journal of Mathematical Physics. - : American Institute of Physics (AIP). - 0022-2488 .- 1089-7658. ; 56:1
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 6. |
- Alho, Artur, et al.
(författare)
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Inflationary alpha-attractor cosmology : A global dynamical systems perspective
- 2017
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Ingår i: Physical Review D. - : American Physical Society. - 2470-0010 .- 2470-0029. ; 95:8
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 7. |
- Alho, Artur, et al.
(författare)
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Multi-body Spherically Symmetric Steady States of Newtonian Self-Gravitating Elastic Matter
- 2019
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Ingår i: Communications in Mathematical Physics. - : Springer Science and Business Media LLC. - 1432-0916 .- 0010-3616. ; 371:3, s. 975-1004
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Tidskriftsartikel (refereegranskat)abstract
- We study the problem of static, spherically symmetric, self-gravitating elastic matter distributions in Newtonian gravity. To this purpose we first introduce a new definition of homogeneous, spherically symmetric (hyper)elastic body in Euler coordinates, i.e., in terms of matter fields defined on the current physical state of the body. We show that our definition is equivalent to the classical one existing in the literature and which is given in Lagrangian coordinates, i.e. in terms of the deformation of the body from a given reference state. After a number of well-known examples of constitutive functions of elastic bodies are re-defined in our new formulation, a detailed study of the Seth model is presented. For this type of material the existence of single and multi-body solutions is established.
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| 8. |
- Alho, Artur, et al.
(författare)
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On dynamical systems approaches and methods in f (R) cosmology
- 2016
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Ingår i: Journal of Cosmology and Astroparticle Physics. - : Institute of Physics Publishing (IOPP). - 1475-7516. ; :8
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Tidskriftsartikel (refereegranskat)abstract
- We discuss dynamical systems approaches and methods applied to flat Robertson Walker models in f(R)-gravity. We argue that a complete description of the solution space of a model requires a global state space analysis that motivates globally covering state space adapted variables. This is shown explicitly by an illustrative example, f(R) = R + alpha R-2, alpha > 0, for which we introduce new regular dynamical systems on global compactly extended state spaces for the Jordan and Einstein frames. This example also allows us to illustrate several local and global dynamical systems techniques involving, e.g., blow ups of nilpotent fixed points, center manifold analysis, averaging, and use of monotone functions. As a result of applying dynamical systems methods to globally state space adapted dynamical systems formulations, we obtain pictures of the entire solution spaces in both the Jordan and the Einstein frames. This shows, e.g., that due to the domain of the conformal transformation between the Jordan and Einstein frames, not all the solutions in the Jordan frame are completely contained in the Einstein frame. We also make comparisons with previous dynamical systems approaches to f (R) cosmology and discuss their advantages and disadvantages.
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| 9. |
- Alho, Artur, et al.
(författare)
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Perturbations of the Lambda-CDM model in a dynamical systems perspective
- 2019
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Ingår i: Journal of Cosmology and Astroparticle Physics. - Bristol, UK : IOP PUBLISHING LTD. - 1475-7516. ; :9
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Tidskriftsartikel (refereegranskat)abstract
- The observational success and simplicity of the ACDM model, and the explicit analytic perturbations thereof, set the standard for any alternative cosmology. It therefore serves as a comparison ground and as a test case for methods which can be extended and applied to other cosmological models. In this paper we introduce dynamical systems and methods to describe linear scalar and tensor perturbations of the ACDM model, which serve as pedagogical examples that show the global illustrative powers of dynamical systems in the context of cosmological perturbations. We also study the asymptotic properties of the shear and Weyl tensors and discuss the validity of the perturbations as approximations to the Einstein field equations. Furthermore, we give a new approximation for the linear growth 5 rate, f (z) = d ln delta/d ln a = Omega(6/11)(m) - 1/70(1-Omega(m))(5/2), where z is the cosmological redshift, Omega(m) = Omega(m)(z), while a is the background scale factor, and show that it is much more accurate than the previous ones in the literature.
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| 10. |
- Alho, Artur, et al.
(författare)
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Quintessence from a state space perspective
- 2023
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Ingår i: Physics of the Dark Universe. - : Elsevier. - 2212-6864. ; 39
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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