Cosmological Dynamical System Research Articles

Continuation of Bianchi spacetimes through the big bang

In this paper we present a framework in which the relational description of general relativity can be used to smoothly continue cosmological dynamical systems through the big bang without invoking quantum gravity effects. Cosmological spacetimes contain as a key dynamical variable a notion of scale through the volume factor ν. However, no cosmological observer is ever able to separate their measuring apparatus from the system they are measuring, in that sense every measurement is a relative one, and measurable dynamical variables are, in fact, dimensionless ratios. This is manifest in the identification of a scaling symmetry or “dynamical similarity” in the Einstein-Hilbert action associated with the volume factor. By quotienting out this scaling symmetry, we form a relational system defined on a contact manifold whose dynamical variables are decoupled from scale. When the phase space is reduced to shape space, we show that there exists unique solutions to the equations of motion that pass smoothly through the initial cosmological singularity in flat Friedmann-Lemaître-Robertson-Walker, Bianchi I, and Quiescent Bianchi IX cosmologies. Published by the American Physical Society 2024

Qualitative study of anisotropic cosmologies with inhomogeneous equation of state

We examine the qualitative features of anisotropic models sourced by the fluid having an inhomogeneous equation of state (EoS). In particular, we study the asymptotic behavior and cosmic dynamics of the locally rotationally symmetric Bianchi I, Bianchi III and the Kantowski–Sachs spacetime by using the dynamical system technique. The dark energy yielded by the inhomogeneous EoS leads to the accelerated expansion of the universe with vanishing shear at late-times. We further extract the geometrical and physical features of the cosmological dynamical system by rewriting the statefinder diagnostic and cosmographic parameters in terms of its dynamical variables. The model parameters are constrained for the quintessence and phantom evolution of universe. On the basis of model parameter values, the Kantowski–Sachs model undergoes synchronous bounce during its evolution.

Cosmic dynamics of isotropic models with inhomogeneous EoS: A dynamical system perspective

We examine the homogeneous and isotropic models sourced by the fluid having an inhomogenous equation of state (EoS) using qualitative methods. The flat Friedmann–Lemaitre–Robertson–Walker (FLRW) model explains the accelerating universe expansion of the present era having past of decelerating era dominated by radiation and matter. The open FLRW model incorporates the radiation, matter and dark energy dominated era in order with the spatial curvature playing role just before the accelerated universe evolution. The dominating behaviors of radiation, matter and spatial curvature affect the evolution trajectories of the deceleration and effective EoS parameter, which are obtained by the numerical solutions of autonomous system. The model parameters affect the nature of dark energy and therefore the EoS parameter may also lie in the quintessence or phantom regions. The closed FLRW model having constant positive curvature may yield the bouncing and turnaround universe evolution with dark energy-dominated phase at late-times. The application of statefinder diagnostic in the cosmological dynamical systems highlights that the universe may approach the [Formula: see text] cold dark matter ([Formula: see text]CDM) universe as a particular case in the future.

Homogeneous and anisotropic cosmologies with affine EoS: a dynamical system perspective

We study a class of homogeneous and anisotropic geometries with affine equation of state (EoS) for different physically plausible scenarios of the universe evolution using dynamical system technique. We analyze the locally rotationally symmetric Bianchi I (LRS BI), Bianchi III (LRS BIII) and Bianchi V (LRS BV) geometry for the exhibition of the effects of affine EoS in the model. The model exhibits stable attractor which is also isotropic and thus, it may explain the late-time accelerated expansion of the universe. The model also possess stiff matter-, radiation- and matter-dominated phases prior to the dark energy assisted accelerating phase which are confirmed by the behaviours of effective equation of state and deceleration parameters. We use the statefinder diagnostic which is a geometrical diagnostic to explore model independent features of the cosmological dynamical system. The LRS BI, BIII and BV geometry based dynamical systems exhibit r=1,s=0(Lambda cold dark matter model) at late-times, which is compatible with the observations. The dynamical system for the Kantowski–Sachs model yields synchronous bounce on the basis of the model parameters. It also yields a late-time attractor which may explain the accelerated expansion of the universe in the model. The qualitative differences between LRS BIII and BV cosmological dynamical systems have also been discussed.

