arXiv subjects

Marek Szydlowski

Publications and source records attributed to Marek Szydlowski.

At least 19 recordsLinked to original sources

arXiv2020

Interpretation of bulk viscosity as the generalized Chaplygin gas

The cosmological observations suggest that the presently accelerating universe should be filled by an exotic form of matter, violating the strong energy condition, of unknown nature and origin. We propose the viscous dark matter of a source of acceleration in the form of Chaplygin gas which is characterized by equation of state in the phenomenological form $p=-\frac{A}{ρ^α}$, where $p$ and $ρ$ are pressure and energy density respectively ($A$ and $α$ are constants). Chaplygin gas is interpreted in terms of viscous matter and without the cosmological constant. The acceleration effect is caused only by viscosity in this class of cosmological models. We show that bulk viscosity effects introduced to the standard FRW cosmology give rise to the natural unification of both dark matter and dark energy. We show that dust viscous cosmological models are structurally stable if $m < 1/2$ ($1+α=1/2-m$).

gr-qc↗

arXiv2020

The evolution of the FRW universe with decaying metastable dark energy --- a dynamical system analysis

We investigate a cosmological model in which dark energy identified with the vacuum energy which is running and decaying. In this model vacuum is metastable and decays into a bare (true) vacuum. This decaying process has a quantum nature and is described by tools of the quantum decay theory of unstable systems. We have found formulas for an asymptotic behavior of the energy density of dark energy in the form of a series of inverse powers of the cosmological time. We investigate the dynamics of FRW models using dynamical system methods as well as searching for exact solutions. From dynamical analysis we obtain different evolutional scenarios admissible for all initial conditions. For the interpretation of the dynamical evolution caused by the decay of the quantum vacuum we study the thermodynamics of the apparent horizon of the model as well as the evolution of the temperature. For the early Universe, we found that the quantum effects modified the evolution of the temperature of the Universe. In our model the adiabatic approximation is valid and the quantum vacuum decay occurs with an adequate unknown particle which constitutes quantum vacuum. We argue that the late-time evolution of metastable energy is the holographic dark energy.

gr-qc↗

arXiv2020

Bifurcations in economic growth model with distributed time delay transformed to ODE

We consider the model of economic growth with time delayed investment function. Assuming the investment is time distributed we can use the linear chain trick technique to transform delay differential equation system to equivalent system of ordinary differential system (ODE). The time delay parameter is a mean time delay of gamma distribution. We reduce the system with distribution delay to both three and four-dimensional ODEs. We study the Hopf bifurcation in these systems with respect to two parameters: the time delay parameter and the rate of growth parameter. We derive the results from the analytical as well as numerical investigations. From the former we obtain the sufficient criteria on the existence and stability of a limit cycle solution through the Hopf bifurcation. In numerical studies with the Dana and Malgrange investment function we found two Hopf bifurcations with respect to the rate growth parameter and detect the existence of stable long-period cycles in the economy. We find that depending on the time delay and adjustment speed parameters the range of admissible values of the rate of growth parameter breaks down into three intervals. First we have stable focus, then the limit cycle and again the stable solution with two Hopf bifurcations. Such behaviour appears for some middle interval of admissible range of values of the rate of growth parameter.

econ.TH↗

arXiv2019

Bifurcations in Ratra-Peebles quintessence models and their extensions

We have used the dynamical system approach in order to investigate the dynamics of cosmological models of the flat Universe with a non-minimally coupled canonical and phantom scalar field and the Ratra-Peebles potential. Applying methods of the bifurcation theory we have found three cases for which the Universe undergoes a generic evolution emerging from either the deSitter or the static Universe state and finishing at the deSitter state, without the presence of the initial singularity. This generic class of solutions explains both the inflation and the late-time acceleration of the Universe. In this class inflation is an endogenous effect of dynamics itself.

gr-qc↗

arXiv2018

Is a pole type singularity an alternative to inflation?

In this paper, we apply a method of reducing the dynamics of FRW cosmological models with the 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 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 mimick 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 the general form of the model contains matter and some kind of dark energy which is parameterized by the potential. We distinguish singularities (by ansatz about the Lagrangian) of the pole type with the inflation and demonstrate that such a singularity can appear in the past.

gr-qc↗

arXiv2018

From quantum unstable systems to the decaying dark energy: Cosmological implications

We consider a cosmology with decaying metastable dark energy and assume that a decay process of this metastable dark energy is a quantum decay process. Such an assumption implies among others that the evolution of the Universe is irreversible and violates the time reversal symmetry. We show that if to replace the cosmological time $t$ appearing in the equation describing the evolution of the Universe by the Hubble cosmological scale time, then we obtain time dependent $Λ(t)$ in the form of the series of even powers of the Hubble parameter $H$: $Λ(t) = Λ(H)$. Out special attention is focused on radioactive like exponential form of the decay process of the dark energy and on the consequences of this type decay.

