Dynamical systems in perturbative scalar field cosmology

The present paper considers a three-fluid cosmological model consisting of noninteracting dark matter, dark energy and baryonic matter in the background of the Friedman–Robertson–Walker–Lemaître flat spacetime. It has been assumed that the dark matter takes the form of dust whereas the dark energy is a quintessence (real) scalar field with exponential potential. It has been further assumed that the baryonic matter is a perfect fluid with barotropic equation of states. The field equations for this model takes the form of an autonomous dynamical system after some suitable changes of variables. Then a complete stability analysis is done considering all possible parameter (the adiabatic index of the baryonic matter and the parameter arising from the dark energy potential) values and for both the cases of hyperbolic and non-hyperbolic critical points. For non-hyperbolic critical points, the invariant manifold theory (center manifold approach) is applied. Finally various topologically different phase planes and vector field diagrams are produced and the cosmological interpretation of this model is presented.

We derive a set of equations monitoring the evolution of covariant and gauge-invariant linear scalar perturbations of Friedman-Lema\^itre-Robertson-Walker models with multiple interacting non-linear scalar fields. We use a dynamical systems' approach in order to perform a stability analysis for some classes of scalar field potentials. In particular, using a recent approximation for the inflationary dynamics of the background solution, we derive conditions under which homogenization occurs for chaotic (quadratic and quartic potentials) and new inflation. We also prove a cosmic no-hair result for power-law inflation and its generalisation for two scalar fields with independent exponential potentials (assisted power-law inflation).

An almost invariant set of a dynamical system is a subset of state space where typical trajectories stay for a long period of time before they enter other parts of state space. These sets are an important characteristic for analyzing the macroscopic behavior of a given dynamical system. For instance, recently the identification of almost invariant sets has successfully been used in the context of the approximation of so-called chemical conformations for molecules. In this paper we propose new numerical and algorithmic tools for the identification of the number and the location of almost invariant sets in state space. These techniques are based on the use of set oriented numerical methods by which a graph is created which models the underlying dynamical behavior. In a second step graph theoretic methods are utilized in order to both identify the number of almost invariant sets and for an approximation of these sets. These algorithmic methods make use of the notion of congestion which is a quantity specifying bottlenecks in the graph. We apply these new techniques to the analysis of the dynamics of the molecules Pentane and Hexane. Our computational results are compared to analytical bounds which again are based on the congestion but also on spectral information on the transition matrix for the underlying graph.

Low energy effective field theories motivated by string theory will likely contain several scalar moduli fields which will be relevant to early Universe cosmology. Some of these fields are expected to couple with non-standard kinetic terms to gravity. In this paper, we study the splitting into adiabatic and isocurvature perturbations for a model with two scalar fields, one of which has a non-standard kinetic term in the Einstein-frame action. Such actions can arise, e.g., in the Pre-Big-Bang and Ekpyrotic scenarios. The presence of a non-standard kinetic term induces a new coupling between adiabatic and isocurvature perturbations which is non-vanishing when the potential for the matter fields is nonzero. This coupling is un-suppressed in the long wavelength limit and thus can lead to an important transfer of power from the entropy to the adiabatic mode on super-Hubble scales. We apply the formalism to the case of a previously found exact solution with an exponential potential and study the resulting mixing of adiabatic and isocurvature fluctuations in this example. We also discuss the possible relevance of the extra coupling in the perturbation equations for the process of generating an adiabatic component of the fluctuations spectrum from isocurvature perturbations without considering a later decay of the isocurvature component.

In the article the collective name «theory just as the theory of scalar elds of dynamical systems or the problem of adaptive control by initial state replaced» by a more informative (in the title of the article). New concepts of dynamical systems theory (scalar field, the center offset of the scalar field) to a single unified space of dimension (3n+2), where 1) n+1 - dimensionality of the state space; 2) n is the dimensionality of the decision space (phase space); 3) (n+1)- dimension of hyperspace-state center displacement (second and third components of a single space called the space of possible States. In a hypothetical example, solved the problem of adaptive control points pulled out of hyperspace. Proved the Pythagorean theorem for everywhere undifferentiated trigonometric functions K. Weierstrass. In the language of classical proportions and sequences of the «Golden section» three new proven formula that helps to identify dynamical chaos as the automorphism (or self-similarity) analytic geometry. The automorphism demonstrated new algorithms for the synthesis of sequences of Lucas and D. Bernoulli (Binet). New mathematical formula synthesized in the language of calculus mathematical operations by means of the exponent (the formalism of tuple and mask), that allows to speak about a new scientific direction, named to the calculation of mathematical formulas. In conclusion, the article made the world's first animation of the motion of the center of displacement of the attractor of Rossler, describing chemical and physical processes of mixing in a mixture, and the specified Internet address Yandex-cloud, on which this animation is available to everyone.