Teleparallel scalar-tensor gravity through cosmological dynamical systems

Scalar-tensor theories offer the prospect of explaining the cosmological evolution of the Universe through an effective description of dark energy as a quantity with a non-trivial evolution. In this work, we investigate this feature of scalar-tensor theories in the teleparallel gravity context. Teleparallel gravity is a novel description of geometric gravity as a torsional- rather than curvature-based quantity which presents a new foundational base for gravity. Our investigation is centered on the impact of a nontrivial input from the kinetic term of the scalar field. We consider a number of model settings in the context of the dynamical system to reveal their evolutionary behavior. We determine the critical points of these systems and discuss their dynamics.

Cosmological global dynamical systems analysis

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.

Cosmological dynamical systems in modified gravity

The field equations of modified gravity theories, when considering a homogeneous and isotropic cosmological model, always become autonomous differential equations. This relies on the fact that in such models all variables only depend on cosmological time, or another suitably chosen time parameter. Consequently, the field equations can always be cast into the form of a dynamical system, a successful approach to study such models. We propose a perspective that is applicable to many different modified gravity models and relies on the standard cosmological density parameters only, making our choice of variables model independent. The drawback of our approach is a more complicated constraint equation. We demonstrate our procedure studying various modified gravity models and show how much generic information can be extracted before a specific model is considered.

Dynamical system approach for the modified Brans–Dicke theory

A modified Brans–Dicke theory (abbreviated as GBD) is proposed by generalizing the Ricci scalar [Formula: see text] to an arbitrary function [Formula: see text] in the original BD action. It can be found that the GBD theory has some interesting properties, such as solving the problem of PPN value without introducing the so-called chameleon mechanism (comparing with the [Formula: see text] modified gravity), making the state parameter to crossover the phantom boundary: [Formula: see text] without introducing the negative kinetic term (comparing with the quintom model). In the GBD theory, the gravitational field equation and the cosmological evolutional equations have been derived. In the framework of cosmology, we apply the dynamical system approach to investigate the stability of the GBD model. A five-variable cosmological dynamical system and three critical points ([Formula: see text], [Formula: see text], [Formula: see text]) are obtained in the GBD model. After calculation, it is shown that the critical point [Formula: see text] corresponds to the radiation dominated universe and it is unstable. The critical point [Formula: see text] is unstable, which corresponds to the geometrical dark energy dominated universe. While for case of [Formula: see text], according to the center manifold theory, this critical point is stable, and it corresponds to geometrical dark energy dominated de Sitter universe ([Formula: see text]).

Autonomous dynamical system description of de Sitter evolution in scalar assisted f(R) − ϕ gravity

In this paper we will study the cosmological dynamical system of an [Formula: see text] gravity in the presence of a canonical scalar field [Formula: see text] with an exponential potential by constructing the dynamical system in a way that it is rendered autonomous. This feature is controlled by a single variable [Formula: see text], which when it is constant, the dynamical system is autonomous. We focus on the [Formula: see text] case which, as we demonstrate by using a numerical analysis approach, leads to an unstable de Sitter attractor, which occurs after [Formula: see text] [Formula: see text]-foldings. This instability can be viewed as a graceful exit from inflation, which is inherent to the dynamics of de Sitter attractors.

Singularities in particle-like description of FRW cosmology

In this paper, we apply the method of reducing the dynamics of FRW cosmological models with a barotropic form of the equation of state to the dynamical system of the Newtonian type to detect the finite scale factor singularities and the finite-time singularities. In this approach all information concerning the dynamics of the system is contained in a diagram of the potential function V(a) of the scale factor. Singularities of the finite scale factor make themselves manifest by poles of the potential function. In our approach the different types of singularities are represented by critical exponents in the power-law approximation of the potential. The classification can be given in terms of these exponents. We have found that the pole singularity can mimic an inflation epoch. We demonstrate that the cosmological singularities can be investigated in terms of the critical exponents of the potential function of the cosmological dynamical systems. We assume that the general form of the model contains matter and some kind of dark energy which is parameterised by the potential. We distinguish singularities (by an ansatz involving the Lagrangian) of the pole type with the inflation and demonstrate that such a singularity can appear in the past.