gr-qc↗

arXiv2017

Polynomial $f(R)$ Palatini cosmology -- dynamical system approach

We investigate cosmological dynamics based on $f(R)$ gravity in the Palatini formulation. In this study we use the dynamical system methods. We show that the evolution of the Friedmann equation reduces to the form of the piece-wise smooth dynamical system. This system is is reduced to a 2D dynamical system of the Newtonian type. We demonstrate how the trajectories can be sewn to guarantee $C^0$ extendibility of the metric similarly as `Milne-like' FLRW spacetimes are $C^0$-extendible. We point out that importance of dynamical system of Newtonian type with non-smooth right-hand sides in the context of Palatini cosmology. In this framework we can investigate singularities which appear in the past and future of the cosmic evolution. We consider cosmological systems in both Einstein and Jordan frames. We show that at each frame the topological structures of phase space are different.

gr-qc↗

arXiv2017

Quantum mechanical look at the radioactive-like decay of metastable dark energy

We derive the Shafieloo, Hazra, Sahni and Starobinsky (SHSS) phenomenological formula for the radioactive-like decay of metastable dark energy directly from the quantum mechanics principles. For this aim we use the Fock-Krylov theory of quantum unstable states. We obtain deeper insight on the decay process as having three basic phases: the phase of radioactive decay, the next phase of damping oscillations, and finally the phase of power law decaying. We consider the cosmological model with matter and dark energy in the form of decaying metastable dark energy and study its dynamics in the framework of non-conservative cosmology with an interacting term determined by the running cosmological parameter. We study cosmological implications of metastable dark energy and estimate the characteristic time of ending of the radioactive-like decay epoch as 22296 of the present age of the Universe. We also confront the model with astronomical data which show that the model is in good agreement with the observations. Our general conclusion is that we are living in the epoch of the radioactive-like decay of metastable dark energy which is a relict of the quantum age of the Universe.

gr-qc↗

arXiv2017

Simple cosmological model with inflation and late times acceleration

In the framework of polynomial Palatini cosmology, we investigate a simple cosmological homogeneous and isotropic model with matter in the Einstein frame. We show that in this model during cosmic evolution, it appears the early inflation and the accelerating phase of the expansion for the late times. In this frame we obtain the Friedmann equation with matter and dark energy in the form of a scalar field with the potential whose form is determined in a covariant way by the Ricci scalar of the FRW metric. The energy density of matter and dark energy are also parametrized through the Ricci scalar. The early inflation is obtained only for an infinitesimally small fraction of energy density of matter. Between the matter and dark energy, there exists interaction because the dark energy is decaying. For characterization of inflation we calculate the slow roll parameters and the constant roll parameter in terms of the Ricci scalar. We have found a characteristic behaviour of the time dependence of density of dark energy on the cosmic time following the logistic-like curve which interpolates two almost constant value phases. From the required numbers of $N$-folds we have found a bound on model parameter.

gr-qc↗

arXiv2017

The phase space structure of the oligopoly dynamical system by means of Darboux integrability

We investigate the dynamical complexity of Cournot oligopoly dynamics of three firms by using the qualitative methods of dynamical systems to study the phase structure of this model. The phase space is organized with one-dimensional and two-dimensional invariant submanifolds (for the monopoly and duopoly) and unique stable node (global attractor) in the positive quadrant of the phase space (Cournot equilibrium). We also study the integrability of the system. We demonstrate the effectiveness of the method of the Darboux polynomials in searching for first integrals of the oligopoly. The general method as well as examples of adopting this method are presented. We study Darboux non-integrability of the oligopoly for linear demand functions and find first integrals of this system for special classes of the system, in particular, rational integrals can be found for a quite general set of model parameters. We show how first integral can be useful in lowering the dimension of the system using the example of $n$ almost identical firms. This first integral also gives information about the structure of the phase space and the behaviour of trajectories in the neighbourhood of a Nash equilibrium

econ.GN↗

arXiv2017

Emergence of dynamical dark energy from polynomial $f(R)$ theory in Palatini formalism

We consider FRW cosmology in $f(R)= R+ γR^2+δR^3$ modified framework. The Palatini approach reduces its dynamics to the simple generalization of Friedmann equation. Thus we study the dynamics in two-dimensional phase space with some details. After reformulation of the model in the Einstein frame, it reduces to the FRW cosmological model with a homogeneous scalar field and vanishing kinetic energy term. This potential determines the running cosmological constant term as a function of the Ricci scalar. As a result we obtain the emergent dark energy parametrization from the covariant theory. We study also singularities of the model and demonstrate that in the Einstein frame some undesirable singularities disappear.