Non-Gaussian fluctuations for structure formation may arise in inflation from the nonlinear interaction of long wavelength gravitational and scalar fields. Long wavelength fields have spatial gradients a -1∇ small compared to the Hubble radius, and they are described in terms of classical random fields that are fed by short wavelength quantum noise. Lattice Langevin calculations are given for a ‘toy model’ with a scalar field interacting with an exponential potential where one can obtain exact analytic solutions of the Fokker-Planck equation. For single scalar field models that are consistent with current microwave background fluctuations, the fluctuations are Gaussian. However, for scales much larger than our observable Universe, one expects large metric fluctuations that are non-Gaussian. This example illuminates non-Gaussian models involving multiple scalar fields which are consistent with current microwave background limits.KeywordsScalar FieldHubble ParameterLapse FunctionExponential PotentialMomentum ConstraintThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We investigate primordial magnetogenesis within a quantum bouncing model driven by a scalar field, focusing on various non-minimal couplings between the electromagnetic field and the scalar field. We test three cases: no coupling, a Cauchy coupling with gradual decay, and a Gaussian coupling with rapid fall-off. By exploring these scenarios, we assess a wide range of coupling strengths across different scales. The scalar field, with an exponential potential, behaves as pressureless matter in the asymptotic past of the contracting phase, as stiff matter around the bounce, and as dark energy during the expanding phase. Our findings reveal that, among the tested cases, only the Gaussian coupling can explain the generation of primordial magnetic fields on cosmological scales.

In this study, the author investigates controllability of discrete‐time bilinear systems by using invariant sets. An invariant set defined in this study is a set on which the system is invariant, namely for any initial state that belongs to the set, the system cannot be steered away from this set. If a system is controllable, then it has no invariant set in its state space other than the state space itself. Invariant sets are thus closely related to controllability. By studying the invariant sets of discrete‐time bilinear systems, the author first presents a necessary and sufficient condition for controllability of the systems in dimension two, which covers a classical result under the same condition. Then the author considers the high‐dimensional systems and improve the recent results on controllability by relaxing a condition concerning invariant sets, where more general controllability criteria are derived. Finally, examples are given to illustrate the obtained results.

In this paper, a new conception of composite cell coordinate system is presented by dividing the continuous state space into the cell state space with different scales. For a dynamical system, attractors, basins of attraction, basin boundaries, saddles, and invariant manifolds can be easily obtained, and any region of the state space can be refined by this method. The global bifurcations, such as crisis and metamorphosis, of the Rayleigh–Duffing oscillator are studied by the composite cell coordinate system method. According to the sudden changes in shapes of the chaotic attractor and the chaotic saddle, we find that three types of crises can all occur, including boundary crisis, interior crisis, and attractor emerging crisis. In addition, the basin boundary metamorphoses, such as fractal-Wada, Wada-Wada, and Wada-fractal, are analyzed through observing the shapes of basin boundaries. These results demonstrate the efficiency and validity of this method in analyzing dynamical systems.

We study integrability by quadrature of a spatially flat Friedmann model containing both a perfect fluid with barotropic equation of state p = (1− h)ρ and a minimally coupled scalar field φ with either a single exponential potential V(φ) ~ exp[−√6σκφ], \(K = \sqrt {8\pi {G_N}}\), of arbitrary sign or a simplest multiple exponential potential V(φ) = W 0 − V 0 sinh (√6σκφ), where the parameters W 0 and V 0 are arbitrary. From the mathematical view point the model is pseudo-Euclidean Toda-like system with 2 degrees of freedom. We apply the methods developed in our previous papers, based on the Minkowsky-like geometry for 2 characteristic vectors depending on the parameters σ and h. For the single exponential potential we present 4 classes of general solutions with the parameters obeying the following relations: A. σ is arbitrary, h = 0; B. σ = 1− h/2, 0 < h < 2; C1. σ =1−h/4, 0 < h ≤ 2; C2. σ = |1−h|, 0 < h ≤ 2, h ≠ 1, 4/3. The properties of the exact solutions near the initial singularity and at the final stage of evolution are analyzed. For the multiple exponential potential the model is integrated with h = 1 and σ = 1/2 and all exact solutions describe the recollapsing universe. We single out the exact solution describing the evolution within the time approximately equal to 2H 0 −1 with the present-day values of the acceleration parameter q 0 = 0.5 and the density parameter Ω ρ 0 = 0.3.