Study of finite-time singularities of loop quantum cosmology interacting multifluids

In this work we study the occurrence of finite-time cosmological singularities in a cosmological system comprising from three fluids. Particularly, the system contains two dark fluids, namely that of dark energy and dark matter, which are interacting, and of a non-interacting baryonic fluid. For the study we adopt the phase space approach by constructing the cosmological dynamical system in such a way so that it rendered to be an autonomous polynomial dynamical system, and in order to achieve this, we appropriately choose the variables of the dynamical system. By employing a rigid mathematical framework, that of dominant balances analysis, we demonstrate that there exist non-singular solutions of the dynamical system, which correspond to a general set of initial conditions, which proves that no Big Rip or Type III finite-time singularities occur in this LQC multifluid dynamical system. Thus the new feature of this work is that we are able to do this using an analytic technique instead of adopting a numerical approach. In addition, we perform a fixed point analysis of the cosmological dynamical system, and we examine the behavior of the total effective equation of state parameter, at the fixed points, as a function of the free parameters of the system. Finally, we investigate the phenomenological implications of the dark energy equation of state which we assumed that it governs the dark energy fluid.

Autonomous dynamical system approach for f(R) gravity

In this work we shall investigate the cosmological dynamical system of $f(R)$ gravity, by constructing it in such a way so that it is rendered autonomous. We shall study the vacuum $f(R)$ gravity case, but also the case that matter and radiation perfect fluids are present along with the $f(R)$ gravity. The dynamical system is constructed in such a way so that the time-dependence of the system is contained in a single parameter which depends on the Hubble rate and it's second derivative. The autonomous structure of the dynamical system is achieved when this parameter is constant, therefore we focus on these cases. For the vacuum $f(R)$ case, we investigate two cases with the first leading to a stable de Sitter attractor fixed point but also to an unstable de Sitter fixed point, and the second is related to a matter dominated era. The stable de Sitter attractor is also found for the $f(R)$ gravity in the presence of matter and radiation perfect fluids. In all the cases we performed a detailed numerical analysis of the dynamical system and we also investigate in detail the stability of the resulting fixed points. Also, we present an exceptional feature of the $R^2$ gravity model, in the absence of perfect fluids. Finally, we investigate what is the approximate form of the $f(R)$ gravities near the stable and the unstable de Sitter fixed points.

Phase space analysis of the accelerating multifluid Universe

We study in detail the phase space of a Friedmann-Robertson-Walker Universe filled with various cosmological fluids which may or may not interact. We use various expressions for the equation of state, and we analyze the physical significance of the resulting fixed points. In addition we discuss the effects of the stability or an instability of some fixed points. Moreover we study an interesting phenomenological scenario for which there is an oscillating interaction between the dark energy and dark matter fluid. As we demonstrate, in the context of the model we use, at early times the interaction is negligible and it starts to grow as the cosmic time approaches the late-time era. Also the cosmological dynamical system is split into two distinct dynamical systems which have two distinct de Sitter fixed points, with the early-time de Sitter point being unstable. This framework gives an explicit example of the unification of the early-time with late-time acceleration. Finally, we discuss in some detail the physical interpretation of the various models we present in this work.

Lyapunov function for cosmological dynamical system

Abstract We prove the asymptotic global stability of the de Sitter solution in the Friedmann-Robertson-Walker conservative and dissipative cosmology. In the proof we construct a Lyapunov function in an exact form and establish its relationship with the first integral of dynamical system determining evolution of the flat Universe. Our result is that de-Sitter solution is asymptotically stable solution for general form of equation of state p = (ρ, H), where dependence on the Hubble function H means that the effect of dissipation are included.

Noether symmetry approach for teleparallel-curvature cosmology

We consider curvature-teleparallel F(R,T) gravity, where the gravitational Lagrangian density is given by an arbitrary function of the Ricci scalar R and the torsion scalar T. Using the Noether Symmetry Approach, we show that the functional form of the F(R,T) function can be determined by the presence of symmetries. Furthermore, we obtain exact solutions through the presence of conserved quantities and the reduction of cosmological dynamical system. Example of particular cosmological models are considered.