gr-qc↗

arXiv2017

Starobinsky cosmological model in Palatini formalism

We classify singularities in FRW cosmologies, which dynamics can be reduced to the dynamical system of the Newtonian type. This classification is performed in terms of geometry of a potential function if it has poles. At the sewn singularity, which is of a finite scale factor type, the singularity in the past meets the singularity in the future. We show, that such singularities appear in the Starobinsky model in $f(\hat{R})=\hat{R}+γ\hat{R}^2$ in the Palatini formalism, when dynamics is determined by the corresponding piece-wise smooth dynamical system. As an effect we obtain a degenerated singularity. Analytical calculations are given for the cosmological model with matter and the cosmological constant. The dynamics of model is also studied using dynamical system methods. From the phase portraits we find generic evolutionary scenarios of the evolution of the Universe. For this model, the best fit value of $Ω_γ=3γH_0^2$ is equal $9.70\times 10^{-11}$. We consider model in both Jordan and Einstein frames. We show that after transition to the Einstein frame we obtain both form of the potential of the scalar field and the decaying Lambda term.

gr-qc↗

arXiv2017

Cosmological implications of the transition from the false vacuum to the true vacuum state

We study the cosmology with the running dark energy. The parametrization of dark energy with the respect to the redshift is derived from the first principles of quantum mechanics. Energy density of dark energy is obtained from the quantum process of transition from the false vacuum state to the true vacuum state. This is the class of the extended interacting $Λ$CDM models. We consider the energy density of dark energy parametrization $ρ_\text{de}(t)$, which follows from the Breit-Wigner energy distribution function which is used to model the quantum unstable systems. The idea that properties of the process of the quantum mechanical decay of unstable states can help to understand the properties of the observed universe was formulated by Krauss and Dent and this idea was used in our considerations. In the cosmological model with the mentioned parametrization there is an energy transfer between the dark matter and dark energy. In such a evolutional scenario the universe is starting from the false vacuum state and going to the true vacuum state of the present day universe. We find that the intermediate regime during the passage from false to true vacuum states takes place. The intensity of the analyzed process is measured by a parameter $α$. For the small value of $α$ ($0<α<0.4$) this intermediate (quantum) regime is characterized by an oscillatory behavior of the density of dark energy while the for $α> 0.4$ the density of the dark energy simply jumps down. In both cases (independent from the parameter $α$) the today value of density of dark energy is reached at the value of $0.7$. We estimate the cosmological parameters for this model with visible and dark matter. This model becomes in good agreement with the astronomical data and is practically indistinguishable from $Λ$CDM model.

gr-qc↗

arXiv2016

Dynamics of the diffusive DM-DE interaction--dynamical system approach

We discuss dynamics of a model of an energy transfer between dark energy (DE) and dark matter (DM). The energy transfer is determined by a non-conservation law resulting from a diffusion of dark matter in an environment of dark energy. The relativistic invariance defines the diffusion in a unique way. The system can contain baryonic matter and radiation which do not interact with the dark sector. We treat the Friedman equation and the conservation laws as a closed dynamical system. The dynamics of the model is examined using the dynamical systems methods for demonstration how solutions depend on initial conditions. We also fit the model parameters using astronomical observation: SNIa, $H(z)$, BAO and Alcock-Paczynski test. We show that the model with diffuse DM-DE is consistent with the data.

gr-qc↗

arXiv2016

Does the diffusion DM-DE interaction model solve cosmological puzzles?

We study dynamics of cosmological models with diffusion effects modeling dark matter and dark energy interactions. We show the simple model with diffusion between the cosmological constant sector and dark matter, where the canonical scaling law of dark matter $(ρ_{dm,0}a^{-3}(t))$ is modified by an additive $ε(t)=γt a^{-3}(t)$ to the form $ρ_{dm}=ρ_{dm,0}a^{-3}(t)+ε(t)$. We reduced this model to the autonomous dynamical system and investigate it using dynamical system methods. This system possesses a two-dimensional invariant submanifold on which the DM-DE interaction can be analyzed on the phase plane. The state variables are density parameter for matter (dark and visible) and parameter $δ$ characterizing the rate of growth of energy transfer between the dark sectors. A corresponding dynamical system belongs to a general class of jungle type of cosmologies represented by coupled cosmological models in a Lotka-Volterra framework. We demonstrate that the de Sitter solution is a global attractor for all trajectories in the phase space and there are two repellers: the Einstein-de Sitter universe and the de Sitter universe state dominating by the diffusion effects. We distinguish in the phase space trajectories, which become in good agreement with the data. They should intersect a rectangle with sides of $Ω_{m,0}\in [0.2724, 0.3624]$, $δ\in [0.0000, 0.0364]$ at the 95\% CL. Our model could solve some of the puzzles of the $Λ$CDM model, such as the coincidence and fine-tuning problems. In the context of the coincidence problem, our model can explain the present ratio of $ρ_{m}$ to $ρ_{de}$, which is equal $0.4576^{+0.1109}_{-0.0831}$ at a 2$σ$ confidence level.