The problem of the cosmic coincidence is a longstanding puzzle. This conundrum may be solved by introducing a coupling between the two dark sectors. In this Letter, we study two cases of the coupled quintessence scenario. (a) Assume that the mass of dark matter particles depends exponentially on the scalar field associated to dark energy and meanwhile the scalar field evolves in an exponential potential; (b) Assume that the mass of dark matter particles depends on a power law function of the scalar field and meanwhile the scalar field evolves in a power law potential. Since the dynamics of this system is dominated by an attractor solution, the mass of dark matter particles is forced to change with time as to ensure that the ratio between the energy densities of dark matter and dark energy becomes a constant at late times, and one thus solve the cosmic coincidence problem naturally. We perform a statefinder diagnostic to both cases of this coupled quintessence scenario. It is shown that the evolving trajectory of this scenario in the s–r diagram is quite different from those of other dark energy models.

We study a class of minisuperspace models consisting of a homogenous isotropic universe with a minimally coupled homogenous scalar field with a potential alpha cosh(2-phi) + beta sinh(2-phi), where alpha and beta are arbitrary parameters. This includes the case of a pure exponential potential exp(2-phi), which arises in the dimensional reduction to four dimensions of five-dimensional Kaluza-Klein theory. We study the classical Lorentzian solutions for the model and find that they exhibit exponential or power-law inflation. We show that the Wheeler-DeWitt equation for this model is exactly soluble. Concentrating on the two particular cases of potentials cosh(2-phi) and exp(2-phi), we consider the Euclidean minisuperspace path integral for a propagation amplitude between fixed scale factors and scalar-field configurations. In the gauge N = 0 (where N is the rescaled lapse function), the path integral reduces, after some essentially trivial functional integrations, to a single nontrivial ordinary integral over N. Because the Euclidean action is unbounded from below, N must be integrated along a complex contour for convergence. We find all possible complex contours which lead to solutions of the Wheeler-DeWitt equation or Green's functions of the Wheeler-DeWitt operator, and we give an approximate evaluation of the integral along these contours, using the method of steepest descents. The steepest-descent contours may be dominated by saddle points corresponding to exact solutions to the full Einstein-scalar equations which may be real Euclidean, real Lorentzian, or complex. We elucidate the conditions under which each of these different types of solution arise. For the exp(2-phi) potential, we evaluate the path integral exactly. Although we cannot evaluate the path integral in closed form for the cosh(2-phi) potential, we show that for particular N contours the amplitude may be written as a given superposition of exact solutions to the Wheeler-DeWitt equation. By choosing certain initial data for the path-integral amplitude we obtain the amplitude specified by the proposal of Hartle and Hawking. We discuss the nature of the geometries corresponding to the saddle points of the no-boundary amplitude. We identify the set of classical solutions this proposal picks out in the classical limit.

Invariant Set (IS) theory is a locally causal ontic theory of physics based on the Cosmological Invariant Set postulate that the universe $U$ can be considered a deterministic dynamical system evolving precisely on a (suitably constructed) fractal dynamically invariant set in $U$'s state space. IS theory violates the Bell inequalities by violating Measurement Independence. Despite this, IS theory is not fine tuned, is not conspiratorial, does not constrain experimenter free will and does not invoke retrocausality. The reasons behind these claims are discussed in this paper. These arise from properties not found in conventional ontic models: the invariant set has zero measure in its Euclidean embedding space, has Cantor Set structure homeomorphic to the p-adic integers ($p \ggg 0$) and is non-computable. In particular, it is shown that the p-adic metric encapulates the physics of the Cosmological Invariant Set postulate, and provides the technical means to demonstrate no fine tuning or conspiracy. Quantum theory can be viewed as the singular limit of IS theory when when $p$ is set equal to infinity. Since it is based around a top-down constraint from cosmology, IS theory suggests that gravitational and quantum physics will be unified by a gravitational theory of the quantum, rather than a quantum theory of gravity. Some implications arising from such a perspective are discussed.

We shall study spatially homogeneous cosmological models containing a self-interacting scalar field with an exponential potential of the form V(φ)=Λekφ. The asymptotic properties of these models are discussed. In particular, their possible isotropization and inflation are investigated for all values of the parameter k. A particular class of models is analyzed qualitatively using the theory of dynamical systems, illustrating the general asymptotic behavior.

Static spherically symmetric Einstein-æther models II: Integrability and the modified Tolman–Oppenheimer–Volkoff approach