Noether symmetry approach in Gauss–Bonnet cosmology

We discuss the Noether Symmetry Approach in the framework of Gauss–Bonnet cosmology showing that the functional form of the F(R, 𝒢) function, where R is the Ricci scalar and 𝒢 is the Gauss–Bonnet topological invariant, can be determined by the presence of symmetries. Besides, the method allows to find out exact solutions due to the reduction of cosmological dynamical system and the presence of conserved quantities. Some specific cosmological models are worked out.

Cosmological evolution of quintessence with a sign-changing interaction in dark sector

Dark energy and dark matter are only indirectly measured though their gravitational effects. It is possibie that there is some direct, non-gravitational interaction between DE and DM, which can be used to solve (or, at least alleviate) several important theoretical problems. In the present work, by analysing the cosmological dynamical system with a dark-sector interaction which changes its sign during the cosmological evolution, we find a scaling attractor to help to alleviate the cosmic-coincidence problem. This result shows that this interaction can bring new features to the cosmology.

Can f(T) gravity theories mimic ΛCDM cosmic history

Recently the teleparallel Lagrangian density described by the torsion scalar T has been extended to a function of T. The f(T) modified teleparallel gravity has been proposed as the natural gravitational alternative for dark energy to explain the late time acceleration of the universe. In order to reconstruct the function f(T) by demanding a background ΛCDM cosmology we assume that, (i) the background cosmic history provided by the flat ΛCDM (the radiation ere with ωeff = (1/3), matter and de Sitter eras with ωeff = 0 and ωeff = −1, respectively) (ii) the radiation dominate in the radiation era with Ω0r = 1 and the matter dominate during the matter phases when Ω0m = 1. We find the cosmological dynamical system which can obey the ΛCDM cosmic history. In each era, we find a critical lines that, the radiation dominated and the matter dominated are one points of them in the radiation and matter phases, respectively. Also, we drive the cosmologically viability condition for these models. We investigate the stability condition with respect to the homogeneous scalar perturbations in each era and we obtain the stability conditions for the fixed points in each eras. Finally, we reconstruct the function f(T) which mimics cosmic expansion history.

Reconstruction of the scalar–tensor Lagrangian from aΛCDM background and Noether symmetry

We consider scalar–tensor theories and reconstruct their potentialU(Φ) and couplingF(Φ) by demandinga background ΛCDM cosmology. In particular we impose a background cosmic historyH(z) provided by the usualflat ΛCDM parameterizationthrough the radiation (weff = 1/3), matter (weff = 0)and de Sitter (weff = −1) eras. The cosmological dynamical system which is constrained to obey theΛCDM cosmic history presents five critical points in each era, one of whichcorresponding to the standard General Relativity (GR). In the cases thatdiffer from GR, the reconstructed coupling and potential are of the formF(Φ)∼Φ2 andU(Φ)∼F(Φ)m, wherem is a constant relatingthe two functions F(Φ) and U(Φ): this assumption is necessary to find out the fixed points of the dynamical system. Thisclass of scalar tensor theories is also theoretically motivated by a completely independentapproach: imposing maximal Noether symmetry on the scalar–tensor Lagrangian. Thisapproach independently (i) provides the form of the coupling and the potential asF(Φ)∼Φ2 and U(Φ)∼F(Φ)m, (ii) provides a conserved charge related to the potential and the coupling, and (iii) allowsthe derivation of exact solutions by first integrals of motion.

Canf(R)modified gravity theories mimic aΛCDMcosmology?

We consider f(R) modified gravity theories in the metric variation formalism and attempt to reconstruct the function f(R) by demanding a background {lambda}CDM cosmology. In particular we impose the following requirements: (a) A background cosmic history H(z) provided by the usual flat {lambda}CDM parametrization though the radiation (w{sub eff}=1/3), matter (w{sub eff}=0), and de Sitter (w{sub eff}=-1) eras. (b) Matter and radiation dominate during the ''matter'' and ''radiation'' eras, respectively, i.e. {omega}{sub m}=1 when w{sub eff}=0 and {omega}{sub r}=1 when w{sub eff}=1/3. We have found that the cosmological dynamical system constrained to obey the {lambda}CDM cosmic history has four critical points in each era which correspondingly lead to four forms of f(R). One of them is the usual general relativistic form f(R)=R-2{lambda}. The other three forms in each era, reproduce the {lambda}CDM cosmic history but they do not satisfy requirement (b) stated above.