gr-qc↗

arXiv2016

Do sewn singularities falsify the Palatini cosmology?

We investigate further (cf. arXiv:1512.01199, JCAP01 (2016) 040) Starobinsky cosmological model $R+γR^2$ in the Palatini formalism with Chaplygin gas and baryonic matter as a source. For this aim we use dynamical system theory. The dynamics is reduced to the 2D sewn dynamical system of a Newtonian type (a piecewise-smooth dynamical system). We classify all evolutional paths in the model as well as trajectories in the phase space. We demonstrate the presence of a degenerate freeze singularity (glued freeze type singularities) for the positive $γ$. In this case it is a generic feature of early evolution of the universe. We point out that a degenerate type III of singularity can be considered as an endogenous model of inflation between the matter dominating epoch and the dark energy phase. We also investigate cosmological models with negative $γ$. It is demonstrated that $γ$ equal zero is a bifurcation parameter and dynamics qualitatively changes in comparison to positive $γ$. Instead of the big bang the sudden bounce singularity of a finite scale factor appears and there is a generic class of bouncing solutions sewn along the line $a=a_{\text{sing}}$. And we argue that the presence of sudden singularities in an evolutional scenario of the Universe falsifies the negative $γ$ in the Palatini cosmology. Only very small values of $Ω_γ$ parameter are admissible if we requires that agreements physics with the $Λ$CDM model. From the statistical analysis of astronomical observations, we deduce that the case of negative values of $Ω_γ$ can be rejected even if it may fit better to the data.

gr-qc↗

arXiv2016

Inflationary cosmology with Chaplygin gas in Palatini formalism

We present a simple generalisation of the $Λ$CDM model which on the one hand reaches very good agreement with the present day experimental data and provides an internal inflationary mechanism on the other hand. It is based on Palatini modified gravity with quadratic Starobinsky term and generalized Chaplygin gas as a matter source providing, besides a current accelerated expansion, the epoch of endogenous inflation driven by type III freeze singularity. It follows from our statistical analysis that astronomical data favors negative value of the parameter coupling quadratic term into Einstein-Hilbert Lagrangian and as a consequence the bounce instead of initial Big-Bang singularity is preferred.

gr-qc↗

arXiv2016

Dynamical system approach to running $Λ$ cosmological models

We discussed the dynamics of cosmological models in which the cosmological constant term is a time dependent function through the scale factor $a(t)$, Hubble function $H(t)$, Ricci scalar $R(t)$ and scalar field $ϕ(t)$. We considered five classes of models; two non-covariant parametrization of $Λ$: 1) $Λ(H)$CDM cosmologies where $H(t)$ is the Hubble parameter, 2) $Λ(a)$CDM cosmologies where $a(t)$ is the scale factor, and three covariant parametrization of $Λ$: 3) $Λ(R)$CDM cosmologies, where $R(t)$ is the Ricci scalar, 4) $Λ(ϕ)$-cosmologies with diffusion, 5) $Λ(X)$-cosmologies, where $X=\frac{1}{2}g^{αβ}\nabla_α\nabla_βϕ$ is a kinetic part of density of the scalar field. We also considered the case of an emergent $Λ(a)$ relation obtained from the behavior of trajectories in a neighborhood of an invariant submanifold. In study of dynamics we use dynamical system methods for investigating how a evolutional scenario can depend on the choice of special initial conditions. We showed that methods of dynamical systems offer the possibility of investigation all admissible solutions of a running $Λ$ cosmology for all initial conditions, their stability, asymptotic states as well as a nature of the evolution in the early universe (singularity or bounce) and a long term behavior at the large times. We also formulated an idea of the emergent cosmological term derived directly from an approximation of exact dynamics. We show that some non-covariant parametrizations of Lambda term like $Λ(a)$, $Λ(H)$ give rise to pathological and nonphysical behaviour of trajectories in the phase space. This behaviour disappears if the term $Λ(a)$ is emergent from the covariant parametrization.

astro-ph.CO↗