| 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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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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| 4. |
- 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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| 5. |
- 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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| 6. |
- 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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| 7. |
- 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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| 8. |
- Alho, Artur, et al.
(författare)
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Quintessential α-attractor inflation : a dynamical systems analysis
- 2023
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Ingår i: Journal of Cosmology and Astroparticle Physics. - : Institute of Physics (IOP). - 1475-7516. ; :11
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 9. |
- Alho, Artur, et al.
(författare)
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Tracking Quintessence
- 2024
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Ingår i: Physics of the Dark Universe. - : Elsevier. - 2212-6864. ; 44
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Tidskriftsartikel (refereegranskat)abstract
- Tracking quintessence, in a spatially flat and isotropic space–time with a minimally coupled canonical scalar field and an asymptotically inverse power-law potential V(φ)∝φ−p, p>0, as φ→0, is investigated. This is done by introducing a new three-dimensional regular dynamical system, which enables a rigorous explanation of the tracking feature: (1) The dynamical system has a tracker fixed point T with a two-dimensional stable manifold that pushes an open set of nearby solutions toward a single tracker solution originating from T. (2) All solutions, including the tracker solution and the solutions that track/shadow it, end at a common future attractor fixed point that depends on the potential. Thus, the open set of solutions that shadow the tracker solution share its properties during the tracking quintessence epoch. We also discuss similarities and differences of underlying mechanisms for tracking, thawing and scaling freezing quintessence, and, moreover, we illustrate with state space pictures that all of these types of quintessence exist simultaneously for certain potentials.
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| 10. |
- Heinzle, J. Mark, et al.
(författare)
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A new proof of the Bianchi type IX attractor theorem
- 2009
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Ingår i: Classical and quantum gravity. - : IOP Publishing. - 0264-9381 .- 1361-6382. ; 26:7, s. 1-28
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Tidskriftsartikel (refereegranskat)abstract
- 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
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